-/* origin: FreeBSD /usr/src/lib/msun/src/e_exp.c */
/*
- * ====================================================
- * Copyright (C) 2004 by Sun Microsystems, Inc. All rights reserved.
+ * Double-precision e^x function.
*
- * Permission to use, copy, modify, and distribute this
- * software is freely granted, provided that this notice
- * is preserved.
- * ====================================================
- */
-/* exp(x)
- * Returns the exponential of x.
- *
- * Method
- * 1. Argument reduction:
- * Reduce x to an r so that |r| <= 0.5*ln2 ~ 0.34658.
- * Given x, find r and integer k such that
- *
- * x = k*ln2 + r, |r| <= 0.5*ln2.
- *
- * Here r will be represented as r = hi-lo for better
- * accuracy.
- *
- * 2. Approximation of exp(r) by a special rational function on
- * the interval [0,0.34658]:
- * Write
- * R(r**2) = r*(exp(r)+1)/(exp(r)-1) = 2 + r*r/6 - r**4/360 + ...
- * We use a special Remez algorithm on [0,0.34658] to generate
- * a polynomial of degree 5 to approximate R. The maximum error
- * of this polynomial approximation is bounded by 2**-59. In
- * other words,
- * R(z) ~ 2.0 + P1*z + P2*z**2 + P3*z**3 + P4*z**4 + P5*z**5
- * (where z=r*r, and the values of P1 to P5 are listed below)
- * and
- * | 5 | -59
- * | 2.0+P1*z+...+P5*z - R(z) | <= 2
- * | |
- * The computation of exp(r) thus becomes
- * 2*r
- * exp(r) = 1 + ----------
- * R(r) - r
- * r*c(r)
- * = 1 + r + ----------- (for better accuracy)
- * 2 - c(r)
- * where
- * 2 4 10
- * c(r) = r - (P1*r + P2*r + ... + P5*r ).
- *
- * 3. Scale back to obtain exp(x):
- * From step 1, we have
- * exp(x) = 2^k * exp(r)
- *
- * Special cases:
- * exp(INF) is INF, exp(NaN) is NaN;
- * exp(-INF) is 0, and
- * for finite argument, only exp(0)=1 is exact.
- *
- * Accuracy:
- * according to an error analysis, the error is always less than
- * 1 ulp (unit in the last place).
- *
- * Misc. info.
- * For IEEE double
- * if x > 709.782712893383973096 then exp(x) overflows
- * if x < -745.133219101941108420 then exp(x) underflows
+ * Copyright (c) 2018, Arm Limited.
+ * SPDX-License-Identifier: MIT
*/
+#include <math.h>
+#include <stdint.h>
#include "libm.h"
+#include "exp_data.h"
-static const double
-half[2] = {0.5,-0.5},
-ln2hi = 6.93147180369123816490e-01, /* 0x3fe62e42, 0xfee00000 */
-ln2lo = 1.90821492927058770002e-10, /* 0x3dea39ef, 0x35793c76 */
-invln2 = 1.44269504088896338700e+00, /* 0x3ff71547, 0x652b82fe */
-P1 = 1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */
-P2 = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */
-P3 = 6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */
-P4 = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */
-P5 = 4.13813679705723846039e-08; /* 0x3E663769, 0x72BEA4D0 */
+#define N (1 << EXP_TABLE_BITS)
+#define InvLn2N __exp_data.invln2N
+#define NegLn2hiN __exp_data.negln2hiN
+#define NegLn2loN __exp_data.negln2loN
+#define Shift __exp_data.shift
+#define T __exp_data.tab
+#define C2 __exp_data.poly[5 - EXP_POLY_ORDER]
+#define C3 __exp_data.poly[6 - EXP_POLY_ORDER]
+#define C4 __exp_data.poly[7 - EXP_POLY_ORDER]
+#define C5 __exp_data.poly[8 - EXP_POLY_ORDER]
-double exp(double x)
+/* Handle cases that may overflow or underflow when computing the result that
+ is scale*(1+TMP) without intermediate rounding. The bit representation of
+ scale is in SBITS, however it has a computed exponent that may have
+ overflown into the sign bit so that needs to be adjusted before using it as
+ a double. (int32_t)KI is the k used in the argument reduction and exponent
+ adjustment of scale, positive k here means the result may overflow and
+ negative k means the result may underflow. */
+static inline double specialcase(double_t tmp, uint64_t sbits, uint64_t ki)
{
- double_t hi, lo, c, xx, y;
- int k, sign;
- uint32_t hx;
-
- GET_HIGH_WORD(hx, x);
- sign = hx>>31;
- hx &= 0x7fffffff; /* high word of |x| */
+ double_t scale, y;
- /* special cases */
- if (hx >= 0x4086232b) { /* if |x| >= 708.39... */
- if (isnan(x))
- return x;
- if (x > 709.782712893383973096) {
- /* overflow if x!=inf */
- x *= 0x1p1023;
- return x;
- }
- if (x < -708.39641853226410622) {
- /* underflow if x!=-inf */
- FORCE_EVAL((float)(-0x1p-149/x));
- if (x < -745.13321910194110842)
