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/* Used by sinf, cosf and sincosf functions.
Copyright (C) 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"
/* 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;
}
/* Return the sine of inputs X and X2 (X squared) using the polynomial P.
N is the quadrant, and if odd the cosine polynomial is used. */
static inline float
sinf_poly (double x, double x2, const sincos_t *p, int n)
{
double x3, x4, x6, x7, s, c, c1, c2, s1;
if ((n & 1) == 0)
{
x3 = x * x2;
s1 = p->s2 + x2 * p->s3;
x7 = x3 * x2;
s = x + x3 * p->s1;
return s + x7 * s1;
}
else
{
x4 = x2 * x2;
c2 = p->c3 + x2 * p->c4;
c1 = p->c0 + x2 * p->c1;
x6 = x4 * x2;
c = c1 + x4 * p->c2;
return 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;
}
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