d2s.c

// Copyright 2018 Ulf Adams
//
// The contents of this file may be used under the terms of the Apache License,
// Version 2.0.
//
//    (See accompanying file LICENSE-Apache or copy at
//     http://www.apache.org/licenses/LICENSE-2.0)
//
// Alternatively, the contents of this file may be used under the terms of
// the Boost Software License, Version 1.0.
//    (See accompanying file LICENSE-Boost or copy at
//     https://www.boost.org/LICENSE_1_0.txt)
//
// Unless required by applicable law or agreed to in writing, this software
// is distributed on an "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY
// KIND, either express or implied.

// Runtime compiler options:
// -DRYU_DEBUG Generate verbose debugging output to stdout.
//
// -DRYU_ONLY_64_BIT_OPS Avoid using uint128_t or 64-bit intrinsics. Slower,
//     depending on your compiler.
//
// -DRYU_OPTIMIZE_SIZE Use smaller lookup tables. Instead of storing every
//     required power of 5, only store every 26th entry, and compute
//     intermediate values with a multiplication. This reduces the lookup table
//     size by about 10x (only one case, and only double) at the cost of some
//     performance. Currently requires MSVC intrinsics.

#include "ryu/ryu.h"

#include <assert.h>
#include <stdbool.h>
#include <stdint.h>
#include <stdlib.h>
#include <string.h>

#ifdef RYU_DEBUG
#include <inttypes.h>
#include <stdio.h>
#endif

// ABSL avoids uint128_t on Win32 even if __SIZEOF_INT128__ is defined.
// Let's do the same for now.
#if defined(__SIZEOF_INT128__) && !defined(_MSC_VER) && !defined(RYU_ONLY_64_BIT_OPS)
#define HAS_UINT128
#elif defined(_MSC_VER) && !defined(RYU_ONLY_64_BIT_OPS) && defined(_M_X64)
#define HAS_64_BIT_INTRINSICS
#endif

#include "ryu/common.h"
#include "ryu/digit_table.h"
#include "ryu/d2s.h"
#include "ryu/d2s_intrinsics.h"

static inline uint32_t pow5Factor(uint64_t value) {
  uint32_t count = 0;
  for (;;) {
    assert(value != 0);
    const uint64_t q = div5(value);
    const uint32_t r = ((uint32_t) value) - 5 * ((uint32_t) q);
    if (r != 0) {
      break;
    }
    value = q;
    ++count;
  }
  return count;
}

// Returns true if value is divisible by 5^p.
static inline bool multipleOfPowerOf5(const uint64_t value, const uint32_t p) {
  // I tried a case distinction on p, but there was no performance difference.
  return pow5Factor(value) >= p;
}

// Returns true if value is divisible by 2^p.
static inline bool multipleOfPowerOf2(const uint64_t value, const uint32_t p) {
  // return __builtin_ctzll(value) >= p;
  return (value & ((1ull << p) - 1)) == 0;
}

// We need a 64x128-bit multiplication and a subsequent 128-bit shift.
// Multiplication:
//   The 64-bit factor is variable and passed in, the 128-bit factor comes
//   from a lookup table. We know that the 64-bit factor only has 55
//   significant bits (i.e., the 9 topmost bits are zeros). The 128-bit
//   factor only has 124 significant bits (i.e., the 4 topmost bits are
//   zeros).
// Shift:
//   In principle, the multiplication result requires 55 + 124 = 179 bits to
//   represent. However, we then shift this value to the right by j, which is
//   at least j >= 115, so the result is guaranteed to fit into 179 - 115 = 64
//   bits. This means that we only need the topmost 64 significant bits of
//   the 64x128-bit multiplication.
//
// There are several ways to do this:
// 1. Best case: the compiler exposes a 128-bit type.
//    We perform two 64x64-bit multiplications, add the higher 64 bits of the
//    lower result to the higher result, and shift by j - 64 bits.
//
//    We explicitly cast from 64-bit to 128-bit, so the compiler can tell
//    that these are only 64-bit inputs, and can map these to the best
//    possible sequence of assembly instructions.
//    x64 machines happen to have matching assembly instructions for
//    64x64-bit multiplications and 128-bit shifts.
//
// 2. Second best case: the compiler exposes intrinsics for the x64 assembly
//    instructions mentioned in 1.
//
// 3. We only have 64x64 bit instructions that return the lower 64 bits of
//    the result, i.e., we have to use plain C.
//    Our inputs are less than the full width, so we have three options:
//    a. Ignore this fact and just implement the intrinsics manually.
//    b. Split both into 31-bit pieces, which guarantees no internal overflow,
//       but requires extra work upfront (unless we change the lookup table).
//    c. Split only the first factor into 31-bit pieces, which also guarantees
//       no internal overflow, but requires extra work since the intermediate
//       results are not perfectly aligned.
#if defined(HAS_UINT128)

