Fast algorithm for double->string conversion.

src/grisu3.cc

+// Copyright 2010 the V8 project authors. All rights reserved.
+// Redistribution and use in source and binary forms, with or without
+// modification, are permitted provided that the following conditions are
+// met:
+//
+//     * Redistributions of source code must retain the above copyright
+//       notice, this list of conditions and the following disclaimer.
+//     * Redistributions in binary form must reproduce the above
+//       copyright notice, this list of conditions and the following
+//       disclaimer in the documentation and/or other materials provided
+//       with the distribution.
+//     * Neither the name of Google Inc. nor the names of its
+//       contributors may be used to endorse or promote products derived
+//       from this software without specific prior written permission.
+//
+// THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS
+// "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT
+// LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR
+// A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT
+// OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
+// SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT
+// LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE,
+// DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY
+// THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
+// (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE
+// OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
+
+#include "v8.h"
+
+#include "grisu3.h"
+#include "cached_powers.h"
+#include "diy_fp.h"
+#include "double.h"
+
+namespace v8 {
+namespace internal {
+
+template <int alpha = -60, int gamma = -32>
+class Grisu3 {
+ public:
+  // Provides a decimal representation of v.
+  // Returns true if it succeeds, otherwise the result can not be trusted.
+  // There will be *length digits inside the buffer (not null-terminated).
+  // If the function returns true then v == (double) (buffer * 10^K).
+  // The digits in the buffer are the shortest representation possible: no
+  // 0.099999999999 instead of 0.1.
+  // The last digit will be closest to the actual v. That is, even if several
+  // digits might correctly yield 'v' when read again, the closest will be
+  // computed.
+  static bool grisu3(double v, char* buffer, int* length, int* K);
+
+ private:
+  static bool RoundWeed(char* buffer, int len, uint64_t wp_W, uint64_t Delta,
+                        uint64_t rest, uint64_t ten_kappa, uint64_t ulp);
+  static bool DigitGen(DiyFp low, DiyFp w, DiyFp high,
+                       char* buffer, int* len, int* kappa);
+  static bool DigitGen_m60_m32(DiyFp low, DiyFp w, DiyFp high,
+                               char* buffer, int* length, int* kappa);
+};
+
+
+template<int alpha, int gamma>
+bool Grisu3<alpha, gamma>::grisu3(double v, char* buffer, int* length, int* K) {
+  DiyFp w = Double(v).AsNormalizedDiyFp();
+  // m_minus and m_plus are the boundaries between w and its neighbors. Any
+  // number x such that m_minus < x < m_plus will round to v when read as
+  // double. When m_minus == x or m_plus == y then the rounding direction
+  // depends on v. Grisu3 does not need to deal with this case, as its precision
+  // is not sufficient for this case anyways.
+  DiyFp m_minus, m_plus;
+  Double(v).NormalizedBoundaries(&m_minus, &m_plus);
+  ASSERT(m_plus.e() == w.e());
+  DiyFp ten_mk;  // Cached power of ten: 10^-k
+  int mk;        // -k
+  GetCachedPower(w.e() + DiyFp::kSignificandSize, alpha, gamma, &mk, &ten_mk);
+  ASSERT(alpha <= w.e() + ten_mk.e() + DiyFp::kSignificandSize &&
+         gamma >= w.e() + ten_mk.e() + DiyFp::kSignificandSize);
+  // Note that ten_mk is only an approximation of 10^-k. A DiyFp only contains a
+  // 64 bit significand and ten_mk is thus only precise up to 64 bits.
+
+  // The DiyFp::Times procedure rounds its result, and ten_mk is approximated
+  // too. The variable scaled_w (as well as scaled_m_minus/plus) are now off
+  // by a small amount.
+  // In fact: scaled_w - w*10^k < 1ulp (unit in the last place) of scaled_w.
+  // In other words: let f = scaled_w.f() and e = scaled_w.e(), then
+  //           (f-1) * 2^e < w*10^k < (f+1) * 2^e
+  DiyFp scaled_w = DiyFp::Times(w, ten_mk);
+  ASSERT(scaled_w.e() == m_plus.e() + ten_mk.e() + DiyFp::kSignificandSize);
+  // In theory it would be possible to avoid some recomputations by computing
+  // the difference between w and m_minus/plus (a power of 2) and to compute
+  // scaled_m_minus/plus by subtracting/adding from scaled_w. However the
+  // code becomes much less readable and the speed enhancements are not
+  // terrible.
+  DiyFp scaled_m_minus = DiyFp::Times(m_minus, ten_mk);
+  DiyFp scaled_m_plus  = DiyFp::Times(m_plus,  ten_mk);
+
+  // DigitGen will generate the digits of scaled_w. Therefore we have
+  // v == (double) (scaled_w * 10^-mk).
+  // Set K == -mk and pass it to DigitGen. If scaled_w is a comma-number it will
+  // be updated.
+  int kappa;
+  bool result =
+      DigitGen(scaled_m_minus, scaled_w, scaled_m_plus, buffer, length, &kappa);
+  *K = -mk + kappa;
+  return result;
+}
+
+// Generates the digits of input number w.
+// w is a floating-point number (DiyFp), consisting of a significand and an
+// exponent. Its exponent is bounded by alpha and gamma. Typically alpha >= -63
+// and gamma <= 3.
+// Returns false if it fails, in which case the generated digits in the buffer
+// should not be used.
+// Preconditions:
+//  * low, w and high are correct up to 1 ulp (unit in the last place). That
+//    is, their error must be less that a unit of their last digits.
+//  * low.e() == w.e() == high.e()
+//  * low < w < high, and taking into account their error: low~ <= high~
+//  * alpha <= w.e() <= gamma
+// Postconditions: returns false if procedure fails.
+//   otherwise:
+//     * buffer is not null-terminated, but len contains the number of digits.
+//     * buffer contains the shortest possible decimal digit-sequence
+//       such that LOW < buffer * 10^kappa < HIGH, where LOW and HIGH are the
+//       correct values of low and high (without their error).
+//     * if more than one decimal representation gives the minimal number of
+//       decimal digits then the one closest to W (where W is the correct value
+//       of w) is chosen.
+// Remark: this procedure takes into account the imprecission of its input

