Characteristics

This package provides extended precision versions of Float64, Float32, Float16.

Representation

Double64 is a magnitude ordered, nonoverlapping pair of Float64

Double32 is a magnitude ordered, nonoverlapping pair of Float32

Double16 is a magnitude ordered, nonoverlapping pair of Float16

• (+, -, *) are error-free, (/, sqrt) are least-error
• elementary functions are quite accurate
  • often better than C "double-double" libraries

ComplexDF64 is a (real, imag) pair of Double64

ComplexDF32 is a (real, imag) pair of Double32

ComplexDF16 is a (real, imag) pair of Double16

• elementary functions are quite accurate
  • functions and their inverses round-trip well

Accuracy

For Double64 arguments within 0.0..2.0

• expect the abserr of elementary functions to be 1e-30 or better
• expect the relerr of elementary functions to be 1e-28 or better

When used with reasonably sized values, expect successive DoubleFloat ops to add no more than 10⋅𝘂² to the cumulative relative error (𝘂 is the relative rounding unit, usually 𝘂 = eps(x)/2). Relative error can accrue steadily. After 100,000 DoubleFloat ops with reasonably sized values, the relerr could approach 100,000 * 10⋅𝘂². In practice these functions are considerably more resiliant: our algorithms come frome seminal papers and extensive numeric investigation.

Absolute and Relative Error

results for f(x), x in 0..1

results for f(x), x in 1..2