% This file is part of the MMIXware package (c) Donald E Knuth 1999
@i boilerplate.w %<< legal stuff: PLEASE READ IT BEFORE MAKING ANY CHANGES!

\def\title{MMIX-ARITH}

\def\MMIX{\.{MMIX}}
\def\MMIXAL{\.{MMIXAL}}
\def\Hex#1{\hbox{$^{\scriptscriptstyle\#}$\tt#1}} % experimental hex constant
\def\dts{\mathinner{\ldotp\ldotp}}
\def\<#1>{\hbox{$\langle\,$#1$\,\rangle$}}\let\is=\longrightarrow
\def\ff{\\{ff\kern-.05em}}
@s ff TeX
@s bool normal @q unreserve a C++ keyword @>
@s xor normal @q unreserve a C++ keyword @>
@s bignum int

@* Introduction. The subroutines below are used to simulate 64-bit \MMIX\
arithmetic on an old-fashioned 32-bit computer---like the one the author
had when he wrote \MMIXAL\ and the first \MMIX\ simulators in 1998 and 1999.
All operations are fabricated from 32-bit arithmetic, including
a full implementation of the IEEE floating point standard,
assuming only that the \CEE/ compiler has a 32-bit unsigned integer type.

Some day 64-bit machines will be commonplace and the awkward manipulations of
the present program will look quite archaic. Interested readers who have such
computers will be able to convert the code to a pure 64-bit form without
difficulty, thereby obtaining much faster and simpler routines. Meanwhile,
however, we can simulate the future and hope for continued progress.

This program module has a simple structure, intended to make it
suitable for loading with \MMIX\ simulators and assemblers.

@c
#include <stdio.h>
#include <string.h>
#include <ctype.h>
@<Stuff for \CEE/ preprocessor@>@;
typedef enum{@+false,true@+} bool;
@<Tetrabyte and octabyte type definitions@>@;
@<Other type definitions@>@;
@<Global variables@>@;
@<Subroutines@>

@ Subroutines of this program are declared first with a prototype,
as in {\mc ANSI C}, then with an old-style \CEE/ function definition.
Here are some preprocessor commands that make this work correctly with both
new-style and old-style compilers.
@^prototypes for functions@>

@<Stuff for \CEE/ preprocessor@>=
#ifdef __STDC__
#define ARGS(list) list
#else
#define ARGS(list) ()
#endif

@ The definition of type \&{tetra} should be changed, if necessary, so that
it represents an unsigned 32-bit integer.
@^system dependencies@>

@<Tetra...@>=
typedef unsigned int tetra;
 /* for systems conforming to the LP-64 data model */
typedef struct { tetra h,l;} octa; /* two tetrabytes make one octabyte */

@ @d sign_bit ((unsigned)0x80000000)

@<Glob...@>=
octa zero_octa; /* |zero_octa.h=zero_octa.l=0| */
octa neg_one={-1,-1}; /* |neg_one.h=neg_one.l=-1| */
octa inf_octa={0x7ff00000,0}; /* floating point $+\infty$ */
octa standard_NaN={0x7ff80000,0}; /* floating point NaN(.5) */

@ It's easy to add and subtract octabytes, if we aren't terribly
worried about speed.

@<Subr...@>=
octa oplus @,@,@[ARGS((octa,octa))@];@+@t}\6{@>
octa oplus(y,z) /* compute $y+z$ */
  octa y,z;
{@+ octa x;
  x.h=y.h+z.h;@+
  x.l=y.l+z.l;
  if (x.l<y.l) x.h++;
  return x;
}
@#
octa ominus @,@,@[ARGS((octa,octa))@];@+@t}\6{@>
octa ominus(y,z) /* compute $y-z$ */
  octa y,z;
{@+ octa x;
  x.h=y.h-z.h;@+
  x.l=y.l-z.l;
  if (x.l>y.l) x.h--;
  return x;
}

@ In the following subroutine, |delta| is a signed quantity that is
assumed to fit in a signed tetrabyte.

@<Subr...@>=
octa incr @,@,@[ARGS((octa,int))@];@+@t}\6{@>
octa incr(y,delta) /* compute $y+\delta$ */
  octa y;
  int delta;
{@+ octa x;
  x.h=y.h;@+ x.l=y.l+delta;
  if (delta>=0) {
    if (x.l<y.l) x.h++;
  }@+else if (x.l>y.l) x.h--;
  return x;
}

@ Left and right shifts are only a bit more difficult.

@<Subr...@>=
octa shift_left @,@,@[ARGS((octa,int))@];@+@t}\6{@>
octa shift_left(y,s) /* shift left by $s$ bits, where $0\le s\le64$ */
  octa y;
  int s;
{
  while (s>=32) y.h=y.l,y.l=0,s-=32;
  if (s) {@+register tetra yhl=y.h<<s,ylh=y.l>>(32-s);
    y.h=yhl+ylh;@+ y.l<<=s;
  }
  return y;
}
@#
octa shift_right @,@,@[ARGS((octa,int,int))@];@+@t}\6{@>
octa shift_right(y,s,u) /* shift right, arithmetically if $u=0$ */
  octa y;
  int s,u;
{
  while (s>=32) y.l=y.h, y.h=(u?0: -(y.h>>31)), s-=32;
  if (s) {@+register tetra yhl=y.h<<(32-s),ylh=y.l>>s;
    y.h=(u? 0:(-(y.h>>31))<<(32-s))+(y.h>>s);@+ y.l=yhl+ylh;
  }
  return y;
}

@* Multiplication. We need to multiply two unsigned 64-bit integers, obtaining
an unsigned 128-bit product. It is easy to do this on a 32-bit machine
by using Algorithm 4.3.1M of {\sl Seminumerical Algorithms}, with $b=2^{16}$.
@^multiprecision multiplication@>

The following subroutine returns the lower half of the product, and
puts the upper half into a global octabyte called |aux|.

@<Subr...@>=
octa omult @,@,@[ARGS((octa,octa))@];@+@t}\6{@>
octa omult(y,z)
  octa y,z;
{
  register int i,j,k;
  tetra u[4],v[4],w[8];
  register tetra t;
  octa acc;
  @<Unpack the multiplier and multiplicand to |u| and |v|@>;
  for (j=0;j<4;j++) w[j]=0;
  for (j=0;j<4;j++)
    if (!v[j]) w[j+4]=0;
    else {
      for (i=k=0;i<4;i++) {
        t=u[i]*v[j]+w[i+j]+k;
        w[i+j]=t&0xffff, k=t>>16;
      }
      w[j+4]=k;
    }
  @<Pack |w| into the outputs |aux| and |acc|@>;
  return acc;
}

@ @<Glob...@>=
octa aux; /* secondary output of subroutines with multiple outputs */
bool overflow; /* set by certain subroutines for signed arithmetic */

@ @<Unpack the mult...@>=
u[3]=y.h>>16, u[2]=y.h&0xffff, u[1]= y.l>>16, u[0]=y.l&0xffff;
v[3]=z.h>>16, v[2]=z.h&0xffff, v[1]= z.l>>16, v[0]=z.l&0xffff;

@ @<Pack |w| into the outputs |aux| and |acc|@>=
aux.h=(w[7]<<16)+w[6], aux.l=(w[5]<<16)+w[4];
acc.h=(w[3]<<16)+w[2], acc.l=(w[1]<<16)+w[0];

@ Signed multiplication has the same lower half product as unsigned
multiplication. The signed upper half product is obtained with at most two
further subtractions, after which the result has overflowed if and only if
the upper half is unequal to 64 copies of the sign bit in the lower half.

@<Subr...@>=
octa signed_omult @,@,@[ARGS((octa,octa))@];@+@t}\6{@>
octa signed_omult(y,z)
  octa y,z;
{
  octa acc;
  acc=omult(y,z);
  if (y.h&sign_bit) aux=ominus(aux,z);
  if (z.h&sign_bit) aux=ominus(aux,y);
  overflow=(aux.h!=aux.l || (aux.h^(aux.h>>1)^(acc.h&sign_bit)));
  return acc;
}  

@* Division. Long division of an unsigned 128-bit integer by an unsigned
64-bit integer is, of course, one of the most challenging routines
needed for \MMIX\ arithmetic. The following program, based on
Algorithm 4.3.1D of {\sl Seminumerical Algorithms}, computes
octabytes $q$ and $r$ such that $(2^{64}x+y)=qz+r$ and $0\le r<z$,
given octabytes $x$, $y$, and~$z$, assuming that $x<z$.
(If $x\ge z$, it simply sets $q=x$ and $r=y$.)
The quotient~$q$ is returned by the subroutine;
the remainder~$r$ is stored in~|aux|.
@^multiprecision division@>

@<Subr...@>=
octa odiv @,@,@[ARGS((octa,octa,octa))@];@+@t}\6{@>
octa odiv(x,y,z)
  octa x,y,z;
{
  register int i,j,k,n,d;
  tetra u[8],v[4],q[4],mask,qhat,rhat,vh,vmh;
  register tetra t;
  octa acc;
  @<Check that |x<z|; otherwise give trivial answer@>;
  @<Unpack the dividend and divisor to |u| and |v|@>;
  @<Determine the number of significant places |n| in the divisor |v|@>;
  @<Normalize the divisor@>;
  for (j=3;j>=0;j--) @<Determine the quotient digit |q[j]|@>;
  @<Unnormalize the remainder@>;
  @<Pack |q| and |u| to |acc| and |aux|@>;
  return acc;
}

@ @<Check that |x<z|; otherwise give trivial answer@>=
if (x.h>z.h || (x.h==z.h && x.l>=z.l)) {
  aux=y;@+ return x;
}

@ @<Unpack the div...@>=
u[7]=x.h>>16, u[6]=x.h&0xffff, u[5]=x.l>>16, u[4]=x.l&0xffff;
u[3]=y.h>>16, u[2]=y.h&0xffff, u[1]=y.l>>16, u[0]=y.l&0xffff;
v[3]=z.h>>16, v[2]=z.h&0xffff, v[1]=z.l>>16, v[0]=z.l&0xffff;

@ @<Determine the number of significant places |n| in the divisor |v|@>=
for (n=4;v[n-1]==0;n--);  

@ We shift |u| and |v| left by |d| places, where |d| is chosen to
make $2^{15}\le v_{n-1}<2^{16}$.