[fschneider, 2010/02/22 16:25:24]

Typo: -> precision

[Florian Loitsch, 2010/02/23 09:05:10]

Done.

+//   numbers. If the precision is not enough to guarantee all the postconditions
+//   then false is returned. This usually happens rarely (~0.5%).
+template<int alpha, int gamma>
+bool Grisu3<alpha, gamma>::DigitGen(DiyFp low, DiyFp w, DiyFp high,
+                                    char* buffer, int* len, int* kappa) {
+  ASSERT(low.e() == w.e() && w.e() == high.e());
+  ASSERT(low.f() + 1 <= high.f() - 1);
+  ASSERT(alpha <= w.e() && w.e() <= gamma);
+  // The following tests use alpha and gamma to avoid unnecessary dynamic tests.
+  if ((alpha >= -60 && gamma <= -32) ||  // -60 <= w.e() <= -32
+      (alpha <= -32 && gamma >= -60 &&   // Alpha/gamma overlaps -60/-32 region.
+       -60 <= w.e() && w.e() <= -32)) {
+    return DigitGen_m60_m32(low, w, high, buffer, len, kappa);
+  } else {
+    // A simple adaption of the special case -60/-32 would allow greater ranges
+    // of alpha/gamma and thus reduce the number of precomputed cached powers of
+    // ten.
+    UNIMPLEMENTED();
+    return false;
+  }
+}
+
+static const uint32_t kTen4 = 10000;
+static const uint32_t kTen5 = 100000;
+static const uint32_t kTen6 = 1000000;
+static const uint32_t kTen7 = 10000000;
+static const uint32_t kTen8 = 100000000;
+static const uint32_t kTen9 = 1000000000;
+
+// Returns the biggest power of ten that is <= than the given number. We
+// furthermore receive the maximum number of bits 'number' has.
+// If number_bits == 0 then 0^-1 is returned
+// The number of bits must be <= 32.
+static void BiggestPowerTen(uint32_t number, int number_bits, uint32_t* power, int* exponent) {
+  switch (number_bits) {
+    case 30:
+    case 31:
+    case 32:

[Lasse Reichstein, 2010/02/22 11:31:36]

For readability, could you reverse the order of the cases in each group, so they
come in decreasing order

[Florian Loitsch, 2010/02/22 15:52:53]

Done.

