emacs/src/floatfns.c

/* Primitive operations on floating point for GNU Emacs Lisp interpreter.

Copyright (C) 1988, 1993-1994, 1999, 2001-2021 Free Software Foundation,
Inc.

Author: Wolfgang Rupprecht (according to ack.texi)

This file is part of GNU Emacs.

GNU Emacs is free software: you can redistribute it and/or modify
it under the terms of the GNU General Public License as published by
the Free Software Foundation, either version 3 of the License, or (at
your option) any later version.

GNU Emacs is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
GNU General Public License for more details.

You should have received a copy of the GNU General Public License
along with GNU Emacs.  If not, see <https://www.gnu.org/licenses/>.  */


/* C89 requires only the following math.h functions, and Emacs omits
   the starred functions since we haven't found a use for them:
   acos, asin, atan, atan2, ceil, cos, *cosh, exp, fabs, floor, fmod,
   frexp, ldexp, log, log10 [via (log X 10)], *modf, pow, sin, *sinh,
   sqrt, tan, *tanh.

   C99 and C11 require the following math.h functions in addition to
   the C89 functions.  Of these, Emacs currently exports only the
   starred ones to Lisp, since we haven't found a use for the others:
   acosh, atanh, cbrt, *copysign, erf, erfc, exp2, expm1, fdim, fma,
   fmax, fmin, fpclassify, hypot, ilogb, isfinite, isgreater,
   isgreaterequal, isinf, isless, islessequal, islessgreater, *isnan,
   isnormal, isunordered, lgamma, log1p, *log2 [via (log X 2)], *logb
   (approximately), lrint/llrint, lround/llround, nan, nearbyint,
   nextafter, nexttoward, remainder, remquo, *rint, round, scalbln,
   scalbn, signbit, tgamma, *trunc.
 */

#include <config.h>

#include "lisp.h"
#include "bignum.h"

#include <math.h>

#include <count-leading-zeros.h>

/* Emacs needs proper handling of +/-inf; correct printing as well as
   important packages depend on it.  Make sure the user didn't specify
   -ffinite-math-only, either directly or implicitly with -Ofast or
   -ffast-math.  */
#if defined __FINITE_MATH_ONLY__ && __FINITE_MATH_ONLY__
 #error Emacs cannot be built with -ffinite-math-only
#endif

/* Check that X is a floating point number.  */

static void
CHECK_FLOAT (Lisp_Object x)
{
  CHECK_TYPE (FLOATP (x), Qfloatp, x);
}

/* Extract a Lisp number as a `double', or signal an error.  */

double
extract_float (Lisp_Object num)
{
  CHECK_NUMBER (num);
  return XFLOATINT (num);
}

/* Trig functions.  */

DEFUN ("acos", Facos, Sacos, 1, 1, 0,
       doc: /* Return the inverse cosine of ARG.  */)
  (Lisp_Object arg)
{
  double d = extract_float (arg);
  d = acos (d);
  return make_float (d);
}

DEFUN ("asin", Fasin, Sasin, 1, 1, 0,
       doc: /* Return the inverse sine of ARG.  */)
  (Lisp_Object arg)
{
  double d = extract_float (arg);
  d = asin (d);
  return make_float (d);
}

DEFUN ("atan", Fatan, Satan, 1, 2, 0,
       doc: /* Return the inverse tangent of the arguments.
If only one argument Y is given, return the inverse tangent of Y.
If two arguments Y and X are given, return the inverse tangent of Y
divided by X, i.e. the angle in radians between the vector (X, Y)
and the x-axis.  */)
  (Lisp_Object y, Lisp_Object x)
{
  double d = extract_float (y);

  if (NILP (x))
    d = atan (d);
  else
    {
      double d2 = extract_float (x);
      d = atan2 (d, d2);
    }
  return make_float (d);
}

DEFUN ("cos", Fcos, Scos, 1, 1, 0,
       doc: /* Return the cosine of ARG.  */)
  (Lisp_Object arg)
{
  double d = extract_float (arg);
  d = cos (d);
  return make_float (d);
}

DEFUN ("sin", Fsin, Ssin, 1, 1, 0,
       doc: /* Return the sine of ARG.  */)
  (Lisp_Object arg)
{
  double d = extract_float (arg);
  d = sin (d);
  return make_float (d);
}

DEFUN ("tan", Ftan, Stan, 1, 1, 0,
       doc: /* Return the tangent of ARG.  */)
  (Lisp_Object arg)
{
  double d = extract_float (arg);
  d = tan (d);
  return make_float (d);
}