@<Normalize the divisor@>=
vh=v[n-1];
for (d=0;vh<0x8000;d++,vh<<=1);
for (j=k=0; j<n+4; j++) {
  t=(u[j]<<d)+k;
  u[j]=t&0xffff, k=t>>16;
}
for (j=k=0; j<n; j++) {
  t=(v[j]<<d)+k;
  v[j]=t&0xffff, k=t>>16;
}
vh=v[n-1];
vmh=(n>1? v[n-2]: 0);

@ @<Unnormalize the remainder@>=
mask=(1<<d)-1;
for (j=3; j>=n; j--) u[j]=0;
for (k=0;j>=0;j--) {
  t=(k<<16)+u[j];
  u[j]=t>>d, k=t&mask;
}

@ @<Pack |q| and |u| to |acc| and |aux|@>=
acc.h=(q[3]<<16)+q[2], acc.l=(q[1]<<16)+q[0];
aux.h=(u[3]<<16)+u[2], aux.l=(u[1]<<16)+u[0];

@ @<Determine the quotient digit |q[j]|@>=
{
  @<Find the trial quotient, $\hat q$@>;
  @<Subtract $b^j\hat q v$ from |u|@>;
  @<If the result was negative, decrease $\hat q$ by 1@>;
  q[j]=qhat;
}

@ @<Find the trial quotient, $\hat q$@>=
t=(u[j+n]<<16)+u[j+n-1];
qhat=t/vh, rhat=t-vh*qhat;
if (n>1) while (qhat==0x10000 || qhat*vmh>(rhat<<16)+u[j+n-2]) {
  qhat--, rhat+=vh;
  if (rhat>=0x10000) break;
}

@ After this step, |u[j+n]| will either equal |k| or |k-1|. The
true value of~|u| would be obtained by subtracting~|k| from |u[j+n]|;
but we don't have to fuss over |u[j+n]|, because it won't be examined later.

@<Subtract $b^j\hat q v$ from |u|@>=
for (i=k=0; i<n; i++) {
  t=u[i+j]+0xffff0000-k-qhat*v[i];
  u[i+j]=t&0xffff, k=0xffff-(t>>16);
}

@ The correction here occurs only rarely, but it can be necessary---for
example, when dividing the number \Hex{7fff800100000000} by \Hex{800080020005}.

@<If the result was negative, decrease $\hat q$ by 1@>=
if (u[j+n]!=(tetra)k) {
  qhat--;
  for (i=k=0; i<n; i++) {
    t=u[i+j]+v[i]+k;
    u[i+j]=t&0xffff, k=t>>16;
  }
}

@ Signed division can be reduced to unsigned division in a tedious
but straightforward manner. We assume that the divisor isn't zero.

@<Subr...@>=
octa signed_odiv @,@,@[ARGS((octa,octa))@];@+@t}\6{@>
octa signed_odiv(y,z)
  octa y,z;
{
  octa yy,zz,q;
  register int sy,sz;
  if (y.h&sign_bit) sy=2, yy=ominus(zero_octa,y);
  else sy=0, yy=y;
  if (z.h&sign_bit) sz=1, zz=ominus(zero_octa,z);
  else sz=0, zz=z;
  q=odiv(zero_octa,yy,zz);
  overflow=false;
  switch (sy+sz) {
 case 2+1: aux=ominus(zero_octa,aux);
  if (q.h==sign_bit) overflow=true;
 case 0+0: default: return q;
 case 2+0:@+ if (aux.h || aux.l) aux=ominus(zz,aux);
  goto negate_q;
 case 0+1:@+ if (aux.h || aux.l) aux=ominus(aux,zz);
 negate_q:@+ if (aux.h || aux.l) return ominus(neg_one,q);
  else return ominus(zero_octa,q);
  }
}

@* Bit fiddling. The bitwise operators of \MMIX\ are fairly easy to
implement directly, but three of them occur often enough to deserve
packaging as subroutines.

@<Subr...@>=
octa oand @,@,@[ARGS((octa,octa))@];@+@t}\6{@>
octa oand(y,z) /* compute $y\land z$ */
  octa y,z;
{@+ octa x;
  x.h=y.h&z.h;@+ x.l=y.l&z.l;
  return x;
}
@#
octa oandn @,@,@[ARGS((octa,octa))@];@+@t}\6{@>
octa oandn(y,z) /* compute $y\land\bar z$ */
  octa y,z;
{@+ octa x;
  x.h=y.h&~z.h;@+ x.l=y.l&~z.l;
  return x;
}
@#
octa oxor @,@,@[ARGS((octa,octa))@];@+@t}\6{@>
octa oxor(y,z) /* compute $y\oplus z$ */
  octa y,z;
{@+ octa x;
  x.h=y.h^z.h;@+ x.l=y.l^z.l;
  return x;
}

@ Here's a fun way to count the number of bits in a tetrabyte.
[This classical trick is called the ``Gillies--Miller method
for sideways addition'' in {\sl The Preparation of Programs
for an Electronic Digital Computer\/} by Wilkes, Wheeler, and
Gill, second edition (Reading, Mass.:\ Addison--Wesley, 1957),
191--193. Some of the tricks used here were suggested by
Balbir Singh, Peter Rossmanith, and Stefan Schwoon.]
@^Gillies, Donald Bruce@>
@^Miller, Jeffrey Charles Percy@>
@^Wilkes, Maurice Vincent@>
@^Wheeler, David John@>
@^Gill, Stanley@>
@^Singh, Balbir@>
@^Rossmanith, Peter@>
@^Schwoon, Stefan@>

@<Subr...@>=
int count_bits @,@,@[ARGS((tetra))@];@+@t}\6{@>
int count_bits(x)
  tetra x;
{
  register int xx=x;
  xx=xx-((xx>>1)&0x55555555);
  xx=(xx&0x33333333)+((xx>>2)&0x33333333);
  xx=(xx+(xx>>4))&0x0f0f0f0f;
  xx=xx+(xx>>8);
  return (xx+(xx>>16)) & 0xff;
}

@ To compute the nonnegative byte differences of two given tetrabytes,
we can carry out the following 20-step branchless computation:

@<Subr...@>=
tetra byte_diff @,@,@[ARGS((tetra,tetra))@];@+@t}\6{@>
tetra byte_diff(y,z)
  tetra y,z;
{
  register tetra d=(y&0x00ff00ff)+0x01000100-(z&0x00ff00ff);
  register tetra m=d&0x01000100;
  register tetra x=d&(m-(m>>8));
  d=((y>>8)&0x00ff00ff)+0x01000100-((z>>8)&0x00ff00ff);
  m=d&0x01000100;
  return x+((d&(m-(m>>8)))<<8);
}

@ To compute the nonnegative wyde differences of two tetrabytes,
another trick leads to a 15-step branchless computation.
(Research problem: Can |count_bits|, |byte_diff|, or |wyde_diff| be done
with fewer operations?)

@<Subr...@>=
tetra wyde_diff @,@,@[ARGS((tetra,tetra))@];@+@t}\6{@>
tetra wyde_diff(y,z)
  tetra y,z;
{
  register tetra a=((y>>16)-(z>>16))&0x10000;
  register tetra b=((y&0xffff)-(z&0xffff))&0x10000;
  return y-(z^((y^z)&(b-a-(b>>16))));
}

@ The last bitwise subroutine we need is the most interesting:
It implements \MMIX's \.{MOR} and \.{MXOR} operations.

@<Subr...@>=
octa bool_mult @,@,@[ARGS((octa,octa,bool))@];@+@t}\6{@>
octa bool_mult(y,z,xor)
  octa y,z; /* the operands */
  bool xor; /* do we do xor instead of or? */
{
  octa o,x;
  register tetra a,b,c;
  register int k;
  for (k=0,o=y,x=zero_octa;o.h||o.l;k++,o=shift_right(o,8,1))
    if (o.l&0xff) {
      a=((z.h>>k)&0x01010101)*0xff;
      b=((z.l>>k)&0x01010101)*0xff;
      c=(o.l&0xff)*0x01010101;
      if (xor) x.h^=a&c, x.l^=b&c;
      else x.h|=a&c, x.l|=b&c;
    }
  return x;
}  

@* Floating point packing and unpacking. Standard IEEE floating binary
numbers pack a sign, exponent, and fraction into a tetrabyte
or octabyte. In this section we consider basic subroutines that
convert between IEEE format and the separate unpacked components.

@d ROUND_OFF 1
@d ROUND_UP 2
@d ROUND_DOWN 3
@d ROUND_NEAR 4

@<Glob...@>=
int cur_round; /* the current rounding mode */

@ The |fpack| routine takes an octabyte $f$, a raw exponent~$e$,
and a sign~|s|, and packs them
into the floating binary number that corresponds to
$\pm2^{e-1076}f$, using a given rounding mode.
The value of $f$ should satisfy $2^{54}\le f\le 2^{55}$.

Thus, for example, the floating binary number $+1.0=\Hex{3ff0000000000000}$
is obtained when $f=2^{54}$, $e=\Hex{3fe}$, and |s='+'|.
The raw exponent~$e$ is usually one less than
the final exponent value; the leading bit of~$f$ is essentially added
to the exponent. (This trick works nicely for subnormal numbers, when
$e<0$, or in cases where the value of $f$ is rounded upwards to $2^{55}$.)

