Improve performance of sinf/cosf/sincosf

Here is the correct patch with both filenames and int cast 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.

Simple benchmark showing speedup factor on AArch64 for various ranges:

range	0.7853982	sinf	1.7	cosf	2.2	sincosf	2.8
range	1.570796	sinf	1.9	cosf	1.9	sincosf	2.7
range	3.141593	sinf	2.0	cosf	2.0	sincosf	3.5
range	6.283185	sinf	2.3	cosf	2.3	sincosf	4.2
range	125.6637	sinf	2.9	cosf	3.0	sincosf	5.1
range	1.1259e15	sinf	26.8	cosf	26.8	sincosf	45.2

ChangeLog:
2018-05-18  Wilco Dijkstra  <wdijkstr@arm.com>

        * newlib/libm/common/Makefile.in: Regenerated.
        * newlib/libm/common/Makefile.am: Add sinf.c, cosf.c, sincosf.c
        sincosf.h, sincosf_data.c. Add -fbuiltin -fno-math-errno to CFLAGS.
        * newlib/libm/common/math_config.h: Add HAVE_FAST_ROUND, HAVE_FAST_LROUND,
        roundtoint, converttoint, force_eval_float, force_eval_double, eval_as_float,
        eval_as_double, likely, unlikely.
        * newlib/libm/common/cosf.c: New file.
        * newlib/libm/common/sinf.c: Likewise.
        * newlib/libm/common/sincosf.h: Likewise.
        * newlib/libm/common/sincosf.c: Likewise.
        * newlib/libm/common/sincosf_data.c: Likewise.
        * newlib/libm/math/sf_cos.c: Add #if to build conditionally.
        * newlib/libm/math/sf_sin.c: Likewise.
        * newlib/libm/math/wf_sincos.c: Likewise.

--

1 files changed, 172 insertions, 0 deletions

diff --git a/newlib/libm/common/sincosf.h b/newlib/libm/common/sincosf.h
new file mode 100644
index 000000000..955a73312
--- /dev/null
+++ b/newlib/libm/common/sincosf.h
@@ -0,0 +1,172 @@
+/* Header for single-precision sin/cos/sincos functions.
+   Copyright (c) 2018 Arm Ltd.  All rights reserved.
+
+   Redistribution and use in source and binary forms, with or without
+   modification, are permitted provided that the following conditions
+   are met:
+   1. Redistributions of source code must retain the above copyright
+      notice, this list of conditions and the following disclaimer.
+   2. Redistributions in binary form must reproduce the above copyright
+      notice, this list of conditions and the following disclaimer in the
+      documentation and/or other materials provided with the distribution.
+   3. The name of the company may not be used to endorse or promote
+      products derived from this software without specific prior written
+      permission.
+
+   THIS SOFTWARE IS PROVIDED BY ARM LTD ``AS IS AND ANY EXPRESS OR IMPLIED
+   WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF
+   MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED.
+   IN NO EVENT SHALL ARM LTD BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
+   SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED
+   TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR
+   PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF
+   LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING
+   NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
+   SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. */
+
+#include <stdint.h>
+#include <math.h>
+#include "math_config.h"
+
+/* PI * 2^-64.  */
+static const double pi64 = 0x1.921FB54442D18p-62;
+/* PI / 4.  */
+static const double pio4 = 0x1.921FB54442D18p-1;
+
+typedef const struct
+{
+  double sign[4];
+  double hpi_inv, hpi, c0, c1, c2, c3, c4, s1, s2, s3;
+} sincos_t;
+
+extern sincos_t sincosf_table[2] HIDDEN;
+
+extern const uint32_t inv_pio4[] 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, 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, 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,
+   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, 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;
+}