DEFUN ("isnan", Fisnan, Sisnan, 1, 1, 0,
       doc: /* Return non-nil if argument X is a NaN.  */)
  (Lisp_Object x)
{
  CHECK_FLOAT (x);
  return isnan (XFLOAT_DATA (x)) ? Qt : Qnil;
}

/* Although the substitute does not work on NaNs, it is good enough
   for platforms lacking the signbit macro.  */
#ifndef signbit
# define signbit(x) ((x) < 0 || (IEEE_FLOATING_POINT && !(x) && 1 / (x) < 0))
#endif

DEFUN ("copysign", Fcopysign, Scopysign, 2, 2, 0,
       doc: /* Copy sign of X2 to value of X1, and return the result.
Cause an error if X1 or X2 is not a float.  */)
  (Lisp_Object x1, Lisp_Object x2)
{
  double f1, f2;

  CHECK_FLOAT (x1);
  CHECK_FLOAT (x2);

  f1 = XFLOAT_DATA (x1);
  f2 = XFLOAT_DATA (x2);

  /* Use signbit instead of copysign, to avoid calling make_float when
     the result is X1.  */
  return signbit (f1) != signbit (f2) ? make_float (-f1) : x1;
}

DEFUN ("frexp", Ffrexp, Sfrexp, 1, 1, 0,
       doc: /* Get significand and exponent of a floating point number.
Breaks the floating point number X into its binary significand SGNFCAND
\(a floating point value between 0.5 (included) and 1.0 (excluded))
and an integral exponent EXP for 2, such that:

  X = SGNFCAND * 2^EXP

The function returns the cons cell (SGNFCAND . EXP).
If X is zero, both parts (SGNFCAND and EXP) are zero.  */)
  (Lisp_Object x)
{
  double f = extract_float (x);
  int exponent;
  double sgnfcand = frexp (f, &exponent);
  return Fcons (make_float (sgnfcand), make_fixnum (exponent));
}

DEFUN ("ldexp", Fldexp, Sldexp, 2, 2, 0,
       doc: /* Return SGNFCAND * 2**EXPONENT, as a floating point number.
EXPONENT must be an integer.   */)
  (Lisp_Object sgnfcand, Lisp_Object exponent)
{
  CHECK_FIXNUM (exponent);
  int e = min (max (INT_MIN, XFIXNUM (exponent)), INT_MAX);
  return make_float (ldexp (extract_float (sgnfcand), e));
}

DEFUN ("exp", Fexp, Sexp, 1, 1, 0,
       doc: /* Return the exponential base e of ARG.  */)
  (Lisp_Object arg)
{
  double d = extract_float (arg);
  d = exp (d);
  return make_float (d);
}

DEFUN ("expt", Fexpt, Sexpt, 2, 2, 0,
       doc: /* Return the exponential ARG1 ** ARG2.  */)
  (Lisp_Object arg1, Lisp_Object arg2)
{
  CHECK_NUMBER (arg1);
  CHECK_NUMBER (arg2);

  /* Common Lisp spec: don't promote if both are integers, and if the
     result is not fractional.  */
  if (INTEGERP (arg1) && !NILP (Fnatnump (arg2)))
    return expt_integer (arg1, arg2);

  return make_float (pow (XFLOATINT (arg1), XFLOATINT (arg2)));
}

DEFUN ("log", Flog, Slog, 1, 2, 0,
       doc: /* Return the natural logarithm of ARG.
If the optional argument BASE is given, return log ARG using that base.  */)
  (Lisp_Object arg, Lisp_Object base)
{
  double d = extract_float (arg);

  if (NILP (base))
    d = log (d);
  else
    {
      double b = extract_float (base);

      if (b == 10.0)
	d = log10 (d);
#if HAVE_LOG2
      else if (b == 2.0)
	d = log2 (d);
#endif
      else
	d = log (d) / log (b);
    }
  return make_float (d);
}

DEFUN ("sqrt", Fsqrt, Ssqrt, 1, 1, 0,
       doc: /* Return the square root of ARG.  */)
  (Lisp_Object arg)
{
  double d = extract_float (arg);
  d = sqrt (d);
  return make_float (d);
}

