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

Average Accuracy: 99.3% → 99.7%

Time: 17.5s

Precision: binary64

Cost: 13248

Math

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

↓

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

FPCore

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

↓

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

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 fma((y - x), fma(z, -6.0, 4.0), x);
}

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 fma(Float64(y - x), fma(z, -6.0, 4.0), x)
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[(y - x), $MachinePrecision] * N[(z * -6.0 + 4.0), $MachinePrecision] + x), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

2. Simplified 99.7%

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

   [Start] 99.3	\[ x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
   +-commutative [=>] 99.3	\[ \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) + x} \]
   associate-*l* [=>] 99.7	\[ \color{blue}{\left(y - x\right) \cdot \left(6 \cdot \left(\frac{2}{3} - z\right)\right)} + x \]
   fma-def [=>] 99.7	\[ \color{blue}{\mathsf{fma}\left(y - x, 6 \cdot \left(\frac{2}{3} - z\right), x\right)} \]
   sub-neg [=>] 99.7	\[ \mathsf{fma}\left(y - x, 6 \cdot \color{blue}{\left(\frac{2}{3} + \left(-z\right)\right)}, x\right) \]
   distribute-lft-in [=>] 99.7	\[ \mathsf{fma}\left(y - x, \color{blue}{6 \cdot \frac{2}{3} + 6 \cdot \left(-z\right)}, x\right) \]
   +-commutative [=>] 99.7	\[ \mathsf{fma}\left(y - x, \color{blue}{6 \cdot \left(-z\right) + 6 \cdot \frac{2}{3}}, x\right) \]
   neg-mul-1 [=>] 99.7	\[ \mathsf{fma}\left(y - x, 6 \cdot \color{blue}{\left(-1 \cdot z\right)} + 6 \cdot \frac{2}{3}, x\right) \]
   associate-*r* [=>] 99.7	\[ \mathsf{fma}\left(y - x, \color{blue}{\left(6 \cdot -1\right) \cdot z} + 6 \cdot \frac{2}{3}, x\right) \]
   *-commutative [=>] 99.7	\[ \mathsf{fma}\left(y - x, \color{blue}{z \cdot \left(6 \cdot -1\right)} + 6 \cdot \frac{2}{3}, x\right) \]
   fma-def [=>] 99.7	\[ \mathsf{fma}\left(y - x, \color{blue}{\mathsf{fma}\left(z, 6 \cdot -1, 6 \cdot \frac{2}{3}\right)}, x\right) \]
   metadata-eval [=>] 99.7	\[ \mathsf{fma}\left(y - x, \mathsf{fma}\left(z, \color{blue}{-6}, 6 \cdot \frac{2}{3}\right), x\right) \]
   metadata-eval [=>] 99.7	\[ \mathsf{fma}\left(y - x, \mathsf{fma}\left(z, -6, 6 \cdot \color{blue}{0.6666666666666666}\right), x\right) \]
   metadata-eval [=>] 99.7	\[ \mathsf{fma}\left(y - x, \mathsf{fma}\left(z, -6, \color{blue}{4}\right), x\right) \]

3. Final simplification 99.7%

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

Alternatives

Alternative 1
Accuracy	49.1%
Cost	2036

\[\begin{array}{l} t_0 := 6 \cdot \left(x \cdot z\right)\\ t_1 := -6 \cdot \left(y \cdot z\right)\\ \mathbf{if}\;z \leq -1.12 \cdot 10^{+124}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -4 \cdot 10^{+70}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -6.5 \cdot 10^{+34}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -8.6 \cdot 10^{-9}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -4.2 \cdot 10^{-60}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq -4.4 \cdot 10^{-176}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq -2.3 \cdot 10^{-231}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq -6.2 \cdot 10^{-268}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq 8.2 \cdot 10^{-308}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq 8.6 \cdot 10^{-235}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{-179}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq 5 \cdot 10^{-154}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq 3.5 \cdot 10^{-7}:\\ \;\;\;\;y \cdot 4\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Accuracy	49.0%
Cost	2036

