s_expm1.c 7.5 KB

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  1. /* @(#)s_expm1.c 5.1 93/09/24 */
  2. /*
  3. * ====================================================
  4. * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
  5. *
  6. * Developed at SunPro, a Sun Microsystems, Inc. business.
  7. * Permission to use, copy, modify, and distribute this
  8. * software is freely granted, provided that this notice
  9. * is preserved.
  10. * ====================================================
  11. */
  12. #if defined(LIBM_SCCS) && !defined(lint)
  13. static char rcsid[] = "$NetBSD: s_expm1.c,v 1.8 1995/05/10 20:47:09 jtc Exp $";
  14. #endif
  15. /* expm1(x)
  16. * Returns exp(x)-1, the exponential of x minus 1.
  17. *
  18. * Method
  19. * 1. Argument reduction:
  20. * Given x, find r and integer k such that
  21. *
  22. * x = k*ln2 + r, |r| <= 0.5*ln2 ~ 0.34658
  23. *
  24. * Here a correction term c will be computed to compensate
  25. * the error in r when rounded to a floating-point number.
  26. *
  27. * 2. Approximating expm1(r) by a special rational function on
  28. * the interval [0,0.34658]:
  29. * Since
  30. * r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 - r^4/360 + ...
  31. * we define R1(r*r) by
  32. * r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 * R1(r*r)
  33. * That is,
  34. * R1(r**2) = 6/r *((exp(r)+1)/(exp(r)-1) - 2/r)
  35. * = 6/r * ( 1 + 2.0*(1/(exp(r)-1) - 1/r))
  36. * = 1 - r^2/60 + r^4/2520 - r^6/100800 + ...
  37. * We use a special Reme algorithm on [0,0.347] to generate
  38. * a polynomial of degree 5 in r*r to approximate R1. The
  39. * maximum error of this polynomial approximation is bounded
  40. * by 2**-61. In other words,
  41. * R1(z) ~ 1.0 + Q1*z + Q2*z**2 + Q3*z**3 + Q4*z**4 + Q5*z**5
  42. * where Q1 = -1.6666666666666567384E-2,
  43. * Q2 = 3.9682539681370365873E-4,
  44. * Q3 = -9.9206344733435987357E-6,
  45. * Q4 = 2.5051361420808517002E-7,
  46. * Q5 = -6.2843505682382617102E-9;
  47. * (where z=r*r, and the values of Q1 to Q5 are listed below)
  48. * with error bounded by
  49. * | 5 | -61
  50. * | 1.0+Q1*z+...+Q5*z - R1(z) | <= 2
  51. * | |
  52. *
  53. * expm1(r) = exp(r)-1 is then computed by the following
  54. * specific way which minimize the accumulation rounding error:
  55. * 2 3
  56. * r r [ 3 - (R1 + R1*r/2) ]
  57. * expm1(r) = r + --- + --- * [--------------------]
  58. * 2 2 [ 6 - r*(3 - R1*r/2) ]
  59. *
  60. * To compensate the error in the argument reduction, we use
  61. * expm1(r+c) = expm1(r) + c + expm1(r)*c
  62. * ~ expm1(r) + c + r*c
  63. * Thus c+r*c will be added in as the correction terms for
  64. * expm1(r+c). Now rearrange the term to avoid optimization
  65. * screw up:
  66. * ( 2 2 )
  67. * ({ ( r [ R1 - (3 - R1*r/2) ] ) } r )
  68. * expm1(r+c)~r - ({r*(--- * [--------------------]-c)-c} - --- )
  69. * ({ ( 2 [ 6 - r*(3 - R1*r/2) ] ) } 2 )
  70. * ( )
  71. *
  72. * = r - E
  73. * 3. Scale back to obtain expm1(x):
  74. * From step 1, we have
  75. * expm1(x) = either 2^k*[expm1(r)+1] - 1
  76. * = or 2^k*[expm1(r) + (1-2^-k)]
  77. * 4. Implementation notes:
  78. * (A). To save one multiplication, we scale the coefficient Qi
  79. * to Qi*2^i, and replace z by (x^2)/2.
  80. * (B). To achieve maximum accuracy, we compute expm1(x) by
  81. * (i) if x < -56*ln2, return -1.0, (raise inexact if x!=inf)
  82. * (ii) if k=0, return r-E
  83. * (iii) if k=-1, return 0.5*(r-E)-0.5
  84. * (iv) if k=1 if r < -0.25, return 2*((r+0.5)- E)
  85. * else return 1.0+2.0*(r-E);
  86. * (v) if (k<-2||k>56) return 2^k(1-(E-r)) - 1 (or exp(x)-1)
  87. * (vi) if k <= 20, return 2^k((1-2^-k)-(E-r)), else
  88. * (vii) return 2^k(1-((E+2^-k)-r))
  89. *
  90. * Special cases:
  91. * expm1(INF) is INF, expm1(NaN) is NaN;
  92. * expm1(-INF) is -1, and
  93. * for finite argument, only expm1(0)=0 is exact.
  94. *
  95. * Accuracy:
  96. * according to an error analysis, the error is always less than
  97. * 1 ulp (unit in the last place).
  98. *
  99. * Misc. info.
  100. * For IEEE double
  101. * if x > 7.09782712893383973096e+02 then expm1(x) overflow
  102. *
  103. * Constants:
  104. * The hexadecimal values are the intended ones for the following
  105. * constants. The decimal values may be used, provided that the
  106. * compiler will convert from decimal to binary accurately enough
  107. * to produce the hexadecimal values shown.