[floitsch, 2012/04/10 15:54:34]

Done.

+      if (kTen9 <= number) {
+        *power = kTen9;
+        *exponent = 9;
+        break;
+      }  // else fallthrough
+    case 27:
+    case 28:
+    case 29:
+      if (kTen8 <= number) {
+        *power = kTen8;
+        *exponent = 8;
+        break;
+      }  // else fallthrough
+    case 24:
+    case 25:
+    case 26:
+      if (kTen7 <= number) {
+        *power = kTen7;
+        *exponent = 7;
+        break;
+      }  // else fallthrough
+    case 20:
+    case 21:
+    case 22:
+    case 23:
+      if (kTen6 <= number) {
+        *power = kTen6;
+        *exponent = 6;
+        break;
+      }  // else fallthrough
+    case 17:
+    case 18:
+    case 19:
+      if (kTen5 <= number) {
+        *power = kTen5;
+        *exponent = 5;
+        break;
+      }  // else fallthrough
+    case 14:
+    case 15:
+    case 16:
+      if (kTen4 <= number) {
+        *power = kTen4;
+        *exponent = 4;
+        break;
+      }  // else fallthrough
+    case 10:
+    case 11:
+    case 12:
+    case 13:
+      if (1000 <= number) {
+        *power = 1000;
+        *exponent = 3;
+        break;
+      }  // else fallthrough
+    case 7:
+    case 8:
+    case 9:
+      if (100 <= number) {
+        *power = 100;
+        *exponent = 2;
+        break;
+      }  // else fallthrough
+    case 4:
+    case 5:
+    case 6:
+      if (10 <= number) {
+        *power = 10;
+        *exponent = 1;
+        break;
+      }  // else fallthrough
+    case 1:
+    case 2:
+    case 3:
+      if (1 <= number) {
+        *power = 1;
+        *exponent = 0;
+        break;
+      }  // else fallthrough
+    case 0:
+      *power = 0;
+      *exponent = -1;
+      break;
+    default:
+      // Following assignments are here to silence compiler warnings.
+      *power = 0;
+      *exponent = 0;
+      UNREACHABLE();
+  }
+}
+
+
+// Same comments as for DigitGen but with additional precondition:
+//    -60 <= w.e() <= -32
+//
+// Say, for the sake of example, that
+//   w.e() == -48, and w.f() == 0x1234567890abcdef
+// w's value can be computed by w.f() * 2^w.e()
+// We can obtain w's integral by simply shifting w.f() by -w.e().
+//  -> w's integral is 0x1234
+//  w's fractional part is therefore 0x567890abcdef.
+// Printing w's integral part is easy (simply print 0x1234 in decimal).
+// In order to print its fraction we repeatedly multiply the fraction by 10 and
+// get each digit. Example the first digit after the comma would be computed by
+//   (0x567890abcdef * 10) >> 48. -> 3
+// The whole thing becomes slightly more complicated because we want to stop
+// once we have enough digits. That is, once the digits inside the buffer
+// represent 'w' we can stop. Everything inside the interval low - high
+// represents w. However we have to pay attention to low, high and w's
+// imprecision.
+template<int alpha, int gamma>
+bool Grisu3<alpha, gamma>::DigitGen_m60_m32(
+    DiyFp low, DiyFp w, DiyFp high, char* buffer, int* length, int* kappa) {
+  // low, w and high are imprecise, but by less than one ulp (unit in the last
+  // place).
+  // If we remove (resp. add) 1 ulp from low (resp. high) we are certain that
+  // the new numbers are outside of the interval we want the final
+  // representation to lie in.
+  // Inversely adding (resp. removing) 1 ulp from low (resp. high) would yield
+  // numbers that are certain to lie in the interval. We will use this fact
+  // later on.
+  // We will now start by generating the digits within the uncertain
+  // interval. Later we will weed out representations that lie outside the safe
+  // interval and thus _might_ lie outside the correct interval.
+  uint64_t unit = 1;
+  DiyFp too_low = DiyFp(low.f() - unit, low.e());