// Best case: use 128-bit type.
static inline uint64_t mulShift(const uint64_t m, const uint64_t* const mul, const int32_t j) {
  const uint128_t b0 = ((uint128_t) m) * mul[0];
  const uint128_t b2 = ((uint128_t) m) * mul[1];
  return (uint64_t) (((b0 >> 64) + b2) >> (j - 64));
}

static inline uint64_t mulShiftAll(const uint64_t m, const uint64_t* const mul, const int32_t j,
  uint64_t* const vp, uint64_t* const vm, const uint32_t mmShift) {
//  m <<= 2;
//  uint128_t b0 = ((uint128_t) m) * mul[0]; // 0
//  uint128_t b2 = ((uint128_t) m) * mul[1]; // 64
//
//  uint128_t hi = (b0 >> 64) + b2;
//  uint128_t lo = b0 & 0xffffffffffffffffull;
//  uint128_t factor = (((uint128_t) mul[1]) << 64) + mul[0];
//  uint128_t vpLo = lo + (factor << 1);
//  *vp = (uint64_t) ((hi + (vpLo >> 64)) >> (j - 64));
//  uint128_t vmLo = lo - (factor << mmShift);
//  *vm = (uint64_t) ((hi + (vmLo >> 64) - (((uint128_t) 1ull) << 64)) >> (j - 64));
//  return (uint64_t) (hi >> (j - 64));
  *vp = mulShift(4 * m + 2, mul, j);
  *vm = mulShift(4 * m - 1 - mmShift, mul, j);
  return mulShift(4 * m, mul, j);
}

#elif defined(HAS_64_BIT_INTRINSICS)

static inline uint64_t mulShift(const uint64_t m, const uint64_t* const mul, const int32_t j) {
  // m is maximum 55 bits
  uint64_t high1;                                   // 128
  const uint64_t low1 = umul128(m, mul[1], &high1); // 64
  uint64_t high0;                                   // 64
  umul128(m, mul[0], &high0);                       // 0
  const uint64_t sum = high0 + low1;
  if (sum < high0) {
    ++high1; // overflow into high1
  }
  return shiftright128(sum, high1, j - 64);
}

static inline uint64_t mulShiftAll(const uint64_t m, const uint64_t* const mul, const int32_t j,
  uint64_t* const vp, uint64_t* const vm, const uint32_t mmShift) {
  *vp = mulShift(4 * m + 2, mul, j);
  *vm = mulShift(4 * m - 1 - mmShift, mul, j);
  return mulShift(4 * m, mul, j);
}

#else // !defined(HAS_UINT128) && !defined(HAS_64_BIT_INTRINSICS)

static inline uint64_t mulShiftAll(uint64_t m, const uint64_t* const mul, const int32_t j,
  uint64_t* const vp, uint64_t* const vm, const uint32_t mmShift) {
  m <<= 1;
  // m is maximum 55 bits
  uint64_t tmp;
  const uint64_t lo = umul128(m, mul[0], &tmp);
  uint64_t hi;
  const uint64_t mid = tmp + umul128(m, mul[1], &hi);
  hi += mid < tmp; // overflow into hi

  const uint64_t lo2 = lo + mul[0];
  const uint64_t mid2 = mid + mul[1] + (lo2 < lo);
  const uint64_t hi2 = hi + (mid2 < mid);
  *vp = shiftright128(mid2, hi2, (uint32_t) (j - 64 - 1));

