5f63fbd67f
* ld128/k_expl.h:
. Split out a computational kernel,__k_expl(x, &hi, &lo, &k) from expl(x).
x must be finite and not tiny or huge. The kernel returns hi and lo
values for extra precision and an exponent k for a 2**k scale factor.
. Define additional kernels k_hexpl() and hexpl() that include a 1/2
scaling and are used by the hyperbolic functions.
* ld80/s_expl.c:
* ld128/s_expl.c:
. Use the __k_expl() kernel.
Obtained from: bde
285 lines
8.3 KiB
C
285 lines
8.3 KiB
C
/*-
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* Copyright (c) 2009-2013 Steven G. Kargl
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* All rights reserved.
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*
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* Redistribution and use in source and binary forms, with or without
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* modification, are permitted provided that the following conditions
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* are met:
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* 1. Redistributions of source code must retain the above copyright
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* notice unmodified, this list of conditions, and the following
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* disclaimer.
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* 2. Redistributions in binary form must reproduce the above copyright
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* notice, this list of conditions and the following disclaimer in the
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* documentation and/or other materials provided with the distribution.
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*
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* THIS SOFTWARE IS PROVIDED BY THE AUTHOR ``AS IS'' AND ANY EXPRESS OR
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* IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES
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* OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED.
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* IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY DIRECT, INDIRECT,
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* INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT
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* NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE,
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* DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY
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* THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
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* (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF
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* THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
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*
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* Optimized by Bruce D. Evans.
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*/
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#include <sys/cdefs.h>
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__FBSDID("$FreeBSD$");
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/**
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* Compute the exponential of x for Intel 80-bit format. This is based on:
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*
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* PTP Tang, "Table-driven implementation of the exponential function
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* in IEEE floating-point arithmetic," ACM Trans. Math. Soft., 15,
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* 144-157 (1989).
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*
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* where the 32 table entries have been expanded to INTERVALS (see below).
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*/
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#include <float.h>
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#ifdef __i386__
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#include <ieeefp.h>
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#endif
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#include "fpmath.h"
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#include "math.h"
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#include "math_private.h"
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#include "k_expl.h"
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/* XXX Prevent compilers from erroneously constant folding these: */
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static const volatile long double
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huge = 0x1p10000L,
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tiny = 0x1p-10000L;
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static const long double
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twom10000 = 0x1p-10000L;
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static const union IEEEl2bits
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/* log(2**16384 - 0.5) rounded towards zero: */
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/* log(2**16384 - 0.5 + 1) rounded towards zero for expm1l() is the same: */
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o_thresholdu = LD80C(0xb17217f7d1cf79ab, 13, 11356.5234062941439488L),
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#define o_threshold (o_thresholdu.e)
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/* log(2**(-16381-64-1)) rounded towards zero: */
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u_thresholdu = LD80C(0xb21dfe7f09e2baa9, 13, -11399.4985314888605581L);
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#define u_threshold (u_thresholdu.e)
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long double
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expl(long double x)
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{
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union IEEEl2bits u;
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long double hi, lo, t, twopk;
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int k;
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uint16_t hx, ix;
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DOPRINT_START(&x);
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/* Filter out exceptional cases. */
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u.e = x;
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hx = u.xbits.expsign;
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ix = hx & 0x7fff;
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if (ix >= BIAS + 13) { /* |x| >= 8192 or x is NaN */
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if (ix == BIAS + LDBL_MAX_EXP) {
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if (hx & 0x8000) /* x is -Inf, -NaN or unsupported */
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RETURNP(-1 / x);
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RETURNP(x + x); /* x is +Inf, +NaN or unsupported */
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}
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if (x > o_threshold)
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RETURNP(huge * huge);
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if (x < u_threshold)
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RETURNP(tiny * tiny);
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} else if (ix < BIAS - 75) { /* |x| < 0x1p-75 (includes pseudos) */
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RETURN2P(1, x); /* 1 with inexact iff x != 0 */
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}
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ENTERI();
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twopk = 1;
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__k_expl(x, &hi, &lo, &k);
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t = SUM2P(hi, lo);
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/* Scale by 2**k. */
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if (k >= LDBL_MIN_EXP) {
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if (k == LDBL_MAX_EXP)
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RETURNI(t * 2 * 0x1p16383L);
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SET_LDBL_EXPSIGN(twopk, BIAS + k);
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RETURNI(t * twopk);
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} else {
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SET_LDBL_EXPSIGN(twopk, BIAS + k + 10000);
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RETURNI(t * twopk * twom10000);
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}
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}
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/**
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* Compute expm1l(x) for Intel 80-bit format. This is based on:
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*
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* PTP Tang, "Table-driven implementation of the Expm1 function
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* in IEEE floating-point arithmetic," ACM Trans. Math. Soft., 18,
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* 211-222 (1992).
