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

Average Error: 0.4 → 0.2

Time: 13.9s

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 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
   (fma.f64 (-.f64 y x) (fma.f64 z -6 4) x): 0 points increase in error, 0 points decrease in error
   (fma.f64 (-.f64 y x) (fma.f64 z (Rewrite<= metadata-eval (*.f64 6 -1)) 4) x): 0 points increase in error, 0 points decrease in error
   (fma.f64 (-.f64 y x) (fma.f64 z (*.f64 6 -1) (Rewrite<= metadata-eval (*.f64 2/3 6))) x): 0 points increase in error, 0 points decrease in error
   (fma.f64 (-.f64 y x) (fma.f64 z (*.f64 6 -1) (*.f64 (Rewrite<= metadata-eval (/.f64 2 3)) 6)) x): 0 points increase in error, 0 points decrease in error
   (fma.f64 (-.f64 y x) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 z (*.f64 6 -1)) (*.f64 (/.f64 2 3) 6))) x): 4 points increase in error, 8 points decrease in error
   (fma.f64 (-.f64 y x) (+.f64 (*.f64 z (Rewrite=> metadata-eval -6)) (*.f64 (/.f64 2 3) 6)) x): 0 points increase in error, 0 points decrease in error
   (fma.f64 (-.f64 y x) (+.f64 (*.f64 z (Rewrite<= metadata-eval (neg.f64 6))) (*.f64 (/.f64 2 3) 6)) x): 0 points increase in error, 0 points decrease in error
   (fma.f64 (-.f64 y x) (+.f64 (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 z 6))) (*.f64 (/.f64 2 3) 6)) x): 0 points increase in error, 0 points decrease in error
   (fma.f64 (-.f64 y x) (+.f64 (Rewrite<= distribute-lft-neg-out_binary64 (*.f64 (neg.f64 z) 6)) (*.f64 (/.f64 2 3) 6)) x): 0 points increase in error, 0 points decrease in error
   (fma.f64 (-.f64 y x) (Rewrite<= distribute-rgt-in_binary64 (*.f64 6 (+.f64 (neg.f64 z) (/.f64 2 3)))) x): 8 points increase in error, 3 points decrease in error
   (fma.f64 (-.f64 y x) (*.f64 6 (Rewrite<= +-commutative_binary64 (+.f64 (/.f64 2 3) (neg.f64 z)))) x): 0 points increase in error, 0 points decrease in error
   (fma.f64 (-.f64 y x) (*.f64 6 (Rewrite<= sub-neg_binary64 (-.f64 (/.f64 2 3) z))) x): 0 points increase in error, 0 points decrease in error
   (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (-.f64 y x) (*.f64 6 (-.f64 (/.f64 2 3) z))) x)): 1 points increase in error, 2 points decrease in error
   (+.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (-.f64 y x) 6) (-.f64 (/.f64 2 3) z))) x): 69 points increase in error, 22 points decrease in error
   (Rewrite<= +-commutative_binary64 (+.f64 x (*.f64 (*.f64 (-.f64 y x) 6) (-.f64 (/.f64 2 3) z)))): 0 points increase in error, 0 points decrease in error

3. Final simplification 0.2

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

Alternatives

Alternative 1
Error	32.3
Cost	1376

\[\begin{array}{l} t_0 := z \cdot \left(x \cdot 6\right)\\ t_1 := y \cdot \left(z \cdot -6\right)\\ \mathbf{if}\;z \leq -1.2383895444300558 \cdot 10^{+26}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -0.0002542958208316335:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -7.528080747597161 \cdot 10^{-268}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq -6.46212628449796 \cdot 10^{-302}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq 8.766806990476859 \cdot 10^{-147}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq 9.613897233162174 \cdot 10^{-68}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq 0.011881861933520465:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq 1.68 \cdot 10^{+193}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	32.5
Cost	1112