DEFUN ("abs", Fabs, Sabs, 1, 1, 0,
       doc: /* Return the absolute value of ARG.  */)
  (Lisp_Object arg)
{
  CHECK_NUMBER (arg);

  if (FIXNUMP (arg))
    {
      if (XFIXNUM (arg) < 0)
	arg = make_int (-XFIXNUM (arg));
    }
  else if (FLOATP (arg))
    {
      if (signbit (XFLOAT_DATA (arg)))
	arg = make_float (- XFLOAT_DATA (arg));
    }
  else
    {
      if (mpz_sgn (*xbignum_val (arg)) < 0)
	{
	  mpz_neg (mpz[0], *xbignum_val (arg));
	  arg = make_integer_mpz ();
	}
    }

  return arg;
}

DEFUN ("float", Ffloat, Sfloat, 1, 1, 0,
       doc: /* Return the floating point number equal to ARG.  */)
  (register Lisp_Object arg)
{
  CHECK_NUMBER (arg);
  /* If ARG is a float, give 'em the same float back.  */
  return FLOATP (arg) ? arg : make_float (XFLOATINT (arg));
}

static int
ecount_leading_zeros (EMACS_UINT x)
{
  return (EMACS_UINT_WIDTH == UINT_WIDTH ? count_leading_zeros (x)
	  : EMACS_UINT_WIDTH == ULONG_WIDTH ? count_leading_zeros_l (x)
	  : count_leading_zeros_ll (x));
}

DEFUN ("logb", Flogb, Slogb, 1, 1, 0,
       doc: /* Returns largest integer <= the base 2 log of the magnitude of ARG.
This is the same as the exponent of a float.  */)
  (Lisp_Object arg)
{
  EMACS_INT value;
  CHECK_NUMBER (arg);

  if (FLOATP (arg))
    {
      double f = XFLOAT_DATA (arg);
      if (f == 0)
	return make_float (-HUGE_VAL);
      if (!isfinite (f))
	return f < 0 ? make_float (-f) : arg;
      int ivalue;
      frexp (f, &ivalue);
      value = ivalue - 1;
    }
  else if (!FIXNUMP (arg))
    value = mpz_sizeinbase (*xbignum_val (arg), 2) - 1;
  else
    {
      EMACS_INT i = XFIXNUM (arg);
      if (i == 0)
	return make_float (-HUGE_VAL);
      value = EMACS_UINT_WIDTH - 1 - ecount_leading_zeros (eabs (i));
    }

  return make_fixnum (value);
}

/* Return the integer exponent E such that D * FLT_RADIX**E (i.e.,
   scalbn (D, E)) is an integer that has precision equal to D and is
   representable as a double.

   Return DBL_MANT_DIG - DBL_MIN_EXP (the maximum possible valid
   scale) if D is zero or tiny.  Return one greater than that if
   D is infinite, and two greater than that if D is a NaN.  */

int
double_integer_scale (double d)
{
  int exponent = ilogb (d);
#ifdef HAIKU
  /* On Haiku, the values returned by ilogb are nonsensical when
     confronted with tiny numbers, inf, or NaN, which breaks the trick
     used by code on other platforms, so we have to test for each case
     manually, and return the appropriate value.  */
  if (exponent == FP_ILOGB0)
    {
      if (isnan (d))
	return (DBL_MANT_DIG - DBL_MIN_EXP) + 2;
      if (isinf (d))
	return (DBL_MANT_DIG - DBL_MIN_EXP) + 1;

      return (DBL_MANT_DIG - DBL_MIN_EXP);
    }
#endif
  return (DBL_MIN_EXP - 1 <= exponent && exponent < INT_MAX
	  ? DBL_MANT_DIG - 1 - exponent
	  : (DBL_MANT_DIG - DBL_MIN_EXP
	     + (isnan (d) ? 2 : exponent == INT_MAX)));
}

/* Convert the Lisp number N to an integer and return a pointer to the
   converted integer, represented as an mpz_t *.  Use *T as a
   temporary; the returned value might be T.  Scale N by the maximum
   of NSCALE and DSCALE while converting.  If NSCALE is nonzero, N
   must be a float; signal an overflow if NSCALE is greater than
   DBL_MANT_DIG - DBL_MIN_EXP, otherwise scalbn (XFLOAT_DATA (N), NSCALE)
   must return an integer value, without rounding or overflow.  */

static mpz_t const *
rescale_for_division (Lisp_Object n, mpz_t *t, int nscale, int dscale)
{
  mpz_t const *pn;

  if (FLOATP (n))
    {
      if (DBL_MANT_DIG - DBL_MIN_EXP < nscale)
	overflow_error ();
      mpz_set_d (*t, scalbn (XFLOAT_DATA (n), nscale));
      pn = t;
    }
  else
    pn = bignum_integer (t, n);

  if (nscale < dscale)
    {
      emacs_mpz_mul_2exp (*t, *pn, (dscale - nscale) * LOG2_FLT_RADIX);
      pn = t;
    }
  return pn;
}