- return 0;
- }
+ if ((ki & 0x80000000) == 0) {
+ /* k > 0, the exponent of scale might have overflowed by <= 460. */
+ sbits -= 1009ull << 52;
+ scale = asdouble(sbits);
+ y = 0x1p1009 * (scale + scale * tmp);
+ return eval_as_double(y);
+ }
+ /* k < 0, need special care in the subnormal range. */
+ sbits += 1022ull << 52;
+ scale = asdouble(sbits);
+ y = scale + scale * tmp;
+ if (y < 1.0) {
+ /* Round y to the right precision before scaling it into the subnormal
+ range to avoid double rounding that can cause 0.5+E/2 ulp error where
+ E is the worst-case ulp error outside the subnormal range. So this
+ is only useful if the goal is better than 1 ulp worst-case error. */
+ double_t hi, lo;
+ lo = scale - y + scale * tmp;
+ hi = 1.0 + y;
+ lo = 1.0 - hi + y + lo;
+ y = eval_as_double(hi + lo) - 1.0;
+ /* Avoid -0.0 with downward rounding. */
+ if (WANT_ROUNDING && y == 0.0)
+ y = 0.0;
+ /* The underflow exception needs to be signaled explicitly. */
+ fp_force_eval(fp_barrier(0x1p-1022) * 0x1p-1022);
}
+ y = 0x1p-1022 * y;
+ return eval_as_double(y);
+}
- /* argument reduction */
- if (hx > 0x3fd62e42) { /* if |x| > 0.5 ln2 */
- if (hx >= 0x3ff0a2b2) /* if |x| >= 1.5 ln2 */
- k = (int)(invln2*x + half[sign]);
- else
- k = 1 - sign - sign;
- hi = x - k*ln2hi; /* k*ln2hi is exact here */
- lo = k*ln2lo;
- x = hi - lo;
- } else if (hx > 0x3e300000) { /* if |x| > 2**-28 */
- k = 0;
- hi = x;
- lo = 0;
- } else {
- /* inexact if x!=0 */
- FORCE_EVAL(0x1p1023 + x);
- return 1 + x;
+/* Top 12 bits of a double (sign and exponent bits). */
+static inline uint32_t top12(double x)
+{
+ return asuint64(x) >> 52;
+}
+
+double exp(double x)
+{
+ uint32_t abstop;
+ uint64_t ki, idx, top, sbits;
+ double_t kd, z, r, r2, scale, tail, tmp;
+
+ abstop = top12(x) & 0x7ff;
+ if (predict_false(abstop - top12(0x1p-54) >= top12(512.0) - top12(0x1p-54))) {
+ if (abstop - top12(0x1p-54) >= 0x80000000)
+ /* Avoid spurious underflow for tiny x. */
+ /* Note: 0 is common input. */
+ return WANT_ROUNDING ? 1.0 + x : 1.0;
+ if (abstop >= top12(1024.0)) {
+ if (asuint64(x) == asuint64(-INFINITY))
+ return 0.0;
+ if (abstop >= top12(INFINITY))
+ return 1.0 + x;
+ if (asuint64(x) >> 63)
+ return __math_uflow(0);
+ else
+ return __math_oflow(0);
+ }
+ /* Large x is special cased below. */
+ abstop = 0;
}
- /* x is now in primary range */
- xx = x*x;
- c = x - xx*(P1+xx*(P2+xx*(P3+xx*(P4+xx*P5))));
- y = 1 + (x*c/(2-c) - lo + hi);
- if (k == 0)
- return y;
- return scalbn(y, k);
+ /* exp(x) = 2^(k/N) * exp(r), with exp(r) in [2^(-1/2N),2^(1/2N)]. */
+ /* x = ln2/N*k + r, with int k and r in [-ln2/2N, ln2/2N]. */
+ z = InvLn2N * x;
+#if TOINT_INTRINSICS
+ kd = roundtoint(z);
+ ki = converttoint(z);
+#elif EXP_USE_TOINT_NARROW
+ /* z - kd is in [-0.5-2^-16, 0.5] in all rounding modes. */
+ kd = eval_as_double(z + Shift);
+ ki = asuint64(kd) >> 16;
+ kd = (double_t)(int32_t)ki;
+#else
+ /* z - kd is in [-1, 1] in non-nearest rounding modes. */
+ kd = eval_as_double(z + Shift);
+ ki = asuint64(kd);
+ kd -= Shift;
+#endif
+ r = x + kd * NegLn2hiN + kd * NegLn2loN;
+ /* 2^(k/N) ~= scale * (1 + tail). */
+ idx = 2 * (ki % N);
+ top = ki << (52 - EXP_TABLE_BITS);
+ tail = asdouble(T[idx]);
+ /* This is only a valid scale when -1023*N < k < 1024*N. */
+ sbits = T[idx + 1] + top;
+ /* exp(x) = 2^(k/N) * exp(r) ~= scale + scale * (tail + exp(r) - 1). */
+ /* Evaluation is optimized assuming superscalar pipelined execution. */
+ r2 = r * r;
+ /* Without fma the worst case error is 0.25/N ulp larger. */
+ /* Worst case error is less than 0.5+1.11/N+(abs poly error * 2^53) ulp. */
+ tmp = tail + r + r2 * (C2 + r * C3) + r2 * r2 * (C4 + r * C5);
+ if (predict_false(abstop == 0))
+ return specialcase(tmp, sbits, ki);
+ scale = asdouble(sbits);
+ /* Note: tmp == 0 or |tmp| > 2^-200 and scale > 2^-739, so there
+ is no spurious underflow here even without fma. */
+ return eval_as_double(scale + scale * tmp);
}
-/* origin: FreeBSD /usr/src/lib/msun/src/s_exp2.c */
-/*-
- * Copyright (c) 2005 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.
+/*
+ * Double-precision 2^x function.
*
- * 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.
+ * Copyright (c) 2018, Arm Limited.