+  DiyFp too_high = DiyFp(high.f() + unit, high.e());
+  // too_low and too_high are guaranteed to lie outside the interval we want the
+  // generated number in.
+  DiyFp unsafe_interval = DiyFp::Minus(too_high, too_low);
+  // We now cut the input number into two parts: the integrals and the
+  // fractionals. We will not write any decimal separator though, but adapt
+  // kappa instead.
+  // Reminder: we are currently computing the digits (stored inside the buffer)
+  // such that:   too_low < buffer * 10^kappa < too_high
+  // We use too_high for the digit_generation and stop as soon as possible.
+  // If we stop early we effectively round down.
+  DiyFp one = DiyFp(static_cast<uint64_t>(1) << -w.e(), w.e());
+  uint32_t integrals = too_high.f() >> -one.e();        // Division by one.
+  uint64_t fractionals = too_high.f() & (one.f() - 1);  // Modulo by one.
+  uint32_t divider;
+  int divider_exponent;
+  BiggestPowerTen(integrals, DiyFp::kSignificandSize - (-one.e()),
+                  &divider, &divider_exponent);
+  *kappa = divider_exponent + 1;
+  *length = 0;
+  // Loop invariant: buffer = too_high / 10^kappa  (integer division)
+  // The invariant holds for the first iteration: kappa has been initialized
+  // with the divider exponent + 1. And the divider is the biggest power of ten
+  // that fits into the bits that had been reserved for the integrals.
+  while (*kappa > 0) {
+    int digit = integrals / divider;
+    buffer[*length] = '0' + digit;
+    (*length)++;
+    integrals %= divider;
+    (*kappa)--;
+    // Note that kappa now equals the exponent of the divider and that the
+    // invariant thus holds again.
+    uint64_t rest =
+        (static_cast<uint64_t>(integrals) << -one.e()) + fractionals;
+    // Invariant: too_high = buffer * 10^kappa + DiyFp(rest, one.e())
+    // Reminder: unsafe_interval.e() == one.e()
+    if (rest < unsafe_interval.f()) {
+      // Rounding down (by not emitting the remaining digits) yields a number
+      // that lies within the unsafe interval.
+      return RoundWeed(buffer, *length, DiyFp::Minus(too_high, w).f(),
+                       unsafe_interval.f(), rest,
+                       static_cast<uint64_t>(divider) << -one.e(), unit);
+    }
+    divider /= 10;
+  }
+  // The integrals have been generated. We are at the point of the decimal
+  // separator. In the following loop we simply multiply the remaining digits by
+  // 10 and divide by one. We just need to pay attention to multiply associated
+  // data (like the interval or 'unit'), too.
+  // Instead of multiplying by 10 we multiply by 5 (cheaper operation) and
+  // increase its (imaginary) exponent. At the same time we decrease the
+  // divider's (one's) exponent and shift its significand.
+  // Basically, if fractionals was a DiyFp (with fractionals.e == one.e):
+  //      fractionals.f *= 10;
+  //      fractionals.f >>= 1; fractionals.e++; // value remains unchanged.
+  //      one.f >>= 1; one.e++;                 // value remains unchanged.
+  //      and we have again fractionals.e == one.e which allows us to divide
+  //           fractionals.f() by one.f()
+  // We simply combine the *= 10 and the >>= 1.
+  while (true) {
+    fractionals *= 5;
+    unit *= 5;
+    unsafe_interval.set_f(unsafe_interval.f() * 5);
+    unsafe_interval.set_e(unsafe_interval.e() + 1);  // Will be optimized out.
+    one.set_f(one.f() >> 1);
+    one.set_e(one.e() + 1);
+    int digit = fractionals >> -one.e();  // Integer division by one.
+    buffer[*length] = '0' + digit;
+    (*length)++;
+    fractionals &= one.f() - 1;  // Modulo by one.
+    (*kappa)--;
+    if (fractionals < unsafe_interval.f()) {
+      return RoundWeed(buffer, *length, DiyFp::Minus(too_high, w).f() * unit,
+                       unsafe_interval.f(), fractionals, one.f(), unit);