\[\begin{array}{l} t_0 := 6 \cdot \left(x \cdot z\right)\\ t_1 := z \cdot \left(y \cdot -6\right)\\ \mathbf{if}\;z \leq -6 \cdot 10^{+125}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -3.8 \cdot 10^{+68}:\\ \;\;\;\;-6 \cdot \left(y \cdot z\right)\\ \mathbf{elif}\;z \leq -1.02 \cdot 10^{+36}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -8.6 \cdot 10^{-9}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -2.4 \cdot 10^{-60}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq -1.7 \cdot 10^{-175}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq -4.1 \cdot 10^{-231}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq -6.6 \cdot 10^{-269}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq -8.5 \cdot 10^{-306}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq 4 \cdot 10^{-233}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq 1.28 \cdot 10^{-179}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq 5 \cdot 10^{-154}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq 3.5 \cdot 10^{-7}:\\ \;\;\;\;y \cdot 4\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Accuracy	49.1%
Cost	2036

\[\begin{array}{l} t_0 := x \cdot \left(z \cdot 6\right)\\ t_1 := z \cdot \left(y \cdot -6\right)\\ \mathbf{if}\;z \leq -1.1 \cdot 10^{+123}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -3.1 \cdot 10^{+71}:\\ \;\;\;\;-6 \cdot \left(y \cdot z\right)\\ \mathbf{elif}\;z \leq -1.35 \cdot 10^{+36}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -8.6 \cdot 10^{-9}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -4.5 \cdot 10^{-62}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq -2.3 \cdot 10^{-176}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq -2.4 \cdot 10^{-231}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq -1.12 \cdot 10^{-269}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq -2.7 \cdot 10^{-308}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq 3.1 \cdot 10^{-233}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq 1.12 \cdot 10^{-182}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq 5 \cdot 10^{-154}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq 3.5 \cdot 10^{-7}:\\ \;\;\;\;y \cdot 4\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Accuracy	67.0%
Cost	1768

\[\begin{array}{l} t_0 := -6 \cdot \left(\left(y - x\right) \cdot z\right)\\ \mathbf{if}\;z \leq -8.6 \cdot 10^{-9}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -2 \cdot 10^{-61}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq -1.95 \cdot 10^{-175}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq -4.5 \cdot 10^{-231}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq -1.5 \cdot 10^{-269}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq -1.35 \cdot 10^{-303}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq 3.4 \cdot 10^{-235}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq 7 \cdot 10^{-181}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq 5 \cdot 10^{-154}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq 0.66:\\ \;\;\;\;y \cdot 4\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Accuracy	67.4%
Cost	1768

\[\begin{array}{l} t_0 := 6 \cdot \left(y \cdot \left(0.6666666666666666 - z\right)\right)\\ t_1 := -6 \cdot \left(\left(y - x\right) \cdot z\right)\\ \mathbf{if}\;z \leq -2.4 \cdot 10^{+15}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -4.8 \cdot 10^{-62}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -2.05 \cdot 10^{-175}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq -4.3 \cdot 10^{-231}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq -1.45 \cdot 10^{-269}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq -1.7 \cdot 10^{-302}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq 7.5 \cdot 10^{-232}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq 7.8 \cdot 10^{-176}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq 5 \cdot 10^{-154}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq 62000000000000:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Accuracy	67.4%
Cost	1768

\[\begin{array}{l} t_0 := 6 \cdot \left(y \cdot \left(0.6666666666666666 - z\right)\right)\\ t_1 := -6 \cdot \left(\left(y - x\right) \cdot z\right)\\ \mathbf{if}\;z \leq -2.4 \cdot 10^{+15}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -5.4 \cdot 10^{-60}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -2.7 \cdot 10^{-175}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq -2.2 \cdot 10^{-231}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq -1.7 \cdot 10^{-269}:\\ \;\;\;\;x \cdot \left(-3 + z \cdot 6\right)\\ \mathbf{elif}\;z \leq 9 \cdot 10^{-308}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{-232}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq 1.02 \cdot 10^{-180}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq 5 \cdot 10^{-154}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq 62000000000000:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Accuracy	67.4%
Cost	1768

