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Improve performance of sincosf

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This patch is a complete rewrite of sincosf.  The new version is
significantly faster, as well as simple and accurate.
The worst-case ULP is 0.5607, maximum relative error is 0.5303 * 2^-23 over
all 4 billion inputs.  In non-nearest rounding modes the error is 1ULP.

The algorithm uses 3 main cases: small inputs which don't need argument
reduction, small inputs which need a simple range reduction and large inputs
requiring complex range reduction.  The code uses approximate integer
comparisons to quickly decide between these cases.

The small range reducer uses a single reduction step to handle values up to
120.0.  It is fastest on targets which support inlined round instructions.

The large range reducer uses integer arithmetic for simplicity.  It does a
32x96 bit multiply to compute a 64-bit modulo result.  This is more than
accurate enough to handle the worst-case cancellation for values close to
an integer multiple of PI/4.  It could be further optimized, however it is
already much faster than necessary.

sincosf throughput gains on Cortex-A72:
* |x| < 0x1p-12 : 1.6x
* |x| < M_PI_4  : 1.7x
* |x| < 2 * M_PI: 1.5x
* |x| < 120.0   : 1.8x
* |x| < Inf     : 2.3x

	* math/Makefile: Add s_sincosf_data.c.
	* sysdeps/ia64/fpu/s_sincosf_data.c: New file.
	* sysdeps/ieee754/flt-32/s_sincosf.h (abstop12): Add new function.
	(sincosf_poly): Likewise.
	(reduce_small): Likewise.
	(reduce_large): Likewise.
	* sysdeps/ieee754/flt-32/s_sincosf.c (sincosf): Rewrite.
	* sysdeps/ieee754/flt-32/s_sincosf_data.c: New file with sincosf data.
	* sysdeps/m68k/m680x0/fpu/s_sincosf_data.c: New file.
	* sysdeps/x86_64/fpu/s_sincosf_data.c: New file.
This commit is contained in:
Wilco Dijkstra 2018-08-10 17:31:30 +01:00
parent 43cfdf8f48
commit ea5c662c62
8 changed files with 274 additions and 133 deletions
Show Stats Download Patch File Download Diff File Expand all files Collapse all files

14
ChangeLog
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@ -1,3 +1,17 @@
2018-08-10 Wilco Dijkstra <wdijkstr@arm.com>
Szabolcs Nagy <szabolcs.nagy@arm.com>
* math/Makefile: Add s_sincosf_data.c.
* sysdeps/ia64/fpu/s_sincosf_data.c: New file.
* sysdeps/ieee754/flt-32/s_sincosf.h (abstop12): Add new function.
(sincosf_poly): Likewise.
(reduce_small): Likewise.
(reduce_large): Likewise.
* sysdeps/ieee754/flt-32/s_sincosf.c (sincosf): Rewrite.
* sysdeps/ieee754/flt-32/s_sincosf_data.c: New file with sincosf data.
* sysdeps/m68k/m680x0/fpu/s_sincosf_data.c: New file.
* sysdeps/x86_64/fpu/s_sincosf_data.c: New file.
2018-08-10 Wilco Dijkstra <wdijkstr@arm.com>
Szabolcs Nagy <szabolcs.nagy@arm.com>

2
math/Makefile
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@ -131,7 +131,7 @@ type-double-routines := branred doasin dosincos mpa mpatan2 \
# float support
type-float-suffix := f
type-float-routines := k_rem_pio2f math_errf e_exp2f_data e_logf_data \
e_log2f_data e_powf_log2_data
e_log2f_data e_powf_log2_data s_sincosf_data
# _Float128 support
type-float128-suffix := f128

1
sysdeps/ia64/fpu/s_sincosf_data.c Normal file
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@ -0,0 +1 @@
/* Not needed. */