+ * SPDX-License-Identifier: MIT
*/
+#include <math.h>
+#include <stdint.h>
#include "libm.h"
+#include "exp_data.h"
-#define TBLSIZE 256
+#define N (1 << EXP_TABLE_BITS)
+#define Shift __exp_data.exp2_shift
+#define T __exp_data.tab
+#define C1 __exp_data.exp2_poly[0]
+#define C2 __exp_data.exp2_poly[1]
+#define C3 __exp_data.exp2_poly[2]
+#define C4 __exp_data.exp2_poly[3]
+#define C5 __exp_data.exp2_poly[4]
-static const double
-redux = 0x1.8p52 / TBLSIZE,
-P1 = 0x1.62e42fefa39efp-1,
-P2 = 0x1.ebfbdff82c575p-3,
-P3 = 0x1.c6b08d704a0a6p-5,
-P4 = 0x1.3b2ab88f70400p-7,
-P5 = 0x1.5d88003875c74p-10;
+/* Handle cases that may overflow or underflow when computing the result that
+ is scale*(1+TMP) without intermediate rounding. The bit representation of
+ scale is in SBITS, however it has a computed exponent that may have
+ overflown into the sign bit so that needs to be adjusted before using it as
+ a double. (int32_t)KI is the k used in the argument reduction and exponent
+ adjustment of scale, positive k here means the result may overflow and
+ negative k means the result may underflow. */
+static inline double specialcase(double_t tmp, uint64_t sbits, uint64_t ki)
+{
+ double_t scale, y;
-static const double tbl[TBLSIZE * 2] = {
-/* exp2(z + eps) eps */
- 0x1.6a09e667f3d5dp-1, 0x1.9880p-44,
- 0x1.6b052fa751744p-1, 0x1.8000p-50,
- 0x1.6c012750bd9fep-1, -0x1.8780p-45,
- 0x1.6cfdcddd476bfp-1, 0x1.ec00p-46,
- 0x1.6dfb23c651a29p-1, -0x1.8000p-50,
- 0x1.6ef9298593ae3p-1, -0x1.c000p-52,
- 0x1.6ff7df9519386p-1, -0x1.fd80p-45,
- 0x1.70f7466f42da3p-1, -0x1.c880p-45,
- 0x1.71f75e8ec5fc3p-1, 0x1.3c00p-46,
- 0x1.72f8286eacf05p-1, -0x1.8300p-44,
- 0x1.73f9a48a58152p-1, -0x1.0c00p-47,
- 0x1.74fbd35d7ccfcp-1, 0x1.f880p-45,
- 0x1.75feb564267f1p-1, 0x1.3e00p-47,
- 0x1.77024b1ab6d48p-1, -0x1.7d00p-45,
- 0x1.780694fde5d38p-1, -0x1.d000p-50,
- 0x1.790b938ac1d00p-1, 0x1.3000p-49,
- 0x1.7a11473eb0178p-1, -0x1.d000p-49,
- 0x1.7b17b0976d060p-1, 0x1.0400p-45,
- 0x1.7c1ed0130c133p-1, 0x1.0000p-53,
- 0x1.7d26a62ff8636p-1, -0x1.6900p-45,
- 0x1.7e2f336cf4e3bp-1, -0x1.2e00p-47,
- 0x1.7f3878491c3e8p-1, -0x1.4580p-45,
- 0x1.80427543e1b4ep-1, 0x1.3000p-44,
- 0x1.814d2add1071ap-1, 0x1.f000p-47,
- 0x1.82589994ccd7ep-1, -0x1.1c00p-45,
- 0x1.8364c1eb942d0p-1, 0x1.9d00p-45,
- 0x1.8471a4623cab5p-1, 0x1.7100p-43,
- 0x1.857f4179f5bbcp-1, 0x1.2600p-45,
- 0x1.868d99b4491afp-1, -0x1.2c40p-44,
- 0x1.879cad931a395p-1, -0x1.3000p-45,
- 0x1.88ac7d98a65b8p-1, -0x1.a800p-45,
- 0x1.89bd0a4785800p-1, -0x1.d000p-49,
- 0x1.8ace5422aa223p-1, 0x1.3280p-44,
- 0x1.8be05bad619fap-1, 0x1.2b40p-43,
- 0x1.8cf3216b54383p-1, -0x1.ed00p-45,
- 0x1.8e06a5e08664cp-1, -0x1.0500p-45,
- 0x1.8f1ae99157807p-1, 0x1.8280p-45,
- 0x1.902fed0282c0ep-1, -0x1.cb00p-46,
- 0x1.9145b0b91ff96p-1, -0x1.5e00p-47,
- 0x1.925c353aa2ff9p-1, 0x1.5400p-48,
- 0x1.93737b0cdc64ap-1, 0x1.7200p-46,
- 0x1.948b82b5f98aep-1, -0x1.9000p-47,
- 0x1.95a44cbc852cbp-1, 0x1.5680p-45,
- 0x1.96bdd9a766f21p-1, -0x1.6d00p-44,
- 0x1.97d829fde4e2ap-1, -0x1.1000p-47,
- 0x1.98f33e47a23a3p-1, 0x1.d000p-45,
- 0x1.9a0f170ca0604p-1, -0x1.8a40p-44,
- 0x1.9b2bb4d53ff89p-1, 0x1.55c0p-44,