Exceptional events are noted by oring appropriate bits into
the global variable |exceptions|. Special considerations apply to
underflow, which is not fully specified by Section 7.4 of the IEEE standard:
Implementations of the standard are free to choose between two definitions
of ``tininess'' and two definitions of ``accuracy loss.''
\MMIX\ determines tininess {\it after\/} rounding, hence a result with
$e<0$ is not necessarily tiny; \MMIX\ treats accuracy loss as equivalent
to inexactness. Thus, a result underflows if and only if
it is tiny and either (i)~it is inexact or (ii)~the underflow trap is enabled.
The |fpack| routine sets |U_BIT| in |exceptions| if and only if the result is
tiny, |X_BIT| if and only if the result is inexact.
@^underflow@>
@^tininess@>
@^accuracy loss@>

@d X_BIT (1<<8) /* floating inexact */
@d Z_BIT (1<<9) /* floating division by zero */
@d U_BIT (1<<10) /* floating underflow */
@d O_BIT (1<<11) /* floating overflow */
@d I_BIT (1<<12) /* floating invalid operation */
@d W_BIT (1<<13) /* float-to-fix overflow */
@d V_BIT (1<<14) /* integer overflow */
@d D_BIT (1<<15) /* integer divide check */
@d E_BIT (1<<18) /* external (dynamic) trap bit */

@<Subr...@>=
octa fpack @,@,@[ARGS((octa,int,int,int))@];@+@t}\6{@>
octa fpack(f,e,s,r)
  octa f; /* the normalized fraction part */
  int e; /* the raw exponent */
  int s; /* the sign */
  int r; /* the rounding mode */
{
  octa o;
  if (e>0x7fd) e=0x7ff, o=zero_octa;
  else {
    if (e<0) {
      if (e<-54) o.h=0, o.l=1;
      else {@+octa oo;
        o=shift_right(f,-e,1);
        oo=shift_left(o,-e);
        if (oo.l!=f.l || oo.h!=f.h) o.l |= 1; /* sticky bit */
@^sticky bit@>
      }
      e=0;
    }@+else o=f;
  }
  @<Round and return the result@>;
}

@ @<Glob...@>=
int exceptions; /* bits possibly destined for rA */

@ Everything falls together so nicely here, it's almost too good to be true!

@<Round and return the result@>=
if (o.l&3) exceptions |= X_BIT;
switch (r) {
 case ROUND_DOWN:@+ if (s=='-') o=incr(o,3);@+break;
 case ROUND_UP:@+ if (s!='-') o=incr(o,3);
 case ROUND_OFF: break;
 case ROUND_NEAR: o=incr(o, o.l&4? 2: 1);@+break;
}
o = shift_right(o,2,1);
o.h += e<<20;
if (o.h>=0x7ff00000) exceptions |= O_BIT+X_BIT; /* overflow */
else if (o.h<0x100000) exceptions |= U_BIT; /* tininess */
if (s=='-') o.h |= sign_bit;
return o;

@ Similarly, |sfpack| packs a short float, from inputs
having the same conventions as |fpack|.

@<Subr...@>=
tetra sfpack @,@,@[ARGS((octa,int,int,int))@];@+@t}\6{@>
tetra sfpack(f,e,s,r)
  octa f; /* the fraction part */
  int e; /* the raw exponent */
  int s; /* the sign */
  int r; /* the rounding mode */
{
  register tetra o;
  if (e>0x47d) e=0x47f, o=0;
  else {
    o=shift_left(f,3).h;
    if (f.l&0x1fffffff) o|=1;
    if (e<0x380) {
      if (e<0x380-25) o=1;
      else {@+register tetra o0,oo;
        o0 = o;
        o = o>>(0x380-e);
        oo = o<<(0x380-e);
        if (oo!=o0) o |= 1; /* sticky bit */
@^sticky bit@>
      }
      e=0x380;
    }
  }
  @<Round and return the short result@>;
}

@ @<Round and return the short result@>=
if (o&3) exceptions |= X_BIT;
switch (r) {
 case ROUND_DOWN:@+ if (s=='-') o+=3;@+break;
 case ROUND_UP:@+ if (s!='-') o+=3;
 case ROUND_OFF: break;
 case ROUND_NEAR: o+=(o&4? 2: 1);@+break;
}
o = o>>2;
o += (e-0x380)<<23;
if (o>=0x7f800000) exceptions |= O_BIT+X_BIT; /* overflow */
else if (o<0x100000) exceptions |= U_BIT; /* tininess */
if (s=='-') o |= sign_bit;
return o;

@ The |funpack| routine is, roughly speaking, the opposite of |fpack|.
It takes a given floating point number~$x$ and separates out its
fraction part~$f$, exponent~$e$, and sign~$s$. It clears |exceptions|
to zero. It returns the type of value found: |zro|, |num|, |inf|,
or |nan|. When it returns |num|,
it will have set $f$, $e$, and~$s$
to the values from which |fpack| would produce the original number~$x$
without exceptions.

@d zero_exponent (-1000) /* zero is assumed to have this exponent */

@<Other type...@>=
typedef enum {@!zro,@!num,@!inf,@!nan}@+ftype;
 
@ @<Subr...@>=
ftype funpack @,@,@[ARGS((octa,octa*,int*,char*))@];@+@t}\6{@>
ftype funpack(x,f,e,s)
  octa x; /* the given floating point value */
  octa *f; /* address where the fraction part should be stored */
  int *e; /* address where the exponent part should be stored */
  char *s; /* address where the sign should be stored */
{
  register int ee;
  exceptions=0;
  *s=(x.h&sign_bit? '-': '+');
  *f=shift_left(x,2);
  f->h &= 0x3fffff;
  ee=(x.h>>20)&0x7ff;
  if (ee) {
    *e=ee-1;
    f->h |= 0x400000;
    return (ee<0x7ff? num: f->h==0x400000 && !f->l? inf: nan);
  }
  if (!x.l && !f->h) {
    *e=zero_exponent;@+ return zro;
  }
  do {@+ ee--;@+ *f=shift_left(*f,1);@+} while (!(f->h&0x400000));
  *e=ee;@+ return num;
}    

@ @<Subr...@>=
ftype sfunpack @,@,@[ARGS((tetra,octa*,int*,char*))@];@+@t}\6{@>
ftype sfunpack(x,f,e,s)
  tetra x; /* the given floating point value */
  octa *f; /* address where the fraction part should be stored */
  int *e; /* address where the exponent part should be stored */
  char *s; /* address where the sign should be stored */
{
  register int ee;
  exceptions=0;
  *s=(x&sign_bit? '-': '+');
  f->h=(x>>1)&0x3fffff, f->l=x<<31;
  ee=(x>>23)&0xff;
  if (ee) {
    *e=ee+0x380-1;
    f->h |= 0x400000;
    return (ee<0xff? num: (x&0x7fffffff)==0x7f800000? inf: nan);
  }
  if (!(x&0x7fffffff)) {
    *e=zero_exponent;@+return zro;
  }
  do {@+ ee--;@+ *f=shift_left(*f,1);@+} while (!(f->h&0x400000));
  *e=ee+0x380;@+ return num;
}    

@ Since \MMIX\ downplays 32-bit operations, it uses |sfpack| and |sfunpack|
only when loading and storing short floats, or when converting
from fixed point to floating point.

@<Subr...@>=
octa load_sf @,@,@[ARGS((tetra))@];@+@t}\6{@>
octa load_sf(z)
  tetra z; /* 32 bits to be loaded into a 64-bit register */
{
  octa f,x;@+int e;@+char s;@+ftype t;
  t=sfunpack(z,&f,&e,&s);
  switch (t) {
 case zro: x=zero_octa;@+break;
 case num: return fpack(f,e,s,ROUND_OFF);
 case inf: x=inf_octa;@+break;
 case nan: x=shift_right(f,2,1);@+x.h|=0x7ff00000;@+break;
  }
  if (s=='-') x.h|=sign_bit;
  return x;
}

@ @<Subr...@>=
tetra store_sf @,@,@[ARGS((octa))@];@+@t}\6{@>
tetra store_sf(x)
  octa x; /* 64 bits to be loaded into a 32-bit word */
{
  octa f;@+tetra z;@+int e;@+char s;@+ftype t;
  t=funpack(x,&f,&e,&s);
  switch (t) {
 case zro: z=0;@+break;
 case num: return sfpack(f,e,s,cur_round);
 case inf: z=0x7f800000;@+break;
 case nan:@+ if (!(f.h&0x200000)) {
      f.h|=0x200000;@+exceptions|=I_BIT; /* NaN was signaling */
    }
    z=0x7f800000|(f.h<<1)|(f.l>>31);@+break;
  }
  if (s=='-') z|=sign_bit;
  return z;
}

@* Floating multiplication and division.
The hardest fixed point operations were multiplication and division;
but these two operations are the {\it easiest\/} to implement in floating point
arithmetic, once their fixed point counterparts are available.

