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

Average Error: 0.28% → 0.27%

Time: 5.7s

Precision: binary64

Cost: 6720

Math

\[\left(\left(x - \frac{16}{116}\right) \cdot 3\right) \cdot y \]

↓

\[\mathsf{fma}\left(x, 3, -0.41379310344827586\right) \cdot y \]

FPCore

(FPCore (x y) :precision binary64 (* (* (- x (/ 16.0 116.0)) 3.0) y))

↓

(FPCore (x y) :precision binary64 (* (fma x 3.0 -0.41379310344827586) y))

C

double code(double x, double y) {
	return ((x - (16.0 / 116.0)) * 3.0) * y;
}

↓

double code(double x, double y) {
	return fma(x, 3.0, -0.41379310344827586) * y;
}

Julia

function code(x, y)
	return Float64(Float64(Float64(x - Float64(16.0 / 116.0)) * 3.0) * y)
end

↓

function code(x, y)
	return Float64(fma(x, 3.0, -0.41379310344827586) * y)
end

Wolfram

code[x_, y_] := N[(N[(N[(x - N[(16.0 / 116.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision] * y), $MachinePrecision]

↓

code[x_, y_] := N[(N[(x * 3.0 + -0.41379310344827586), $MachinePrecision] * y), $MachinePrecision]

Target

Original	0.28%
Target	0.27%
Herbie	0.27%

\[y \cdot \left(x \cdot 3 - 0.41379310344827586\right) \]

Derivation

1. Initial program 0.28

   \[\left(\left(x - \frac{16}{116}\right) \cdot 3\right) \cdot y \]

2. Simplified 0.27

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, 3, -0.41379310344827586\right) \cdot y} \]
   Proof

   [Start] 0.28	\[ \left(\left(x - \frac{16}{116}\right) \cdot 3\right) \cdot y \]
   *-commutative [=>] 0.28	\[ \color{blue}{\left(3 \cdot \left(x - \frac{16}{116}\right)\right)} \cdot y \]
   sub-neg [=>] 0.28	\[ \left(3 \cdot \color{blue}{\left(x + \left(-\frac{16}{116}\right)\right)}\right) \cdot y \]
   distribute-lft-in [=>] 0.27	\[ \color{blue}{\left(3 \cdot x + 3 \cdot \left(-\frac{16}{116}\right)\right)} \cdot y \]
   *-commutative [=>] 0.27	\[ \left(\color{blue}{x \cdot 3} + 3 \cdot \left(-\frac{16}{116}\right)\right) \cdot y \]
   fma-def [=>] 0.27	\[ \color{blue}{\mathsf{fma}\left(x, 3, 3 \cdot \left(-\frac{16}{116}\right)\right)} \cdot y \]
   metadata-eval [=>] 0.27	\[ \mathsf{fma}\left(x, 3, 3 \cdot \left(-\color{blue}{0.13793103448275862}\right)\right) \cdot y \]
   metadata-eval [=>] 0.27	\[ \mathsf{fma}\left(x, 3, 3 \cdot \color{blue}{-0.13793103448275862}\right) \cdot y \]
   metadata-eval [=>] 0.27	\[ \mathsf{fma}\left(x, 3, \color{blue}{-0.41379310344827586}\right) \cdot y \]

3. Final simplification 0.27

   \[\leadsto \mathsf{fma}\left(x, 3, -0.41379310344827586\right) \cdot y \]

Alternatives

Alternative 1
Error	2.85%
Cost	585

\[\begin{array}{l} \mathbf{if}\;x \leq -0.14 \lor \neg \left(x \leq 0.14\right):\\ \;\;\;\;3 \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;-0.41379310344827586 \cdot y\\ \end{array} \]

Alternative 2
Error	2.85%
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -0.14:\\ \;\;\;\;3 \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;x \leq 0.14:\\ \;\;\;\;-0.41379310344827586 \cdot y\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot 3\right)\\ \end{array} \]

Alternative 3
Error	0.28%
Cost	576

\[y \cdot \left(x \cdot 3\right) + -0.41379310344827586 \cdot y \]

Alternative 4
Error	0.48%
Cost	448

\[3 \cdot \left(y \cdot \left(x + -0.13793103448275862\right)\right) \]

Alternative 5
Error	0.47%
Cost	448

\[\left(3 \cdot y\right) \cdot \left(x + -0.13793103448275862\right) \]

Alternative 6
Error	0.27%
Cost	448

\[y \cdot \left(-0.41379310344827586 + x \cdot 3\right) \]

Alternative 7
Error	42.79%
Cost	192

\[-0.41379310344827586 \cdot y \]

Reproduce

herbie shell --seed 2023102 
(FPCore (x y)
  :name "Data.Colour.CIE:cieLAB from colour-2.3.3, A"
  :precision binary64

  :herbie-target
  (* y (- (* x 3.0) 0.41379310344827586))

  (* (* (- x (/ 16.0 116.0)) 3.0) y))