+    }
+  }
+}
+
+
+// Rounds the given generated digits in the buffer and weeds out generated
+// digits that are not in the safe interval, or where we cannot find a rounded
+// representation.
+// Input: * buffer containing the digits of too_high / 10^kappa
+//        * the buffer's length
+//        * distance_too_high_w == (too_high - w).f() * unit
+//        * unsafe_interval == (too_high - too_low).f() * unit
+//        * rest = (too_high - buffer * 10^kappa).f() * unit
+//        * ten_kappa = 10^kappa * unit
+//        * unit = the common multiplier
+// Output: returns true on success.
+//    Modifies the generated digits in the buffer to approach (round towards) w.
+template<int alpha, int gamma>
+bool Grisu3<alpha, gamma>::RoundWeed(
+    char* buffer, int length, uint64_t distance_too_high_w,
+    uint64_t unsafe_interval, uint64_t rest, uint64_t ten_kappa,
+    uint64_t unit) {
+  uint64_t small_distance = distance_too_high_w - unit;
+  uint64_t big_distance = distance_too_high_w + unit;
+  // Let w- = too_high - big_distance, and
+  //     w+ = too_high - small_distance.
+  // Note: w- < w < w+
+  //
+  // The real w (* unit) must lie somewhere inside the interval
+  // ]w-; w+[ (also often written as (w-; w+))
+
+  // Basically the buffer currently contains a number in the unsafe interval
+  // ]too_low; too_high[ with too_low < w < too_high
+  //
+  // By generating the digits of too_high we got the biggest last digit.
+  // In the case that w+ < buffer < too_high we try to decrement the buffer.
+  // This way the buffer approaches (rounds towards) w.
+  // There are 3 conditions that stop the decrementation process:
+  //   1) the buffer is already below w+
+  //   2) decrementing the buffer would make it leave the unsafe interval
+  //   3) decrementing the buffer would yield a number below w+ and farther away
+  //      than the current number. In other words:
+  //                       (buffer{-1} < w+) && w+ - buffer{-1} > buffer - w+
+  // Instead of using the buffer directly we use its distance to too_high.
+  // Conceptually rest ~= too_high - buffer
+  while (rest < small_distance &&  // condition 1
+         unsafe_interval - rest >= ten_kappa &&  // condition 2
+         (rest + ten_kappa < small_distance ||  // buffer{-1} > w+
+          small_distance - rest >= rest + ten_kappa - small_distance)) {
+    buffer[length - 1]--;
+    rest += ten_kappa;
+  }
+
+  // We have approached w+ as much as possible. We now test if approaching w-
+  // would require changing the buffer. If yes, then we have two possible
+  // representations close to w, but we cannot decide which one is closer.
+  if (rest < big_distance &&
+      unsafe_interval - rest >= ten_kappa &&
+      (rest + ten_kappa < big_distance ||
+       big_distance - rest > rest + ten_kappa - big_distance)) {
+    return false;
+  }
+
+  // Weeding test.
+  //   The safe interval is [too_low + 2 ulp; too_high - 2 ulp]
+  //   Since too_low = too_high - unsafe_interval this is equivalent too
+  //      [too_high - unsafe_interval + 4 ulp; too_high - 2 ulp]
+  //   Conceptually we have: rest ~= too_high - buffer
+  return (2 * unit <= rest) && (rest <= unsafe_interval - 4 * unit);
+}
+
+
+bool grisu3(double v, char* buffer, int* sign, int* length, int* decimal_point) {
+  if (v < 0) {
+    v = -v;
+    *sign = 1;
+  } else {
+    *sign = 0;
+  }
+  int K;

[Lasse Reichstein, 2010/02/22 11:31:36]

Lower case variable names (and preferably not single-letter names).

[Florian Loitsch, 2010/02/22 15:52:53]

Done.

[floitsch, 2012/04/10 15:54:34]

Done.

+  bool result = Grisu3<-60, -32>::grisu3(v, buffer, length, &K);
+  *decimal_point = *length + K;
+  buffer[*length] = '\0';
+  return result;
+}
+
+} }  // namespace v8::internal