Improve performance of sinf and cosf

The second patch improves performance of sinf and cosf using the same
algorithms and polynomials.  The returned values are identical to sincosf
for the same input.  ULP definitions for AArch64 and x64 are updated.

sinf/cosf througput gains on Cortex-A72:
* |x| < 0x1p-12 : 1.2x
* |x| < M_PI_4  : 1.8x
* |x| < 2 * M_PI: 1.7x
* |x| < 120.0   : 2.3x
* |x| < Inf     : 3.0x

	* NEWS: Mention sinf, cosf, sincosf.
	* sysdeps/aarch64/libm-test-ulps: Update ULP for sinf, cosf, sincosf.
	* sysdeps/x86_64/fpu/libm-test-ulps: Update ULP for sinf and cosf.
	* sysdeps/x86_64/fpu/multiarch/s_sincosf-fma.c: Add definitions of
	constants rather than including generic sincosf.h.
	* sysdeps/x86_64/fpu/s_sincosf_data.c: Remove.
	* sysdeps/ieee754/flt-32/s_cosf.c (cosf): Rewrite.
	* sysdeps/ieee754/flt-32/s_sincosf.h (reduced_sin): Remove.
	(reduced_cos): Remove.
	(sinf_poly): New function.
	* sysdeps/ieee754/flt-32/s_sinf.c (sinf): Rewrite.

3 files changed, 134 insertions, 370 deletions

diff --git a/sysdeps/ieee754/flt-32/s_cosf.c b/sysdeps/ieee754/flt-32/s_cosf.c
index 061264d259..13b5ffead8 100644
--- a/sysdeps/ieee754/flt-32/s_cosf.c
+++ b/sysdeps/ieee754/flt-32/s_cosf.c
@@ -1,5 +1,5 @@
 /* Compute 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
@@ -16,10 +16,11 @@
    License along with the GNU C Library; if not, see
    <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 COSF
@@ -28,121 +29,57 @@
 # define COSF_FUNC COSF
 #endif
 
+/* Fast cosf 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.
+*/
 float
-COSF_FUNC (float x)
+COSF_FUNC (float y)
 {
-  double theta = x;
-  double abstheta = fabs (theta);
-  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)))
+	return 1.0f;
+
+      return sinf_poly (x, x2, p, 1);
+    }
+  else if (__glibc_likely (abstop12 (y) < abstop12 (120.0f)))
     {
-      double cx;
-      if (abstheta >= 0x1p-5)
-	{
-	  const double theta2 = theta * theta;
-	  /* Chebyshev polynomial of the form for cos:
-	   * 1 + 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 cx;
-	}
-      else if (abstheta >= 0x1p-27)
-	{
-	  /* A simpler Chebyshev approximation is close enough for this range:
-	   * 1 + x^2 (CC0 + x^3 * CC1).  */
-	  const double theta2 = theta * theta;
-	  cx = CC0 + theta * theta2 * CC1;
-	  cx = 1.0 + theta2 * cx;
-	  return cx;
-	}
-      else
-	{
-	  /* For small enough |theta|, this is close enough.  */
-	  return 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];
+
+      return sinf_poly (x * s, x * x, p, n ^ 1);
     }
-  else /* |theta| >= Pi/4.  */
+  else if (abstop12 (y) < abstop12 (INFINITY))
     {
-      if (isless (abstheta, 9 * M_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];
-	  return reduced_cos (theta, n);
-	}
-      else if (isless (abstheta, INFINITY))
-	{
-	  if (abstheta < 0x1p+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.  */
-	      return reduced_cos (theta, n);
-	    }
-	  else /* |theta| >= 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;
-		  return reduced_cos (e, l + 1);
-		}
-	      else
-		{
-		  e += b;
-		  e += c;
-		  e += d;
-		  if (e <= 1.0)
-		    {
-		      e *= M_PI_4;
-		      return reduced_cos (e, l + 1);
-		    }
-		  else
-		    {
-		      l++;
-		      e -= 2.0;
-		      e *= M_PI_4;
-		      return reduced_cos (e, l + 1);
-		    }
-		}
-	    }
-	}
-      else
-	{
-	  int32_t ix;
-	  GET_FLOAT_WORD (ix, abstheta);
-	  /* cos(Inf or NaN) is NaN.  */
-	  if (ix == 0x7f800000) /* Inf.  */
-	    __set_errno (EDOM);
-	  return x - x;
-	}
+      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];
+
+      return sinf_poly (x * s, x * x, p, n ^ 1);
     }
+  else
+    return __math_invalidf (y);
 }
 
