decimal_to_binary.h

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#ifndef FASTFLOAT_DECIMAL_TO_BINARY_H
#define FASTFLOAT_DECIMAL_TO_BINARY_H
 
#include "float_common.h"
#include "fast_table.h"
#include <cfloat>
#include <cinttypes>
#include <cmath>
#include <cstdint>
#include <cstdlib>
#include <cstring>
 
namespace fast_float {
 
// This will compute or rather approximate w * 5**q and return a pair of 64-bit words approximating
// the result, with the "high" part corresponding to the most significant bits and the
// low part corresponding to the least significant bits.
//
template <int bit_precision>
fastfloat_really_inline
value128 compute_product_approximation(int64_t q, uint64_t w) {
  const int index = 2 * int(q - powers::smallest_power_of_five);
  // For small values of q, e.g., q in [0,27], the answer is always exact because
  // The line value128 firstproduct = full_multiplication(w, power_of_five_128[index]);
  // gives the exact answer.
  value128 firstproduct = full_multiplication(w, powers::power_of_five_128[index]);
  static_assert((bit_precision >= 0) && (bit_precision <= 64), " precision should  be in (0,64]");
  constexpr uint64_t precision_mask = (bit_precision < 64) ?
               (uint64_t(0xFFFFFFFFFFFFFFFF) >> bit_precision)
               : uint64_t(0xFFFFFFFFFFFFFFFF);
  if((firstproduct.high & precision_mask) == precision_mask) { // could further guard with  (lower + w < lower)
    // regarding the second product, we only need secondproduct.high, but our expectation is that the compiler will optimize this extra work away if needed.
    value128 secondproduct = full_multiplication(w, powers::power_of_five_128[index + 1]);
    firstproduct.low += secondproduct.high;
    if(secondproduct.high > firstproduct.low) {
      firstproduct.high++;
    }
  }
  return firstproduct;
}
 
namespace detail {
  constexpr fastfloat_really_inline int32_t power(int32_t q)  noexcept  {
    return (((152170 + 65536) * q) >> 16) + 63;
  }
} // namespace detail
 
// create an adjusted mantissa, biased by the invalid power2
// for significant digits already multiplied by 10 ** q.
template <typename binary>
fastfloat_really_inline
adjusted_mantissa compute_error_scaled(int64_t q, uint64_t w, int lz) noexcept  {
  int hilz = int(w >> 63) ^ 1;
  adjusted_mantissa answer;
  answer.mantissa = w << hilz;
  int bias = binary::mantissa_explicit_bits() - binary::minimum_exponent();
  answer.power2 = int32_t(detail::power(int32_t(q)) + bias - hilz - lz - 62 + invalid_am_bias);
  return answer;
}
 
// w * 10 ** q, without rounding the representation up.
// the power2 in the exponent will be adjusted by invalid_am_bias.
template <typename binary>
fastfloat_really_inline
adjusted_mantissa compute_error(int64_t q, uint64_t w)  noexcept  {
  int lz = leading_zeroes(w);
  w <<= lz;
  value128 product = compute_product_approximation<binary::mantissa_explicit_bits() + 3>(q, w);
  return compute_error_scaled<binary>(q, product.high, lz);
}
 
// w * 10 ** q
// The returned value should be a valid ieee64 number that simply need to be packed.
// However, in some very rare cases, the computation will fail. In such cases, we
// return an adjusted_mantissa with a negative power of 2: the caller should recompute
// in such cases.
template <typename binary>
fastfloat_really_inline
adjusted_mantissa compute_float(int64_t q, uint64_t w)  noexcept  {
  adjusted_mantissa answer;
  if ((w == 0) || (q < binary::smallest_power_of_ten())) {
    answer.power2 = 0;
    answer.mantissa = 0;
    // result should be zero
    return answer;
  }
  if (q > binary::largest_power_of_ten()) {
    // we want to get infinity:
    answer.power2 = binary::infinite_power();
    answer.mantissa = 0;
    return answer;
  }
  // At this point in time q is in [powers::smallest_power_of_five, powers::largest_power_of_five].
 