- 0x1.9c49182a3f15bp-1, 0x1.6b80p-45,
- 0x1.9d674194bb8c5p-1, -0x1.c000p-49,
- 0x1.9e86319e3238ep-1, 0x1.7d00p-46,
- 0x1.9fa5e8d07f302p-1, 0x1.6400p-46,
- 0x1.a0c667b5de54dp-1, -0x1.5000p-48,
- 0x1.a1e7aed8eb8f6p-1, 0x1.9e00p-47,
- 0x1.a309bec4a2e27p-1, 0x1.ad80p-45,
- 0x1.a42c980460a5dp-1, -0x1.af00p-46,
- 0x1.a5503b23e259bp-1, 0x1.b600p-47,
- 0x1.a674a8af46213p-1, 0x1.8880p-44,
- 0x1.a799e1330b3a7p-1, 0x1.1200p-46,
- 0x1.a8bfe53c12e8dp-1, 0x1.6c00p-47,
- 0x1.a9e6b5579fcd2p-1, -0x1.9b80p-45,
- 0x1.ab0e521356fb8p-1, 0x1.b700p-45,
- 0x1.ac36bbfd3f381p-1, 0x1.9000p-50,
- 0x1.ad5ff3a3c2780p-1, 0x1.4000p-49,
- 0x1.ae89f995ad2a3p-1, -0x1.c900p-45,
- 0x1.afb4ce622f367p-1, 0x1.6500p-46,
- 0x1.b0e07298db790p-1, 0x1.fd40p-45,
- 0x1.b20ce6c9a89a9p-1, 0x1.2700p-46,
- 0x1.b33a2b84f1a4bp-1, 0x1.d470p-43,
- 0x1.b468415b747e7p-1, -0x1.8380p-44,
- 0x1.b59728de5593ap-1, 0x1.8000p-54,
- 0x1.b6c6e29f1c56ap-1, 0x1.ad00p-47,
- 0x1.b7f76f2fb5e50p-1, 0x1.e800p-50,
- 0x1.b928cf22749b2p-1, -0x1.4c00p-47,
- 0x1.ba5b030a10603p-1, -0x1.d700p-47,
- 0x1.bb8e0b79a6f66p-1, 0x1.d900p-47,
- 0x1.bcc1e904bc1ffp-1, 0x1.2a00p-47,
- 0x1.bdf69c3f3a16fp-1, -0x1.f780p-46,
- 0x1.bf2c25bd71db8p-1, -0x1.0a00p-46,
- 0x1.c06286141b2e9p-1, -0x1.1400p-46,
- 0x1.c199bdd8552e0p-1, 0x1.be00p-47,
- 0x1.c2d1cd9fa64eep-1, -0x1.9400p-47,
- 0x1.c40ab5fffd02fp-1, -0x1.ed00p-47,
- 0x1.c544778fafd15p-1, 0x1.9660p-44,
- 0x1.c67f12e57d0cbp-1, -0x1.a100p-46,
- 0x1.c7ba88988c1b6p-1, -0x1.8458p-42,
- 0x1.c8f6d9406e733p-1, -0x1.a480p-46,
- 0x1.ca3405751c4dfp-1, 0x1.b000p-51,
- 0x1.cb720dcef9094p-1, 0x1.1400p-47,
- 0x1.ccb0f2e6d1689p-1, 0x1.0200p-48,
- 0x1.cdf0b555dc412p-1, 0x1.3600p-48,
- 0x1.cf3155b5bab3bp-1, -0x1.6900p-47,
- 0x1.d072d4a0789bcp-1, 0x1.9a00p-47,
- 0x1.d1b532b08c8fap-1, -0x1.5e00p-46,
- 0x1.d2f87080d8a85p-1, 0x1.d280p-46,
- 0x1.d43c8eacaa203p-1, 0x1.1a00p-47,
- 0x1.d5818dcfba491p-1, 0x1.f000p-50,
- 0x1.d6c76e862e6a1p-1, -0x1.3a00p-47,
- 0x1.d80e316c9834ep-1, -0x1.cd80p-47,
- 0x1.d955d71ff6090p-1, 0x1.4c00p-48,
- 0x1.da9e603db32aep-1, 0x1.f900p-48,
- 0x1.dbe7cd63a8325p-1, 0x1.9800p-49,
- 0x1.dd321f301b445p-1, -0x1.5200p-48,
- 0x1.de7d5641c05bfp-1, -0x1.d700p-46,
- 0x1.dfc97337b9aecp-1, -0x1.6140p-46,
- 0x1.e11676b197d5ep-1, 0x1.b480p-47,
- 0x1.e264614f5a3e7p-1, 0x1.0ce0p-43,
- 0x1.e3b333b16ee5cp-1, 0x1.c680p-47,
- 0x1.e502ee78b3fb4p-1, -0x1.9300p-47,
- 0x1.e653924676d68p-1, -0x1.5000p-49,
- 0x1.e7a51fbc74c44p-1, -0x1.7f80p-47,
- 0x1.e8f7977cdb726p-1, -0x1.3700p-48,
- 0x1.ea4afa2a490e8p-1, 0x1.5d00p-49,
- 0x1.eb9f4867ccae4p-1, 0x1.61a0p-46,
- 0x1.ecf482d8e680dp-1, 0x1.5500p-48,
- 0x1.ee4aaa2188514p-1, 0x1.6400p-51,
- 0x1.efa1bee615a13p-1, -0x1.e800p-49,
- 0x1.f0f9c1cb64106p-1, -0x1.a880p-48,
- 0x1.f252b376bb963p-1, -0x1.c900p-45,
- 0x1.f3ac948dd7275p-1, 0x1.a000p-53,
- 0x1.f50765b6e4524p-1, -0x1.4f00p-48,
- 0x1.f6632798844fdp-1, 0x1.a800p-51,
- 0x1.f7bfdad9cbe38p-1, 0x1.abc0p-48,
- 0x1.f91d802243c82p-1, -0x1.4600p-50,
- 0x1.fa7c1819e908ep-1, -0x1.b0c0p-47,
- 0x1.fbdba3692d511p-1, -0x1.0e00p-51,
- 0x1.fd3c22b8f7194p-1, -0x1.0de8p-46,
- 0x1.fe9d96b2a23eep-1, 0x1.e430p-49,