196
sysdeps/ieee754/flt-32/s_sincosf.c
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@ -1,5 +1,5 @@
/* Compute sine and cosine of argument.
Copyright (C) 2017-2018 Free Software Foundation, Inc.
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
@ -17,9 +17,11 @@
<http://www.gnu.org/licenses/>. */
#include <errno.h>
#include <stdint.h>
#include <math.h>
#include <math_private.h>
#include <math-barriers.h>
#include <libm-alias-float.h>
#include "math_config.h"
#include "s_sincosf.h"
#ifndef SINCOSF
@ -28,141 +30,71 @@
# define SINCOSF_FUNC SINCOSF
#endif
/* Fast sincosf implementation. Worst-case ULP is 0.5607, maximum relative
error is 0.5303 * 2^-23. A single-step range reduction is used for
small values. Large inputs have their range reduced using fast integer
arithmetic. */
void
SINCOSF_FUNC (float x, float *sinx, float *cosx)
SINCOSF_FUNC (float y, float *sinp, float *cosp)
{
double cx;
double theta = x;
double abstheta = fabs (theta);
/* If |x|< Pi/4. */
if (isless (abstheta, M_PI_4))
double x = y;
double s;
int n;
const sincos_t *p = &__sincosf_table[0];
if (abstop12 (y) < abstop12 (pio4))
{
if (abstheta >= 0x1p-5) /* |x| >= 2^-5. */
{
const double theta2 = theta * theta;
/* Chebyshev polynomial of the form for sin and cos. */
cx = C3 + theta2 * C4;
cx = C2 + theta2 * cx;
cx = C1 + theta2 * cx;
cx = C0 + theta2 * cx;
cx = 1.0 + theta2 * cx;
*cosx = cx;
cx = S3 + theta2 * S4;
cx = S2 + theta2 * cx;
cx = S1 + theta2 * cx;
cx = S0 + theta2 * cx;
cx = theta + theta * theta2 * cx;
*sinx = cx;
}
else if (abstheta >= 0x1p-27) /* |x| >= 2^-27. */
{
/* A simpler Chebyshev approximation is close enough for this range:
for sin: x+x^3*(SS0+x^2*SS1)
for cos: 1.0+x^2*(CC0+x^3*CC1). */
const double theta2 = theta * theta;
cx = CC0 + theta * theta2 * CC1;
cx = 1.0 + theta2 * cx;
*cosx = cx;
cx = SS0 + theta2 * SS1;
cx = theta + theta * theta2 * cx;
*sinx = cx;
}
else
{
/* Handle some special cases. */
if (theta)
*sinx = theta - (theta * SMALL);
else
*sinx = theta;
*cosx = 1.0 - abstheta;
}
double x2 = x * x;
if (__glibc_unlikely (abstop12 (y) < abstop12 (0x1p-12f)))
{
/* Force underflow for tiny y. */
if (__glibc_unlikely (abstop12 (y) < abstop12 (0x1p-126f)))
math_force_eval ((float)x2);
*sinp = y;
*cosp = 1.0f;
return;
}
sincosf_poly (x, x2, p, 0, sinp, cosp);
}
else /* |x| >= Pi/4. */
else if (abstop12 (y) < abstop12 (120.0f))
{
unsigned int signbit = isless (x, 0);
if (isless (abstheta, 9 * M_PI_4)) /* |x| < 9*Pi/4. */
{
/* There are cases where FE_UPWARD rounding mode can
produce a result of abstheta * inv_PI_4 == 9,
where abstheta < 9pi/4, so the domain for
pio2_table must go to 5 (9 / 2 + 1). */
unsigned int n = (abstheta * inv_PI_4) + 1;
theta = abstheta - pio2_table[n / 2];
*sinx = reduced_sin (theta, n, signbit);
*cosx = reduced_cos (theta, n);
}
else if (isless (abstheta, INFINITY))
{
if (abstheta < 0x1p+23) /* |x| < 2^23. */
{
unsigned int n = ((unsigned int) (abstheta * inv_PI_4)) + 1;
double x = n / 2;
theta = (abstheta - x * PI_2_hi) - x * PI_2_lo;
/* Argument reduction needed. */
*sinx = reduced_sin (theta, n, signbit);
*cosx = reduced_cos (theta, n);
}
else /* |x| >= 2^23. */
{
x = fabsf (x);
int exponent;
GET_FLOAT_WORD (exponent, x);
exponent
= (exponent >> FLOAT_EXPONENT_SHIFT) - FLOAT_EXPONENT_BIAS;
exponent += 3;
exponent /= 28;
double a = invpio4_table[exponent] * x;
double b = invpio4_table[exponent + 1] * x;
double c = invpio4_table[exponent + 2] * x;
double d = invpio4_table[exponent + 3] * x;
uint64_t l = a;
l &= ~0x7;
a -= l;
double e = a + b;
l = e;
e = a - l;
if (l & 1)
{
e -= 1.0;
e += b;
e += c;
e += d;
e *= M_PI_4;
*sinx = reduced_sin (e, l + 1, signbit);
*cosx = reduced_cos (e, l + 1);
}
else
{
e += b;
e += c;
e += d;
if (e <= 1.0)
{
e *= M_PI_4;
*sinx = reduced_sin (e, l + 1, signbit);
*cosx = reduced_cos (e, l + 1);
}
else
{
l++;
e -= 2.0;
e *= M_PI_4;
*sinx = reduced_sin (e, l + 1, signbit);
*cosx = reduced_cos (e, l + 1);
}
}
}
}
else
{
int32_t ix;
/* High word of x. */
GET_FLOAT_WORD (ix, abstheta);
/* sin/cos(Inf or NaN) is NaN. */
*sinx = *cosx = x - x;
if (ix == 0x7f800000)
__set_errno (EDOM);
}
x = reduce_fast (x, p, &n);
/* Setup the signs for sin and cos. */
s = p->sign[n & 3];
if (n & 2)
p = &__sincosf_table[1];
sincosf_poly (x * s, x * x, p, n, sinp, cosp);
}
else if (__glibc_likely (abstop12 (y) < abstop12 (INFINITY)))
{
uint32_t xi = asuint (y);
int sign = xi >> 31;
x = reduce_large (xi, &n);
/* Setup signs for sin and cos - include original sign. */
s = p->sign[(n + sign) & 3];
if ((n + sign) & 2)
p = &__sincosf_table[1];
sincosf_poly (x * s, x * x, p, n, sinp, cosp);
}
else
{
/* Return NaN if Inf or NaN for both sin and cos. */
*sinp = *cosp = y - y;
#if WANT_ERRNO
/* Needed to set errno for +-Inf, the add is a hack to work
around a gcc register allocation issue: just passing y
affects code generation in the fast path (PR86901). */
__math_invalidf (y + y);
#endif
}
}

