pack-priv Class FastDoubleMath

Largest power of 10 value of the exponent.

Any number of form w * 10^309 where w >= 1 is going to be infinite in a double, so we never need to worry about powers of 10 greater than 308.

Smallest power of 10 value of the exponent.

The smallest non-zero double is 2^−1074.

We take as input numbers of the form w x 10^q where w < 2^64.

We have that w * 10^-343 < 2^(63-343) * 5^-343 < 2^-1076.

However, we have that (2^64-1) * 10^-342 = (2^64 - 1) * 2^-342 * 5^-342 > 2^−1074. Thus, it is possible for a number of the form w * 10^-342 where w is a 64-bit value to be a non-zero double.

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If we are solely interested in the *normal* numbers then the smallest value is 2^-1022. We can generate a value larger than 2^-1022 with expressions of the form w * 10^-326. Thus, we need to pick SMALLEST_POWER_OF_TEN >= -326.

A complement to mantissa_64 complete to a 128-bit mantissa.

Uses about 5KB but is rarely accessed.

UInt128 getMantissa128(int q) {
    return new UInt128(
       MANTISSA_64[q - SMALLEST_POWER_OF_TEN],
       MANTISSA_128[q - SMALLEST_POWER_OF_TEN];
    );
}

When mapping numbers from decimal to binary, we go from w * 10^q to m * 2^p, but we have 10^q = 5^q * 2^q, so effectively we are trying to match w * 2^q * 5^q to m * 2^p.

Thus, the powers of two are not a concern since they can be represented exactly using the binary notation, only the powers of five affect the binary significand.

The mantissas of powers of ten from -308 to 308, extended out to sixty-four bits. The array contains the powers of ten approximated as a 64-bit mantissa. It goes from 10^-325 to 10^308 (inclusively). The mantissa is truncated, and never rounded up. Uses about 5 KB.

long getMantissaHigh(int q) {
 MANTISSA_64[q - SMALLEST_POWER_OF_TEN];
}