| /*- |
| * Copyright (c) 2009-2013 Steven G. Kargl |
| * 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 unmodified, 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 ``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 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. |
| * |
| * Optimized by Bruce D. Evans. |
| */ |
| |
| #include <sys/cdefs.h> |
| __FBSDID("$FreeBSD$"); |
| |
| /* |
| * ld128 version of s_expl.c. See ../ld80/s_expl.c for most comments. |
| */ |
| |
| #include <float.h> |
| |
| #include "fpmath.h" |
| #include "math.h" |
| #include "math_private.h" |
| #include "k_expl.h" |
| |
| /* XXX Prevent compilers from erroneously constant folding these: */ |
| static const volatile long double |
| huge = 0x1p10000L, |
| tiny = 0x1p-10000L; |
| |
| static const long double |
| twom10000 = 0x1p-10000L; |
| |
| static const long double |
| /* log(2**16384 - 0.5) rounded towards zero: */ |
| /* log(2**16384 - 0.5 + 1) rounded towards zero for expm1l() is the same: */ |
| o_threshold = 11356.523406294143949491931077970763428L, |
| /* log(2**(-16381-64-1)) rounded towards zero: */ |
| u_threshold = -11433.462743336297878837243843452621503L; |
| |
| long double |
| expl(long double x) |
| { |
| union IEEEl2bits u; |
| long double hi, lo, t, twopk; |
| int k; |
| uint16_t hx, ix; |
| |
| DOPRINT_START(&x); |
| |
| /* Filter out exceptional cases. */ |
| u.e = x; |
| hx = u.xbits.expsign; |
| ix = hx & 0x7fff; |
| if (ix >= BIAS + 13) { /* |x| >= 8192 or x is NaN */ |
| if (ix == BIAS + LDBL_MAX_EXP) { |
| if (hx & 0x8000) /* x is -Inf or -NaN */ |
| RETURNP(-1 / x); |
| RETURNP(x + x); /* x is +Inf or +NaN */ |
| } |
| if (x > o_threshold) |
| RETURNP(huge * huge); |
| if (x < u_threshold) |
| RETURNP(tiny * tiny); |
| } else if (ix < BIAS - 114) { /* |x| < 0x1p-114 */ |
| RETURN2P(1, x); /* 1 with inexact iff x != 0 */ |
| } |
| |
| ENTERI(); |
| |
| twopk = 1; |
| __k_expl(x, &hi, &lo, &k); |
| t = SUM2P(hi, lo); |
| |
| /* Scale by 2**k. */ |
| /* XXX sparc64 multiplication is so slow that scalbnl() is faster. */ |
| if (k >= LDBL_MIN_EXP) { |
| if (k == LDBL_MAX_EXP) |
| RETURNI(t * 2 * 0x1p16383L); |
| SET_LDBL_EXPSIGN(twopk, BIAS + k); |
| RETURNI(t * twopk); |
| } else { |
| SET_LDBL_EXPSIGN(twopk, BIAS + k + 10000); |
| RETURNI(t * twopk * twom10000); |
| } |
| } |
| |
| /* |
| * Our T1 and T2 are chosen to be approximately the points where method |
| * A and method B have the same accuracy. Tang's T1 and T2 are the |
| * points where method A's accuracy changes by a full bit. For Tang, |
| * this drop in accuracy makes method A immediately less accurate than |
| * method B, but our larger INTERVALS makes method A 2 bits more |
| * accurate so it remains the most accurate method significantly |
| * closer to the origin despite losing the full bit in our extended |
| * range for it. |
| * |
| * Split the interval [T1, T2] into two intervals [T1, T3] and [T3, T2]. |
| * Setting T3 to 0 would require the |x| < 0x1p-113 condition to appear |
| * in both subintervals, so set T3 = 2**-5, which places the condition |
| * into the [T1, T3] interval. |
| * |
| * XXX we now do this more to (partially) balance the number of terms |
| * in the C and D polys than to avoid checking the condition in both |
| * intervals. |
| * |
| * XXX these micro-optimizations are excessive. |
