@@ -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
new file mode 100644
@@ -0,0 +1 @@
+/* Not needed. */
@@ -42,6 +42,9 @@
the semantics documented below. */
# define TOINT_INTRINSICS 0
#endif
+#ifndef PREFER_FLOAT_COMPARISON
+# define PREFER_FLOAT_COMPARISON 0
+#endif
#if TOINT_INTRINSICS
/* Round x to nearest int in all rounding modes, ties have to be rounded
@@ -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
}
}
@@ -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;
+}
new file mode 100644
@@ -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
+};
new file mode 100644
@@ -0,0 +1 @@
+/* Not needed. */
new file mode 100644
@@ -0,0 +1 @@
+/* Not needed. */