  if (mmShift == 1) {
    const uint64_t lo3 = lo - mul[0];
    const uint64_t mid3 = mid - mul[1] - (lo3 > lo);
    const uint64_t hi3 = hi - (mid3 > mid);
    *vm = shiftright128(mid3, hi3, (uint32_t) (j - 64 - 1));
  } else {
    const uint64_t lo3 = lo + lo;
    const uint64_t mid3 = mid + mid + (lo3 < lo);
    const uint64_t hi3 = hi + hi + (mid3 < mid);
    const uint64_t lo4 = lo3 - mul[0];
    const uint64_t mid4 = mid3 - mul[1] - (lo4 > lo3);
    const uint64_t hi4 = hi3 - (mid4 > mid3);
    *vm = shiftright128(mid4, hi4, (uint32_t) (j - 64));
  }

  return shiftright128(mid, hi, (uint32_t) (j - 64 - 1));
}

#endif // HAS_64_BIT_INTRINSICS

static inline uint32_t decimalLength(const uint64_t v) {
  // This is slightly faster than a loop.
  // The average output length is 16.38 digits, so we check high-to-low.
  // Function precondition: v is not an 18, 19, or 20-digit number.
  // (17 digits are sufficient for round-tripping.)
  assert(v < 100000000000000000L);
  if (v >= 10000000000000000L) { return 17; }
  if (v >= 1000000000000000L) { return 16; }
  if (v >= 100000000000000L) { return 15; }
  if (v >= 10000000000000L) { return 14; }
  if (v >= 1000000000000L) { return 13; }
  if (v >= 100000000000L) { return 12; }
  if (v >= 10000000000L) { return 11; }
  if (v >= 1000000000L) { return 10; }
  if (v >= 100000000L) { return 9; }
  if (v >= 10000000L) { return 8; }
  if (v >= 1000000L) { return 7; }
  if (v >= 100000L) { return 6; }
  if (v >= 10000L) { return 5; }
  if (v >= 1000L) { return 4; }
  if (v >= 100L) { return 3; }
  if (v >= 10L) { return 2; }
  return 1;
}

// A floating decimal representing m * 10^e.
typedef struct floating_decimal_64 {
  uint64_t mantissa;
  int32_t exponent;
} floating_decimal_64;

static inline floating_decimal_64 d2d(const uint64_t ieeeMantissa, const uint32_t ieeeExponent) {
  int32_t e2;
  uint64_t m2;
  if (ieeeExponent == 0) {
    // We subtract 2 so that the bounds computation has 2 additional bits.
    e2 = 1 - DOUBLE_BIAS - DOUBLE_MANTISSA_BITS - 2;
    m2 = ieeeMantissa;
  } else {
    e2 = (int32_t) ieeeExponent - DOUBLE_BIAS - DOUBLE_MANTISSA_BITS - 2;
    m2 = (1ull << DOUBLE_MANTISSA_BITS) | ieeeMantissa;
  }
  const bool even = (m2 & 1) == 0;
  const bool acceptBounds = even;

#ifdef RYU_DEBUG
  printf("-> %" PRIu64 " * 2^%d\n", m2, e2 + 2);
#endif

  // Step 2: Determine the interval of valid decimal representations.
  const uint64_t mv = 4 * m2;
  // Implicit bool -> int conversion. True is 1, false is 0.
  const uint32_t mmShift = ieeeMantissa != 0 || ieeeExponent <= 1;
  // We would compute mp and mm like this:
  // uint64_t mp = 4 * m2 + 2;
  // uint64_t mm = mv - 1 - mmShift;

