--- /dev/null
+/* origin: FreeBSD /usr/src/lib/msun/src/k_exp.c */
+/*-
+ * Copyright (c) 2011 David Schultz <das@FreeBSD.ORG>
+ * 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.
+ *
+ * THIS SOFTWARE IS PROVIDED BY THE AUTHOR AND CONTRIBUTORS ``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 THE AUTHOR OR CONTRIBUTORS 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 "libm.h"
+
+static const uint32_t k = 1799; /* constant for reduction */
+static const double kln2 = 1246.97177782734161156; /* k * ln2 */
+
+/*
+ * Compute exp(x), scaled to avoid spurious overflow. An exponent is
+ * returned separately in 'expt'.
+ *
+ * Input: ln(DBL_MAX) <= x < ln(2 * DBL_MAX / DBL_MIN_DENORM) ~= 1454.91
+ * Output: 2**1023 <= y < 2**1024
+ */
+static double __frexp_exp(double x, int *expt)
+{
+ double exp_x;
+ uint32_t hx;
+
+ /*
+ * We use exp(x) = exp(x - kln2) * 2**k, carefully chosen to
+ * minimize |exp(kln2) - 2**k|. We also scale the exponent of
+ * exp_x to MAX_EXP so that the result can be multiplied by
+ * a tiny number without losing accuracy due to denormalization.
+ */
+ exp_x = exp(x - kln2);
+ GET_HIGH_WORD(hx, exp_x);
+ *expt = (hx >> 20) - (0x3ff + 1023) + k;
+ SET_HIGH_WORD(exp_x, (hx & 0xfffff) | ((0x3ff + 1023) << 20));
+ return exp_x;
+}
+
+/*
+ * __ldexp_cexp(x, expt) compute exp(x) * 2**expt.
+ * It is intended for large arguments (real part >= ln(DBL_MAX))
+ * where care is needed to avoid overflow.
+ *
+ * The present implementation is narrowly tailored for our hyperbolic and
+ * exponential functions. We assume expt is small (0 or -1), and the caller
+ * has filtered out very large x, for which overflow would be inevitable.
+ */
+double complex __ldexp_cexp(double complex z, int expt)
+{
+ double x, y, exp_x, scale1, scale2;
+ int ex_expt, half_expt;
+
+ x = creal(z);
+ y = cimag(z);
+ exp_x = __frexp_exp(x, &ex_expt);
+ expt += ex_expt;
+
+ /*
+ * Arrange so that scale1 * scale2 == 2**expt. We use this to
+ * compensate for scalbn being horrendously slow.
+ */
+ half_expt = expt / 2;
+ INSERT_WORDS(scale1, (0x3ff + half_expt) << 20, 0);
+ half_expt = expt - half_expt;
+ INSERT_WORDS(scale2, (0x3ff + half_expt) << 20, 0);
+
+ return cpack(cos(y) * exp_x * scale1 * scale2, sin(y) * exp_x * scale1 * scale2);
+}
--- /dev/null
+/* origin: FreeBSD /usr/src/lib/msun/src/k_expf.c */
+/*-
+ * Copyright (c) 2011 David Schultz <das@FreeBSD.ORG>
+ * 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.
+ *
+ * THIS SOFTWARE IS PROVIDED BY THE AUTHOR AND CONTRIBUTORS ``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 THE AUTHOR OR CONTRIBUTORS 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 "libm.h"
+
+static const uint32_t k = 235; /* constant for reduction */
+static const float kln2 = 162.88958740F; /* k * ln2 */
+
+/*
+ * See __cexp.c for details.