\[\begin{array}{l} t_0 := 6 \cdot \left(y \cdot \left(0.6666666666666666 - z\right)\right)\\ \mathbf{if}\;z \leq -2.4 \cdot 10^{+15}:\\ \;\;\;\;-6 \cdot \left(\left(y - x\right) \cdot z\right)\\ \mathbf{elif}\;z \leq -1.45 \cdot 10^{-59}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -8 \cdot 10^{-176}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq -2.6 \cdot 10^{-231}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq -3.4 \cdot 10^{-268}:\\ \;\;\;\;x \cdot \left(-3 + z \cdot 6\right)\\ \mathbf{elif}\;z \leq 4.3 \cdot 10^{-308}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq 2.8 \cdot 10^{-235}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq 7 \cdot 10^{-173}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq 5 \cdot 10^{-154}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq 62000000000000:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(\left(y - x\right) \cdot -6\right)\\ \end{array} \]

Alternative 8
Accuracy	67.4%
Cost	1768

\[\begin{array}{l} t_0 := y \cdot \left(4 + z \cdot -6\right)\\ \mathbf{if}\;z \leq -2.4 \cdot 10^{+15}:\\ \;\;\;\;-6 \cdot \left(\left(y - x\right) \cdot z\right)\\ \mathbf{elif}\;z \leq -2.3 \cdot 10^{-60}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -1.1 \cdot 10^{-175}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq -2.2 \cdot 10^{-231}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq -1.32 \cdot 10^{-270}:\\ \;\;\;\;x \cdot \left(-3 + z \cdot 6\right)\\ \mathbf{elif}\;z \leq 4 \cdot 10^{-308}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq 3.7 \cdot 10^{-235}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq 3.45 \cdot 10^{-178}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq 5 \cdot 10^{-154}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq 62000000000000:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(\left(y - x\right) \cdot -6\right)\\ \end{array} \]

Alternative 9
Accuracy	49.0%
Cost	1640

\[\begin{array}{l} t_0 := -6 \cdot \left(y \cdot z\right)\\ \mathbf{if}\;z \leq -8.6 \cdot 10^{-9}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -4.1 \cdot 10^{-61}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq -3.8 \cdot 10^{-176}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq -2.2 \cdot 10^{-231}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq -3.9 \cdot 10^{-268}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq -9.2 \cdot 10^{-304}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{-234}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{-171}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq 5 \cdot 10^{-154}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq 3.5 \cdot 10^{-7}:\\ \;\;\;\;y \cdot 4\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 10
Accuracy	97.3%
Cost	841

\[\begin{array}{l} \mathbf{if}\;z \leq -0.56 \lor \neg \left(z \leq 0.66\right):\\ \;\;\;\;z \cdot \left(x \cdot 6 + y \cdot -6\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot 4 + x \cdot -3\\ \end{array} \]

Alternative 11
Accuracy	99.3%
Cost	832

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

Alternative 12
Accuracy	99.5%
Cost	832

\[x + \frac{\left(y - x\right) \cdot 6}{\frac{1}{0.6666666666666666 - z}} \]

Alternative 13
Accuracy	97.3%
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -0.55:\\ \;\;\;\;-6 \cdot \left(\left(y - x\right) \cdot z\right)\\ \mathbf{elif}\;z \leq 0.6:\\ \;\;\;\;y \cdot 4 + x \cdot -3\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(\left(y - x\right) \cdot -6\right)\\ \end{array} \]

Alternative 14
Accuracy	99.3%
Cost	704

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

Alternative 15
Accuracy	47.8%
Cost	456

\[\begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{+39}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;y \leq 11.5:\\ \;\;\;\;x \cdot -3\\ \mathbf{else}:\\ \;\;\;\;y \cdot 4\\ \end{array} \]

Alternative 16
Accuracy	32.9%
Cost	192

\[y \cdot 4 \]

Reproduce

herbie shell --seed 2023138 
(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))))