[builtins] Introduce proper Float64Log1p operator.

src/base/ieee754.cc

 // The following is adapted from fdlibm (http://www.netlib.org/fdlibm).
 //
 // ====================================================
 // Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
 //
 // Developed at SunSoft, a Sun Microsystems, Inc. business.
 // Permission to use, copy, modify, and distribute this
 // software is freely granted, provided that this notice
 // is preserved.
 // ====================================================
@@ skipping 145 matching lines @@
 /* Set the less significant 32 bits of a double from an int.  */
 
 #define SET_LOW_WORD(d, v)       \
   do {                           \
     ieee_double_shape_type sl_u; \
     sl_u.value = (d);            \
     sl_u.parts.lsw = (v);        \
     (d) = sl_u.value;            \
   } while (0)
 
+/* Support macro. */
+
+#define STRICT_ASSIGN(type, lval, rval) ((lval) = (rval))
+
 }  // namespace
 
 /* log(x)
  * Return the logrithm of x
  *
  * Method :
  *   1. Argument Reduction: find k and f such that
  *      x = 2^k * (1+f),
  *     where  sqrt(2)/2 < 1+f < sqrt(2) .
  *
@@ skipping 111 matching lines @@
     else
       return dk * ln2_hi - ((hfsq - (s * (hfsq + R) + dk * ln2_lo)) - f);
   } else {
     if (k == 0)
       return f - s * (f - R);
     else
       return dk * ln2_hi - ((s * (f - R) - dk * ln2_lo) - f);
   }
 }
 