  // Step 3: Convert to a decimal power base using 128-bit arithmetic.
  uint64_t vr, vp, vm;
  int32_t e10;
  bool vmIsTrailingZeros = false;
  bool vrIsTrailingZeros = false;
  if (e2 >= 0) {
    // I tried special-casing q == 0, but there was no effect on performance.
    // This expression is slightly faster than max(0, log10Pow2(e2) - 1).
    const uint32_t q = log10Pow2(e2) - (e2 > 3);
    e10 = (int32_t) q;
    const int32_t k = DOUBLE_POW5_INV_BITCOUNT + pow5bits((int32_t) q) - 1;
    const int32_t i = -e2 + (int32_t) q + k;
#if defined(RYU_OPTIMIZE_SIZE)
    uint64_t pow5[2];
    double_computeInvPow5(q, pow5);
    vr = mulShiftAll(m2, pow5, i, &vp, &vm, mmShift);
#else
    vr = mulShiftAll(m2, DOUBLE_POW5_INV_SPLIT[q], i, &vp, &vm, mmShift);
#endif
#ifdef RYU_DEBUG
    printf("%" PRIu64 " * 2^%d / 10^%u\n", mv, e2, q);
    printf("V+=%" PRIu64 "\nV =%" PRIu64 "\nV-=%" PRIu64 "\n", vp, vr, vm);
#endif
    if (q <= 21) {
      // This should use q <= 22, but I think 21 is also safe. Smaller values
      // may still be safe, but it's more difficult to reason about them.
      // Only one of mp, mv, and mm can be a multiple of 5, if any.
      const uint32_t mvMod5 = ((uint32_t) mv) - 5 * ((uint32_t) div5(mv));
      if (mvMod5 == 0) {
        vrIsTrailingZeros = multipleOfPowerOf5(mv, q);
      } else if (acceptBounds) {
        // Same as min(e2 + (~mm & 1), pow5Factor(mm)) >= q
        // <=> e2 + (~mm & 1) >= q && pow5Factor(mm) >= q
        // <=> true && pow5Factor(mm) >= q, since e2 >= q.
        vmIsTrailingZeros = multipleOfPowerOf5(mv - 1 - mmShift, q);
      } else {
        // Same as min(e2 + 1, pow5Factor(mp)) >= q.
        vp -= multipleOfPowerOf5(mv + 2, q);
      }
    }
  } else {
    // This expression is slightly faster than max(0, log10Pow5(-e2) - 1).
    const uint32_t q = log10Pow5(-e2) - (-e2 > 1);
    e10 = (int32_t) q + e2;
    const int32_t i = -e2 - (int32_t) q;
    const int32_t k = pow5bits(i) - DOUBLE_POW5_BITCOUNT;
    const int32_t j = (int32_t) q - k;
#if defined(RYU_OPTIMIZE_SIZE)
    uint64_t pow5[2];
    double_computePow5(i, pow5);
    vr = mulShiftAll(m2, pow5, j, &vp, &vm, mmShift);
#else
    vr = mulShiftAll(m2, DOUBLE_POW5_SPLIT[i], j, &vp, &vm, mmShift);
#endif
#ifdef RYU_DEBUG
    printf("%" PRIu64 " * 5^%d / 10^%u\n", mv, -e2, q);
    printf("%u %d %d %d\n", q, i, k, j);
    printf("V+=%" PRIu64 "\nV =%" PRIu64 "\nV-=%" PRIu64 "\n", vp, vr, vm);
#endif
    if (q <= 1) {
      // {vr,vp,vm} is trailing zeros if {mv,mp,mm} has at least q trailing 0 bits.
      // mv = 4 * m2, so it always has at least two trailing 0 bits.
      vrIsTrailingZeros = true;
      if (acceptBounds) {
        // mm = mv - 1 - mmShift, so it has 1 trailing 0 bit iff mmShift == 1.
        vmIsTrailingZeros = mmShift == 1;
      } else {
        // mp = mv + 2, so it always has at least one trailing 0 bit.
        --vp;
      }
    } else if (q < 63) { // TODO(ulfjack): Use a tighter bound here.
      // We need to compute min(ntz(mv), pow5Factor(mv) - e2) >= q - 1
      // <=> ntz(mv) >= q - 1 && pow5Factor(mv) - e2 >= q - 1
      // <=> ntz(mv) >= q - 1 (e2 is negative and -e2 >= q)
      // <=> (mv & ((1 << (q - 1)) - 1)) == 0
      // We also need to make sure that the left shift does not overflow.
      vrIsTrailingZeros = multipleOfPowerOf2(mv, q - 1);
#ifdef RYU_DEBUG
      printf("vr is trailing zeros=%s\n", vrIsTrailingZeros ? "true" : "false");
#endif
    }
  }
#ifdef RYU_DEBUG
  printf("e10=%d\n", e10);
  printf("V+=%" PRIu64 "\nV =%" PRIu64 "\nV-=%" PRIu64 "\n", vp, vr, vm);
  printf("vm is trailing zeros=%s\n", vmIsTrailingZeros ? "true" : "false");
  printf("vr is trailing zeros=%s\n", vrIsTrailingZeros ? "true" : "false");
#endif

