# Characteristics

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

type namesignificandexponentbase typesignificandexponent
Double64106 bits11 bitsFloat6453 bits11 bits
Double3248 bits8 bitsFloat3224 bits8 bits
Double1622 bits5 bitsFloat1611 bits5 bits

## 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

functionabserrrelerr
exp1.0e-311.0e-31
log1.0e-311.0e-31
sin1.0e-311.0e-31
cos1.0e-311.0e-31
tan1.0e-311.0e-31
asin1.0e-311.0e-31
acos1.0e-311.0e-31
atan1.0e-311.0e-31
sinh1.0e-311.0e-29
cosh1.0e-311.0e-31
tanh1.0e-311.0e-29
asinh1.0e-311.0e-29
atanh1.0e-311.0e-30

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

functionabserrrelerr
exp1.0e-301.0e-31
log1.0e-311.0e-31
sin1.0e-311.0e-31
cos1.0e-311.0e-28
tan1.0e-301.0e-30
atan1.0e-311.0e-31
sinh1.0e-301.0e-31
cosh1.0e-301.0e-31
tanh1.0e-311.0e-31
asinh1.0e-311.0e-31