/* the rounding functions  */

static Lisp_Object
rounding_driver (Lisp_Object n, Lisp_Object d,
		 double (*double_round) (double),
		 void (*int_divide) (mpz_t, mpz_t const, mpz_t const),
		 EMACS_INT (*fixnum_divide) (EMACS_INT, EMACS_INT))
{
  CHECK_NUMBER (n);

  if (NILP (d))
    return FLOATP (n) ? double_to_integer (double_round (XFLOAT_DATA (n))) : n;

  CHECK_NUMBER (d);

  int dscale = 0;
  if (FIXNUMP (d))
    {
      if (XFIXNUM (d) == 0)
	xsignal0 (Qarith_error);

      /* Divide fixnum by fixnum specially, for speed.  */
      if (FIXNUMP (n))
	return make_int (fixnum_divide (XFIXNUM (n), XFIXNUM (d)));
    }
  else if (FLOATP (d))
    {
      if (XFLOAT_DATA (d) == 0)
	xsignal0 (Qarith_error);
      dscale = double_integer_scale (XFLOAT_DATA (d));
    }

  int nscale = FLOATP (n) ? double_integer_scale (XFLOAT_DATA (n)) : 0;

  /* If the numerator is finite and the denominator infinite, the
     quotient is zero and there is no need to try the impossible task
     of rescaling the denominator.  */
  if (dscale == DBL_MANT_DIG - DBL_MIN_EXP + 1 && nscale < dscale)
    return make_fixnum (0);

  int_divide (mpz[0],
	      *rescale_for_division (n, &mpz[0], nscale, dscale),
	      *rescale_for_division (d, &mpz[1], dscale, nscale));
  return make_integer_mpz ();
}

static EMACS_INT
ceiling2 (EMACS_INT n, EMACS_INT d)
{
  return n / d + ((n % d != 0) & ((n < 0) == (d < 0)));
}

static EMACS_INT
floor2 (EMACS_INT n, EMACS_INT d)
{
  return n / d - ((n % d != 0) & ((n < 0) != (d < 0)));
}

static EMACS_INT
truncate2 (EMACS_INT n, EMACS_INT d)
{
  return n / d;
}

static EMACS_INT
round2 (EMACS_INT n, EMACS_INT d)
{
  /* The C language's division operator gives us the remainder R
     corresponding to truncated division, but we want the remainder R1
     on the other side of 0 if R1 is closer to 0 than R is; because we
     want to round to even, we also want R1 if R and R1 are the same
     distance from 0 and if the truncated quotient is odd.  */
  EMACS_INT q = n / d;
  EMACS_INT r = n % d;
  bool neg_d = d < 0;
  bool neg_r = r < 0;
  EMACS_INT abs_r = eabs (r);
  EMACS_INT abs_r1 = eabs (d) - abs_r;
  if (abs_r1 < abs_r + (q & 1))
    q += neg_d == neg_r ? 1 : -1;
  return q;
}

static void
rounddiv_q (mpz_t q, mpz_t const n, mpz_t const d)
{
  /* Mimic the source code of round2, using mpz_t instead of EMACS_INT.  */
  mpz_t *r = &mpz[2], *abs_r = r, *abs_r1 = &mpz[3];
  mpz_tdiv_qr (q, *r, n, d);
  bool neg_d = mpz_sgn (d) < 0;
  bool neg_r = mpz_sgn (*r) < 0;
  mpz_abs (*abs_r, *r);
  mpz_abs (*abs_r1, d);
  mpz_sub (*abs_r1, *abs_r1, *abs_r);
  if (mpz_cmp (*abs_r1, *abs_r) < (mpz_odd_p (q) != 0))
    (neg_d == neg_r ? mpz_add_ui : mpz_sub_ui) (q, q, 1);
}

/* The code uses emacs_rint, so that it works to undefine HAVE_RINT
   if `rint' exists but does not work right.  */
#ifdef HAVE_RINT
#define emacs_rint rint
#else
static double
emacs_rint (double d)
{
  double d1 = d + 0.5;
  double r = floor (d1);
  return r - (r == d1 && fmod (r, 2) != 0);
}
#endif

#ifndef HAVE_TRUNC
double
trunc (double d)
{
  return (d < 0 ? ceil : floor) (d);
}
#endif