@<Subr...@>=
octa fmult @,@,@[ARGS((octa,octa))@];@+@t}\6{@>
octa fmult(y,z)
  octa y,z;
{
  ftype yt,zt;
  int ye,ze;
  char ys,zs;
  octa x,xf,yf,zf;
  register int xe;
  register char xs;
  yt=funpack(y,&yf,&ye,&ys);
  zt=funpack(z,&zf,&ze,&zs);
  xs=ys+zs-'+'; /* will be |'-'| when the result is negative */
  switch (4*yt+zt) {
 @t\4@>@<The usual NaN cases@>;
 default: case 4*zro+zro: case 4*zro+num: case 4*num+zro: x=zero_octa;@+break;
 case 4*num+inf: case 4*inf+num: case 4*inf+inf: x=inf_octa;@+break;
 case 4*zro+inf: case 4*inf+zro: x=standard_NaN;
  exceptions|=I_BIT;@+break;
 case 4*num+num: @<Multiply nonzero numbers and |return|@>;
  }
  if (xs=='-') x.h|=sign_bit;
  return x;
}

@ @<The usual NaN cases@>=
case 4*nan+nan:@+if (!(y.h&0x80000)) exceptions|=I_BIT; /* |y| is signaling */
case 4*zro+nan: case 4*num+nan: case 4*inf+nan:
  if (!(z.h&0x80000)) exceptions|=I_BIT, z.h|=0x80000;
  return z;
case 4*nan+zro: case 4*nan+num: case 4*nan+inf:
  if (!(y.h&0x80000)) exceptions|=I_BIT, y.h|=0x80000;
  return y;

@ @<Multiply nonzero numbers and |return|@>=
xe=ye+ze-0x3fd; /* the raw exponent */
x=omult(yf,shift_left(zf,9));
if (aux.h>=0x400000) xf=aux;
else xf=shift_left(aux,1), xe--;
if (x.h||x.l) xf.l|=1; /* adjust the sticky bit */
return fpack(xf,xe,xs,cur_round);

@ @<Subr...@>=
octa fdivide @,@,@[ARGS((octa,octa))@];@+@t}\6{@>
octa fdivide(y,z)
  octa y,z;
{
  ftype yt,zt;
  int ye,ze;
  char ys,zs;
  octa x,xf,yf,zf;
  register int xe;
  register char xs;
  yt=funpack(y,&yf,&ye,&ys);
  zt=funpack(z,&zf,&ze,&zs);
  xs=ys+zs-'+'; /* will be |'-'| when the result is negative */
  switch (4*yt+zt) {
 @t\4@>@<The usual NaN cases@>;
 case 4*zro+inf: case 4*zro+num: case 4*num+inf: x=zero_octa;@+break;
 case 4*num+zro: exceptions|=Z_BIT;
 case 4*inf+num: case 4*inf+zro: x=inf_octa;@+break;
 default: case 4*zro+zro: case 4*inf+inf: x=standard_NaN;
  exceptions|=I_BIT;@+break;
 case 4*num+num: @<Divide nonzero numbers and |return|@>;
  }
  if (xs=='-') x.h|=sign_bit;
  return x;
}

@ @<Divide nonzero numbers...@>=
xe=ye-ze+0x3fd; /* the raw exponent */
xf=odiv(yf,zero_octa,shift_left(zf,9));
if (xf.h>=0x800000) {
  aux.l|=xf.l&1;
  xf=shift_right(xf,1,1);
  xe++;
}
if (aux.h||aux.l) xf.l|=1; /* adjust the sticky bit */
return fpack(xf,xe,xs,cur_round);

@*Floating addition and subtraction. Now for the bread-and-butter
operation, the sum of two floating point numbers.
It is not terribly difficult, but many cases need to be handled carefully.

@<Subr...@>=
octa fplus @,@,@[ARGS((octa,octa))@];@+@t}\6{@>
octa fplus(y,z)
  octa y,z;
{
  ftype yt,zt;
  int ye,ze;
  char ys,zs;
  octa x,xf,yf,zf;
  register int xe,d;
  register char xs;
  yt=funpack(y,&yf,&ye,&ys);
  zt=funpack(z,&zf,&ze,&zs);
  switch (4*yt+zt) {
 @t\4@>@<The usual NaN cases@>;
 case 4*zro+num: return fpack(zf,ze,zs,ROUND_OFF);@+break; /* may underflow */
 case 4*num+zro: return fpack(yf,ye,ys,ROUND_OFF);@+break; /* may underflow */
 case 4*inf+inf:@+if (ys!=zs) {
    exceptions|=I_BIT;@+x=standard_NaN;@+xs=zs;@+break;
  }
 case 4*num+inf: case 4*zro+inf: x=inf_octa;@+xs=zs;@+break;
 case 4*inf+num: case 4*inf+zro: x=inf_octa;@+xs=ys;@+break;
 case 4*num+num:@+ if (y.h!=(z.h^0x80000000) || y.l!=z.l) 
   @<Add nonzero numbers and |return|@>;
 default: case 4*zro+zro: x=zero_octa;
  xs=(ys==zs? ys: cur_round==ROUND_DOWN? '-': '+');@+break;
  }
  if (xs=='-') x.h|=sign_bit;
  return x;
}

@ @<Add nonzero numbers...@>=
{@+octa o,oo;
  if (ye<ze || (ye==ze && (yf.h<zf.h || (yf.h==zf.h && yf.l<zf.l))))
    @<Exchange |y| with |z|@>;
  d=ye-ze;
  xs=ys, xe=ye;
  if (d) @<Adjust for difference in exponents@>;
  if (ys==zs) {
    xf=oplus(yf,zf);
    if (xf.h>=0x800000) xe++, d=xf.l&1, xf=shift_right(xf,1,1), xf.l|=d;
  }@+else {
    xf=ominus(yf,zf);
    if (xf.h>=0x800000) xe++, d=xf.l&1, xf=shift_right(xf,1,1), xf.l|=d;
    else@+ while (xf.h<0x400000) xe--, xf=shift_left(xf,1);
  }
  return fpack(xf,xe,xs,cur_round);
}

@ @<Exchange |y| with |z|@>=
{
  o=yf, yf=zf, zf=o;
  d=ye, ye=ze, ze=d;
  d=ys, ys=zs, zs=d;
}

@ Proper rounding requires two bits to the right of the fraction delivered
to~|fpack|. The first is the true next bit of the result;
the other is a ``sticky'' bit, which is nonzero if any further bits of the
true result are nonzero. Sticky rounding to an integer takes
$x$ into the number $\lfloor x/2\rfloor+\lceil x/2\rceil$.
@^sticky bit@>

Some subtleties need to be observed here, in order to
prevent the sticky bit from being shifted left. If we did not
shift |yf| left~1 before shifting |zf| to the right, an incorrect
answer would be obtained in certain cases---for example, if
$|yf|=2^{54}$, $|zf|=2^{54}+2^{53}-4$, $d=52$.

@<Adjust for difference in exponents@>=
{
  if (d<=2) zf=shift_right(zf,d,1); /* exact result */
  else if (d>54) zf.h=0, zf.l=1; /* tricky but OK */
  else {
    if (ys!=zs) d--,xe--,yf=shift_left(yf,1);
    o=zf;
    zf=shift_right(o,d,1);
    oo=shift_left(zf,d);
    if (oo.l!=o.l || oo.h!=o.h) zf.l|=1;
  }
}

@ The comparison of floating point numbers with respect to $\epsilon$
shares some of the characteristics of floating point addition/subtraction.
In some ways it is simpler, and in other ways it is more difficult;
we might as well deal with it now. % anyways

Subroutine |fepscomp(y,z,e,s)| returns 2 if |y|, |z|, or |e| is a NaN
or |e| is negative. It returns 1 if |s=0| and $y\approx z\ (e)$ or if
|s!=0| and $y\sim z\ (e)$,
as defined in Section~4.2.2 of {\sl Seminumerical Algorithms\/};
otherwise it returns~0.

@<Subr...@>=
int fepscomp @,@,@[ARGS((octa,octa,octa,int))@];@+@t}\6{@>
int fepscomp(y,z,e,s)
  octa y,z,e; /* the operands */
  int s; /* test similarity? */
{
  octa yf,zf,ef,o,oo;
  int ye,ze,ee;
  char ys,zs,es;
  register int yt,zt,et,d;
  et=funpack(e,&ef,&ee,&es);
  if (es=='-') return 2;
  switch (et) {
 case nan: return 2;
 case inf: ee=10000;
 case num: case zro: break;
  }
  yt=funpack(y,&yf,&ye,&ys);
  zt=funpack(z,&zf,&ze,&zs);
  switch (4*yt+zt) {
 case 4*nan+nan: case 4*nan+inf: case 4*nan+num: case 4*nan+zro:
 case 4*inf+nan: case 4*num+nan: case 4*zro+nan: return 2;
 case 4*inf+inf: return (ys==zs || ee>=1023);
 case 4*inf+num: case 4*inf+zro: case 4*num+inf: case 4*zro+inf:
   return (s && ee>=1022);
 case 4*zro+zro: return 1;
 case 4*zro+num: case 4*num+zro:@+ if (!s) return 0;
 case 4*num+num: break;
  }
  @<Compare two numbers with respect to epsilon and |return|@>;
}

@ The relation $y\approx z\ (\epsilon)$ reduces to 
$y\sim z\ (\epsilon/2^d)$, if $d$~is the difference between the
larger and smaller exponents of $y$ and~$z$.

@<Compare two numbers with respect to epsilon and |return|@>=
@<Unsubnormalize |y| and |z|, if they are subnormal@>;
if (ye<ze || (ye==ze && (yf.h<zf.h || (yf.h==zf.h && yf.l<zf.l))))
  @<Exchange |y| with |z|@>;
if (ze==zero_exponent) ze=ye;
d=ye-ze;
if (!s) ee-=d;
if (ee>=1023) return 1; /* if $\epsilon\ge2$, $z\in N_\epsilon(y)$ */
@<Compute the difference of fraction parts, |o|@>;
if (!o.h && !o.l) return 1;
if (ee<968) return 0; /* if $y\ne z$ and $\epsilon<2^{-54}$, $y\not\sim z$ */
if (ee>=1021) ef=shift_left(ef,ee-1021);
else ef=shift_right(ef,1021-ee,1);
return o.h<ef.h || (o.h==ef.h && o.l<=ef.l);

@ @<Unsubnormalize |y| and |z|, if they are subnormal@>=
if (ye<0 && yt!=zro) yf=shift_left(y,2), ye=0;
if (ze<0 && zt!=zro) zf=shift_left(z,2), ze=0;

@ At this point $y\sim z$ if and only if
$$|yf|+(-1)^{[ys=zs]}|zf|/2^d\le 2^{ee-1021}|ef|=2^{55}\epsilon.$$
We need to evaluate this relation without overstepping the bounds of
our simulated 64-bit registers.