\[\begin{array}{l} t_0 := y \cdot \left(z \cdot -6\right)\\ \mathbf{if}\;z \leq -0.0002542958208316335:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -7.528080747597161 \cdot 10^{-268}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq -6.46212628449796 \cdot 10^{-302}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq 8.766806990476859 \cdot 10^{-147}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq 9.613897233162174 \cdot 10^{-68}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq 0.011881861933520465:\\ \;\;\;\;y \cdot 4\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	32.5
Cost	1112

\[\begin{array}{l} \mathbf{if}\;z \leq -0.0002542958208316335:\\ \;\;\;\;y \cdot \left(z \cdot -6\right)\\ \mathbf{elif}\;z \leq -7.528080747597161 \cdot 10^{-268}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq -6.46212628449796 \cdot 10^{-302}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq 8.766806990476859 \cdot 10^{-147}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq 9.613897233162174 \cdot 10^{-68}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq 0.011881861933520465:\\ \;\;\;\;y \cdot 4\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(y \cdot -6\right)\\ \end{array} \]

Alternative 4
Error	0.3
Cost	1088

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

Alternative 5
Error	1.5
Cost	968

\[\begin{array}{l} \mathbf{if}\;z \leq -598726434163.7137:\\ \;\;\;\;x + z \cdot \left(\left(y - x\right) \cdot -6\right)\\ \mathbf{elif}\;z \leq 717315.7439188524:\\ \;\;\;\;x \cdot -3 - 6 \cdot \left(y \cdot \left(z + -0.6666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y - x\right) \cdot \left(z \cdot -6\right)\\ \end{array} \]

Alternative 6
Error	13.1
Cost	848

\[\begin{array}{l} t_0 := z \cdot \left(x \cdot 6\right)\\ t_1 := y \cdot \left(z \cdot -6\right)\\ \mathbf{if}\;z \leq -1.2383895444300558 \cdot 10^{+26}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -232907.12960139074:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 0.011881861933520465:\\ \;\;\;\;x + \left(y - x\right) \cdot 4\\ \mathbf{elif}\;z \leq 1.68 \cdot 10^{+193}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	1.9
Cost	712

\[\begin{array}{l} t_0 := \left(y - x\right) \cdot \left(z \cdot -6\right)\\ \mathbf{if}\;z \leq -232907.12960139074:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 0.011881861933520465:\\ \;\;\;\;x + \left(y - x\right) \cdot 4\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	1.9
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -232907.12960139074:\\ \;\;\;\;-6 \cdot \left(\left(y - x\right) \cdot z\right)\\ \mathbf{elif}\;z \leq 0.011881861933520465:\\ \;\;\;\;x + \left(y - x\right) \cdot 4\\ \mathbf{else}:\\ \;\;\;\;\left(y - x\right) \cdot \left(z \cdot -6\right)\\ \end{array} \]

Alternative 9
Error	1.9
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -232907.12960139074:\\ \;\;\;\;x + z \cdot \left(\left(y - x\right) \cdot -6\right)\\ \mathbf{elif}\;z \leq 0.011881861933520465:\\ \;\;\;\;x + \left(y - x\right) \cdot 4\\ \mathbf{else}:\\ \;\;\;\;\left(y - x\right) \cdot \left(z \cdot -6\right)\\ \end{array} \]

Alternative 10
Error	0.4
Cost	704

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

Alternative 11
Error	0.4
Cost	704

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

Alternative 12
Error	34.6
Cost	456

\[\begin{array}{l} \mathbf{if}\;x \leq -8.131675494172207 \cdot 10^{-41}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;x \leq 1.0178195895650086 \cdot 10^{-95}:\\ \;\;\;\;y \cdot 4\\ \mathbf{else}:\\ \;\;\;\;x \cdot -3\\ \end{array} \]

Alternative 13
Error	43.1
Cost	192

\[y \cdot 4 \]

Alternative 14
Error	62.4
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022300 
(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))))