/* Copyright JS Foundation and other contributors, http://js.foundation * * Licensed under the Apache License, Version 2.0 (the "License"); * you may not use this file except in compliance with the License. * You may obtain a copy of the License at * * http://www.apache.org/licenses/LICENSE-2.0 * * Unless required by applicable law or agreed to in writing, software * distributed under the License is distributed on an "AS IS" BASIS * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. * See the License for the specific language governing permissions and * limitations under the License. * * This file is based on work under the following copyright and permission * notice: * * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. * * Permission to use, copy, modify, and distribute this * software is freely granted, provided that this notice * is preserved. * * @(#)s_expm1.c 5.1 93/09/24 */ #include "jerry-libm-internal.h" /* expm1(x) * Returns exp(x)-1, the exponential of x minus 1. * * Method * 1. Argument 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; * z = r*r, * 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. */ #define one 1.0 #define huge 1.0e+300 #define tiny 1.0e-300 #define o_threshold 7.09782712893383973096e+02 /* 0x40862E42, 0xFEFA39EF */ #define ln2_hi 6.93147180369123816490e-01 /* 0x3fe62e42, 0xfee00000 */ #define ln2_lo 1.90821492927058770002e-10 /* 0x3dea39ef, 0x35793c76 */ #define invln2 1.44269504088896338700e+00 /* 0x3ff71547, 0x652b82fe */ /* Scaled Q's: Qn_here = 2**n * Qn_above, for R(2*z) where z = hxs = x*x/2: */ #define Q1 -3.33333333333331316428e-02 /* BFA11111 111110F4 */ #define Q2 1.58730158725481460165e-03 /* 3F5A01A0 19FE5585 */ #define Q3 -7.93650757867487942473e-05 /* BF14CE19 9EAADBB7 */ #define Q4 4.00821782732936239552e-06 /* 3ED0CFCA 86E65239 */ #define Q5 -2.01099218183624371326e-07 /* BE8AFDB7 6E09C32D */ double expm1 (double x) { double y, hi, lo, c, e, hxs, hfx, r1; double_accessor t, twopk; int k, xsb; unsigned int hx; hx = __HI (x); xsb = hx & 0x80000000; /* sign bit of x */ hx &= 0x7fffffff; /* high word of |x| */ /* filter out huge and non-finite argument */ if (hx >= 0x4043687A) { /* if |x|>=56*ln2 */ if (hx >= 0x40862E42) { /* if |x|>=709.78... */ if (hx >= 0x7ff00000) { unsigned int low; low = __LO (x); if (((hx & 0xfffff) | low) != 0) { /* NaN */ return x + x; } else { /* exp(+-inf)-1={inf,-1} */ return (xsb == 0) ? x : -1.0; } } if (x > o_threshold) { /* overflow */ return huge * huge; } } if (xsb != 0) { /* x < -56*ln2, return -1.0 with inexact */ if (x + tiny < 0.0) /* raise inexact */ { /* return -1 */ return tiny - one; } } } /* argument reduction */ if (hx > 0x3fd62e42) { /* if |x| > 0.5 ln2 */ if (hx < 0x3FF0A2B2) { /* and |x| < 1.5 ln2 */ if (xsb == 0) { hi = x - ln2_hi; lo = ln2_lo; k = 1; } else { hi = x + ln2_hi; lo = -ln2_lo; k = -1; } } else { k = (int) (invln2 * x + ((xsb == 0) ? 0.5 : -0.5)); t.dbl = k; hi = x - t.dbl * ln2_hi; /* t*ln2_hi is exact here */ lo = t.dbl * ln2_lo; } x = hi - lo; c = (hi - x) - lo; } else if (hx < 0x3c900000) { /* when |x|<2**-54, return x */ return x; } else { 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.dbl = 3.0 - r1 * hfx; e = hxs * ((r1 - t.dbl) / (6.0 - x * t.dbl)); if (k == 0) { /* c is 0 */ return x - (x * e - hxs); } else { twopk.as_int.hi = 0x3ff00000 + ((unsigned int) k << 20); /* 2^k */ twopk.as_int.lo = 0; 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); if (k == 1024) { y = y * 2.0 * 0x1p1023; } else { y = y * twopk.dbl; } return y - one; } t.dbl = one; if (k < 20) { t.as_int.hi = (0x3ff00000 - (0x200000 >> k)); /* t=1-2^-k */ y = t.dbl - (e - x); y = y * twopk.dbl; } else { t.as_int.hi = ((0x3ff - k) << 20); /* 2^-k */ y = x - (e + t.dbl); y += one; y = y * twopk.dbl; } } return y; } /* expm1 */ #undef one #undef huge #undef tiny #undef o_threshold #undef ln2_hi #undef ln2_lo #undef invln2 #undef Q1 #undef Q2 #undef Q3 #undef Q4 #undef Q5