Patchwork [PATCH1/2] Improve performance of sincosf

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Submitter Wilco Dijkstra
Date June 30, 2018, 2:49 p.m.
Message ID <DB5PR08MB10306DAF22F63FAF0C1C014D834D0@DB5PR08MB1030.eurprd08.prod.outlook.com>
Download mbox | patch
Permalink /patch/28173/
State New
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Comments

Wilco Dijkstra - June 30, 2018, 2:49 p.m.
Joseph Myers wrote:

> You then need to add a dummy s_sincosf_data.c for architectures with their 
> own sincosf implementation, to avoid wasting space in libm.so with unused 
> data from s_sincosf_data.c.  That looks like ia64, m68k, x86_64.

It's about 256 bytes so it's not a huge waste, I'll do it as a separate enhancement
when I get back from holiday.

Here is v3 with spaces and namespace fixed:

This patch is a complete rewrite of sinf, cosf and sincosf.  The new version
is significantly faster, as well as simple and accurate. 
The worst-case ULP is 0.56072, maximum relative error is 0.5303p-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 - on some targets this may
be slow, so this can be configured to use floating point comparisons.

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

On a benchmark with significant use of sincosf the overall speedup is >33%.

ChangeLog:
2018-06-30  Wilco Dijkstra  <wdijkstr@arm.com>

	* math/Makefile: Add s_sincosf_data.c.
	* sysdeps/aarch64/fpu/math_private.h (roundtoint): Use round.
	(converttoint): Use lround.
	* sysdeps/ieee754/flt-32/math_config.h (PREFER_FLOAT_COMPARISON): Define.
	* 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.
--
Joseph Myers - June 30, 2018, 8:25 p.m.
On Sat, 30 Jun 2018, Wilco Dijkstra wrote:

> Joseph Myers wrote:
> 
> > You then need to add a dummy s_sincosf_data.c for architectures with their 
> > own sincosf implementation, to avoid wasting space in libm.so with unused 
> > data from s_sincosf_data.c.  That looks like ia64, m68k, x86_64.
> 
> It's about 256 bytes so it's not a huge waste, I'll do it as a separate 
> enhancement when I get back from holiday.

I think the patch that adds a file only relevant to some targets should be 
the same patch that also avoids it building any code for other targets.

Patch

diff --git a/math/Makefile b/math/Makefile
index 90b3b68916e12d8563ed57772013f4420415928b..e73ceb8d4e8d520c3047f63bf6c6d05856a89186 100644
--- a/math/Makefile
+++ b/math/Makefile
@@ -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
diff --git a/sysdeps/aarch64/fpu/math_private.h b/sysdeps/aarch64/fpu/math_private.h
index fcd02c06543c0841cac9949c7ca052c042b3d85b..657f654176760125b96348004129af9b544b7206 100644
--- a/sysdeps/aarch64/fpu/math_private.h
+++ b/sysdeps/aarch64/fpu/math_private.h
@@ -21,6 +21,8 @@ 
 
 #include <fenv.h>
 #include <fpu_control.h>
+#include <stdint.h>
+#include <math.h>
 
 static __always_inline void
 libc_feholdexcept_aarch64 (fenv_t *envp)
@@ -298,25 +300,20 @@  libc_feresetround_noex_aarch64_ctx (struct rm_ctx *ctx)
 #define libc_feresetround_noexf_ctx	libc_feresetround_noex_aarch64_ctx
 #define libc_feresetround_noexl_ctx	libc_feresetround_noex_aarch64_ctx
 
-/* Hack: only include the large arm_neon.h when needed.  */
-#ifdef _MATH_CONFIG_H
-# include <arm_neon.h>
-
-/* ACLE intrinsics for frintn and fcvtns instructions.  */
-# define TOINT_INTRINSICS 1
+/* Use inline round and lround instructions.  */
+#define TOINT_INTRINSICS 1
 
 static inline double_t
 roundtoint (double_t x)
 {
-  return vget_lane_f64 (vrndn_f64 (vld1_f64 (&x)), 0);
+  return round (x);
 }
 
 static inline uint64_t
 converttoint (double_t x)
 {
-  return vcvtnd_s64_f64 (x);
+  return (uint64_t) lround (x);
 }
-#endif
 