  // We want the most significant bit of i to be 1. Shift if needed.
  int lz = leading_zeroes(w);
  w <<= lz;
 
  // The required precision is binary::mantissa_explicit_bits() + 3 because
  // 1. We need the implicit bit
  // 2. We need an extra bit for rounding purposes
  // 3. We might lose a bit due to the "upperbit" routine (result too small, requiring a shift)
 
  value128 product = compute_product_approximation<binary::mantissa_explicit_bits() + 3>(q, w);
  if(product.low == 0xFFFFFFFFFFFFFFFF) { //  could guard it further
    // In some very rare cases, this could happen, in which case we might need a more accurate
    // computation that what we can provide cheaply. This is very, very unlikely.
    //
    const bool inside_safe_exponent = (q >= -27) && (q <= 55); // always good because 5**q <2**128 when q>=0, 
    // and otherwise, for q<0, we have 5**-q<2**64 and the 128-bit reciprocal allows for exact computation.
    if(!inside_safe_exponent) {
      return compute_error_scaled<binary>(q, product.high, lz);
    }
  }
  // The "compute_product_approximation" function can be slightly slower than a branchless approach:
  // value128 product = compute_product(q, w);
  // but in practice, we can win big with the compute_product_approximation if its additional branch
  // is easily predicted. Which is best is data specific.
  int upperbit = int(product.high >> 63);
 
  answer.mantissa = product.high >> (upperbit + 64 - binary::mantissa_explicit_bits() - 3);
 
  answer.power2 = int32_t(detail::power(int32_t(q)) + upperbit - lz - binary::minimum_exponent());
  if (answer.power2 <= 0) { // we have a subnormal?
    // Here have that answer.power2 <= 0 so -answer.power2 >= 0
    if(-answer.power2 + 1 >= 64) { // if we have more than 64 bits below the minimum exponent, you have a zero for sure.
      answer.power2 = 0;
      answer.mantissa = 0;
      // result should be zero
      return answer;
    }
    // next line is safe because -answer.power2 + 1 < 64
    answer.mantissa >>= -answer.power2 + 1;
    // Thankfully, we can't have both "round-to-even" and subnormals because
    // "round-to-even" only occurs for powers close to 0.
    answer.mantissa += (answer.mantissa & 1); // round up
    answer.mantissa >>= 1;
    // There is a weird scenario where we don't have a subnormal but just.
    // Suppose we start with 2.2250738585072013e-308, we end up
    // with 0x3fffffffffffff x 2^-1023-53 which is technically subnormal
    // whereas 0x40000000000000 x 2^-1023-53  is normal. Now, we need to round
    // up 0x3fffffffffffff x 2^-1023-53  and once we do, we are no longer
    // subnormal, but we can only know this after rounding.
    // So we only declare a subnormal if we are smaller than the threshold.
    answer.power2 = (answer.mantissa < (uint64_t(1) << binary::mantissa_explicit_bits())) ? 0 : 1;
    return answer;
  }
 
  // usually, we round *up*, but if we fall right in between and and we have an
  // even basis, we need to round down
  // We are only concerned with the cases where 5**q fits in single 64-bit word.
  if ((product.low <= 1) &&  (q >= binary::min_exponent_round_to_even()) && (q <= binary::max_exponent_round_to_even()) &&
      ((answer.mantissa & 3) == 1) ) { // we may fall between two floats!
    // To be in-between two floats we need that in doing
    //   answer.mantissa = product.high >> (upperbit + 64 - binary::mantissa_explicit_bits() - 3);
    // ... we dropped out only zeroes. But if this happened, then we can go back!!!
    if((answer.mantissa  << (upperbit + 64 - binary::mantissa_explicit_bits() - 3)) ==  product.high) {
      answer.mantissa &= ~uint64_t(1);          // flip it so that we do not round up
    }
  }
 
  answer.mantissa += (answer.mantissa & 1); // round up
  answer.mantissa >>= 1;
  if (answer.mantissa >= (uint64_t(2) << binary::mantissa_explicit_bits())) {
    answer.mantissa = (uint64_t(1) << binary::mantissa_explicit_bits());
    answer.power2++; // undo previous addition
  }
 
  answer.mantissa &= ~(uint64_t(1) << binary::mantissa_explicit_bits());
  if (answer.power2 >= binary::infinite_power()) { // infinity
    answer.power2 = binary::infinite_power();
    answer.mantissa = 0;
  }
  return answer;
}
 
} // namespace fast_float
 
#endif