/* * CDDL HEADER START * * The contents of this file are subject to the terms of the * Common Development and Distribution License (the "License"). * You may not use this file except in compliance with the License. * * You can obtain a copy of the license at usr/src/OPENSOLARIS.LICENSE * or http://www.opensolaris.org/os/licensing. * See the License for the specific language governing permissions * and limitations under the License. * * When distributing Covered Code, include this CDDL HEADER in each * file and include the License file at usr/src/OPENSOLARIS.LICENSE. * If applicable, add the following below this CDDL HEADER, with the * fields enclosed by brackets "[]" replaced with your own identifying * information: Portions Copyright [yyyy] [name of copyright owner] * * CDDL HEADER END */ /* * Copyright 2011 Nexenta Systems, Inc. All rights reserved. */ /* * Copyright 2006 Sun Microsystems, Inc. All rights reserved. * Use is subject to license terms. */ #pragma weak __expm1 = expm1 /* INDENT OFF */ /* * expm1(x) * Returns exp(x)-1, the exponential of x minus 1. * * Method * 1. Arugment reduction: * Given x, find r and integer k such that * * x = k*ln2 + r, |r| <= 0.5*ln2 ~ 0.34658 * * Here a correction term c will be computed to compensate * the error in r when rounded to a floating-point number. * * 2. Approximating expm1(r) by a special rational function on * the interval [0,0.34658]: * Since * r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 - r^4/360 + ... * we define R1(r*r) by * r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 * R1(r*r) * That is, * R1(r**2) = 6/r *((exp(r)+1)/(exp(r)-1) - 2/r) * = 6/r * ( 1 + 2.0*(1/(exp(r)-1) - 1/r)) * = 1 - r^2/60 + r^4/2520 - r^6/100800 + ... * We use a special Reme algorithm on [0,0.347] to generate * a polynomial of degree 5 in r*r to approximate R1. The * maximum error of this polynomial approximation is bounded * by 2**-61. In other words, * R1(z) ~ 1.0 + Q1*z + Q2*z**2 + Q3*z**3 + Q4*z**4 + Q5*z**5 * where Q1 = -1.6666666666666567384E-2, * Q2 = 3.9682539681370365873E-4, * Q3 = -9.9206344733435987357E-6, * Q4 = 2.5051361420808517002E-7, * Q5 = -6.2843505682382617102E-9; * (where z=r*r, and the values of Q1 to Q5 are listed below) * with error bounded by * | 5 | -61 * | 1.0+Q1*z+...+Q5*z - R1(z) | <= 2 * | | * * expm1(r) = exp(r)-1 is then computed by the following * specific way which minimize the accumulation rounding error: * 2 3 * r r [ 3 - (R1 + R1*r/2) ] * expm1(r) = r + --- + --- * [--------------------] * 2 2 [ 6 - r*(3 - R1*r/2) ] * * To compensate the error in the argument reduction, we use * expm1(r+c) = expm1(r) + c + expm1(r)*c * ~ expm1(r) + c + r*c * Thus c+r*c will be added in as the correction terms for * expm1(r+c). Now rearrange the term to avoid optimization * screw up: * ( 2 2 ) * ({ ( r [ R1 - (3 - R1*r/2) ] ) } r ) * expm1(r+c)~r - ({r*(--- * [--------------------]-c)-c} - --- ) * ({ ( 2 [ 6 - r*(3 - R1*r/2) ] ) } 2 ) * ( ) * * = r - E * 3. Scale back to obtain expm1(x): * From step 1, we have * expm1(x) = either 2^k*[expm1(r)+1] - 1 * = or 2^k*[expm1(r) + (1-2^-k)] * 4. Implementation notes: * (A). To save one multiplication, we scale the coefficient Qi * to Qi*2^i, and replace z by (x^2)/2. * (B). To achieve maximum accuracy, we compute expm1(x) by * (i) if x < -56*ln2, return -1.0, (raise inexact if x != inf) * (ii) if k=0, return r-E * (iii) if k=-1, return 0.5*(r-E)-0.5 * (iv) if k=1 if r < -0.25, return 2*((r+0.5)- E) * else return 1.0+2.0*(r-E); * (v) if (k<-2||k>56) return 2^k(1-(E-r)) - 1 (or exp(x)-1) * (vi) if k <= 20, return 2^k((1-2^-k)-(E-r)), else * (vii) return 2^k(1-((E+2^-k)-r)) * * Special cases: * expm1(INF) is INF, expm1(NaN) is NaN; * expm1(-INF) is -1, and * for finite argument, only expm1(0)=0 