| */ |
| static const double |
| T1 = -0.1659, /* ~-30.625/128 * log(2) */ |
| T2 = 0.1659, /* ~30.625/128 * log(2) */ |
| T3 = 0.03125; |
| |
| /* |
| * Domain [-0.1659, 0.03125], range ~[2.9134e-44, 1.8404e-37]: |
| * |(exp(x)-1-x-x**2/2)/x - p(x)| < 2**-122.03 |
| * |
| * XXX none of the long double C or D coeffs except C10 is correctly printed. |
| * If you re-print their values in %.35Le format, the result is always |
| * different. For example, the last 2 digits in C3 should be 59, not 67. |
| * 67 is apparently from rounding an extra-precision value to 36 decimal |
| * places. |
| */ |
| static const long double |
| C3 = 1.66666666666666666666666666666666667e-1L, |
| C4 = 4.16666666666666666666666666666666645e-2L, |
| C5 = 8.33333333333333333333333333333371638e-3L, |
| C6 = 1.38888888888888888888888888891188658e-3L, |
| C7 = 1.98412698412698412698412697235950394e-4L, |
| C8 = 2.48015873015873015873015112487849040e-5L, |
| C9 = 2.75573192239858906525606685484412005e-6L, |
| C10 = 2.75573192239858906612966093057020362e-7L, |
| C11 = 2.50521083854417203619031960151253944e-8L, |
| C12 = 2.08767569878679576457272282566520649e-9L, |
| C13 = 1.60590438367252471783548748824255707e-10L; |
| |
| /* |
| * XXX this has 1 more coeff than needed. |
| * XXX can start the double coeffs but not the double mults at C10. |
| * With my coeffs (C10-C17 double; s = best_s): |
| * Domain [-0.1659, 0.03125], range ~[-1.1976e-37, 1.1976e-37]: |
| * |(exp(x)-1-x-x**2/2)/x - p(x)| ~< 2**-122.65 |
| */ |
| static const double |
| C14 = 1.1470745580491932e-11, /* 0x1.93974a81dae30p-37 */ |
| C15 = 7.6471620181090468e-13, /* 0x1.ae7f3820adab1p-41 */ |
| C16 = 4.7793721460260450e-14, /* 0x1.ae7cd18a18eacp-45 */ |
| C17 = 2.8074757356658877e-15, /* 0x1.949992a1937d9p-49 */ |
| C18 = 1.4760610323699476e-16; /* 0x1.545b43aabfbcdp-53 */ |
| |
| /* |
| * Domain [0.03125, 0.1659], range ~[-2.7676e-37, -1.0367e-38]: |
| * |(exp(x)-1-x-x**2/2)/x - p(x)| < 2**-121.44 |
| */ |
| static const long double |
| D3 = 1.66666666666666666666666666666682245e-1L, |
| D4 = 4.16666666666666666666666666634228324e-2L, |
| D5 = 8.33333333333333333333333364022244481e-3L, |
| D6 = 1.38888888888888888888887138722762072e-3L, |
| D7 = 1.98412698412698412699085805424661471e-4L, |
| D8 = 2.48015873015873015687993712101479612e-5L, |
| D9 = 2.75573192239858944101036288338208042e-6L, |
| D10 = 2.75573192239853161148064676533754048e-7L, |
| D11 = 2.50521083855084570046480450935267433e-8L, |
| D12 = 2.08767569819738524488686318024854942e-9L, |
| D13 = 1.60590442297008495301927448122499313e-10L; |
| |
| /* |
| * XXX this has 1 more coeff than needed. |
| * XXX can start the double coeffs but not the double mults at D11. |
| * With my coeffs (D11-D16 double): |
| * Domain [0.03125, 0.1659], range ~[-1.1980e-37, 1.1980e-37]: |
| * |(exp(x)-1-x-x**2/2)/x - p(x)| ~< 2**-122.65 |
| */ |
| static const double |
| D14 = 1.1470726176204336e-11, /* 0x1.93971dc395d9ep-37 */ |
| D15 = 7.6478532249581686e-13, /* 0x1.ae892e3D16fcep-41 */ |
| D16 = 4.7628892832607741e-14, /* 0x1.ad00Dfe41feccp-45 */ |
| D17 = 3.0524857220358650e-15; /* 0x1.D7e8d886Df921p-49 */ |
| |
| long double |
| expm1l(long double x) |
| { |
| union IEEEl2bits u, v; |
| long double hx2_hi, hx2_lo, q, r, r1, t, twomk, twopk, x_hi; |
| long double x_lo, x2; |
| double dr, dx, fn, r2; |