When $d>2$, the difference of fraction parts might not fit exactly
in an octabyte;
in that case the numbers are not similar unless $\epsilon>3/8$,
and we replace the difference by the ceiling of the
true result. When $\epsilon<1/8$, our program essentially replaces
$2^{55}\epsilon$ by $\lfloor2^{55}\epsilon\rfloor$. These
truncations are not needed simultaneously. Therefore the logic
is justified by the facts that, if $n$ is an integer, we have
$x\le n$ if and only if $\lceil x\rceil\le n$;
$n\le x$ if and only if $n\le\lfloor x\rfloor$. (Notice that the
concept of ``sticky bit'' is {\it not\/} appropriate here.)
@^sticky bit@>

@<Compute the difference of fraction parts, |o|@>=
if (d>54) o=zero_octa,oo=zf;
else o=shift_right(zf,d,1),oo=shift_left(o,d);
if (oo.h!=zf.h || oo.l!=zf.l) { /* truncated result, hence $d>2$ */
  if (ee<1020) return 0; /* difference is too large for similarity */
  o=incr(o,ys==zs? 0: 1); /* adjust for ceiling */
}
o=(ys==zs? ominus(yf,o): oplus(yf,o));

@*Floating point output conversion.
The |print_float| routine converts an octabyte to a floating decimal
representation that will be input as precisely the same value.
@^binary-to-decimal conversion@>
@^radix conversion@>
@^multiprecision conversion@>

@<Subr...@>=
static void bignum_times_ten @,@,@[ARGS((bignum*))@];
static void bignum_dec @,@,@[ARGS((bignum*,bignum*,tetra))@];
static int bignum_compare @,@,@[ARGS((bignum*,bignum*))@];
void print_float @,@,@[ARGS((octa))@];@+@t}\6{@>
void print_float(x)
  octa x;
{
  @<Local variables for |print_float|@>;
  if (x.h&sign_bit) printf("-");
  @<Extract the exponent |e| and determine the
       fraction interval $[f\dts g]$ or $(f\dts g)$@>;
  @<Store $f$ and $g$ as multiprecise integers@>;
  @<Compute the significant digits |s| and decimal exponent |e|@>;
  @<Print the significant digits with proper context@>;
}

@ One way to visualize the problem being solved here is to consider
the vastly simpler case in which there are only 2-bit exponents
and 2-bit fractions. Then the sixteen possible 4-bit combinations
have the following interpretations:
$$\def\\{\;\dts\;}
\vbox{\halign{#\qquad&$#$\hfil\cr
0000&[0\\0.125]\cr
0001&(0.125\\0.375)\cr
0010&[0.375\\0.625]\cr
0011&(0.625\\0.875)\cr
0100&[0.875\\1.125]\cr
0101&(1.125\\1.375)\cr
0110&[1.375\\1.625]\cr
0111&(1.625\\1.875)\cr
1000&[1.875\\2.25]\cr
1001&(2.25\\2.75)\cr
1010&[2.75\\3.25]\cr
1011&(3.25\\3.75)\cr
1100&[3.75\\\infty]\cr
1101&\rm NaN(0\\0.375)\cr
1110&\rm NaN[0.375\\0.625]\cr
1111&\rm NaN(0.625\\1)\cr}}$$
Notice that the interval is closed, $[f\dts g]$, when the fraction part
is even; it is open, $(f\dts g)$, when the fraction part is odd.
The printed outputs for these sixteen values, if we actually were
dealing with such short exponents and fractions, would be
\.{0.}, \.{.2}, \.{.5}, \.{.7}, \.{1.}, \.{1.2}, \.{1.5}, \.{1.7},
\.{2.}, \.{2.5}, \.{3.}, \.{3.5}, \.{Inf}, \.{NaN.2}, \.{NaN}, \.{NaN.8},
respectively.

@<Extract the exponent |e|...@>=
f=shift_left(x,1);
e=f.h>>21;
f.h&=0x1fffff;
if (!f.h && !f.l) @<Handle the special case when the fraction part is zero@>@;
else {
  g=incr(f,1);
  f=incr(f,-1);
  if (!e) e=1; /* subnormal */
  else if (e==0x7ff) {
    printf("NaN");
    if (g.h==0x100000 && g.l==1) return; /* the ``standard'' NaN */
    e=0x3ff; /* extreme NaNs come out OK even without adjusting |f| or |g| */
  }@+else f.h|=0x200000, g.h|=0x200000;
}

@ @<Local variables for |print_float|@>=
octa f,g; /* lower and upper bounds on the fraction part */
register int e; /* exponent part */
register int j,k; /* all purpose indices */

@ The transition points between exponents correspond to powers of~2. At
such points the interval extends only half as far to the left of that
power of~2 as it does to the right. For example, in the 4-bit minifloat numbers
considered above, case 1000 corresponds to the interval $[1.875\;\dts\;2.25]$.

@<Handle the special case when the fraction part is zero@>=
{
  if (!e) {
    printf("0.");@+return;
  }
  if (e==0x7ff) {
    printf("Inf");@+return;
  }
  e--;
  f.h=0x3fffff, f.l=0xffffffff;
  g.h=0x400000, g.l=2;
}

@ We want to find the ``simplest'' value in the interval corresponding
to the given number, in the sense that it has fewest significant
digits when expressed in decimal notation. Thus, for example,
if the floating point number can be described by a relatively
short string such as `\.{.1}' or `\.{37e100}', we want to discover that
representation.

The basic idea is to generate the decimal representations of the
two endpoints of the interval, outputting the leading digits where
both endpoints agree, then making a final decision at the first place where
they disagree.

The ``simplest'' value is not always unique. For example, in the
case of 4-bit minifloat numbers we could represent the bit pattern 0001 as
either \.{.2} or \.{.3}, and we could represent 1001 in five equally short
ways: \.{2.3} or \.{2.4} or \.{2.5} or \.{2.6} or \.{2.7}. The
algorithm below tries to choose the middle possibility in such cases.

[A solution to the analogous problem for fixed-point representations,
without the additional complication of round-to-even, was used by
the author in the program for \TeX; see {\sl Beauty is Our Business\/}
(Springer, 1990), 233--242.]
@^Knuth, Donald Ervin@>

Suppose we are given two fractions $f$ and $g$, where $0\le f<g<1$, and
we want to compute the shortest decimal in the closed interval $[f\dts g]$.
If $f=0$, we are done. Otherwise let $10f=d+f'$ and $10g=e+g'$, where
$0\le f'<1$ and $0\le g'<1$. If $d<e$, we can terminate by outputting
any of the digits $d+1$, \dots,~$e$; otherwise we output the
common digit $d=e$, and repeat the process on the fractions $0\le f'<g'<1$.
A similar procedure works with respect to the open interval $(f\dts g)$.

@ The program below carries out the stated algorithm by using multiprecision
arithmetic on 77-place integers with 28 bits each. This choice
facilitates multiplication by~10, and allows us to deal with the whole range of
floating binary numbers using fixed point arithmetic. We keep track of
the leading and trailing digit positions so that trivial operations on
zeros are avoided.

If |f| points to a \&{bignum}, its radix-$2^{28}$ digits are
|f->dat[0]| through |f->dat[76]|, from most significant to least significant.
We assume that all digit positions are zero unless they lie in the
subarray between indices |f->a| and |f->b|, inclusive.
Furthermore, both |f->dat[f->a]| and |f->dat[f->b]| are nonzero,
unless |f->a=f->b=bignum_prec-1|.

The \&{bignum} data type can be used with any radix less than
$2^{32}$; we will use it later with radix~$10^9$. The |dat| array
is made large enough to accommodate both applications.

@d bignum_prec 157 /* would be 77 if we cared only about |print_float| */

@<Other type...@>=
typedef struct {
  int a; /* index of the most significant digit */
  int b; /* index of the least significant digit; must be $\ge a$ */
  tetra dat[bignum_prec]; /* the digits; undefined except between |a| and |b| */
} bignum;

@ Here, for example, is how we go from $f$ to $10f$, assuming that
overflow will not occur and that the radix is $2^{28}$:

@<Subr...@>=
static void bignum_times_ten(f)
  bignum *f;
{
  register tetra *p,*q; register tetra x,carry;
  for (p=&f->dat[f->b],q=&f->dat[f->a],carry=0; p>=q; p--) {
    x=*p*10+carry;
    *p=x&0xfffffff;
    carry=x>>28;
  }
  *p=carry;
  if (carry) f->a--;
  if (f->dat[f->b]==0 && f->b>f->a) f->b--;
}

@ And here is how we test whether $f<g$, $f=g$, or $f>g$, using any
radix whatever:

@<Subr...@>=
static int bignum_compare(f,g)
  bignum *f,*g;
{
  register tetra *p,*pp,*q,*qq;
  if (f->a!=g->a) return f->a > g->a? -1: 1;
  pp=&f->dat[f->b], qq=&g->dat[g->b];
  for (p=&f->dat[f->a],q=&g->dat[g->a]; p<=pp; p++,q++) {
    if (*p!=*q) return *p<*q? -1: 1;
    if (q==qq) return p<pp;
  }
  return -1;
}

@ The following subroutine subtracts $g$ from~$f$, assuming that
$f\ge g>0$ and using a given radix.

