path: root/sysdeps/ieee754/flt-32/s_sincosf.h

/* Used by sinf, cosf and sincosf functions.
   Copyright (C) 2017-2018 Free Software Foundation, Inc.
   This file is part of the GNU C Library.

   The GNU C Library is free software; you can redistribute it and/or
   modify it under the terms of the GNU Lesser General Public
   License as published by the Free Software Foundation; either
   version 2.1 of the License, or (at your option) any later version.

   The GNU C Library 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
   Lesser General Public License for more details.

   You should have received a copy of the GNU Lesser General Public
   License along with the GNU C Library; if not, see
   <http://www.gnu.org/licenses/>.  */

#include <stdint.h>
#include <math.h>
#include "math_config.h"

/* Chebyshev constants for cos, range -PI/4 - PI/4.  */
static const double C0 = -0x1.ffffffffe98aep-2;
static const double C1 =  0x1.55555545c50c7p-5;
static const double C2 = -0x1.6c16b348b6874p-10;
static const double C3 =  0x1.a00eb9ac43ccp-16;
static const double C4 = -0x1.23c97dd8844d7p-22;

/* Chebyshev constants for sin, range -PI/4 - PI/4.  */
static const double S0 = -0x1.5555555551cd9p-3;
static const double S1 =  0x1.1111110c2688bp-7;
static const double S2 = -0x1.a019f8b4bd1f9p-13;
static const double S3 =  0x1.71d7264e6b5b4p-19;
static const double S4 = -0x1.a947e1674b58ap-26;

/* Chebyshev constants for sin, range 2^-27 - 2^-5.  */
static const double SS0 = -0x1.555555543d49dp-3;
static const double SS1 =  0x1.110f475cec8c5p-7;

/* Chebyshev constants for cos, range 2^-27 - 2^-5.  */
static const double CC0 = -0x1.fffffff5cc6fdp-2;
static const double CC1 =  0x1.55514b178dac5p-5;

/* PI/2 with 98 bits of accuracy.  */
static const double PI_2_hi = 0x1.921fb544p+0;
static const double PI_2_lo = 0x1.0b4611a626332p-34;

static const double SMALL = 0x1p-50; /* 2^-50.  */
static const double inv_PI_4 = 0x1.45f306dc9c883p+0; /* 4/PI.  */

#define FLOAT_EXPONENT_SHIFT 23
#define FLOAT_EXPONENT_BIAS 127

static const double pio2_table[] = {
  0 * M_PI_2,
  1 * M_PI_2,
  2 * M_PI_2,
  3 * M_PI_2,
  4 * M_PI_2,
  5 * M_PI_2
};

static const double invpio4_table[] = {
  0x0p+0,
  0x1.45f306cp+0,
  0x1.c9c882ap-28,
  0x1.4fe13a8p-58,
  0x1.f47d4dp-85,
  0x1.bb81b6cp-112,
  0x1.4acc9ep-142,
  0x1.0e4107cp-169
};

static const double ones[] = { 1.0, -1.0 };

/* Compute the sine value using Chebyshev polynomials where
   THETA is the range reduced absolute value of the input
   and it is less than Pi/4,
   N is calculated as trunc(|x|/(Pi/4)) + 1 and it is used to decide
   whether a sine or cosine approximation is more accurate and
   SIGNBIT is used to add the correct sign after the Chebyshev
   polynomial is computed.  */
static inline float
reduced_sin (const double theta, const unsigned int n,
	 const unsigned int signbit)
{
  double sx;
  const double theta2 = theta * theta;
  /* We are operating on |x|, so we need to add back the original
     signbit for sinf.  */
  double sign;
  /* Determine positive or negative primary interval.  */
  sign = ones[((n >> 2) & 1) ^ signbit];
  /* Are we in the primary interval of sin or cos?  */
  if ((n & 2) == 0)
    {
      /* Here sinf() is calculated using sin Chebyshev polynomial:
	x+x^3*(S0+x^2*(S1+x^2*(S2+x^2*(S3+x^2*S4)))).  */
      sx = S3 + theta2 * S4;     /* S3+x^2*S4.  */
      sx = S2 + theta2 * sx;     /* S2+x^2*(S3+x^2*S4).  */
      sx = S1 + theta2 * sx;     /* S1+x^2*(S2+x^2*(S3+x^2*S4)).  */
      sx = S0 + theta2 * sx;     /* S0+x^2*(S1+x^2*(S2+x^2*(S3+x^2*S4))).  */
      sx = theta + theta * theta2 * sx;
    }
  else
    {
     /* Here sinf() is calculated using cos Chebyshev polynomial:
	1.0+x^2*(C0+x^2*(C1+x^2*(C2+x^2*(C3+x^2*C4)))).  */
      sx = C3 + theta2 * C4;     /* C3+x^2*C4.  */
      sx = C2 + theta2 * sx;     /* C2+x^2*(C3+x^2*C4).  */
      sx = C1 + theta2 * sx;     /* C1+x^2*(C2+x^2*(C3+x^2*C4)).  */
      sx = C0 + theta2 * sx;     /* C0+x^2*(C1+x^2*(C2+x^2*(C3+x^2*C4))).  */
      sx = 1.0 + theta2 * sx;
    }

  /* Add in the signbit and assign the result.  */
  return sign * sx;
}

/* Compute the cosine value using Chebyshev polynomials where
   THETA is the range reduced absolute value of the input
   and it is less than Pi/4,
   N is calculated as trunc(|x|/(Pi/4)) + 1 and it is used to decide
   whether a sine or cosine approximation is more accurate and
   the sign of the result.  */
static inline float
reduced_cos (double theta, unsigned int n)
{
  double sign, cx;
  const double theta2 = theta * theta;