  108. */
  109. #include "math.h"
  110. #include "math_private.h"
  111. #ifdef __STDC__
  112. static const double
  113. #else
  114. static double
  115. #endif
  116. one = 1.0,
  117. huge = 1.0e+300,
  118. tiny = 1.0e-300,
  119. o_threshold = 7.09782712893383973096e+02,/* 0x40862E42, 0xFEFA39EF */
  120. ln2_hi = 6.93147180369123816490e-01,/* 0x3fe62e42, 0xfee00000 */
  121. ln2_lo = 1.90821492927058770002e-10,/* 0x3dea39ef, 0x35793c76 */
  122. invln2 = 1.44269504088896338700e+00,/* 0x3ff71547, 0x652b82fe */
  123. /* scaled coefficients related to expm1 */
  124. Q1 = -3.33333333333331316428e-02, /* BFA11111 111110F4 */
  125. Q2 = 1.58730158725481460165e-03, /* 3F5A01A0 19FE5585 */
  126. Q3 = -7.93650757867487942473e-05, /* BF14CE19 9EAADBB7 */
  127. Q4 = 4.00821782732936239552e-06, /* 3ED0CFCA 86E65239 */
  128. Q5 = -2.01099218183624371326e-07; /* BE8AFDB7 6E09C32D */
  129. #ifdef __STDC__
  130. double expm1(double x)
  131. #else
  132. double expm1(x)
  133. double x;
  134. #endif
  135. {
  136. double y,hi,lo,c=0.0,t,e,hxs,hfx,r1;
  137. int32_t k,xsb;
  138. u_int32_t hx;
  139. GET_HIGH_WORD(hx,x);
  140. xsb = hx&0x80000000; /* sign bit of x */
  141. if(xsb==0) y=x; else y= -x; /* y = |x| */
  142. hx &= 0x7fffffff; /* high word of |x| */
  143. /* filter out huge and non-finite argument */
  144. if(hx >= 0x4043687A) { /* if |x|>=56*ln2 */
  145. if(hx >= 0x40862E42) { /* if |x|>=709.78... */
  146. if(hx>=0x7ff00000) {
  147. u_int32_t low;
  148. GET_LOW_WORD(low,x);
  149. if(((hx&0xfffff)|low)!=0)
  150. return x+x; /* NaN */
  151. else return (xsb==0)? x:-1.0;/* exp(+-inf)={inf,-1} */
  152. }
  153. if(x > o_threshold) return huge*huge; /* overflow */
  154. }
  155. if(xsb!=0) { /* x < -56*ln2, return -1.0 with inexact */
  156. if(x+tiny<0.0) /* raise inexact */
  157. return tiny-one; /* return -1 */
  158. }
  159. }
  160. /* argument reduction */
  161. if(hx > 0x3fd62e42) { /* if |x| > 0.5 ln2 */
  162. if(hx < 0x3FF0A2B2) { /* and |x| < 1.5 ln2 */
  163. if(xsb==0)
  164. {hi = x - ln2_hi; lo = ln2_lo; k = 1;}
  165. else
  166. {hi = x + ln2_hi; lo = -ln2_lo; k = -1;}
  167. } else {
  168. k = invln2*x+((xsb==0)?0.5:-0.5);
  169. t = k;
  170. hi = x - t*ln2_hi; /* t*ln2_hi is exact here */
  171. lo = t*ln2_lo;
  172. }
  173. x = hi - lo;
  174. c = (hi-x)-lo;
  175. }
  176. else if(hx < 0x3c900000) { /* when |x|<2**-54, return x */
  177. t = huge+x; /* return x with inexact flags when x!=0 */
  178. return x - (t-(huge+x));
  179. }
  180. else k = 0;
  181. /* x is now in primary range */
  182. hfx = 0.5*x;
  183. hxs = x*hfx;
  184. r1 = one+hxs*(Q1+hxs*(Q2+hxs*(Q3+hxs*(Q4+hxs*Q5))));
  185. t = 3.0-r1*hfx;
  186. e = hxs*((r1-t)/(6.0 - x*t));
  187. if(k==0) return x - (x*e-hxs); /* c is 0 */
  188. else {
  189. e = (x*(e-c)-c);
  190. e -= hxs;
  191. if(k== -1) return 0.5*(x-e)-0.5;
  192. if(k==1) {
  193. if(x < -0.25) return -2.0*(e-(x+0.5));
  194. else return one+2.0*(x-e);
  195. }
  196. if (k <= -2 || k>56) { /* suffice to return exp(x)-1 */
  197. u_int32_t high;
  198. y = one-(e-x);
  199. GET_HIGH_WORD(high,y);
  200. SET_HIGH_WORD(y,high+(k<<20)); /* add k to y's exponent */
  201. return y-one;
  202. }
  203. t = one;
  204. if(k<20) {
  205. u_int32_t high;
  206. SET_HIGH_WORD(t,0x3ff00000 - (0x200000>>k)); /* t=1-2^-k */
  207. y = t-(e-x);
  208. GET_HIGH_WORD(high,y);
  209. SET_HIGH_WORD(y,high+(k<<20)); /* add k to y's exponent */
  210. } else {
  211. u_int32_t high;
  212. SET_HIGH_WORD(t,((0x3ff-k)<<20)); /* 2^-k */
  213. y = x-(e+t);
  214. y += one;
  215. GET_HIGH_WORD(high,y);
  216. SET_HIGH_WORD(y,high+(k<<20)); /* add k to y's exponent */
  217. }
  218. }
  219. return y;
  220. }
  221. libm_hidden_def(expm1)