@<Subr...@>=
static void bignum_dec(f,g,r)
  bignum *f,*g;
  tetra r; /* the radix */
{
  register tetra *p,*q,*qq;
  register int x,borrow;
  while (g->b>f->b) f->dat[++f->b]=0;
  qq=&g->dat[g->a];
  for (p=&f->dat[g->b],q=&g->dat[g->b],borrow=0;q>=qq;p--,q--) {
    x=*p - *q - borrow;
    if (x>=0) borrow=0, *p=x;
    else borrow=1, *p=x+r;
  }
  for (;borrow;p--)
    if (*p) borrow=0, *p=*p-1;
    else *p=r-1;
  while (f->dat[f->a]==0) {
    if (f->a==f->b) { /* the result is zero */
      f->a=f->b=bignum_prec-1, f->dat[bignum_prec-1]=0;
      return;
    }
    f->a++;
  }
  while (f->dat[f->b]==0) f->b--;
}

@ Armed with these subroutines, we are ready to solve the problem.
The first task is to put the numbers into \&{bignum} form.
If the exponent is |e|, the number destined for digit |dat[k]| will
consist of the rightmost 28 bits of the given fraction after it has
been shifted right $c-e-28k$ bits, for some constant~$c$.
We choose $c$ so that,
when $e$ has its maximum value \Hex{7ff}, the leading digit will
go into position |dat[1]|, and so that when the number to be printed
is exactly~1 the integer part of~$g$ will also be exactly~1.

@d magic_offset 2112 /* the constant $c$ that makes it work */
@d origin 37 /* the radix point follows |dat[37]| */

@<Store $f$ and $g$ as multiprecise integers@>=
k=(magic_offset-e)/28;
ff.dat[k-1]=shift_right(f,magic_offset+28-e-28*k,1).l&0xfffffff;
gg.dat[k-1]=shift_right(g,magic_offset+28-e-28*k,1).l&0xfffffff;
ff.dat[k]=shift_right(f,magic_offset-e-28*k,1).l&0xfffffff;
gg.dat[k]=shift_right(g,magic_offset-e-28*k,1).l&0xfffffff;
ff.dat[k+1]=shift_left(f,e+28*k-(magic_offset-28)).l&0xfffffff;
gg.dat[k+1]=shift_left(g,e+28*k-(magic_offset-28)).l&0xfffffff;
ff.a=(ff.dat[k-1]? k-1: k);
ff.b=(ff.dat[k+1]? k+1: k);
gg.a=(gg.dat[k-1]? k-1: k);
gg.b=(gg.dat[k+1]? k+1: k);

@ If $e$ is sufficiently small, the fractions $f$ and $g$ will be less than~1,
and we can use the stated algorithm directly. Of course, if $e$ is
extremely small, a lot of leading zeros need to be lopped off; in the
worst case, we may have to multiply $f$ and~$g$ by~10 more than 300 times.
But hey, we don't need to do that extremely often, and computers are
pretty fast nowadays.

In the small-exponent case, the computation always terminates before
$f$ becomes zero, because the interval endpoints are fractions with
denominator $2^t$ for some $t>50$.

The invariant relations |ff.dat[ff.a]!=0| and |gg.dat[gg.a]!=0| are
not maintained by the computation here, when |ff.a=origin| or |gg.a=origin|.
But no harm is done, because |bignum_compare| is not used.

@<Compute the significant digits |s|...@>=
if (e>0x401) @<Compute the significant digits in the large-exponent case@>@;
else@+{ /* if |e<=0x401| we have |gg.a>=origin| and |gg.dat[origin]<=8| */
  if (ff.a>origin) ff.dat[origin]=0;
  for (e=1, p=s; gg.a>origin || ff.dat[origin]==gg.dat[origin]; ) {
    if (gg.a>origin) e--;
    else *p++=ff.dat[origin]+'0', ff.dat[origin]=0, gg.dat[origin]=0;
    bignum_times_ten(&ff);
    bignum_times_ten(&gg);
  }
  *p++=((ff.dat[origin]+1+gg.dat[origin])>>1)+'0'; /* the middle digit */
}
*p='\0'; /* terminate the string |s| */

@ When |e| is large, we use the stated algorithm by considering $f$ and
$g$ to be fractions whose denominator is a power of~10.

An interesting case arises when the number to be converted is
\Hex{44ada56a4b0835bf}, since the interval turns out to be
$$ (69999999999999991611392\ \ \dts\ \ 70000000000000000000000).$$
If this were a closed interval, we could simply give the answer
\.{7e22}; but the number \.{7e22} actually corresponds to
\Hex{44ada56a4b0835c0}
because of the round-to-even rule. Therefore the correct answer is, say,
\.{6.9999999999999995e22}. This example shows that we need a slightly
different strategy in the case of open intervals; we cannot simply
look at the first position in which the endpoints have different
decimal digits. Therefore we change the invariant relation to $0\le f<g\le 1$,
when open intervals are involved,
and we do not terminate the process when $f=0$ or $g=1$.

@<Compute the significant digits in the large-exponent case@>=
{@+register int open=x.l&1;
  tt.dat[origin]=10;
  tt.a=tt.b=origin;
  for (e=1;bignum_compare(&gg,&tt)>=open;e++)
    bignum_times_ten(&tt);
  p=s;
  while (1) {
    bignum_times_ten(&ff);
    bignum_times_ten(&gg);
    for (j='0';bignum_compare(&ff,&tt)>=0;j++)
      bignum_dec(&ff,&tt,0x10000000),bignum_dec(&gg,&tt,0x10000000);
    if (bignum_compare(&gg,&tt)>=open) break;
    *p++=j;
    if (ff.a==bignum_prec-1 && !open)
      goto done; /* $f=0$ in a closed interval */
  }
  for (k=j;bignum_compare(&gg,&tt)>=open;k++) bignum_dec(&gg,&tt,0x10000000);
  *p++=(j+1+k)>>1; /* the middle digit */
 done:;
}

@ The length of string~|s| will be at most 17. For if $f$ and $g$
agree to 17 places, we have $g/f<1+10^{-16}$; but the
ratio $g/f$ is always $\ge(1+2^{-52}+2^{-53})/(1+2^{-52}-2^{-53})
>1+2\times10^{-16}$.

@<Local variables for |print_float|@>=
bignum ff,gg; /* fractions or numerators of fractions */
bignum tt; /* power of ten (used as the denominator) */
char s[18];
register char *p;

@ At this point the significant digits are in string |s|, and |s[0]!='0'|.
If we put a decimal point at the left of~|s|, the result should
be multiplied by $10^e$.

We prefer the output `\.{300.}' to the form `\.{3e2}', and we prefer
`\.{.03}' to `\.{3e-2}'. In general, the output will use an
explicit exponent only if the alternative would take more than
18~characters.

@<Print the significant digits with proper context@>=
if (e>17 || e<(int)strlen(s)-17)
  printf("%c%s%se%d",s[0],(s[1]? ".": ""),s+1,e-1);
else if (e<0) printf(".%0*d%s",-e,0,s);
else if ((int)strlen(s)>=e) printf("%.*s.%s",e,s,s+e);
else printf("%s%0*d.",s,e-(int)strlen(s),0);

@*Floating point input conversion. Going the other way, we want to
be able to convert a given decimal number into its floating binary
@^decimal-to-binary conversion@>
@^radix conversion@>
@^multiprecision conversion@>
equivalent. The following syntax is supported:
$$\vbox{\halign{$#$\hfil\cr
\<digit>\is\.0\mid\.1\mid\.2\mid\.3\mid\.4\mid
        \.5\mid\.6\mid\.7\mid\.8\mid\.9\cr
\<digit string>\is\<digit>\mid\<digit string>\<digit>\cr
\<decimal string>\is\<digit string>\..\mid\..\<digit string>\mid
                      \<digit string>\..\<digit string>\cr
\<optional sign>\is\<empty>\mid\.+\mid\.-\cr
\<exponent>\is\.e\<optional sign>\<digit string>\cr
\<optional exponent>\is\<empty>\mid\<exponent>\cr
\<floating magnitude>\is\<digit string>\<exponent>\mid
                    \<decimal string>\<optional exponent>\mid\cr
\hskip12em          \.{Inf}\mid\.{NaN}\mid\.{NaN.}\<digit string>\cr
\<floating constant>\is\<optional sign>\<floating magnitude>\cr
\<decimal constant>\is\<optional sign>\<digit string>\cr
}}$$
For example, `\.{-3.}' is the floating constant \Hex{c008000000000000}\thinspace;
`\.{1e3}' and `\.{1000}' are both equivalent to \Hex{408f400000000000}\thinspace;
`\.{NaN}' and `\.{+NaN.5}' are both equivalent to \Hex{7ff8000000000000}.

The |scan_const| routine looks at a given string and finds the
longest initial substring that matches the syntax of either \<decimal
constant> or \<floating constant>. It puts the corresponding value
into the global octabyte variable~|val|; it also puts the position of the first
unscanned character in the global pointer variable |next_char|.
It returns 1 if a floating constant was found, 0~if a decimal constant
was found, $-1$ if nothing was found. A decimal constant that doesn't
fit in an octabyte is computed modulo~$2^{64}$.
@^syntax of floating point constants@>

The value of |exceptions| set by |scan_const| is not necessarily correct.

