Data.Colour.CIE:cieLABView from colour-2.3.3, A

Average Error: 0.1 → 0.1

Time: 4.6s

Precision: binary64

Cost: 6592

Math

\[\frac{841}{108} \cdot x + \frac{4}{29} \]

↓

\[\mathsf{fma}\left(7.787037037037037, x, 0.13793103448275862\right) \]

FPCore

(FPCore (x) :precision binary64 (+ (* (/ 841.0 108.0) x) (/ 4.0 29.0)))

↓

(FPCore (x) :precision binary64 (fma 7.787037037037037 x 0.13793103448275862))

C

double code(double x) {
	return ((841.0 / 108.0) * x) + (4.0 / 29.0);
}

↓

double code(double x) {
	return fma(7.787037037037037, x, 0.13793103448275862);
}

Julia

function code(x)
	return Float64(Float64(Float64(841.0 / 108.0) * x) + Float64(4.0 / 29.0))
end

↓

function code(x)
	return fma(7.787037037037037, x, 0.13793103448275862)
end

Wolfram

code[x_] := N[(N[(N[(841.0 / 108.0), $MachinePrecision] * x), $MachinePrecision] + N[(4.0 / 29.0), $MachinePrecision]), $MachinePrecision]

↓

code[x_] := N[(7.787037037037037 * x + 0.13793103448275862), $MachinePrecision]

Derivation

1. Initial program 0.1

   \[\frac{841}{108} \cdot x + \frac{4}{29} \]

2. Simplified 0.1

   \[\leadsto \color{blue}{\mathsf{fma}\left(7.787037037037037, x, 0.13793103448275862\right)} \]
   Proof
   (fma.f64 841/108 x 4/29): 0 points increase in error, 0 points decrease in error
   (fma.f64 (Rewrite<= metadata-eval (/.f64 841 108)) x 4/29): 0 points increase in error, 0 points decrease in error
   (fma.f64 (/.f64 841 108) x (Rewrite<= metadata-eval (/.f64 4 29))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (/.f64 841 108) x) (/.f64 4 29))): 1 points increase in error, 0 points decrease in error

3. Final simplification 0.1

   \[\leadsto \mathsf{fma}\left(7.787037037037037, x, 0.13793103448275862\right) \]

Alternatives

Alternative 1
Error	1.6
Cost	456

\[\begin{array}{l} \mathbf{if}\;x \leq -0.6582595272765548:\\ \;\;\;\;7.787037037037037 \cdot x\\ \mathbf{elif}\;x \leq 4.332061406564082 \cdot 10^{-17}:\\ \;\;\;\;0.13793103448275862\\ \mathbf{else}:\\ \;\;\;\;7.787037037037037 \cdot x\\ \end{array} \]

Alternative 2
Error	0.1
Cost	320

\[0.13793103448275862 + 7.787037037037037 \cdot x \]

Alternative 3
Error	32.1
Cost	64

\[0.13793103448275862 \]

Reproduce

herbie shell --seed 2022295 
(FPCore (x)
  :name "Data.Colour.CIE:cieLABView from colour-2.3.3, A"
  :precision binary64
  (+ (* (/ 841.0 108.0) x) (/ 4.0 29.0)))