- 0x1.0000000000000p+0, 0x0.0000p+0,
- 0x1.00b1afa5abcbep+0, -0x1.3400p-52,
- 0x1.0163da9fb3303p+0, -0x1.2170p-46,
- 0x1.02168143b0282p+0, 0x1.a400p-52,
- 0x1.02c9a3e77806cp+0, 0x1.f980p-49,
- 0x1.037d42e11bbcap+0, -0x1.7400p-51,
- 0x1.04315e86e7f89p+0, 0x1.8300p-50,
- 0x1.04e5f72f65467p+0, -0x1.a3f0p-46,
- 0x1.059b0d315855ap+0, -0x1.2840p-47,
- 0x1.0650a0e3c1f95p+0, 0x1.1600p-48,
- 0x1.0706b29ddf71ap+0, 0x1.5240p-46,
- 0x1.07bd42b72a82dp+0, -0x1.9a00p-49,
- 0x1.0874518759bd0p+0, 0x1.6400p-49,
- 0x1.092bdf66607c8p+0, -0x1.0780p-47,
- 0x1.09e3ecac6f383p+0, -0x1.8000p-54,
- 0x1.0a9c79b1f3930p+0, 0x1.fa00p-48,
- 0x1.0b5586cf988fcp+0, -0x1.ac80p-48,
- 0x1.0c0f145e46c8ap+0, 0x1.9c00p-50,
- 0x1.0cc922b724816p+0, 0x1.5200p-47,
- 0x1.0d83b23395dd8p+0, -0x1.ad00p-48,
- 0x1.0e3ec32d3d1f3p+0, 0x1.bac0p-46,
- 0x1.0efa55fdfa9a6p+0, -0x1.4e80p-47,
- 0x1.0fb66affed2f0p+0, -0x1.d300p-47,
- 0x1.1073028d7234bp+0, 0x1.1500p-48,
- 0x1.11301d0125b5bp+0, 0x1.c000p-49,
- 0x1.11edbab5e2af9p+0, 0x1.6bc0p-46,
- 0x1.12abdc06c31d5p+0, 0x1.8400p-49,
- 0x1.136a814f2047dp+0, -0x1.ed00p-47,
- 0x1.1429aaea92de9p+0, 0x1.8e00p-49,
- 0x1.14e95934f3138p+0, 0x1.b400p-49,
- 0x1.15a98c8a58e71p+0, 0x1.5300p-47,
- 0x1.166a45471c3dfp+0, 0x1.3380p-47,
- 0x1.172b83c7d5211p+0, 0x1.8d40p-45,
- 0x1.17ed48695bb9fp+0, -0x1.5d00p-47,
- 0x1.18af9388c8d93p+0, -0x1.c880p-46,
- 0x1.1972658375d66p+0, 0x1.1f00p-46,
- 0x1.1a35beb6fcba7p+0, 0x1.0480p-46,
- 0x1.1af99f81387e3p+0, -0x1.7390p-43,
- 0x1.1bbe084045d54p+0, 0x1.4e40p-45,
- 0x1.1c82f95281c43p+0, -0x1.a200p-47,
- 0x1.1d4873168b9b2p+0, 0x1.3800p-49,
- 0x1.1e0e75eb44031p+0, 0x1.ac00p-49,
- 0x1.1ed5022fcd938p+0, 0x1.1900p-47,
- 0x1.1f9c18438cdf7p+0, -0x1.b780p-46,
- 0x1.2063b88628d8fp+0, 0x1.d940p-45,
- 0x1.212be3578a81ep+0, 0x1.8000p-50,
- 0x1.21f49917ddd41p+0, 0x1.b340p-45,
- 0x1.22bdda2791323p+0, 0x1.9f80p-46,
- 0x1.2387a6e7561e7p+0, -0x1.9c80p-46,
- 0x1.2451ffb821427p+0, 0x1.2300p-47,
- 0x1.251ce4fb2a602p+0, -0x1.3480p-46,
- 0x1.25e85711eceb0p+0, 0x1.2700p-46,
- 0x1.26b4565e27d16p+0, 0x1.1d00p-46,
- 0x1.2780e341de00fp+0, 0x1.1ee0p-44,
- 0x1.284dfe1f5633ep+0, -0x1.4c00p-46,
- 0x1.291ba7591bb30p+0, -0x1.3d80p-46,
- 0x1.29e9df51fdf09p+0, 0x1.8b00p-47,
- 0x1.2ab8a66d10e9bp+0, -0x1.27c0p-45,
- 0x1.2b87fd0dada3ap+0, 0x1.a340p-45,
- 0x1.2c57e39771af9p+0, -0x1.0800p-46,
- 0x1.2d285a6e402d9p+0, -0x1.ed00p-47,
- 0x1.2df961f641579p+0, -0x1.4200p-48,
- 0x1.2ecafa93e2ecfp+0, -0x1.4980p-45,
- 0x1.2f9d24abd8822p+0, -0x1.6300p-46,
- 0x1.306fe0a31b625p+0, -0x1.2360p-44,
- 0x1.31432edeea50bp+0, -0x1.0df8p-40,
- 0x1.32170fc4cd7b8p+0, -0x1.2480p-45,
- 0x1.32eb83ba8e9a2p+0, -0x1.5980p-45,
- 0x1.33c08b2641766p+0, 0x1.ed00p-46,
- 0x1.3496266e3fa27p+0, -0x1.c000p-50,
- 0x1.356c55f929f0fp+0, -0x1.0d80p-44,
- 0x1.36431a2de88b9p+0, 0x1.2c80p-45,
- 0x1.371a7373aaa39p+0, 0x1.0600p-45,
- 0x1.37f26231e74fep+0, -0x1.6600p-46,
- 0x1.38cae6d05d838p+0, -0x1.ae00p-47,
- 0x1.39a401b713ec3p+0, -0x1.4720p-43,
- 0x1.3a7db34e5a020p+0, 0x1.8200p-47,
- 0x1.3b57fbfec6e95p+0, 0x1.e800p-44,
- 0x1.3c32dc313a8f2p+0, 0x1.f800p-49,
- 0x1.3d0e544ede122p+0, -0x1.7a00p-46,
- 0x1.3dea64c1234bbp+0, 0x1.6300p-45,
