Characteristics
This package provides extended precision versions of Float64
, Float32
, Float16
.
type name | significand | exponent | ◊ | base type | significand | exponent |
---|---|---|---|---|---|---|
Double64 | 106 bits | 11 bits | ◊ | Float64 | 53 bits | 11 bits |
Double32 | 48 bits | 8 bits | ◊ | Float32 | 24 bits | 8 bits |
Double16 | 22 bits | 5 bits | ◊ | Float16 | 11 bits | 5 bits |
Representation
Double64
is a magnitude ordered, nonoverlapping pair ofFloat64
Double32
is a magnitude ordered, nonoverlapping pair ofFloat32
Double16
is a magnitude ordered, nonoverlapping pair ofFloat16
- (
+
,-
,*
) 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 ofDouble64
ComplexDF32
is a (real, imag) pair ofDouble32
ComplexDF16
is a (real, imag) pair ofDouble16
- 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
function | abserr | relerr |
---|---|---|
exp | 1.0e-31 | 1.0e-31 |
log | 1.0e-31 | 1.0e-31 |
sin | 1.0e-31 | 1.0e-31 |
cos | 1.0e-31 | 1.0e-31 |
tan | 1.0e-31 | 1.0e-31 |
asin | 1.0e-31 | 1.0e-31 |
acos | 1.0e-31 | 1.0e-31 |
atan | 1.0e-31 | 1.0e-31 |
sinh | 1.0e-31 | 1.0e-29 |
cosh | 1.0e-31 | 1.0e-31 |
tanh | 1.0e-31 | 1.0e-29 |
asinh | 1.0e-31 | 1.0e-29 |
atanh | 1.0e-31 | 1.0e-30 |
results for f(x), x in 1..2
function | abserr | relerr |
---|---|---|
exp | 1.0e-30 | 1.0e-31 |
log | 1.0e-31 | 1.0e-31 |
sin | 1.0e-31 | 1.0e-31 |
cos | 1.0e-31 | 1.0e-28 |
tan | 1.0e-30 | 1.0e-30 |
atan | 1.0e-31 | 1.0e-31 |
sinh | 1.0e-30 | 1.0e-31 |
cosh | 1.0e-30 | 1.0e-31 |
tanh | 1.0e-31 | 1.0e-31 |
asinh | 1.0e-31 | 1.0e-31 |