  // Step 4: Find the shortest decimal representation in the interval of valid representations.
  int32_t removed = 0;
  uint8_t lastRemovedDigit = 0;
  uint64_t output;
  // On average, we remove ~2 digits.
  if (vmIsTrailingZeros || vrIsTrailingZeros) {
    // General case, which happens rarely (~0.7%).
    for (;;) {
      const uint64_t vpDiv10 = div10(vp);
      const uint64_t vmDiv10 = div10(vm);
      if (vpDiv10 <= vmDiv10) {
        break;
      }
      const uint32_t vmMod10 = ((uint32_t) vm) - 10 * ((uint32_t) vmDiv10);
      const uint64_t vrDiv10 = div10(vr);
      const uint32_t vrMod10 = ((uint32_t) vr) - 10 * ((uint32_t) vrDiv10);
      vmIsTrailingZeros &= vmMod10 == 0;
      vrIsTrailingZeros &= lastRemovedDigit == 0;
      lastRemovedDigit = (uint8_t) vrMod10;
      vr = vrDiv10;
      vp = vpDiv10;
      vm = vmDiv10;
      ++removed;
    }
#ifdef RYU_DEBUG
    printf("V+=%" PRIu64 "\nV =%" PRIu64 "\nV-=%" PRIu64 "\n", vp, vr, vm);
    printf("d-10=%s\n", vmIsTrailingZeros ? "true" : "false");
#endif
    if (vmIsTrailingZeros) {
      for (;;) {
        const uint64_t vmDiv10 = div10(vm);
        const uint32_t vmMod10 = ((uint32_t) vm) - 10 * ((uint32_t) vmDiv10);
        if (vmMod10 != 0) {
          break;
        }
        const uint64_t vpDiv10 = div10(vp);
        const uint64_t vrDiv10 = div10(vr);
        const uint32_t vrMod10 = ((uint32_t) vr) - 10 * ((uint32_t) vrDiv10);
        vrIsTrailingZeros &= lastRemovedDigit == 0;
        lastRemovedDigit = (uint8_t) vrMod10;
        vr = vrDiv10;
        vp = vpDiv10;
        vm = vmDiv10;
        ++removed;
      }
    }
#ifdef RYU_DEBUG
    printf("%" PRIu64 " %d\n", vr, lastRemovedDigit);
    printf("vr is trailing zeros=%s\n", vrIsTrailingZeros ? "true" : "false");
#endif
    if (vrIsTrailingZeros && lastRemovedDigit == 5 && vr % 2 == 0) {
      // Round even if the exact number is .....50..0.
      lastRemovedDigit = 4;
    }
    // We need to take vr + 1 if vr is outside bounds or we need to round up.
    output = vr + ((vr == vm && (!acceptBounds || !vmIsTrailingZeros)) || lastRemovedDigit >= 5);
  } else {
    // Specialized for the common case (~99.3%). Percentages below are relative to this.
    bool roundUp = false;
    const uint64_t vpDiv100 = div100(vp);
    const uint64_t vmDiv100 = div100(vm);
    if (vpDiv100 > vmDiv100) { // Optimization: remove two digits at a time (~86.2%).
      const uint64_t vrDiv100 = div100(vr);
      const uint32_t vrMod100 = ((uint32_t) vr) - 100 * ((uint32_t) vrDiv100);
      roundUp = vrMod100 >= 50;
      vr = vrDiv100;
      vp = vpDiv100;
      vm = vmDiv100;
      removed += 2;
    }
    // Loop iterations below (approximately), without optimization above:
    // 0: 0.03%, 1: 13.8%, 2: 70.6%, 3: 14.0%, 4: 1.40%, 5: 0.14%, 6+: 0.02%
    // Loop iterations below (approximately), with optimization above:
    // 0: 70.6%, 1: 27.8%, 2: 1.40%, 3: 0.14%, 4+: 0.02%
    for (;;) {
      const uint64_t vpDiv10 = div10(vp);
      const uint64_t vmDiv10 = div10(vm);
      if (vpDiv10 <= vmDiv10) {
        break;
      }
      const uint64_t vrDiv10 = div10(vr);
      const uint32_t vrMod10 = ((uint32_t) vr) - 10 * ((uint32_t) vrDiv10);
      roundUp = vrMod10 >= 5;
      vr = vrDiv10;
      vp = vpDiv10;
      vm = vmDiv10;
      ++removed;
    }
#ifdef RYU_DEBUG
    printf("%" PRIu64 " roundUp=%s\n", vr, roundUp ? "true" : "false");
    printf("vr is trailing zeros=%s\n", vrIsTrailingZeros ? "true" : "false");
#endif
    // We need to take vr + 1 if vr is outside bounds or we need to round up.
    output = vr + (vr == vm || roundUp);
  }
  const int32_t exp = e10 + removed;