- 0x1.3ec70df1c4eccp+0, -0x1.8a60p-43,
- 0x1.3fa4504ac7e8cp+0, -0x1.cdc0p-44,
- 0x1.40822c367a0bbp+0, 0x1.5b80p-45,
- 0x1.4160a21f72e95p+0, 0x1.ec00p-46,
- 0x1.423fb27094646p+0, -0x1.3600p-46,
- 0x1.431f5d950a920p+0, 0x1.3980p-45,
- 0x1.43ffa3f84b9ebp+0, 0x1.a000p-48,
- 0x1.44e0860618919p+0, -0x1.6c00p-48,
- 0x1.45c2042a7d201p+0, -0x1.bc00p-47,
- 0x1.46a41ed1d0016p+0, -0x1.2800p-46,
- 0x1.4786d668b3326p+0, 0x1.0e00p-44,
- 0x1.486a2b5c13c00p+0, -0x1.d400p-45,
- 0x1.494e1e192af04p+0, 0x1.c200p-47,
- 0x1.4a32af0d7d372p+0, -0x1.e500p-46,
- 0x1.4b17dea6db801p+0, 0x1.7800p-47,
- 0x1.4bfdad53629e1p+0, -0x1.3800p-46,
- 0x1.4ce41b817c132p+0, 0x1.0800p-47,
- 0x1.4dcb299fddddbp+0, 0x1.c700p-45,
- 0x1.4eb2d81d8ab96p+0, -0x1.ce00p-46,
- 0x1.4f9b2769d2d02p+0, 0x1.9200p-46,
- 0x1.508417f4531c1p+0, -0x1.8c00p-47,
- 0x1.516daa2cf662ap+0, -0x1.a000p-48,
- 0x1.5257de83f51eap+0, 0x1.a080p-43,
- 0x1.5342b569d4edap+0, -0x1.6d80p-45,
- 0x1.542e2f4f6ac1ap+0, -0x1.2440p-44,
- 0x1.551a4ca5d94dbp+0, 0x1.83c0p-43,
- 0x1.56070dde9116bp+0, 0x1.4b00p-45,
- 0x1.56f4736b529dep+0, 0x1.15a0p-43,
- 0x1.57e27dbe2c40ep+0, -0x1.9e00p-45,
- 0x1.58d12d497c76fp+0, -0x1.3080p-45,
- 0x1.59c0827ff0b4cp+0, 0x1.dec0p-43,
- 0x1.5ab07dd485427p+0, -0x1.4000p-51,
- 0x1.5ba11fba87af4p+0, 0x1.0080p-44,
- 0x1.5c9268a59460bp+0, -0x1.6c80p-45,
- 0x1.5d84590998e3fp+0, 0x1.69a0p-43,
- 0x1.5e76f15ad20e1p+0, -0x1.b400p-46,
- 0x1.5f6a320dcebcap+0, 0x1.7700p-46,
- 0x1.605e1b976dcb8p+0, 0x1.6f80p-45,
- 0x1.6152ae6cdf715p+0, 0x1.1000p-47,
- 0x1.6247eb03a5531p+0, -0x1.5d00p-46,
- 0x1.633dd1d1929b5p+0, -0x1.2d00p-46,
- 0x1.6434634ccc313p+0, -0x1.a800p-49,
- 0x1.652b9febc8efap+0, -0x1.8600p-45,
- 0x1.6623882553397p+0, 0x1.1fe0p-40,
- 0x1.671c1c708328ep+0, -0x1.7200p-44,
- 0x1.68155d44ca97ep+0, 0x1.6800p-49,
- 0x1.690f4b19e9471p+0, -0x1.9780p-45,
-};
+ if ((ki & 0x80000000) == 0) {
+ /* k > 0, the exponent of scale might have overflowed by 1. */
+ sbits -= 1ull << 52;
+ scale = asdouble(sbits);
+ y = 2 * (scale + scale * tmp);
+ return eval_as_double(y);
+ }
+ /* k < 0, need special care in the subnormal range. */
+ sbits += 1022ull << 52;
+ scale = asdouble(sbits);
+ y = scale + scale * tmp;
+ if (y < 1.0) {
+ /* Round y to the right precision before scaling it into the subnormal
+ range to avoid double rounding that can cause 0.5+E/2 ulp error where
+ E is the worst-case ulp error outside the subnormal range. So this
+ is only useful if the goal is better than 1 ulp worst-case error. */
+ double_t hi, lo;
+ lo = scale - y + scale * tmp;
+ hi = 1.0 + y;
+ lo = 1.0 - hi + y + lo;
+ y = eval_as_double(hi + lo) - 1.0;
+ /* Avoid -0.0 with downward rounding. */
+ if (WANT_ROUNDING && y == 0.0)
+ y = 0.0;
+ /* The underflow exception needs to be signaled explicitly. */
+ fp_force_eval(fp_barrier(0x1p-1022) * 0x1p-1022);
+ }
+ y = 0x1p-1022 * y;
+ return eval_as_double(y);
+}
+
+/* Top 12 bits of a double (sign and exponent bits). */
+static inline uint32_t top12(double x)
+{
+ return asuint64(x) >> 52;
+}
-/*
- * exp2(x): compute the base 2 exponential of x
- *
- * Accuracy: Peak error < 0.503 ulp for normalized results.