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*/
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/*
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* Our T1 and T2 are chosen to be approximately the points where method
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* A and method B have the same accuracy. Tang's T1 and T2 are the
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* points where method A's accuracy changes by a full bit. For Tang,
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* this drop in accuracy makes method A immediately less accurate than
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* method B, but our larger INTERVALS makes method A 2 bits more
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* accurate so it remains the most accurate method significantly
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* closer to the origin despite losing the full bit in our extended
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* range for it.
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*/
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static const double
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T1 = -0.1659, /* ~-30.625/128 * log(2) */
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T2 = 0.1659; /* ~30.625/128 * log(2) */
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/*
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* Domain [-0.1659, 0.1659], range ~[-2.6155e-22, 2.5507e-23]:
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* |(exp(x)-1-x-x**2/2)/x - p(x)| < 2**-71.6
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*
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* XXX the coeffs aren't very carefully rounded, and I get 2.8 more bits,
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* but unlike for ld128 we can't drop any terms.
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*/
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static const union IEEEl2bits
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B3 = LD80C(0xaaaaaaaaaaaaaaab, -3, 1.66666666666666666671e-1L),
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B4 = LD80C(0xaaaaaaaaaaaaaaac, -5, 4.16666666666666666712e-2L);
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static const double
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B5 = 8.3333333333333245e-3, /* 0x1.111111111110cp-7 */
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B6 = 1.3888888888888861e-3, /* 0x1.6c16c16c16c0ap-10 */
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B7 = 1.9841269841532042e-4, /* 0x1.a01a01a0319f9p-13 */
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B8 = 2.4801587302069236e-5, /* 0x1.a01a01a03cbbcp-16 */
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B9 = 2.7557316558468562e-6, /* 0x1.71de37fd33d67p-19 */
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B10 = 2.7557315829785151e-7, /* 0x1.27e4f91418144p-22 */
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B11 = 2.5063168199779829e-8, /* 0x1.ae94fabdc6b27p-26 */
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B12 = 2.0887164654459567e-9; /* 0x1.1f122d6413fe1p-29 */
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long double
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expm1l(long double x)
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{
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union IEEEl2bits u, v;
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long double fn, hx2_hi, hx2_lo, q, r, r1, r2, t, twomk, twopk, x_hi;
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long double x_lo, x2, z;
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long double x4;
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int k, n, n2;
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uint16_t hx, ix;
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DOPRINT_START(&x);
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/* Filter out exceptional cases. */
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u.e = x;
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hx = u.xbits.expsign;
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ix = hx & 0x7fff;
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if (ix >= BIAS + 6) { /* |x| >= 64 or x is NaN */
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if (ix == BIAS + LDBL_MAX_EXP) {
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if (hx & 0x8000) /* x is -Inf, -NaN or unsupported */
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RETURNP(-1 / x - 1);
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RETURNP(x + x); /* x is +Inf, +NaN or unsupported */
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}
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if (x > o_threshold)
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RETURNP(huge * huge);
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/*
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* expm1l() never underflows, but it must avoid
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* unrepresentable large negative exponents. We used a
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* much smaller threshold for large |x| above than in
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* expl() so as to handle not so large negative exponents
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* in the same way as large ones here.