@<Subr...@>=
static void bignum_double @,@,@[ARGS((bignum*))@];
int scan_const @,@,@[ARGS((char*))@];@+@t}\6{@>
int scan_const(s)
  char *s;
{
  @<Local variables for |scan_const|@>;
  val.h=val.l=0;
  p=s;
  if (*p=='+' || *p=='-') sign=*p++;@+else sign='+';
  if (strncmp(p,"NaN",3)==0) NaN=true, p+=3;
  else NaN=false;
  if ((isdigit(*p)&&!NaN) || (*p=='.' && isdigit(*(p+1))))
    @<Scan a number and |return|@>;
  if (NaN) @<Return the standard NaN@>;
  if (strncmp(p,"Inf",3)==0) @<Return infinity@>;
  next_char=s;@+return -1;
}

@ @<Glob...@>=
octa val; /* value returned by |scan_const| */
char *next_char; /* pointer returned by |scan_const| */

@ @<Local variables for |scan_const|@>=
register char *p,*q; /* for string manipulations */
register bool NaN; /* are we processing a NaN? */
int sign; /* |'+'| or |'-'| */

@ @<Return the standard NaN@>=
{
  next_char=p;
  val.h=0x600000, exp=0x3fe;
  goto packit;
}

@ @<Return infinity@>=
{
  next_char=p+3;
  goto make_it_infinite;
}

@ We saw above that a string of at most 17 digits is enough to characterize
a floating point number, for purposes of output. But a much longer buffer
for digits is needed when we're doing input. For example, consider the
borderline quantity $(1+2^{-53})/2^{1022}$; its decimal expansion, when
written out exactly, is a number with more than 750 significant digits:
\.{2.2250738585...8125e-308}.
If {\it any one\/} of those digits is increased, or if
additional nonzero digits are added as in
\.{2.2250738585...81250000001e-308},
the rounded value is supposed to change from \Hex{0010000000000000}
to \Hex{0010000000000001}.

We assume here that the user prefers a perfectly correct answer to
a speedy almost-correct one, so we implement the most general case.

@<Scan a number...@>=
{
  for (q=buf0,dec_pt=(char*)0;isdigit(*p);p++) {
    val=oplus(val,shift_left(val,2)); /* multiply by 5 */
    val=incr(shift_left(val,1),*p-'0');
    if (q>buf0 || *p!='0') {
       if (q<buf_max) *q++=*p;
       else if (*(q-1)=='0') *(q-1)=*p;@+
    }
  }
  if (NaN) *q++='1';
  if (*p=='.') @<Scan a fraction part@>;
  next_char=p;
  exp=0;
  if (*p=='e' && !NaN) @<Scan an exponent@>;
  if (dec_pt) @<Return a floating point constant@>;
  if (sign=='-') val=ominus(zero_octa,val);
  return 0;
}

@ @<Scan a fraction part@>=
{
  dec_pt=q;
  p++;
  for (zeros=0;isdigit(*p);p++)
    if (*p=='0' && q==buf0) zeros++;
    else if (q<buf_max) *q++=*p;
    else if (*(q-1)=='0') *(q-1)=*p;
}

@ The buffer needs room for eight digits of padding at the left, followed
by up to $1022+53-307$ significant digits, followed by a ``sticky'' digit
at position |buf_max-1|, and eight more digits of padding.

@d buf0 (buf+8)
@d buf_max (buf+777)

@<Glob...@>=
static char buf[785]="00000000"; /* where we put significant input digits */

@ @<Local variables for |scan_const|@>=
register char* dec_pt; /* position of decimal point in |buf| */
register int exp; /* scanned exponent; later used for raw binary exponent */
register int zeros=0; /* leading zeros removed after decimal point */

@ Here we don't advance |next_char| and force a decimal point until we
know that a syntactically correct exponent exists.

The code here will convert extra-large inputs like
`\.{9e+9999999999999999}' into $\infty$ and extra-small inputs into zero.
Strange inputs like `\.{-00.0e9999999}' must also be accommodated.
(But we {\it don't\/} try to deliver precise answers when there are
a billion or more leading zeros.)

@<Scan an exponent@>=
{@+register char exp_sign;
  p++;
  if (*p=='+' || *p=='-') exp_sign=*p++;@+else exp_sign='+';
  if (isdigit(*p)) {
    for (exp=*p++ -'0';isdigit(*p);p++)
      if (exp<100000000) exp = 10*exp + *p - '0';
    if (!dec_pt) dec_pt=q, zeros=0;
    if (exp_sign=='-') exp=-exp;
    next_char=p;
  }  
}

@ @<Return a floating point constant@>=
{
  @<Move the digits from |buf| to |ff|@>;
  @<Determine the binary fraction and binary exponent@>;
packit:  @<Pack and round the answer@>;
  return 1;
}

@ Now we get ready to compute the binary fraction bits, by putting the
scanned input digits into a multiprecision fixed-point
accumulator |ff| that spans the full necessary range.
After this step, the number that we want to convert to floating binary
will appear in |ff.dat[ff.a]|, |ff.dat[ff.a+1]|, \dots,
|ff.dat[ff.b]|.
The radix-$10^9$ digit in ${\it ff}[36-k]$ is understood to be multiplied
by $10^{9k}$, for $36\ge k\ge-120$.

@<Move the digits from |buf| to |ff|@>=
x=buf+341+zeros-dec_pt-exp;
if (q==buf0 || x>=1413) {
  exp=-99999;@+ goto packit;
}
if (x<0) {
 make_it_infinite: exp=99999;@+ goto packit;
}
ff.a=x/9;
for (p=q;p<q+8;p++) *p='0'; /* pad with trailing zeros */
q=q-1-(q+341+zeros-dec_pt-exp)%9; /* compute stopping place in |buf| */
for (p=buf0-x%9,k=ff.a;p<=q && k<=156; p+=9, k++)
  @<Put the 9-digit number |*p|\thinspace\dots\thinspace|*(p+8)|
       into |ff.dat[k]|@>;
ff.b=k-1;
for (x=0;p<=q;p+=9) if (strncmp(p,"000000000",9)!=0) x=1;
ff.dat[156]+=x; /* nonzero digits that fall off the right are sticky */
@^sticky bit@>
while (ff.dat[ff.b]==0) ff.b--;

@ @<Put the 9-digit number...@>=
{
  for (x=*p-'0',pp=p+1;pp<p+9;pp++) x=10*x + *pp - '0';
  ff.dat[k]=x;
}

@ @<Local variables for |scan_const|@>=
register int k,x;
register char *pp;
bignum ff,tt;

@ Here's a subroutine that is dual to |bignum_times_ten|. It changes $f$
to~$2f$, assuming that overflow will not occur and that the radix is $10^9$.

@<Subr...@>=
static void bignum_double(f)
  bignum *f;
{
  register tetra *p,*q; register int x,carry;
  for (p=&f->dat[f->b],q=&f->dat[f->a],carry=0; p>=q; p--) {
    x = *p + *p + carry;
    if (x>=1000000000) carry=1, *p=x-1000000000;
    else carry=0, *p=x;
  }
  *p=carry;
  if (carry) f->a--;
  if (f->dat[f->b]==0 && f->b>f->a) f->b--;
}

@ @<Determine the binary fraction and binary exponent@>=
val=zero_octa;
if (ff.a>36) {
  for (exp=0x3fe;ff.a>36;exp--) bignum_double(&ff);
  for (k=54;k;k--) {
    if (ff.dat[36]) {
      if (k>=32) val.h |= 1<<(k-32);@+else val.l |= 1<<k;
      ff.dat[36]=0;
      if (ff.b==36) break; /* break if |ff| now zero */
    }
    bignum_double(&ff);
  }
}@+else {
  tt.a=tt.b=36, tt.dat[36]=2;
  for (exp=0x3fe;bignum_compare(&ff,&tt)>=0;exp++) bignum_double(&tt);     
  for (k=54;k;k--) {
    bignum_double(&ff);
    if (bignum_compare(&ff,&tt)>=0) {
      if (k>=32) val.h |= 1<<(k-32);@+else val.l |= 1<<k;
      bignum_dec(&ff,&tt,1000000000);
      if (ff.a==bignum_prec-1) break; /* break if |ff| now zero */
    }
  }
}
if (k==0) val.l |= 1; /* add sticky bit if |ff| nonzero */    

@ We need to be careful that the input `\.{NaN.999999999999999999999}' doesn't
get rounded up; it is supposed to yield \Hex{7fffffffffffffff}.

Although the input `\.{NaN.0}' is illegal, strictly speaking, we silently
convert it to \Hex{7ff0000000000001}---a number that would be
output as `\.{NaN.0000000000000002}'.

@<Pack and round the answer@>=
val=fpack(val,exp,sign,ROUND_NEAR);
if (NaN) {
  if ((val.h&0x7fffffff)==0x40000000) val.h |= 0x7fffffff, val.l=0xffffffff;
  else if ((val.h&0x7fffffff)==0x3ff00000 && !val.l) val.h|=0x40000000,val.l=1;
  else val.h |= 0x40000000;
}

@*Floating point remainders. In this section we implement the remainder
of the floating point operations---one of which happens to be the
operation of taking the remainder.

The easiest task remaining is to compare two floating point quantities.
Routine |fcomp| returns $-1$~if~$y<z$, 0~if~$y=z$, $+1$~if~$y>z$, and
$+2$~if $y$ and~$z$ are unordered.

