Data.Colour.RGBSpace.HSL:hsl from colour-2.3.3, D

Average Error: 0.4 → 0.2

Time: 12.9s

Precision: binary64

Cost: 960

Math

\[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]

↓

\[\left(y - x\right) \cdot \left(z \cdot -6\right) + \left(x + \left(y - x\right) \cdot 4\right) \]

FPCore

(FPCore (x y z)
 :precision binary64
 (+ x (* (* (- y x) 6.0) (- (/ 2.0 3.0) z))))

↓

(FPCore (x y z)
 :precision binary64
 (+ (* (- y x) (* z -6.0)) (+ x (* (- y x) 4.0))))

C

double code(double x, double y, double z) {
	return x + (((y - x) * 6.0) * ((2.0 / 3.0) - z));
}

↓

double code(double x, double y, double z) {
	return ((y - x) * (z * -6.0)) + (x + ((y - x) * 4.0));
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x + (((y - x) * 6.0d0) * ((2.0d0 / 3.0d0) - z))
end function

↓

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = ((y - x) * (z * (-6.0d0))) + (x + ((y - x) * 4.0d0))
end function

Java

public static double code(double x, double y, double z) {
	return x + (((y - x) * 6.0) * ((2.0 / 3.0) - z));
}

↓

public static double code(double x, double y, double z) {
	return ((y - x) * (z * -6.0)) + (x + ((y - x) * 4.0));
}

Python

def code(x, y, z):
	return x + (((y - x) * 6.0) * ((2.0 / 3.0) - z))

↓

def code(x, y, z):
	return ((y - x) * (z * -6.0)) + (x + ((y - x) * 4.0))

Julia

function code(x, y, z)
	return Float64(x + Float64(Float64(Float64(y - x) * 6.0) * Float64(Float64(2.0 / 3.0) - z)))
end

↓

function code(x, y, z)
	return Float64(Float64(Float64(y - x) * Float64(z * -6.0)) + Float64(x + Float64(Float64(y - x) * 4.0)))
end

MATLAB

function tmp = code(x, y, z)
	tmp = x + (((y - x) * 6.0) * ((2.0 / 3.0) - z));
end

↓

function tmp = code(x, y, z)
	tmp = ((y - x) * (z * -6.0)) + (x + ((y - x) * 4.0));
end

Wolfram

code[x_, y_, z_] := N[(x + N[(N[(N[(y - x), $MachinePrecision] * 6.0), $MachinePrecision] * N[(N[(2.0 / 3.0), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_] := N[(N[(N[(y - x), $MachinePrecision] * N[(z * -6.0), $MachinePrecision]), $MachinePrecision] + N[(x + N[(N[(y - x), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 0.4

   \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]

2. Simplified 0.2

   \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \mathsf{fma}\left(z, -6, 4\right), x\right)} \]
   Proof

   [Start] 0.4	\[ x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
   +-commutative [=>] 0.4	\[ \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) + x} \]
   associate-*l* [=>] 0.2	\[ \color{blue}{\left(y - x\right) \cdot \left(6 \cdot \left(\frac{2}{3} - z\right)\right)} + x \]
   fma-def [=>] 0.2	\[ \color{blue}{\mathsf{fma}\left(y - x, 6 \cdot \left(\frac{2}{3} - z\right), x\right)} \]
   sub-neg [=>] 0.2	\[ \mathsf{fma}\left(y - x, 6 \cdot \color{blue}{\left(\frac{2}{3} + \left(-z\right)\right)}, x\right) \]
   distribute-lft-in [=>] 0.2	\[ \mathsf{fma}\left(y - x, \color{blue}{6 \cdot \frac{2}{3} + 6 \cdot \left(-z\right)}, x\right) \]
   +-commutative [=>] 0.2	\[ \mathsf{fma}\left(y - x, \color{blue}{6 \cdot \left(-z\right) + 6 \cdot \frac{2}{3}}, x\right) \]
   neg-mul-1 [=>] 0.2	\[ \mathsf{fma}\left(y - x, 6 \cdot \color{blue}{\left(-1 \cdot z\right)} + 6 \cdot \frac{2}{3}, x\right) \]
   associate-*r* [=>] 0.2	\[ \mathsf{fma}\left(y - x, \color{blue}{\left(6 \cdot -1\right) \cdot z} + 6 \cdot \frac{2}{3}, x\right) \]
   *-commutative [=>] 0.2	\[ \mathsf{fma}\left(y - x, \color{blue}{z \cdot \left(6 \cdot -1\right)} + 6 \cdot \frac{2}{3}, x\right) \]
   fma-def [=>] 0.2	\[ \mathsf{fma}\left(y - x, \color{blue}{\mathsf{fma}\left(z, 6 \cdot -1, 6 \cdot \frac{2}{3}\right)}, x\right) \]
   metadata-eval [=>] 0.2	\[ \mathsf{fma}\left(y - x, \mathsf{fma}\left(z, \color{blue}{-6}, 6 \cdot \frac{2}{3}\right), x\right) \]
   metadata-eval [=>] 0.2	\[ \mathsf{fma}\left(y - x, \mathsf{fma}\left(z, -6, 6 \cdot \color{blue}{0.6666666666666666}\right), x\right) \]
   metadata-eval [=>] 0.2	\[ \mathsf{fma}\left(y - x, \mathsf{fma}\left(z, -6, \color{blue}{4}\right), x\right) \]