118
sysdeps/ieee754/flt-32/s_sincosf.h
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@ -16,6 +16,10 @@
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;
@ -153,3 +157,117 @@ reduced_cos (double theta, unsigned int n)
}
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;
}

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sysdeps/ieee754/flt-32/s_sincosf_data.c Normal file
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@ -0,0 +1,74 @@
/* Compute sine and cosine of argument.
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"
#include "s_sincosf.h"
/* The constants and polynomials for sine and cosine. The 2nd entry
computes -cos (x) rather than cos (x) to get negation for free. */
const sincos_t __sincosf_table[2] =
{
{
{ 1.0, -1.0, -1.0, 1.0 },
#if TOINT_INTRINSICS
0x1.45F306DC9C883p-1,
#else
0x1.45F306DC9C883p+23,
#endif
0x1.921FB54442D18p0,
0x1p0,
-0x1.ffffffd0c621cp-2,
0x1.55553e1068f19p-5,
-0x1.6c087e89a359dp-10,
0x1.99343027bf8c3p-16,
-0x1.555545995a603p-3,
0x1.1107605230bc4p-7,
-0x1.994eb3774cf24p-13
},
{
{ 1.0, -1.0, -1.0, 1.0 },
#if TOINT_INTRINSICS
0x1.45F306DC9C883p-1,
#else
0x1.45F306DC9C883p+23,
#endif
0x1.921FB54442D18p0,
-0x1p0,
0x1.ffffffd0c621cp-2,
-0x1.55553e1068f19p-5,
0x1.6c087e89a359dp-10,
-0x1.99343027bf8c3p-16,
-0x1.555545995a603p-3,
0x1.1107605230bc4p-7,
-0x1.994eb3774cf24p-13
}
};
/* Table with 4/PI to 192 bit precision. To avoid unaligned accesses
only 8 new bits are added per entry, making the table 4 times larger. */
const uint32_t __inv_pio4[24] =
{
0xa2, 0xa2f9, 0xa2f983, 0xa2f9836e,
0xf9836e4e, 0x836e4e44, 0x6e4e4415, 0x4e441529,
0x441529fc, 0x1529fc27, 0x29fc2757, 0xfc2757d1,
0x2757d1f5, 0x57d1f534, 0xd1f534dd, 0xf534ddc0,
0x34ddc0db, 0xddc0db62, 0xc0db6295, 0xdb629599,
0x6295993c, 0x95993c43, 0x993c4390, 0x3c439041
};

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sysdeps/m68k/m680x0/fpu/s_sincosf_data.c Normal file
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@ -0,0 +1 @@
/* Not needed. */

1
sysdeps/x86_64/fpu/s_sincosf_data.c Normal file
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@ -0,0 +1 @@
/* Not needed. */