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 > 7.09782712893383973096e+02 then expm1(x) overflow * * Constants: * The hexadecimal values are the intended ones for the following * constants. The decimal values may be used, provided that the * compiler will convert from decimal to binary accurately enough * to produce the hexadecimal values shown. */ /* INDENT ON */ #include "libm_macros.h" #include <math.h> static const double xxx[] = { /* one */ 1.0, /* huge */ 1.0e+300, /* tiny */ 1.0e-300, /* o_threshold */ 7.09782712893383973096e+02, /* 40862E42 FEFA39EF */ /* ln2_hi */ 6.93147180369123816490e-01, /* 3FE62E42 FEE00000 */ /* ln2_lo */ 1.90821492927058770002e-10, /* 3DEA39EF 35793C76 */ /* invln2 */ 1.44269504088896338700e+00, /* 3FF71547 652B82FE */ /* scaled coefficients related to expm1 */ /* Q1 */ -3.33333333333331316428e-02, /* BFA11111 111110F4 */ /* Q2 */ 1.58730158725481460165e-03, /* 3F5A01A0 19FE5585 */ /* Q3 */ -7.93650757867487942473e-05, /* BF14CE19 9EAADBB7 */ /* Q4 */ 4.00821782732936239552e-06, /* 3ED0CFCA 86E65239 */ /* Q5 */ -2.01099218183624371326e-07 /* BE8AFDB7 6E09C32D */ }; #define one xxx[0] #define huge xxx[1] #define tiny xxx[2] #define o_threshold xxx[3] #define ln2_hi xxx[4] #define ln2_lo xxx[5] #define invln2 xxx[6] #define Q1 xxx[7] #define Q2 xxx[8] #define Q3 xxx[9] #define Q4 xxx[10] #define Q5 xxx[11] double expm1(double x) { double y, hi, lo, c = 0.0L, t, e, hxs, hfx, r1; int k, xsb; unsigned hx; hx = ((unsigned *) &x)[HIWORD]; /* high word of x */ xsb = hx & 0x80000000; /* sign bit of x */ if (xsb == 0) y = x; else y = -x; /* y = |x| */ hx &= 0x7fffffff; /* high word of |x| */ /* filter out huge and non-finite argument */ /* for example exp(38)-1 is approximately 3.1855932e+16 */ if (hx >= 0x4043687A) { /* if |x|>=56*ln2 (~38.8162...) */ if (hx >= 0x40862E42) { /* if |x|>=709.78... -> inf */ if (hx >= 0x7ff00000) { if (((hx & 0xfffff) | ((int *) &x)[LOWORD]) != 0) return (x * x); /* + -> * for Cheetah */ else /* exp(+-inf)={inf,-1} */ return (xsb == 0 ? x : -1.0); } if (x > o_threshold) return (huge * huge); /* overflow */ } if (xsb != 0) { /* x < -56*ln2, return -1.0 w/inexact */ if (x + tiny < 0.0) /* raise inexact */ return (tiny - one); /* return -1 */ } } /* argument reduction */ if (hx > 0x3fd62e42) { /* if |x| > 0.5 ln2 */ if (hx < 0x3FF0A2B2) { /* and |x| < 1.5 ln2 */ if (xsb == 0) { /* positive number */ hi = x - ln2_hi; lo = ln2_lo; k = 1; } else { /* negative number */ hi = x + ln2_hi; lo = -ln2_lo; k = -1; } } else { /* |x| > 1.5 ln2 */ k = (int) (invln2 * x + (xsb == 0 ? 0.5 : -0.5)); t = k; hi = x - t * ln2_hi; /* t*ln2_hi is exact here */ lo = t * ln2_lo; } x = hi - lo; c = (hi - x) - lo; /* still at |x| > 0.5 ln2 */ } else if (hx < 0x3c900000) { /* when |x|<2**-54, return x */ t = huge + x; /* return x w/inexact when x != 0 */ return (x - (t - (huge + x))); } else /* |x| <= 0.5 ln2 */ k = 0; /* x is now in primary range */ hfx = 0.5 * x; hxs = x * hfx; r1 = one + hxs * (Q1 + hxs * (Q2 + hxs * (Q3 + hxs * (Q4 + hxs * Q5)))); t = 3.0 - r1 * hfx; e = hxs * ((r1 - t) / (6.0 - x * t)); if (k == 0) /* |x| <= 0.5 ln2 */ return (x - (x * e - hxs)); else { /* |x| > 0.5 ln2 */ e = (x * (e - c) - c); e -= hxs; if (k == -1) return (0.5 * (x - e) - 0.5); if (k == 1) { if (x < -0.25) return (-2.0 * (e - (x + 0.5))); else return (one + 2.0 * (x - e)); } if (k <= -2 || k > 56) { /* suffice to return exp(x)-1 */ y = one - (e - x); ((int *) &y)[HIWORD] += k << 20; return (y - one); } t = one; if (k < 20) { ((int *) &t)[HIWORD] = 0x3ff00000 - (0x200000 >> k); /* t = 1 - 2^-k */ y = t - (e - x); ((int *) &y)[HIWORD] += k << 20; } else { ((int *) &t)[HIWORD] = (0x3ff - k) << 20; /* 2^-k */ y = x - (e + t); y += one; ((int *) &y)[HIWORD] += k << 20; } } return (y); }