3. Applied egg-rr 0.2

   \[\leadsto \color{blue}{\left(y - x\right) \cdot \left(z \cdot -6\right) + \left(\left(y - x\right) \cdot 4 + x\right)} \]

4. Final simplification 0.2

   \[\leadsto \left(y - x\right) \cdot \left(z \cdot -6\right) + \left(x + \left(y - x\right) \cdot 4\right) \]

Alternatives

Alternative 1
Error	21.4
Cost	1108

\[\begin{array}{l} t_0 := -6 \cdot \left(\left(y - x\right) \cdot z\right)\\ \mathbf{if}\;z \leq -4800000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -2.35 \cdot 10^{-33}:\\ \;\;\;\;6 \cdot \left(y \cdot \left(0.6666666666666666 - z\right)\right)\\ \mathbf{elif}\;z \leq -8.5 \cdot 10^{-133}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq 1.75 \cdot 10^{-158}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq 0.5:\\ \;\;\;\;x \cdot -3\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	32.8
Cost	980

\[\begin{array}{l} t_0 := 6 \cdot \left(x \cdot z\right)\\ \mathbf{if}\;z \leq -7 \cdot 10^{+64}:\\ \;\;\;\;-6 \cdot \left(y \cdot z\right)\\ \mathbf{elif}\;z \leq -0.072:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -2.15 \cdot 10^{-134}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq 7 \cdot 10^{-158}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{-10}:\\ \;\;\;\;x \cdot -3\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	32.8
Cost	980

\[\begin{array}{l} \mathbf{if}\;z \leq -6.7 \cdot 10^{+64}:\\ \;\;\;\;-6 \cdot \left(y \cdot z\right)\\ \mathbf{elif}\;z \leq -0.072:\\ \;\;\;\;6 \cdot \left(x \cdot z\right)\\ \mathbf{elif}\;z \leq -1.25 \cdot 10^{-135}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq 4.8 \cdot 10^{-156}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{-10}:\\ \;\;\;\;x \cdot -3\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(z \cdot 6\right)\\ \end{array} \]

Alternative 4
Error	21.7
Cost	976

\[\begin{array}{l} t_0 := -6 \cdot \left(\left(y - x\right) \cdot z\right)\\ \mathbf{if}\;z \leq -2.2 \cdot 10^{-17}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -2.6 \cdot 10^{-132}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq 1.22 \cdot 10^{-157}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq 0.5:\\ \;\;\;\;x \cdot -3\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	33.1
Cost	848

\[\begin{array}{l} t_0 := -6 \cdot \left(y \cdot z\right)\\ \mathbf{if}\;z \leq -2.2 \cdot 10^{-17}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -3.6 \cdot 10^{-133}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq 2.25 \cdot 10^{-156}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq 0.65:\\ \;\;\;\;x \cdot -3\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	33.7
Cost	721

\[\begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{+94}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;x \leq -2.15 \cdot 10^{+69} \lor \neg \left(x \leq -5 \cdot 10^{-30}\right) \land x \leq 2.5 \cdot 10^{-5}:\\ \;\;\;\;y \cdot 4\\ \mathbf{else}:\\ \;\;\;\;x \cdot -3\\ \end{array} \]

Alternative 7
Error	17.0
Cost	713

\[\begin{array}{l} \mathbf{if}\;x \leq -4.9 \cdot 10^{-26} \lor \neg \left(x \leq 0.8\right):\\ \;\;\;\;x \cdot \left(-3 + z \cdot 6\right)\\ \mathbf{else}:\\ \;\;\;\;6 \cdot \left(y \cdot \left(0.6666666666666666 - z\right)\right)\\ \end{array} \]

Alternative 8
Error	1.8
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -0.6:\\ \;\;\;\;\left(y - x\right) \cdot \left(z \cdot -6\right)\\ \mathbf{elif}\;z \leq 0.6:\\ \;\;\;\;x + \left(y - x\right) \cdot 4\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(\left(y - x\right) \cdot -6\right)\\ \end{array} \]

Alternative 9
Error	0.4
Cost	704

\[x + \left(0.6666666666666666 - z\right) \cdot \left(\left(y - x\right) \cdot 6\right) \]

Alternative 10
Error	0.3
Cost	704

\[x + \frac{0.6666666666666666 - z}{\frac{0.16666666666666666}{y - x}} \]

Alternative 11
Error	43.1
Cost	192

\[y \cdot 4 \]

Reproduce

herbie shell --seed 2023039 
(FPCore (x y z)
  :name "Data.Colour.RGBSpace.HSL:hsl from colour-2.3.3, D"
  :precision binary64
  (+ x (* (* (- y x) 6.0) (- (/ 2.0 3.0) z))))