#ifdef RYU_DEBUG
  printf("V+=%" PRIu64 "\nV =%" PRIu64 "\nV-=%" PRIu64 "\n", vp, vr, vm);
  printf("O=%" PRIu64 "\n", output);
  printf("EXP=%d\n", exp);
#endif

  floating_decimal_64 fd;
  fd.exponent = exp;
  fd.mantissa = output;
  return fd;
}

static inline int to_chars(const floating_decimal_64 v, const bool sign, char* const result) {
  // Step 5: Print the decimal representation.
  int index = 0;
  if (sign) {
    result[index++] = '-';
  }

  uint64_t output = v.mantissa;
  const uint32_t olength = decimalLength(output);

#ifdef RYU_DEBUG
  printf("DIGITS=%" PRIu64 "\n", v.mantissa);
  printf("OLEN=%u\n", olength);
  printf("EXP=%u\n", v.exponent + olength);
#endif

  // Print the decimal digits.
  // The following code is equivalent to:
  // for (uint32_t i = 0; i < olength - 1; ++i) {
  //   const uint32_t c = output % 10; output /= 10;
  //   result[index + olength - i] = (char) ('0' + c);
  // }
  // result[index] = '0' + output % 10;

  uint32_t i = 0;
  // We prefer 32-bit operations, even on 64-bit platforms.
  // We have at most 17 digits, and uint32_t can store 9 digits.
  // If output doesn't fit into uint32_t, we cut off 8 digits,
  // so the rest will fit into uint32_t.
  if ((output >> 32) != 0) {
    // Expensive 64-bit division.
    const uint64_t q = div1e8(output);
    uint32_t output2 = ((uint32_t) output) - 100000000 * ((uint32_t) q);
    output = q;

    const uint32_t c = output2 % 10000;
    output2 /= 10000;
    const uint32_t d = output2 % 10000;
    const uint32_t c0 = (c % 100) << 1;
    const uint32_t c1 = (c / 100) << 1;
    const uint32_t d0 = (d % 100) << 1;
    const uint32_t d1 = (d / 100) << 1;
    memcpy(result + index + olength - i - 1, DIGIT_TABLE + c0, 2);
    memcpy(result + index + olength - i - 3, DIGIT_TABLE + c1, 2);
    memcpy(result + index + olength - i - 5, DIGIT_TABLE + d0, 2);
    memcpy(result + index + olength - i - 7, DIGIT_TABLE + d1, 2);
    i += 8;
  }
  uint32_t output2 = (uint32_t) output;
  while (output2 >= 10000) {
#ifdef __clang__ // https://bugs.llvm.org/show_bug.cgi?id=38217
    const uint32_t c = output2 - 10000 * (output2 / 10000);
#else
    const uint32_t c = output2 % 10000;
#endif
    output2 /= 10000;
    const uint32_t c0 = (c % 100) << 1;
    const uint32_t c1 = (c / 100) << 1;
    memcpy(result + index + olength - i - 1, DIGIT_TABLE + c0, 2);
    memcpy(result + index + olength - i - 3, DIGIT_TABLE + c1, 2);
    i += 4;
  }
  if (output2 >= 100) {
    const uint32_t c = (output2 % 100) << 1;
    output2 /= 100;
    memcpy(result + index + olength - i - 1, DIGIT_TABLE + c, 2);
    i += 2;
  }
  if (output2 >= 10) {
    const uint32_t c = output2 << 1;
    // We can't use memcpy here: the decimal dot goes between these two digits.
    result[index + olength - i] = DIGIT_TABLE[c + 1];
    result[index] = DIGIT_TABLE[c];
  } else {
    result[index] = (char) ('0' + output2);
  }

  // Print decimal point if needed.
  if (olength > 1) {
    result[index + 1] = '.';
    index += olength + 1;
  } else {
    ++index;
  }