 #include_next <math_private.h>
 
diff --git a/sysdeps/ieee754/flt-32/math_config.h b/sysdeps/ieee754/flt-32/math_config.h
index c4def9bbc1898d71865572d0084b219034bf3805..6729ea06c9d687452e9891874b2c022348d274e7 100644
--- a/sysdeps/ieee754/flt-32/math_config.h
+++ b/sysdeps/ieee754/flt-32/math_config.h
@@ -46,6 +46,9 @@ 
 #ifndef TOINT_SHIFT
 # define TOINT_SHIFT 1
 #endif
+#ifndef PREFER_FLOAT_COMPARISON
+# define PREFER_FLOAT_COMPARISON 0
+#endif
 
 static inline uint32_t
 asuint (float f)
diff --git a/sysdeps/ieee754/flt-32/s_sincosf.h b/sysdeps/ieee754/flt-32/s_sincosf.h
index 35b5eee536ef0ca55e68e3d3a091486dca39d36a..f1fcf4ce5fa35459808c32d7461f83036716b096 100644
--- a/sysdeps/ieee754/flt-32/s_sincosf.h
+++ b/sysdeps/ieee754/flt-32/s_sincosf.h
@@ -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 +157,118 @@  reduced_cos (double theta, unsigned int n)
     }
   return sign * cx;
 }
+
+
+/* PI * 2^-64.  */
+static const double pi64 = 0x1.921FB54442D18p-62;
+/* PI / 4.  */
+static const double pio4 = 0x1.921FB54442D18p-1;
+
+typedef struct
+{
+  double sign[4];
+  double hpi_inv, hpi, c0, c1, c2, c3, c4, s1, s2, s3;
+} sincos_t;
+
+extern const sincos_t __sincosf_table[2] attribute_hidden;
+
+extern const uint32_t __inv_pio4[] attribute_hidden;
+
+/* abstop12 assumes floating point reinterpret is fast by default.
+   If floating point comparisons are faster, define PREFER_FLOAT_COMPARISON.  */
+#if PREFER_FLOAT_COMPARISON
+static inline float
+abstop12 (float x)
+{
+  return fabsf (x);
+}
+#else
+static inline uint32_t
+abstop12 (float x)
+{
+  return (asuint (x) >> 20) & 0x7ff;
+}
+#endif
+
+/* 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,
+   only 2 multiplies are required and the result is accurate for |X| <= 120.0.
+   Use round/lround if inlined, otherwise convert to int.  To avoid inaccuracies
+   introduced by truncating negative values, compute the quadrant * 2^24.  */
+static inline double
+reduce_fast (double x, const sincos_t *p, int *np)
+{
+  double r;
+#if TOINT_INTRINSICS
+  r = x * p->hpi_inv;
+  *np = converttoint (r);
+  return x - roundtoint (r) * p->hpi;
+#else
+  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/4 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 * pi64;
+}
diff --git a/sysdeps/ieee754/flt-32/s_sincosf.c b/sysdeps/ieee754/flt-32/s_sincosf.c
index d4a5a1b22c2f9a5cf607cd2d31157dd1dbd14e9a..cb3c379f882d6b29b46f32684e57426350770d9a 100644
--- a/sysdeps/ieee754/flt-32/s_sincosf.c
+++ b/sysdeps/ieee754/flt-32/s_sincosf.c
@@ -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 +30,72 @@ 
 # define SINCOSF_FUNC SINCOSF
 #endif
 
+/* Fast sincosf implementation.  Worst-case ULP is 0.56072, maximum relative
+   error is 0.5303p-23.  A single-step signed 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))
+    {
+      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 if (abstop12 (y) < abstop12 (120.0f))
     {
-      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;
-	}
+      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                          /* |x| >= Pi/4.  */
+  else if (__glibc_likely (abstop12 (y) < abstop12 (INFINITY)))
     {
-      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);
-	}
+      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.  */
+      __math_invalidf (y + y);
+#endif
     }
 }
 
diff --git a/sysdeps/ieee754/flt-32/s_sincosf_data.c b/sysdeps/ieee754/flt-32/s_sincosf_data.c
new file mode 100644
index 0000000000000000000000000000000000000000..21fc2b60f9ead7714deedb78c05529a8c58fe096
--- /dev/null
+++ b/sysdeps/ieee754/flt-32/s_sincosf_data.c
@@ -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
+};