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*/
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if (hx & 0x8000) /* x <= -64 */
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RETURN2P(tiny, -1); /* good for x < -65ln2 - eps */
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}
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ENTERI();
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if (T1 < x && x < T2) {
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if (ix < BIAS - 74) { /* |x| < 0x1p-74 (includes pseudos) */
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/* x (rounded) with inexact if x != 0: */
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RETURNPI(x == 0 ? x :
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(0x1p100 * x + fabsl(x)) * 0x1p-100);
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}
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x2 = x * x;
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x4 = x2 * x2;
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q = x4 * (x2 * (x4 *
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/*
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* XXX the number of terms is no longer good for
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* pairwise grouping of all except B3, and the
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* grouping is no longer from highest down.
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*/
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(x2 * B12 + (x * B11 + B10)) +
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(x2 * (x * B9 + B8) + (x * B7 + B6))) +
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(x * B5 + B4.e)) + x2 * x * B3.e;
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x_hi = (float)x;
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x_lo = x - x_hi;
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hx2_hi = x_hi * x_hi / 2;
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hx2_lo = x_lo * (x + x_hi) / 2;
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if (ix >= BIAS - 7)
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RETURN2PI(hx2_hi + x_hi, hx2_lo + x_lo + q);
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else
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RETURN2PI(x, hx2_lo + q + hx2_hi);
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}
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/* Reduce x to (k*ln2 + endpoint[n2] + r1 + r2). */
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/* Use a specialized rint() to get fn. Assume round-to-nearest. */
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fn = x * INV_L + 0x1.8p63 - 0x1.8p63;
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#if defined(HAVE_EFFICIENT_IRINTL)
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n = irintl(fn);
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#elif defined(HAVE_EFFICIENT_IRINT)
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n = irint(fn);
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#else
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n = (int)fn;
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#endif
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n2 = (unsigned)n % INTERVALS;
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k = n >> LOG2_INTERVALS;
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r1 = x - fn * L1;
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r2 = fn * -L2;
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r = r1 + r2;
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/* Prepare scale factor. */
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v.e = 1;
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v.xbits.expsign = BIAS + k;
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twopk = v.e;
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/*
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* Evaluate lower terms of
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* expl(endpoint[n2] + r1 + r2) = tbl[n2] * expl(r1 + r2).
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*/
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z = r * r;
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q = r2 + z * (A2 + r * A3) + z * z * (A4 + r * A5) + z * z * z * A6;
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t = (long double)tbl[n2].lo + tbl[n2].hi;
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if (k == 0) {
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t = SUM2P(tbl[n2].hi - 1, tbl[n2].lo * (r1 + 1) + t * q +
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tbl[n2].hi * r1);
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RETURNI(t);
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}
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if (k == -1) {
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t = SUM2P(tbl[n2].hi - 2, tbl[n2].lo * (r1 + 1) + t * q +
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tbl[n2].hi * r1);
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RETURNI(t / 2);
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}
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if (k < -7) {
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t = SUM2P(tbl[n2].hi, tbl[n2].lo + t * (q + r1));
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RETURNI(t * twopk - 1);
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}
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if (k > 2 * LDBL_MANT_DIG - 1) {
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t = SUM2P(tbl[n2].hi, tbl[n2].lo + t * (q + r1));
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if (k == LDBL_MAX_EXP)
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RETURNI(t * 2 * 0x1p16383L - 1);
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RETURNI(t * twopk - 1);
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}
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v.xbits.expsign = BIAS - k;
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twomk = v.e;
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if (k > LDBL_MANT_DIG - 1)
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t = SUM2P(tbl[n2].hi, tbl[n2].lo - twomk + t * (q + r1));
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else
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t = SUM2P(tbl[n2].hi - twomk, tbl[n2].lo + t * (q + r1));
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RETURNI(t * twopk);
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}
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