  // Print the exponent.
  result[index++] = 'E';
  int32_t exp = v.exponent + (int32_t) olength - 1;
  if (exp < 0) {
    result[index++] = '-';
    exp = -exp;
  }

  if (exp >= 100) {
    const int32_t c = exp % 10;
    memcpy(result + index, DIGIT_TABLE + 2 * (exp / 10), 2);
    result[index + 2] = (char) ('0' + c);
    index += 3;
  } else if (exp >= 10) {
    memcpy(result + index, DIGIT_TABLE + 2 * exp, 2);
    index += 2;
  } else {
    result[index++] = (char) ('0' + exp);
  }

  return index;
}

static inline bool d2d_small_int(const uint64_t ieeeMantissa, const uint32_t ieeeExponent, floating_decimal_64* v) {
  const uint64_t m2 = (1ull << DOUBLE_MANTISSA_BITS) | ieeeMantissa;
  const int32_t e2 = (int32_t)ieeeExponent - DOUBLE_BIAS - DOUBLE_MANTISSA_BITS;

  if (e2 > 0) {
    // f = m2 * 2^e2 >= 2^53 is an integer.
    // Ignore this case for now.
    return false;
  }
  if (e2 < -52) {
    // f < 1.
    return false;
  }

  // Since 2^52 <= m2 < 2^53 and 0 <= -e2 <= 52: 1 <= f = m2 / 2^-e2 < 2^53.
  // Test if the lower -e2 bits of the significand are 0, i.e. whether the fraction is 0.
  const uint64_t mask = (1ull << -e2) - 1;
  const uint64_t fraction = m2 & mask;
  if (fraction != 0) {
    return false;
  }

  // f is an integer in the range [1, 2^53).
  // Note: mantissa might contain trailing (decimal) 0's.
  // Note: since 2^53 < 10^16, there is no need to adjust decimalLength().
  v->mantissa = m2 >> -e2;
  v->exponent = 0;
  return true;
}

int d2s_buffered_n(double f, char* result) {
  // Step 1: Decode the floating-point number, and unify normalized and subnormal cases.
  const uint64_t bits = double_to_bits(f);

#ifdef RYU_DEBUG
  printf("IN=");
  for (int32_t bit = 63; bit >= 0; --bit) {
    printf("%d", (int) ((bits >> bit) & 1));
  }
  printf("\n");
#endif

  // Decode bits into sign, mantissa, and exponent.
  const bool ieeeSign = ((bits >> (DOUBLE_MANTISSA_BITS + DOUBLE_EXPONENT_BITS)) & 1) != 0;
  const uint64_t ieeeMantissa = bits & ((1ull << DOUBLE_MANTISSA_BITS) - 1);
  const uint32_t ieeeExponent = (uint32_t) ((bits >> DOUBLE_MANTISSA_BITS) & ((1u << DOUBLE_EXPONENT_BITS) - 1));
  // Case distinction; exit early for the easy cases.
  if (ieeeExponent == ((1u << DOUBLE_EXPONENT_BITS) - 1u) || (ieeeExponent == 0 && ieeeMantissa == 0)) {
    return copy_special_str(result, ieeeSign, ieeeExponent, ieeeMantissa);
  }

  floating_decimal_64 v;
  const bool isSmallInt = d2d_small_int(ieeeMantissa, ieeeExponent, &v);
  if (isSmallInt) {
    // For small integers in the range [1,2^53), v.mantissa might contain trailing (decimal) zeros.
    // For scientic notation we need to move these zeros into the exponent.
    // (This is not needed for fixed-point notation, so it might be beneficial to trim
    // trailing zeros in to_chars only if needed - once fixed-point notation output is implemented.)
    for (;;) {
      const uint64_t q = div10(v.mantissa);
      const uint32_t r = ((uint32_t) v.mantissa) - 10 * ((uint32_t) q);
      if (r != 0) {
        break;
      }
      v.mantissa = q;
      ++v.exponent;
    }
  } else {
    v = d2d(ieeeMantissa, ieeeExponent);
  }

  return to_chars(v, ieeeSign, result);
}

void d2s_buffered(double f, char* result) {
  const int index = d2s_buffered_n(f, result);

  // Terminate the string.
  result[index] = '\0';
}

char* d2s(double f) {
  char* const result = (char*) malloc(25);
  d2s_buffered(f, result);
  return result;
}