  /* Determine positive or negative primary interval.  */
  n += 2;
  sign = ones[(n >> 2) & 1];

  /* Are we in the primary interval of sin or cos?  */
  if ((n & 2) == 0)
    {
      /* Here cosf() is calculated using sin Chebyshev polynomial:
	x+x^3*(S0+x^2*(S1+x^2*(S2+x^2*(S3+x^2*S4)))).  */
      cx = S3 + theta2 * S4;
      cx = S2 + theta2 * cx;
      cx = S1 + theta2 * cx;
      cx = S0 + theta2 * cx;
      cx = theta + theta * theta2 * cx;
    }
  else
    {
     /* Here cosf() is calculated using cos Chebyshev polynomial:
	1.0+x^2*(C0+x^2*(C1+x^2*(C2+x^2*(C3+x^2*C4)))).  */
      cx = C3 + theta2 * C4;
      cx = C2 + theta2 * cx;
      cx = C1 + theta2 * cx;
      cx = C0 + theta2 * cx;
      cx = 1. + theta2 * cx;
    }
  return sign * cx;
}


/* 2PI * 2^-64.  */
static const double pi63 = 0x1.921FB54442D18p-62;
/* PI / 4.  */
static const double pio4 = 0x1.921FB54442D18p-1;

/* The constants and polynomials for sine and cosine.  */
typedef struct
{
  double sign[4];		/* Sign of sine in quadrants 0..3.  */
  double hpi_inv;		/* 2 / PI ( * 2^24 if !TOINT_INTRINSICS).  */
  double hpi;			/* PI / 2.  */
  double c0, c1, c2, c3, c4;	/* Cosine polynomial.  */
  double s1, s2, s3;		/* Sine polynomial.  */
} sincos_t;

/* Polynomial data (the cosine polynomial is negated in the 2nd entry).  */
extern const sincos_t __sincosf_table[2] attribute_hidden;

/* Table with 4/PI to 192 bit precision.  */
extern const uint32_t __inv_pio4[] attribute_hidden;

/* Top 12 bits of the float representation with the sign bit cleared.  */
static inline uint32_t
abstop12 (float x)
{
  return (asuint (x) >> 20) & 0x7ff;
}

/* Compute the sine and cosine of inputs X and X2 (X squared), using the
   polynomial P and store the results in SINP and COSP.  N is the quadrant,
   if odd the cosine and sine polynomials are swapped.  */
static inline void
sincosf_poly (double x, double x2, const sincos_t *p, int n, float *sinp,
	      float *cosp)
{
  double x3, x4, x5, x6, s, c, c1, c2, s1;

  x4 = x2 * x2;
  x3 = x2 * x;
  c2 = p->c3 + x2 * p->c4;
  s1 = p->s2 + x2 * p->s3;

  /* Swap sin/cos result based on quadrant.  */
  float *tmp = (n & 1 ? cosp : sinp);
  cosp = (n & 1 ? sinp : cosp);
  sinp = tmp;

  c1 = p->c0 + x2 * p->c1;
  x5 = x3 * x2;
  x6 = x4 * x2;

  s = x + x3 * p->s1;
  c = c1 + x4 * p->c2;

  *sinp = s + x5 * s1;
  *cosp = c + x6 * c2;
}

/* Fast range reduction using single multiply-subtract.  Return the modulo of
   X as a value between -PI/4 and PI/4 and store the quadrant in NP.
   The values for PI/2 and 2/PI are accessed via P.  Since PI/2 as a double
   is accurate to 55 bits and the worst-case cancellation happens at 6 * PI/4,
   the result is accurate for |X| <= 120.0.  */
static inline double
reduce_fast (double x, const sincos_t *p, int *np)
{
  double r;
#if TOINT_INTRINSICS
  /* Use fast round and lround instructions when available.  */
  r = x * p->hpi_inv;
  *np = converttoint (r);
  return x - roundtoint (r) * p->hpi;
#else
  /* Use scaled float to int conversion with explicit rounding.
     hpi_inv is prescaled by 2^24 so the quadrant ends up in bits 24..31.
     This avoids inaccuracies introduced by truncating negative values.  */
  r = x * p->hpi_inv;
  int n = ((int32_t)r + 0x800000) >> 24;
  *np = n;
  return x - n * p->hpi;
#endif
}

/* Reduce the range of XI to a multiple of PI/2 using fast integer arithmetic.
   XI is a reinterpreted float and must be >= 2.0f (the sign bit is ignored).
   Return the modulo between -PI/4 and PI/4 and store the quadrant in NP.
   Reduction uses a table of 4/PI with 192 bits of precision.  A 32x96->128 bit
   multiply computes the exact 2.62-bit fixed-point modulo.  Since the result
   can have at most 29 leading zeros after the binary point, the double
   precision result is accurate to 33 bits.  */
static inline double
reduce_large (uint32_t xi, int *np)
{
  const uint32_t *arr = &__inv_pio4[(xi >> 26) & 15];
  int shift = (xi >> 23) & 7;
  uint64_t n, res0, res1, res2;

  xi = (xi & 0xffffff) | 0x800000;
  xi <<= shift;

  res0 = xi * arr[0];
  res1 = (uint64_t)xi * arr[4];
  res2 = (uint64_t)xi * arr[8];
  res0 = (res2 >> 32) | (res0 << 32);
  res0 += res1;

  n = (res0 + (1ULL << 61)) >> 62;
  res0 -= n << 62;
  double x = (int64_t)res0;
  *np = n;
  return x * pi63;
}