+/* double log1p(double x)
+ *
+ * Method :
+ *   1. Argument Reduction: find k and f such that
+ *      1+x = 2^k * (1+f),
+ *     where  sqrt(2)/2 < 1+f < sqrt(2) .
+ *
+ *      Note. If k=0, then f=x is exact. However, if k!=0, then f
+ *  may not be representable exactly. In that case, a correction
+ *  term is need. Let u=1+x rounded. Let c = (1+x)-u, then
+ *  log(1+x) - log(u) ~ c/u. Thus, we proceed to compute log(u),
+ *  and add back the correction term c/u.
+ *  (Note: when x > 2**53, one can simply return log(x))
+ *
+ *   2. Approximation of log1p(f).
+ *  Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
+ *     = 2s + 2/3 s**3 + 2/5 s**5 + .....,
+ *         = 2s + s*R
+ *      We use a special Reme algorithm on [0,0.1716] to generate
+ *  a polynomial of degree 14 to approximate R The maximum error
+ *  of this polynomial approximation is bounded by 2**-58.45. In
+ *  other words,
+ *            2      4      6      8      10      12      14
+ *      R(z) ~ Lp1*s +Lp2*s +Lp3*s +Lp4*s +Lp5*s  +Lp6*s  +Lp7*s
+ *    (the values of Lp1 to Lp7 are listed in the program)
+ *  and
+ *      |      2          14          |     -58.45
+ *      | Lp1*s +...+Lp7*s    -  R(z) | <= 2
+ *      |                             |
+ *  Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2.
+ *  In order to guarantee error in log below 1ulp, we compute log
+ *  by
+ *    log1p(f) = f - (hfsq - s*(hfsq+R)).
+ *
+ *  3. Finally, log1p(x) = k*ln2 + log1p(f).
+ *           = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo)))
+ *     Here ln2 is split into two floating point number:
+ *      ln2_hi + ln2_lo,
+ *     where n*ln2_hi is always exact for |n| < 2000.
+ *
+ * Special cases:
+ *  log1p(x) is NaN with signal if x < -1 (including -INF) ;
+ *  log1p(+INF) is +INF; log1p(-1) is -INF with signal;
+ *  log1p(NaN) is that NaN with no signal.
+ *
+ * Accuracy:
+ *  according to an error analysis, the error is always less than
+ *  1 ulp (unit in the last place).
+ *
+ * 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.
+ *
+ * Note: Assuming log() return accurate answer, the following
+ *   algorithm can be used to compute log1p(x) to within a few ULP:
+ *
+ *    u = 1+x;
+ *    if(u==1.0) return x ; else
+ *         return log(u)*(x/(u-1.0));
+ *
+ *   See HP-15C Advanced Functions Handbook, p.193.
+ */
+double log1p(double x) {
+  static const double                      /* -- */
+      ln2_hi = 6.93147180369123816490e-01, /* 3fe62e42 fee00000 */
+      ln2_lo = 1.90821492927058770002e-10, /* 3dea39ef 35793c76 */
+      two54 = 1.80143985094819840000e+16,  /* 43500000 00000000 */
+      Lp1 = 6.666666666666735130e-01,      /* 3FE55555 55555593 */
+      Lp2 = 3.999999999940941908e-01,      /* 3FD99999 9997FA04 */
+      Lp3 = 2.857142874366239149e-01,      /* 3FD24924 94229359 */
+      Lp4 = 2.222219843214978396e-01,      /* 3FCC71C5 1D8E78AF */
+      Lp5 = 1.818357216161805012e-01,      /* 3FC74664 96CB03DE */
+      Lp6 = 1.531383769920937332e-01,      /* 3FC39A09 D078C69F */
+      Lp7 = 1.479819860511658591e-01;      /* 3FC2F112 DF3E5244 */
+
+  static const double zero = 0.0;
+  static volatile double vzero = 0.0;
+
+  double hfsq, f, c, s, z, R, u;
+  int32_t k, hx, hu, ax;
+
+  GET_HIGH_WORD(hx, x);
+  ax = hx & 0x7fffffff;
+
+  k = 1;
+  if (hx < 0x3FDA827A) {    /* 1+x < sqrt(2)+ */
+    if (ax >= 0x3ff00000) { /* x <= -1.0 */
+      if (x == -1.0)
+        return -two54 / vzero; /* log1p(-1)=+inf */
+      else
+        return (x - x) / (x - x); /* log1p(x<-1)=NaN */
+    }
+    if (ax < 0x3e200000) {    /* |x| < 2**-29 */
+      if (two54 + x > zero    /* raise inexact */
+          && ax < 0x3c900000) /* |x| < 2**-54 */
+        return x;
+      else
+        return x - x * x * 0.5;
+    }
+    if (hx > 0 || hx <= static_cast<int32_t>(0xbfd2bec4)) {
+      k = 0;
+      f = x;
+      hu = 1;
+    } /* sqrt(2)/2- <= 1+x < sqrt(2)+ */
+  }
+  if (hx >= 0x7ff00000) return x + x;
+  if (k != 0) {
+    if (hx < 0x43400000) {
+      STRICT_ASSIGN(double, u, 1.0 + x);
+      GET_HIGH_WORD(hu, u);
+      k = (hu >> 20) - 1023;
+      c = (k > 0) ? 1.0 - (u - x) : x - (u - 1.0); /* correction term */
+      c /= u;
+    } else {
+      u = x;
+      GET_HIGH_WORD(hu, u);
+      k = (hu >> 20) - 1023;
+      c = 0;
+    }
+    hu &= 0x000fffff;
+    /*
+     * The approximation to sqrt(2) used in thresholds is not
+     * critical.  However, the ones used above must give less
+     * strict bounds than the one here so that the k==0 case is
+     * never reached from here, since here we have committed to
+     * using the correction term but don't use it if k==0.
+     */
+    if (hu < 0x6a09e) {                  /* u ~< sqrt(2) */
+      SET_HIGH_WORD(u, hu | 0x3ff00000); /* normalize u */
+    } else {
+      k += 1;
+      SET_HIGH_WORD(u, hu | 0x3fe00000); /* normalize u/2 */
+      hu = (0x00100000 - hu) >> 2;
+    }
+    f = u - 1.0;
+  }
+  hfsq = 0.5 * f * f;
+  if (hu == 0) { /* |f| < 2**-20 */
+    if (f == zero) {
+      if (k == 0) {
+        return zero;
+      } else {
+        c += k * ln2_lo;
+        return k * ln2_hi + c;
+      }
+    }
+    R = hfsq * (1.0 - 0.66666666666666666 * f);
+    if (k == 0)
+      return f - R;
+    else
+      return k * ln2_hi - ((R - (k * ln2_lo + c)) - f);
+  }
+  s = f / (2.0 + f);
+  z = s * s;
+  R = z * (Lp1 +
+           z * (Lp2 + z * (Lp3 + z * (Lp4 + z * (Lp5 + z * (Lp6 + z * Lp7))))));
+  if (k == 0)
+    return f - (hfsq - s * (hfsq + R));
+  else
+    return k * ln2_hi - ((hfsq - (s * (hfsq + R) + (k * ln2_lo + c))) - f);
+}
+
 }  // namespace ieee754
 }  // namespace base
 }  // namespace v8