+ *
+ * Input: ln(FLT_MAX) <= x < ln(2 * FLT_MAX / FLT_MIN_DENORM) ~= 192.7
+ * Output: 2**127 <= y < 2**128
+ */
+static float __frexp_expf(float x, int *expt)
+{
+ float exp_x;
+ uint32_t hx;
+
+ exp_x = expf(x - kln2);
+ GET_FLOAT_WORD(hx, exp_x);
+ *expt = (hx >> 23) - (0x7f + 127) + k;
+ SET_FLOAT_WORD(exp_x, (hx & 0x7fffff) | ((0x7f + 127) << 23));
+ return exp_x;
+}
+
+float complex __ldexp_cexpf(float complex z, int expt)
+{
+ float x, y, exp_x, scale1, scale2;
+ int ex_expt, half_expt;
+
+ x = crealf(z);
+ y = cimagf(z);
+ exp_x = __frexp_expf(x, &ex_expt);
+ expt += ex_expt;
+
+ half_expt = expt / 2;
+ SET_FLOAT_WORD(scale1, (0x7f + half_expt) << 23);
+ half_expt = expt - half_expt;
+ SET_FLOAT_WORD(scale2, (0x7f + half_expt) << 23);
+
+ return cpackf(cosf(y) * exp_x * scale1 * scale2,
+ sinf(y) * exp_x * scale1 * scale2);
+}
double __sin(double,double,int);
double __cos(double,double);
double __tan(double,double,int);
-double __ldexp_exp(double,int);
+double __expo2(double);
double complex __ldexp_cexp(double complex,int);
int __rem_pio2f(float,double*);
float __sindf(double);
float __cosdf(double);
float __tandf(double,int);
-float __ldexp_expf(float,int);
+float __expo2f(float);
float complex __ldexp_cexpf(float complex,int);
long double __sinl(long double, long double, int);
#include "libm.h"
-static const uint32_t k = 1799; /* constant for reduction */
-static const double kln2 = 1246.97177782734161156; /* k * ln2 */
-
-/*
- * Compute exp(x), scaled to avoid spurious overflow. An exponent is
- * returned separately in 'expt'.
- *
- * Input: ln(DBL_MAX) <= x < ln(2 * DBL_MAX / DBL_MIN_DENORM) ~= 1454.91
- * Output: 2**1023 <= y < 2**1024
- */
-static double __frexp_exp(double x, int *expt)
-{
- double exp_x;
- uint32_t hx;
-
- /*
- * We use exp(x) = exp(x - kln2) * 2**k, carefully chosen to
- * minimize |exp(kln2) - 2**k|. We also scale the exponent of
- * exp_x to MAX_EXP so that the result can be multiplied by
- * a tiny number without losing accuracy due to denormalization.
- */
- exp_x = exp(x - kln2);
- GET_HIGH_WORD(hx, exp_x);
- *expt = (hx >> 20) - (0x3ff + 1023) + k;
- SET_HIGH_WORD(exp_x, (hx & 0xfffff) | ((0x3ff + 1023) << 20));
- return exp_x;
-}
-
/*
- * __ldexp_exp(x, expt) and __ldexp_cexp(x, expt) compute exp(x) * 2**expt.
- * They are intended for large arguments (real part >= ln(DBL_MAX))
- * where care is needed to avoid overflow.
- *
- * The present implementation is narrowly tailored for our hyperbolic and
- * exponential functions. We assume expt is small (0 or -1), and the caller
- * has filtered out very large x, for which overflow would be inevitable.
+ * We use exp(x) = exp(x - kln2) * 2**k,
+ * k is carefully chosen to minimize |exp(kln2) - 2**k|
*/
-double __ldexp_exp(double x, int expt)
-{
- double exp_x, scale;
- int ex_expt;
+static const uint32_t k = 1799;
+static const double kln2 = 1246.97177782734161156;
- exp_x = __frexp_exp(x, &ex_expt);
- expt += ex_expt;
- INSERT_WORDS(scale, (0x3ff + expt) << 20, 0);
- return exp_x * scale;
-}
-
-double complex __ldexp_cexp(double complex z, int expt)
+/* exp(x)/2 when x is huge */
+double __expo2(double x)
{
- double x, y, exp_x, scale1, scale2;
- int ex_expt, half_expt;
-
- x = creal(z);
- y = cimag(z);
- exp_x = __frexp_exp(x, &ex_expt);
- expt += ex_expt;
+ double scale;
+ int n;
/*
- * Arrange so that scale1 * scale2 == 2**expt. We use this to
- * compensate for scalbn being horrendously slow.