- *
- * Method: (accurate tables)
- *
- * Reduce x:
- * x = k + y, for integer k and |y| <= 1/2.
- * Thus we have exp2(x) = 2**k * exp2(y).
- *
- * Reduce y:
- * y = i/TBLSIZE + z - eps[i] for integer i near y * TBLSIZE.
- * Thus we have exp2(y) = exp2(i/TBLSIZE) * exp2(z - eps[i]),
- * with |z - eps[i]| <= 2**-9 + 2**-39 for the table used.
- *
- * We compute exp2(i/TBLSIZE) via table lookup and exp2(z - eps[i]) via
- * a degree-5 minimax polynomial with maximum error under 1.3 * 2**-61.
- * The values in exp2t[] and eps[] are chosen such that
- * exp2t[i] = exp2(i/TBLSIZE + eps[i]), and eps[i] is a small offset such
- * that exp2t[i] is accurate to 2**-64.
- *
- * Note that the range of i is +-TBLSIZE/2, so we actually index the tables
- * by i0 = i + TBLSIZE/2. For cache efficiency, exp2t[] and eps[] are
- * virtual tables, interleaved in the real table tbl[].
- *
- * This method is due to Gal, with many details due to Gal and Bachelis:
- *
- * Gal, S. and Bachelis, B. An Accurate Elementary Mathematical Library
- * for the IEEE Floating Point Standard. TOMS 17(1), 26-46 (1991).
- */
double exp2(double x)
{
- double_t r, t, z;
- uint32_t ix, i0;
- union {double f; uint64_t i;} u = {x};
- union {uint32_t u; int32_t i;} k;
+ uint32_t abstop;
+ uint64_t ki, idx, top, sbits;
+ double_t kd, r, r2, scale, tail, tmp;
- /* Filter out exceptional cases. */
- ix = u.i>>32 & 0x7fffffff;
- if (ix >= 0x408ff000) { /* |x| >= 1022 or nan */
- if (ix >= 0x40900000 && u.i>>63 == 0) { /* x >= 1024 or nan */
- /* overflow */
- x *= 0x1p1023;
- return x;
- }
- if (ix >= 0x7ff00000) /* -inf or -nan */
- return -1/x;
- if (u.i>>63) { /* x <= -1022 */
- /* underflow */
- if (x <= -1075 || x - 0x1p52 + 0x1p52 != x)
- FORCE_EVAL((float)(-0x1p-149/x));
- if (x <= -1075)
- return 0;
+ abstop = top12(x) & 0x7ff;
+ if (predict_false(abstop - top12(0x1p-54) >= top12(512.0) - top12(0x1p-54))) {
+ if (abstop - top12(0x1p-54) >= 0x80000000)
+ /* Avoid spurious underflow for tiny x. */
+ /* Note: 0 is common input. */
+ return WANT_ROUNDING ? 1.0 + x : 1.0;
+ if (abstop >= top12(1024.0)) {
+ if (asuint64(x) == asuint64(-INFINITY))
+ return 0.0;
+ if (abstop >= top12(INFINITY))
+ return 1.0 + x;
+ if (!(asuint64(x) >> 63))
+ return __math_oflow(0);
+ else if (asuint64(x) >= asuint64(-1075.0))
+ return __math_uflow(0);
}
- } else if (ix < 0x3c900000) { /* |x| < 0x1p-54 */
- return 1.0 + x;
+ if (2 * asuint64(x) > 2 * asuint64(928.0))
+ /* Large x is special cased below. */
+ abstop = 0;
}
- /* Reduce x, computing z, i0, and k. */
- u.f = x + redux;
- i0 = u.i;
- i0 += TBLSIZE / 2;
- k.u = i0 / TBLSIZE * TBLSIZE;
- k.i /= TBLSIZE;
- i0 %= TBLSIZE;
- u.f -= redux;
- z = x - u.f;
-
- /* Compute r = exp2(y) = exp2t[i0] * p(z - eps[i]). */
- t = tbl[2*i0]; /* exp2t[i0] */
- z -= tbl[2*i0 + 1]; /* eps[i0] */
- r = t + t * z * (P1 + z * (P2 + z * (P3 + z * (P4 + z * P5))));
-
- return scalbn(r, k.i);
+ /* exp2(x) = 2^(k/N) * 2^r, with 2^r in [2^(-1/2N),2^(1/2N)]. */
+ /* x = k/N + r, with int k and r in [-1/2N, 1/2N]. */
+ kd = eval_as_double(x + Shift);
+ ki = asuint64(kd); /* k. */
+ kd -= Shift; /* k/N for int k. */
+ r = x - kd;
+ /* 2^(k/N) ~= scale * (1 + tail). */
+ idx = 2 * (ki % N);
+ top = ki << (52 - EXP_TABLE_BITS);
+ tail = asdouble(T[idx]);
+ /* This is only a valid scale when -1023*N < k < 1024*N. */
+ sbits = T[idx + 1] + top;
+ /* exp2(x) = 2^(k/N) * 2^r ~= scale + scale * (tail + 2^r - 1). */
+ /* Evaluation is optimized assuming superscalar pipelined execution. */
+ r2 = r * r;
+ /* Without fma the worst case error is 0.5/N ulp larger. */
+ /* Worst case error is less than 0.5+0.86/N+(abs poly error * 2^53) ulp. */
+ tmp = tail + r * C1 + r2 * (C2 + r * C3) + r2 * r2 * (C4 + r * C5);
+ if (predict_false(abstop == 0))
+ return specialcase(tmp, sbits, ki);
+ scale = asdouble(sbits);
+ /* Note: tmp == 0 or |tmp| > 2^-65 and scale > 2^-928, so there
+ is no spurious underflow here even without fma. */
+ return eval_as_double(scale + scale * tmp);
}
--- /dev/null
+/*
+ * Shared data between exp, exp2 and pow.