| int k, n, n2; |
| uint16_t hx, ix; |
| |
| DOPRINT_START(&x); |
| |
| /* Filter out exceptional cases. */ |
| u.e = x; |
| hx = u.xbits.expsign; |
| ix = hx & 0x7fff; |
| if (ix >= BIAS + 7) { /* |x| >= 128 or x is NaN */ |
| if (ix == BIAS + LDBL_MAX_EXP) { |
| if (hx & 0x8000) /* x is -Inf or -NaN */ |
| RETURNP(-1 / x - 1); |
| RETURNP(x + x); /* x is +Inf or +NaN */ |
| } |
| if (x > o_threshold) |
| RETURNP(huge * huge); |
| /* |
| * expm1l() never underflows, but it must avoid |
| * unrepresentable large negative exponents. We used a |
| * much smaller threshold for large |x| above than in |
| * expl() so as to handle not so large negative exponents |
| * in the same way as large ones here. |
| */ |
| if (hx & 0x8000) /* x <= -128 */ |
| RETURN2P(tiny, -1); /* good for x < -114ln2 - eps */ |
| } |
| |
| ENTERI(); |
| |
| if (T1 < x && x < T2) { |
| x2 = x * x; |
| dx = x; |
| |
| if (x < T3) { |
| if (ix < BIAS - 113) { /* |x| < 0x1p-113 */ |
| /* x (rounded) with inexact if x != 0: */ |
| RETURNPI(x == 0 ? x : |
| (0x1p200 * x + fabsl(x)) * 0x1p-200); |
| } |
| q = x * x2 * C3 + x2 * x2 * (C4 + x * (C5 + x * (C6 + |
| x * (C7 + x * (C8 + x * (C9 + x * (C10 + |
| x * (C11 + x * (C12 + x * (C13 + |
| dx * (C14 + dx * (C15 + dx * (C16 + |
| dx * (C17 + dx * C18)))))))))))))); |
| } else { |
| q = x * x2 * D3 + x2 * x2 * (D4 + x * (D5 + x * (D6 + |
| x * (D7 + x * (D8 + x * (D9 + x * (D10 + |
| x * (D11 + x * (D12 + x * (D13 + |
| dx * (D14 + dx * (D15 + dx * (D16 + |
| dx * D17))))))))))))); |
| } |
| |
| x_hi = (float)x; |
| x_lo = x - x_hi; |
| hx2_hi = x_hi * x_hi / 2; |
| hx2_lo = x_lo * (x + x_hi) / 2; |
| if (ix >= BIAS - 7) |
| RETURN2PI(hx2_hi + x_hi, hx2_lo + x_lo + q); |
| else |
| RETURN2PI(x, hx2_lo + q + hx2_hi); |
| } |
| |
| /* Reduce x to (k*ln2 + endpoint[n2] + r1 + r2). */ |
| /* Use a specialized rint() to get fn. Assume round-to-nearest. */ |
| fn = (double)x * INV_L + 0x1.8p52 - 0x1.8p52; |
| #if defined(HAVE_EFFICIENT_IRINT) |
| n = irint(fn); |
| #else |
| n = (int)fn; |
| #endif |
| n2 = (unsigned)n % INTERVALS; |
| k = n >> LOG2_INTERVALS; |
| r1 = x - fn * L1; |
| r2 = fn * -L2; |
| r = r1 + r2; |
| |
| /* Prepare scale factor. */ |
| v.e = 1; |
| v.xbits.expsign = BIAS + k; |
| twopk = v.e; |
| |
| /* |
| * Evaluate lower terms of |
| * expl(endpoint[n2] + r1 + r2) = tbl[n2] * expl(r1 + r2). |
| */ |
| dr = r; |
| q = r2 + r * r * (A2 + r * (A3 + r * (A4 + r * (A5 + r * (A6 + |
| dr * (A7 + dr * (A8 + dr * (A9 + dr * A10)))))))); |
| |
| t = tbl[n2].lo + tbl[n2].hi; |
| |
| if (k == 0) { |
| t = SUM2P(tbl[n2].hi - 1, tbl[n2].lo * (r1 + 1) + t * q + |
| tbl[n2].hi * r1); |
| RETURNI(t); |
| } |
| if (k == -1) { |
| t = SUM2P(tbl[n2].hi - 2, tbl[n2].lo * (r1 + 1) + t * q + |
| tbl[n2].hi * r1); |
| RETURNI(t / 2); |
| } |
| if (k < -7) { |
| t = SUM2P(tbl[n2].hi, tbl[n2].lo + t * (q + r1)); |
| RETURNI(t * twopk - 1); |
| } |
| if (k > 2 * LDBL_MANT_DIG - 1) { |
| t = SUM2P(tbl[n2].hi, tbl[n2].lo + t * (q + r1)); |
| if (k == LDBL_MAX_EXP) |
| RETURNI(t * 2 * 0x1p16383L - 1); |
| RETURNI(t * twopk - 1); |
| } |
| |
| v.xbits.expsign = BIAS - k; |
| twomk = v.e; |
| |
| if (k > LDBL_MANT_DIG - 1) |
| t = SUM2P(tbl[n2].hi, tbl[n2].lo - twomk + t * (q + r1)); |
| else |
| t = SUM2P(tbl[n2].hi - twomk, tbl[n2].lo + t * (q + r1)); |
| RETURNI(t * twopk); |
| } |