+ * efficient scalbn(y, k-1):
+ * 2**(k-1) cannot be represented
+ * so we use that k-1 is even and scale in two steps
*/
- half_expt = expt / 2;
- INSERT_WORDS(scale1, (0x3ff + half_expt) << 20, 0);
- half_expt = expt - half_expt;
- INSERT_WORDS(scale2, (0x3ff + half_expt) << 20, 0);
-
- return cpack(cos(y) * exp_x * scale1 * scale2, sin(y) * exp_x * scale1 * scale2);
+ n = (k - 1)/2;
+ INSERT_WORDS(scale, (0x3ff + n) << 20, 0);
+ return exp(x - kln2) * scale * scale;
}
#include "libm.h"
-static const uint32_t k = 235; /* constant for reduction */
-static const float kln2 = 162.88958740F; /* k * ln2 */
-
/*
- * See __exp.c for details.
- *
- * Input: ln(FLT_MAX) <= x < ln(2 * FLT_MAX / FLT_MIN_DENORM) ~= 192.7
- * Output: 2**127 <= y < 2**128
+ * We use exp(x) = exp(x - kln2) * 2**k,
+ * k is carefully chosen to minimize |exp(kln2) - 2**k|
*/
-static float __frexp_expf(float x, int *expt)
-{
- float exp_x;
- uint32_t hx;
-
- exp_x = expf(x - kln2);
- GET_FLOAT_WORD(hx, exp_x);
- *expt = (hx >> 23) - (0x7f + 127) + k;
- SET_FLOAT_WORD(exp_x, (hx & 0x7fffff) | ((0x7f + 127) << 23));
- return exp_x;
-}
-
-float __ldexp_expf(float x, int expt)
-{
- float exp_x, scale;
- int ex_expt;
-
- exp_x = __frexp_expf(x, &ex_expt);
- expt += ex_expt;
- SET_FLOAT_WORD(scale, (0x7f + expt) << 23);
- return exp_x * scale;
-}
+static const uint32_t k = 235;
+static const float kln2 = 162.88958740f;
-float complex __ldexp_cexpf(float complex z, int expt)
+/* expf(x)/2 when x is huge */
+float __expo2f(float x)
{
- float x, y, exp_x, scale1, scale2;
- int ex_expt, half_expt;
-
- x = crealf(z);
- y = cimagf(z);
- exp_x = __frexp_expf(x, &ex_expt);
- expt += ex_expt;
-
- half_expt = expt / 2;
- SET_FLOAT_WORD(scale1, (0x7f + half_expt) << 23);
- half_expt = expt - half_expt;
- SET_FLOAT_WORD(scale2, (0x7f + half_expt) << 23);
-
- return (cpackf(cosf(y) * exp_x * scale1 * scale2,
- sinf(y) * exp_x * scale1 * scale2));
+ float scale;
+ int n;
+
+ /*
+ * efficient scalbnf(y, k-1):
+ * 2**(k-1) cannot be represented
+ * so we use that k-1 is even and scale in two steps
+ */
+ n = (k - 1)/2;
+ SET_FLOAT_WORD(scale, (0x7f + n) << 23);
+ return expf(x - kln2) * scale * scale;
}
/* |x| in [log(maxdouble), overflowthresold] */
if (ix <= 0x408633CE)
- return __ldexp_exp(fabs(x), -1);
+ return __expo2(fabs(x));
/* |x| > overflowthresold, cosh(x) overflow */
return huge*huge;
/* |x| in [log(maxfloat), overflowthresold] */
if (ix <= 0x42b2d4fc)
- return __ldexp_expf(fabsf(x), -1);
+ return __expo2f(fabsf(x));
/* |x| > overflowthresold, cosh(x) overflow */
return huge*huge;
/* |x| in [log(maxdouble), overflowthresold] */
if (ix <= 0x408633CE)
- // FIXME: 0.5 * 2.0 * huge == huge ?
- return h*2.0*__ldexp_exp(fabs(x), -1);
+ return h * 2.0 * __expo2(fabs(x)); /* h is for sign only */
/* |x| > overflowthresold, sinh(x) overflow */
return x*huge;
/* |x| in [logf(maxfloat), overflowthresold] */
if (ix <= 0x42b2d4fc)
- // FIXME: 0.5f * 2.0f * huge == huge ?
- return h*2.0F*__ldexp_expf(fabsf(x), -1);
+ return h * 2.0f * __expo2f(fabsf(x)); /* h is for sign only */
/* |x| > overflowthresold, sinh(x) overflow */
return x*huge;