 #ifndef COSF
diff --git a/sysdeps/ieee754/flt-32/s_sincosf.h b/sysdeps/ieee754/flt-32/s_sincosf.h
index d3d7b4d6f3..1dcb04f235 100644
--- a/sysdeps/ieee754/flt-32/s_sincosf.h
+++ b/sysdeps/ieee754/flt-32/s_sincosf.h
@@ -1,5 +1,5 @@
 /* Used by sinf, cosf and sincosf functions.
-   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
@@ -20,145 +20,6 @@
 #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.  */
@@ -217,6 +78,36 @@ sincosf_poly (double x, double x2, const sincos_t *p, int n, float *sinp,
   *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
diff --git a/sysdeps/ieee754/flt-32/s_sinf.c b/sysdeps/ieee754/flt-32/s_sinf.c
index 138e318dcc..f6964e6022 100644
--- a/sysdeps/ieee754/flt-32/s_sinf.c
+++ b/sysdeps/ieee754/flt-32/s_sinf.c
@@ -1,5 +1,5 @@
 /* Compute sine 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
@@ -16,10 +16,11 @@
    License along with the GNU C Library; if not, see
    <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 SINF
@@ -28,127 +29,62 @@
 # define SINF_FUNC SINF
 #endif
 
+/* Fast sinf 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.
+*/
 float
-SINF_FUNC (float x)
+SINF_FUNC (float y)
 {
-  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))
+    {
+      s = 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)s);
+	return y;
+      }
+
+      return sinf_poly (x, s, p, 0);
+    }
+  else if (__glibc_likely (abstop12 (y) < abstop12 (120.0f)))
     {
-      if (abstheta >= 0x1p-5) /* |x| >= 2^-5.  */
-	{
-	  const double theta2 = theta * theta;
-	  /* Chebyshev polynomial of the form for sin
-	     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;
-	  return 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).  */
-	  const double theta2 = theta * theta;
-	  cx = SS0 + theta2 * SS1;
-	  cx = theta + theta * theta2 * cx;
-	  return cx;
-	}
-      else
-	{
-	  /* Handle some special cases.  */
-	  if (theta)
-	    return theta - (theta * SMALL);
-	  else
-	    return theta;
-	}
+      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];
+
+      return sinf_poly (x * s, x * x, p, n);
     }
-  else                          /* |x| >= Pi/4.  */
+  else if (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];
-	  return reduced_sin (theta, n, signbit);
-	}
-      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.  */
-	      return reduced_sin (theta, n, signbit);
-	    }
-	  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;
-	          return reduced_sin (e, l + 1, signbit);
-	        }
-	      else
-		{
-		  e += b;
-		  e += c;
-		  e += d;
-		  if (e <= 1.0)
-		    {
-		      e *= M_PI_4;
-		      return reduced_sin (e, l + 1, signbit);
-		    }
-		  else
-		    {
-		      l++;
-		      e -= 2.0;
-		      e *= M_PI_4;
-		      return reduced_sin (e, l + 1, signbit);
-		    }
-		}
-	    }
-	}
-      else
-	{
-	  int32_t ix;
-	  /* High word of x.  */
-	  GET_FLOAT_WORD (ix, abstheta);
-	  /* Sin(Inf or NaN) is NaN.  */
-	  if (ix == 0x7f800000)
-	    __set_errno (EDOM);
-	  return x - x;
-	}
+      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];
+
+      return sinf_poly (x * s, x * x, p, n);
     }
+  else
+    return __math_invalidf (y);
 }
 
 #ifndef SINF