+ *
+ * Copyright (c) 2018, Arm Limited.
+ * SPDX-License-Identifier: MIT
+ */
+
+#include "exp_data.h"
+
+#define N (1 << EXP_TABLE_BITS)
+
+const struct exp_data __exp_data = {
+// N/ln2
+.invln2N = 0x1.71547652b82fep0 * N,
+// -ln2/N
+.negln2hiN = -0x1.62e42fefa0000p-8,
+.negln2loN = -0x1.cf79abc9e3b3ap-47,
+// Used for rounding when !TOINT_INTRINSICS
+#if EXP_USE_TOINT_NARROW
+.shift = 0x1800000000.8p0,
+#else
+.shift = 0x1.8p52,
+#endif
+// exp polynomial coefficients.
+.poly = {
+// abs error: 1.555*2^-66
+// ulp error: 0.509 (0.511 without fma)
+// if |x| < ln2/256+eps
+// abs error if |x| < ln2/256+0x1p-15: 1.09*2^-65
+// abs error if |x| < ln2/128: 1.7145*2^-56
+0x1.ffffffffffdbdp-2,
+0x1.555555555543cp-3,
+0x1.55555cf172b91p-5,
+0x1.1111167a4d017p-7,
+},
+.exp2_shift = 0x1.8p52 / N,
+// exp2 polynomial coefficients.
+.exp2_poly = {
+// abs error: 1.2195*2^-65
+// ulp error: 0.507 (0.511 without fma)
+// if |x| < 1/256
+// abs error if |x| < 1/128: 1.9941*2^-56
+0x1.62e42fefa39efp-1,
+0x1.ebfbdff82c424p-3,
+0x1.c6b08d70cf4b5p-5,
+0x1.3b2abd24650ccp-7,
+0x1.5d7e09b4e3a84p-10,
+},
+// 2^(k/N) ~= H[k]*(1 + T[k]) for int k in [0,N)
+// tab[2*k] = asuint64(T[k])
+// tab[2*k+1] = asuint64(H[k]) - (k << 52)/N
+.tab = {
+0x0, 0x3ff0000000000000,
+0x3c9b3b4f1a88bf6e, 0x3feff63da9fb3335,
+0xbc7160139cd8dc5d, 0x3fefec9a3e778061,
+0xbc905e7a108766d1, 0x3fefe315e86e7f85,
+0x3c8cd2523567f613, 0x3fefd9b0d3158574,
+0xbc8bce8023f98efa, 0x3fefd06b29ddf6de,
+0x3c60f74e61e6c861, 0x3fefc74518759bc8,
+0x3c90a3e45b33d399, 0x3fefbe3ecac6f383,
+0x3c979aa65d837b6d, 0x3fefb5586cf9890f,
+0x3c8eb51a92fdeffc, 0x3fefac922b7247f7,
+0x3c3ebe3d702f9cd1, 0x3fefa3ec32d3d1a2,
+0xbc6a033489906e0b, 0x3fef9b66affed31b,
+0xbc9556522a2fbd0e, 0x3fef9301d0125b51,
+0xbc5080ef8c4eea55, 0x3fef8abdc06c31cc,
+0xbc91c923b9d5f416, 0x3fef829aaea92de0,
+0x3c80d3e3e95c55af, 0x3fef7a98c8a58e51,
+0xbc801b15eaa59348, 0x3fef72b83c7d517b,
+0xbc8f1ff055de323d, 0x3fef6af9388c8dea,
+0x3c8b898c3f1353bf, 0x3fef635beb6fcb75,
+0xbc96d99c7611eb26, 0x3fef5be084045cd4,
+0x3c9aecf73e3a2f60, 0x3fef54873168b9aa,
+0xbc8fe782cb86389d, 0x3fef4d5022fcd91d,
+0x3c8a6f4144a6c38d, 0x3fef463b88628cd6,
+0x3c807a05b0e4047d, 0x3fef3f49917ddc96,
+0x3c968efde3a8a894, 0x3fef387a6e756238,
+0x3c875e18f274487d, 0x3fef31ce4fb2a63f,
+0x3c80472b981fe7f2, 0x3fef2b4565e27cdd,
+0xbc96b87b3f71085e, 0x3fef24dfe1f56381,
+0x3c82f7e16d09ab31, 0x3fef1e9df51fdee1,
+0xbc3d219b1a6fbffa, 0x3fef187fd0dad990,
+0x3c8b3782720c0ab4, 0x3fef1285a6e4030b,
+0x3c6e149289cecb8f, 0x3fef0cafa93e2f56,
+0x3c834d754db0abb6, 0x3fef06fe0a31b715,
+0x3c864201e2ac744c, 0x3fef0170fc4cd831,
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+};
projects / oweals / musl.git / commitdiff