DEFUN ("ceiling", Fceiling, Sceiling, 1, 2, 0,
       doc: /* Return the smallest integer no less than ARG.
This rounds the value towards +inf.
With optional DIVISOR, return the smallest integer no less than ARG/DIVISOR.  */)
  (Lisp_Object arg, Lisp_Object divisor)
{
  return rounding_driver (arg, divisor, ceil, mpz_cdiv_q, ceiling2);
}

DEFUN ("floor", Ffloor, Sfloor, 1, 2, 0,
       doc: /* Return the largest integer no greater than ARG.
This rounds the value towards -inf.
With optional DIVISOR, return the largest integer no greater than ARG/DIVISOR.  */)
  (Lisp_Object arg, Lisp_Object divisor)
{
  return rounding_driver (arg, divisor, floor, mpz_fdiv_q, floor2);
}

DEFUN ("round", Fround, Sround, 1, 2, 0,
       doc: /* Return the nearest integer to ARG.
With optional DIVISOR, return the nearest integer to ARG/DIVISOR.

Rounding a value equidistant between two integers may choose the
integer closer to zero, or it may prefer an even integer, depending on
your machine.  For example, (round 2.5) can return 3 on some
systems, but 2 on others.  */)
  (Lisp_Object arg, Lisp_Object divisor)
{
  return rounding_driver (arg, divisor, emacs_rint, rounddiv_q, round2);
}

/* Since rounding_driver truncates anyway, no need to call 'trunc'.  */
static double
identity (double x)
{
  return x;
}

DEFUN ("truncate", Ftruncate, Struncate, 1, 2, 0,
       doc: /* Truncate a floating point number to an int.
Rounds ARG toward zero.
With optional DIVISOR, truncate ARG/DIVISOR.  */)
  (Lisp_Object arg, Lisp_Object divisor)
{
  return rounding_driver (arg, divisor, identity, mpz_tdiv_q, truncate2);
}


Lisp_Object
fmod_float (Lisp_Object x, Lisp_Object y)
{
  double f1 = XFLOATINT (x);
  double f2 = XFLOATINT (y);

  f1 = fmod (f1, f2);

  /* If the "remainder" comes out with the wrong sign, fix it.  */
  if (f2 < 0 ? f1 > 0 : f1 < 0)
    f1 += f2;

  return make_float (f1);
}

DEFUN ("fceiling", Ffceiling, Sfceiling, 1, 1, 0,
       doc: /* Return the smallest integer no less than ARG, as a float.
\(Round toward +inf.)  */)
  (Lisp_Object arg)
{
  CHECK_FLOAT (arg);
  double d = XFLOAT_DATA (arg);
  d = ceil (d);
  return make_float (d);
}

DEFUN ("ffloor", Fffloor, Sffloor, 1, 1, 0,
       doc: /* Return the largest integer no greater than ARG, as a float.
\(Round toward -inf.)  */)
  (Lisp_Object arg)
{
  CHECK_FLOAT (arg);
  double d = XFLOAT_DATA (arg);
  d = floor (d);
  return make_float (d);
}

DEFUN ("fround", Ffround, Sfround, 1, 1, 0,
       doc: /* Return the nearest integer to ARG, as a float.  */)
  (Lisp_Object arg)
{
  CHECK_FLOAT (arg);
  double d = XFLOAT_DATA (arg);
  d = emacs_rint (d);
  return make_float (d);
}

DEFUN ("ftruncate", Fftruncate, Sftruncate, 1, 1, 0,
       doc: /* Truncate a floating point number to an integral float value.
\(Round toward zero.)  */)
  (Lisp_Object arg)
{
  CHECK_FLOAT (arg);
  double d = XFLOAT_DATA (arg);
  d = trunc (d);
  return make_float (d);
}

void
syms_of_floatfns (void)
{
  defsubr (&Sacos);
  defsubr (&Sasin);
  defsubr (&Satan);
  defsubr (&Scos);
  defsubr (&Ssin);
  defsubr (&Stan);
  defsubr (&Sisnan);
  defsubr (&Scopysign);
  defsubr (&Sfrexp);
  defsubr (&Sldexp);
  defsubr (&Sfceiling);
  defsubr (&Sffloor);
  defsubr (&Sfround);
  defsubr (&Sftruncate);
  defsubr (&Sexp);
  defsubr (&Sexpt);
  defsubr (&Slog);
  defsubr (&Ssqrt);

  defsubr (&Sabs);
  defsubr (&Sfloat);
  defsubr (&Slogb);
  defsubr (&Sceiling);
  defsubr (&Sfloor);
  defsubr (&Sround);
  defsubr (&Struncate);
}