s_log1p.c 5.4 KB

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  1. /* @(#)s_log1p.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_log1p.c,v 1.8 1995/05/10 20:47:46 jtc Exp $";
  14. #endif
  15. /* double log1p(double x)
  16. *
  17. * Method :
  18. * 1. Argument Reduction: find k and f such that
  19. * 1+x = 2^k * (1+f),
  20. * where sqrt(2)/2 < 1+f < sqrt(2) .
  21. *
  22. * Note. If k=0, then f=x is exact. However, if k!=0, then f
  23. * may not be representable exactly. In that case, a correction
  24. * term is need. Let u=1+x rounded. Let c = (1+x)-u, then
  25. * log(1+x) - log(u) ~ c/u. Thus, we proceed to compute log(u),
  26. * and add back the correction term c/u.
  27. * (Note: when x > 2**53, one can simply return log(x))
  28. *
  29. * 2. Approximation of log1p(f).
  30. * Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
  31. * = 2s + 2/3 s**3 + 2/5 s**5 + .....,
  32. * = 2s + s*R
  33. * We use a special Reme algorithm on [0,0.1716] to generate
  34. * a polynomial of degree 14 to approximate R The maximum error
  35. * of this polynomial approximation is bounded by 2**-58.45. In
  36. * other words,
  37. * 2 4 6 8 10 12 14
  38. * R(z) ~ Lp1*s +Lp2*s +Lp3*s +Lp4*s +Lp5*s +Lp6*s +Lp7*s
  39. * (the values of Lp1 to Lp7 are listed in the program)
  40. * and
  41. * | 2 14 | -58.45
  42. * | Lp1*s +...+Lp7*s - R(z) | <= 2
  43. * | |
  44. * Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2.
  45. * In order to guarantee error in log below 1ulp, we compute log
  46. * by
  47. * log1p(f) = f - (hfsq - s*(hfsq+R)).
  48. *
  49. * 3. Finally, log1p(x) = k*ln2 + log1p(f).
  50. * = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo)))
  51. * Here ln2 is split into two floating point number:
  52. * ln2_hi + ln2_lo,
  53. * where n*ln2_hi is always exact for |n| < 2000.
  54. *
  55. * Special cases:
  56. * log1p(x) is NaN with signal if x < -1 (including -INF) ;
  57. * log1p(+INF) is +INF; log1p(-1) is -INF with signal;
  58. * log1p(NaN) is that NaN with no signal.
  59. *
  60. * Accuracy:
  61. * according to an error analysis, the error is always less than
  62. * 1 ulp (unit in the last place).
  63. *
  64. * Constants:
  65. * The hexadecimal values are the intended ones for the following
  66. * constants. The decimal values may be used, provided that the
  67. * compiler will convert from decimal to binary accurately enough
  68. * to produce the hexadecimal values shown.
  69. *
  70. * Note: Assuming log() return accurate answer, the following
  71. * algorithm can be used to compute log1p(x) to within a few ULP:
  72. *
  73. * u = 1+x;
  74. * if(u==1.0) return x ; else
  75. * return log(u)*(x/(u-1.0));
  76. *
  77. * See HP-15C Advanced Functions Handbook, p.193.
  78. */
  79. #include "math.h"
  80. #include "math_private.h"
  81. #ifdef __STDC__
  82. static const double
  83. #else
  84. static double
  85. #endif
  86. ln2_hi = 6.93147180369123816490e-01, /* 3fe62e42 fee00000 */
  87. ln2_lo = 1.90821492927058770002e-10, /* 3dea39ef 35793c76 */
  88. two54 = 1.80143985094819840000e+16, /* 43500000 00000000 */
  89. Lp1 = 6.666666666666735130e-01, /* 3FE55555 55555593 */
  90. Lp2 = 3.999999999940941908e-01, /* 3FD99999 9997FA04 */
  91. Lp3 = 2.857142874366239149e-01, /* 3FD24924 94229359 */
  92. Lp4 = 2.222219843214978396e-01, /* 3FCC71C5 1D8E78AF */
  93. Lp5 = 1.818357216161805012e-01, /* 3FC74664 96CB03DE */
  94. Lp6 = 1.531383769920937332e-01, /* 3FC39A09 D078C69F */
  95. Lp7 = 1.479819860511658591e-01; /* 3FC2F112 DF3E5244 */
  96. #ifdef __STDC__
  97. static const double zero = 0.0;
  98. #else
  99. static double zero = 0.0;
  100. #endif
  101. #ifdef __STDC__
  102. double log1p(double x)
  103. #else
  104. double log1p(x)
  105. double x;
  106. #endif
  107. {
  108. double hfsq,f=0,c=0,s,z,R,u;
  109. int32_t k,hx,hu=0,ax;
  110. GET_HIGH_WORD(hx,x);
  111. ax = hx&0x7fffffff;
  112. k = 1;
  113. if (hx < 0x3FDA827A) { /* x < 0.41422 */
  114. if(ax>=0x3ff00000) { /* x <= -1.0 */
  115. if(x==-1.0) return -two54/zero; /* log1p(-1)=+inf */
  116. else return (x-x)/(x-x); /* log1p(x<-1)=NaN */
  117. }
  118. if(ax<0x3e200000) { /* |x| < 2**-29 */
  119. if(two54+x>zero /* raise inexact */
  120. &&ax<0x3c900000) /* |x| < 2**-54 */
  121. return x;
  122. else
  123. return x - x*x*0.5;
  124. }
  125. if(hx>0||hx<=((int32_t)0xbfd2bec3)) {
  126. k=0;f=x;hu=1;} /* -0.2929<x<0.41422 */
  127. }
  128. if (hx >= 0x7ff00000) return x+x;
  129. if(k!=0) {
  130. if(hx<0x43400000) {
  131. u = 1.0+x;
  132. GET_HIGH_WORD(hu,u);
  133. k = (hu>>20)-1023;
  134. c = (k>0)? 1.0-(u-x):x-(u-1.0);/* correction term */
  135. c /= u;
  136. } else {
  137. u = x;
  138. GET_HIGH_WORD(hu,u);
  139. k = (hu>>20)-1023;
  140. c = 0;
  141. }
  142. hu &= 0x000fffff;
  143. if(hu<0x6a09e) {
  144. SET_HIGH_WORD(u,hu|0x3ff00000); /* normalize u */
  145. } else {
  146. k += 1;
  147. SET_HIGH_WORD(u,hu|0x3fe00000); /* normalize u/2 */
  148. hu = (0x00100000-hu)>>2;
  149. }
  150. f = u-1.0;
  151. }
  152. hfsq=0.5*f*f;
  153. if(hu==0) { /* |f| < 2**-20 */
  154. if(f==zero) {if(k==0) return zero;
  155. else {c += k*ln2_lo; return k*ln2_hi+c;}
  156. }
  157. R = hfsq*(1.0-0.66666666666666666*f);
  158. if(k==0) return f-R; else
  159. return k*ln2_hi-((R-(k*ln2_lo+c))-f);
  160. }
  161. s = f/(2.0+f);
  162. z = s*s;
  163. R = z*(Lp1+z*(Lp2+z*(Lp3+z*(Lp4+z*(Lp5+z*(Lp6+z*Lp7))))));
  164. if(k==0) return f-(hfsq-s*(hfsq+R)); else
  165. return k*ln2_hi-((hfsq-(s*(hfsq+R)+(k*ln2_lo+c)))-f);
  166. }