[builtins] Introduce proper base::ieee754::log.

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.
+// ====================================================
+//
+// The original source code covered by the above license above has been
+// modified significantly by Google Inc.
+// Copyright 2016 the V8 project authors. All rights reserved.
+
+#include "src/base/ieee754.h"
+
+#include <limits>
+
+#include "src/base/build_config.h"
+#include "src/base/macros.h"
+
+namespace v8 {
+namespace base {
+namespace ieee754 {
+
+namespace {
+
+union Float64 {
+  double v;
+  uint64_t w;
+  struct {
+#if V8_TARGET_LITTLE_ENDIAN
+    uint32_t lw;
+    uint32_t hw;
+#else
+    uint32_t hw;
+    uint32_t lw;
+#endif
+  } words;
+};
+
+// Extract the less significant 32-bit word from a double.
+V8_INLINE uint32_t extractLowWord32(double v) {
+  Float64 f;
+  f.v = v;
+  return f.words.lw;
+}
+
+// Extract the most significant 32-bit word from a double.
+V8_INLINE uint32_t extractHighWord32(double v) {
+  Float64 f;
+  f.v = v;
+  return f.words.hw;
+}
+
+// Insert the most significant 32-bit word into a double.
+V8_INLINE double insertHighWord32(double v, uint32_t hw) {
+  Float64 f;
+  f.v = v;
+  f.words.hw = hw;
+  return f.v;
+}
+
+double const kLn2Hi = 6.93147180369123816490e-01;  // 3fe62e42 fee00000
+double const kLn2Lo = 1.90821492927058770002e-10;  // 3dea39ef 35793c76
+double const kTwo54 = 1.80143985094819840000e+16;  // 43500000 00000000
+double const kLg1 = 6.666666666666735130e-01;      // 3FE55555 55555593
+double const kLg2 = 3.999999999940941908e-01;      // 3FD99999 9997FA04
+double const kLg3 = 2.857142874366239149e-01;      // 3FD24924 94229359
+double const kLg4 = 2.222219843214978396e-01;      // 3FCC71C5 1D8E78AF
+double const kLg5 = 1.818357216161805012e-01;      // 3FC74664 96CB03DE
+double const kLg6 = 1.531383769920937332e-01;      // 3FC39A09 D078C69F
+double const kLg7 = 1.479819860511658591e-01;      // 3FC2F112 DF3E5244
+
+}  // 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) .
+ *
+ *   2. Approximation of log(1+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) ~ Lg1*s +Lg2*s +Lg3*s +Lg4*s +Lg5*s  +Lg6*s  +Lg7*s
+ *    (the values of Lg1 to Lg7 are listed in the program)
+ *  and
+ *      |      2          14          |     -58.45
+ *      | Lg1*s +...+Lg7*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
+ *    log(1+f) = f - s*(f - R)  (if f is not too large)
+ *    log(1+f) = f - (hfsq - s*(hfsq+R)). (better accuracy)
+ *
+ *  3. Finally,  log(x) = k*ln2 + log(1+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:
+ *  log(x) is NaN with signal if x < 0 (including -INF) ;
+ *  log(+INF) is +INF; log(0) is -INF with signal;
+ *  log(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.
+ */
+double log(double x) {
+  double hfsq, f, s, z, r, w, t1, t2, dk;
+  int32_t k = 0, i, j;
+  int32_t hx = extractHighWord32(x);
+  uint32_t lx = extractLowWord32(x);
+
+  if (hx < 0x00100000) { /* x < 2**-1022  */
+    if (((hx & 0x7fffffff) | lx) == 0) {
+      return -std::numeric_limits<double>::infinity();
+    }
+    if (hx < 0) {
+      return std::numeric_limits<double>::quiet_NaN();
+    }
+    k -= 54;
+    x *= kTwo54; /* subnormal number, scale up x */
+    hx = extractHighWord32(x);
+  }
+  if (hx >= 0x7ff00000) return x + x;
+  k += (hx >> 20) - 1023;
+  hx &= 0x000fffff;
+  i = (hx + 0x95f64) & 0x100000;
+  x = insertHighWord32(x, hx | (i ^ 0x3ff00000)); /* normalize x or x/2 */
+  k += (i >> 20);
+  f = x - 1.0;
+  if ((0x000fffff & (2 + hx)) < 3) { /* -2**-20 <= f < 2**-20 */
+    if (f == 0.0) {
+      if (k == 0) {
+        return 0.0;
+      } else {
+        dk = static_cast<double>(k);
+        return dk * kLn2Hi + dk * kLn2Lo;
+      }
+    }
+    r = f * f * (0.5 - 0.33333333333333333 * f);
+    if (k == 0) {
+      return f - r;
+    } else {
+      dk = static_cast<double>(k);
+      return dk * kLn2Hi - ((r - dk * kLn2Lo) - f);
+    }
+  }
+  s = f / (2.0 + f);
+  dk = static_cast<double>(k);
+  z = s * s;
+  i = hx - 0x6147a;
+  w = z * z;
+  j = 0x6b851 - hx;
+  t1 = w * (kLg2 + w * (kLg4 + w * kLg6));
+  t2 = z * (kLg1 + w * (kLg3 + w * (kLg5 + w * kLg7)));
+  i |= j;
+  r = t2 + t1;
+  if (i > 0) {
+    hfsq = 0.5 * f * f;
+    if (k == 0) {
+      return f - (hfsq - s * (hfsq + r));
+    } else {
+      return dk * kLn2Hi - ((hfsq - (s * (hfsq + r) + dk * kLn2Lo)) - f);
+    }
+  } else {
+    if (k == 0) {
+      return f - s * (f - r);
+    } else {
+      return dk * kLn2Hi - ((s * (f - r) - dk * kLn2Lo) - f);
+    }
+  }
+}
+
+}  // namespace ieee754
+}  // namespace base
+}  // namespace v8