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

Average Error: 0.2 → 0.2

Time: 4.4s

Precision: binary64

Cost: 448

Math

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

↓

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

FPCore

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

↓

(FPCore (x y) :precision binary64 (* (+ (* 3.0 x) -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 ((3.0 * x) + -0.41379310344827586) * y;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = ((x - (16.0d0 / 116.0d0)) * 3.0d0) * y
end function

↓

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = ((3.0d0 * x) + (-0.41379310344827586d0)) * y
end function

Java

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

↓

public static double code(double x, double y) {
	return ((3.0 * x) + -0.41379310344827586) * y;
}

Python

def code(x, y):
	return ((x - (16.0 / 116.0)) * 3.0) * y

↓

def code(x, y):
	return ((3.0 * x) + -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(Float64(Float64(3.0 * x) + -0.41379310344827586) * y)
end

MATLAB

function tmp = code(x, y)
	tmp = ((x - (16.0 / 116.0)) * 3.0) * y;
end

↓

function tmp = code(x, y)
	tmp = ((3.0 * x) + -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[(N[(3.0 * x), $MachinePrecision] + -0.41379310344827586), $MachinePrecision] * y), $MachinePrecision]

Target

Original	0.2
Target	0.2
Herbie	0.2

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

Derivation

1. Initial program 0.2

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

2. Simplified 0.2

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, 3, -0.41379310344827586\right) \cdot y} \]
   Proof
   (*.f64 (fma.f64 x 3 -12/29) y): 0 points increase in error, 0 points decrease in error
   (*.f64 (fma.f64 x 3 (Rewrite<= metadata-eval (*.f64 -4/29 3))) y): 0 points increase in error, 0 points decrease in error
   (*.f64 (fma.f64 x 3 (*.f64 (Rewrite<= metadata-eval (neg.f64 4/29)) 3)) y): 0 points increase in error, 0 points decrease in error
   (*.f64 (fma.f64 x 3 (*.f64 (neg.f64 (Rewrite<= metadata-eval (/.f64 16 116))) 3)) y): 0 points increase in error, 0 points decrease in error
   (*.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 x 3) (*.f64 (neg.f64 (/.f64 16 116)) 3))) y): 6 points increase in error, 9 points decrease in error
   (*.f64 (Rewrite<= distribute-rgt-in_binary64 (*.f64 3 (+.f64 x (neg.f64 (/.f64 16 116))))) y): 4 points increase in error, 0 points decrease in error
   (*.f64 (*.f64 3 (Rewrite<= sub-neg_binary64 (-.f64 x (/.f64 16 116)))) y): 0 points increase in error, 0 points decrease in error
   (*.f64 (Rewrite<= *-commutative_binary64 (*.f64 (-.f64 x (/.f64 16 116)) 3)) y): 0 points increase in error, 0 points decrease in error

3. Taylor expanded in x around 0 0.2

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

4. Final simplification 0.2

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

Alternatives

Alternative 1
Error	1.9
Cost	584

\[\begin{array}{l} t_0 := \frac{y}{\frac{0.3333333333333333}{x}}\\ \mathbf{if}\;x \leq -3.6221462031608493:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 0.02848159413739107:\\ \;\;\;\;y \cdot -0.41379310344827586\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	1.9
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -3.6221462031608493:\\ \;\;\;\;\frac{y}{\frac{0.3333333333333333}{x}}\\ \mathbf{elif}\;x \leq 0.02848159413739107:\\ \;\;\;\;y \cdot -0.41379310344827586\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(x \cdot y\right)\\ \end{array} \]

Alternative 3
Error	1.8
Cost	584

\[\begin{array}{l} t_0 := x \cdot \left(3 \cdot y\right)\\ \mathbf{if}\;x \leq -3.6221462031608493:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 0.02848159413739107:\\ \;\;\;\;y \cdot -0.41379310344827586\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	1.9
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -3.6221462031608493:\\ \;\;\;\;x \cdot \left(3 \cdot y\right)\\ \mathbf{elif}\;x \leq 0.02848159413739107:\\ \;\;\;\;y \cdot -0.41379310344827586\\ \mathbf{else}:\\ \;\;\;\;\left(3 \cdot x\right) \cdot y\\ \end{array} \]

Alternative 5
Error	0.3
Cost	448

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

Alternative 6
Error	27.4
Cost	192

\[\frac{y}{-2.4166666666666665} \]

Alternative 7
Error	27.2
Cost	192

\[y \cdot -0.41379310344827586 \]

Reproduce

herbie shell --seed 2022297 
(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))