@<Subr...@>=
int fcomp @,@,@[ARGS((octa,octa))@];@+@t}\6{@>
int fcomp(y,z)
  octa y,z;
{
  ftype yt,zt;
  int ye,ze;
  char ys,zs;
  octa yf,zf;
  register int x;
  yt=funpack(y,&yf,&ye,&ys);
  zt=funpack(z,&zf,&ze,&zs);
  switch (4*yt+zt) {
 case 4*nan+nan: case 4*zro+nan: case 4*num+nan: case 4*inf+nan:
 case 4*nan+zro: case 4*nan+num: case 4*nan+inf: return 2;
 default: case 4*zro+zro: return 0;
 case 4*zro+num: case 4*num+zro: case 4*zro+inf: case 4*inf+zro:
 case 4*num+num: case 4*num+inf: case 4*inf+num: case 4*inf+inf:
  if (ys!=zs) x=1;
  else if (y.h>z.h) x=1;
  else if (y.h<z.h) x=-1;
  else if (y.l>z.l) x=1;
  else if (y.l<z.l) x=-1;
  else return 0;
  break;
  }
  return (ys=='-'? -x: x);
}

@ Several \MMIX\ operations act on a single floating point number and
accept an arbitrary rounding mode. For example, consider the
operation of rounding to the nearest floating point integer:

@<Subr...@>=
octa fintegerize @,@,@[ARGS((octa,int))@];@+@t}\6{@>
octa fintegerize(z,r)
  octa z; /* the operand */
  int r; /* the rounding mode */
{
  ftype zt;
  int ze;
  char zs;
  octa xf,zf;
  zt=funpack(z,&zf,&ze,&zs);
  if (!r) r=cur_round;
  switch (zt) {
 case nan:@+if (!(z.h&0x80000)) {@+exceptions|=I_BIT;@+z.h|=0x80000;@+}
 case inf: case zro: default: return z;
 case num: @<Integerize and |return|@>;
  }
}

@ @<Integerize...@>=
if (ze>=1074) return fpack(zf,ze,zs,ROUND_OFF); /* already an integer */
if (ze<=1020) xf.h=0,xf.l=1;
else {@+octa oo;
  xf=shift_right(zf,1074-ze,1);
  oo=shift_left(xf,1074-ze);
  if (oo.l!=zf.l || oo.h!=zf.h) xf.l|=1; /* sticky bit */
@^sticky bit@>
}
switch (r) {
 case ROUND_DOWN:@+ if (zs=='-') xf=incr(xf,3);@+break;
 case ROUND_UP:@+ if (zs!='-') xf=incr(xf,3);
 case ROUND_OFF: break;
 case ROUND_NEAR: xf=incr(xf, xf.l&4? 2: 1);@+break;
}
xf.l&=0xfffffffc;
if (ze>=1022) return fpack(shift_left(xf,1074-ze),ze,zs,ROUND_OFF);
if (xf.l) xf.h=0x3ff00000, xf.l=0;
if (zs=='-') xf.h|=sign_bit;
return xf;

@ To convert floating point to fixed point, we use |fixit|.

@<Subr...@>=
octa fixit @,@,@[ARGS((octa,int))@];@+@t}\6{@>
octa fixit(z,r)
  octa z; /* the operand */
  int r; /* the rounding mode */
{
  ftype zt;
  int ze;
  char zs;
  octa zf,o;
  zt=funpack(z,&zf,&ze,&zs);
  if (!r) r=cur_round;
  switch (zt) {
 case nan: case inf: exceptions|=I_BIT;@+return z;
 case zro: default: return zero_octa;
 case num:@+if (funpack(fintegerize(z,r),&zf,&ze,&zs)==zro) return zero_octa;
   if (ze<=1076) o=shift_right(zf,1076-ze,1);
   else {
     if (ze>1085 || (ze==1085 && (zf.h>0x400000 || @|
           (zf.h==0x400000 && (zf.l || zs!='-'))))) exceptions|=W_BIT;
     if (ze>=1140) return zero_octa;
     o=shift_left(zf,ze-1076);
    }
  return (zs=='-'? ominus(zero_octa,o): o);
  }
}

@ Going the other way, we can specify not only a rounding mode but whether
the given fixed point octabyte is signed or unsigned, and whether the
result should be rounded to short precision.

@<Subr...@>=
octa floatit @,@,@[ARGS((octa,int,int,int))@];@+@t}\6{@>
octa floatit(z,r,u,p)
  octa z; /* octabyte to float */
  int r; /* rounding mode */
  int u; /* unsigned? */
  int p; /* short precision? */
{
  int e;@+char s;
  register int t;
  exceptions=0;
  if (!z.h && !z.l) return zero_octa;
  if (!r) r=cur_round;
  if (!u && (z.h&sign_bit)) s='-', z=ominus(zero_octa,z);@+ else s='+';
  e=1076;
  while (z.h<0x400000) e--,z=shift_left(z,1);
  while (z.h>=0x800000) {
    e++;
    t=z.l&1;
    z=shift_right(z,1,1);
    z.l|=t;
  }
  if (p) @<Convert to short float@>;
  return fpack(z,e,s,r);
} 

@ @<Convert to short float@>=
{
  register int ex;@+register tetra t;
  t=sfpack(z,e,s,r);
  ex=exceptions;
  sfunpack(t,&z,&e,&s);
  exceptions=ex;
}

@ The square root operation is more interesting.

@<Subr...@>=
octa froot @,@,@[ARGS((octa,int))@];@+@t}\6{@>
octa froot(z,r)
  octa z; /* the operand */
  int r; /* the rounding mode */
{
  ftype zt;
  int ze;
  char zs;
  octa x,xf,rf,zf;
  register int xe,k;
  if (!r) r=cur_round;
  zt=funpack(z,&zf,&ze,&zs);
  if (zs=='-' && zt!=zro) exceptions|=I_BIT, x=standard_NaN;
  else@+switch (zt) {
 case nan:@+ if (!(z.h&0x80000)) exceptions|=I_BIT, z.h|=0x80000;
  return z;
 default: case inf: case zro: x=z;@+break;
 case num: @<Take the square root and |return|@>;
  }
  if (zs=='-') x.h|=sign_bit;
  return x;
}

@ The square root can be found by an adaptation of the old pencil-and-paper
method. If $n=\lfloor\sqrt s\rfloor$, where $s$ is an integer,
we have $s=n^2+r$ where $0\le r\le2n$;
this invariant can be maintained if we replace $s$ by $4s+(0,1,2,3)$
and $n$ by $2n+(0,1)$. The following code implements this idea with
$2n$ in~|xf| and $r$ in~|rf|. (It could easily be made to run about
twice as fast.)

@<Take the square root and |return|@>=
xf.h=0, xf.l=2;
xe=(ze+0x3fe)>>1;
if (ze&1) zf=shift_left(zf,1);
rf.h=0, rf.l=(zf.h>>22)-1;
for (k=53;k;k--) {
  rf=shift_left(rf,2);@+ xf=shift_left(xf,1);
  if (k>=43) rf=incr(rf,(zf.h>>(2*(k-43)))&3);
  else if (k>=27) rf=incr(rf,(zf.l>>(2*(k-27)))&3);
  if ((rf.l>xf.l && rf.h>=xf.h) || rf.h>xf.h) {
    xf.l++;@+rf=ominus(rf,xf);@+xf.l++;
  }
}
if (rf.h || rf.l) xf.l++; /* sticky bit */
return fpack(xf,xe,'+',r);

@ And finally, the genuine floating point remainder. Subroutine |fremstep|
either calculates $y\,{\rm rem}\,z$ or reduces $y$ to a smaller number
having the same remainder with respect to~$z$. In the latter case
the |E_BIT| is set in |exceptions|. A third parameter, |delta|,
gives a decrease in exponent that is acceptable for incomplete results;
if |delta| is sufficiently large, say 2500, the correct result will
always be obtained in one step of |fremstep|.

@<Subr...@>=
octa fremstep @,@,@[ARGS((octa,octa,int))@];@+@t}\6{@>
octa fremstep(y,z,delta)
  octa y,z;
  int delta;
{
  ftype yt,zt;
  int ye,ze;
  char xs,ys,zs;
  octa x,xf,yf,zf;
  register int xe,thresh,odd;
  yt=funpack(y,&yf,&ye,&ys);
  zt=funpack(z,&zf,&ze,&zs);
  switch (4*yt+zt) {
 @t\4@>@<The usual NaN cases@>;
 case 4*zro+zro: case 4*num+zro: case 4*inf+zro:
 case 4*inf+num: case 4*inf+inf: x=standard_NaN;
  exceptions|=I_BIT;@+break;
 case 4*zro+num: case 4*zro+inf: case 4*num+inf: return y;
 case 4*num+num: @<Remainderize nonzero numbers and |return|@>;
 default: zero_out: x=zero_octa;
  }
  if (ys=='-') x.h|=sign_bit;
  return x;
}

@ If there's a huge difference in exponents and the remainder is nonzero,
this computation will take a long time. One could compute 
$(2^ny)\,{\rm rem}\,z$ much more quickly for large~$n$ by using $O(\log n)$
multiplications modulo~$z$, but the floating remainder operation isn't
important enough to justify such expensive hardware.

Results of floating remainder are always exact, so the rounding mode
is immaterial.

@<Remainderize...@>=
odd=0; /* will be 1 if we've subtracted an odd multiple of~$z$ from $y$ */
thresh=ye-delta;
if (thresh<ze) thresh=ze;
while (ye>=thresh) @<Reduce |(ye,yf)| by a multiple of |zf|;
   |goto zero_out| if the remainder is zero,
   |goto try_complement| if appropriate@>;
if (ye>=ze) {
  exceptions|=E_BIT;@+return fpack(yf,ye,ys,ROUND_OFF);
}
if (ye<ze-1) return fpack(yf,ye,ys,ROUND_OFF);
yf=shift_right(yf,1,1);
try_complement: xf=ominus(zf,yf), xe=ze, xs='+' + '-' - ys;
if (xf.h>yf.h || (xf.h==yf.h && (xf.l>yf.l || (xf.l==yf.l && !odd))))
  xf=yf, xs=ys;
while (xf.h<0x400000) xe--, xf=shift_left(xf,1);
return fpack(xf,xe,xs,ROUND_OFF);

@ Here we are careful not to change the sign of |y|, because a remainder
of~0 is supposed to inherit the original sign of~|y|.

@<Reduce |(ye,yf)| by a multiple of |zf|...@>=
{
  if (yf.h==zf.h && yf.l==zf.l) goto zero_out;
  if (yf.h<zf.h || (yf.h==zf.h && yf.l<zf.l)) {
    if (ye==ze) goto try_complement;
    ye--, yf=shift_left(yf,1);
  }
  yf=ominus(yf,zf);
  if (ye==ze) odd=1;
  while (yf.h<0x400000) ye--, yf=shift_left(yf,1);
}

@* Index.