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

Average Error: 0.4 → 0.2

Time: 15.6s

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): 7 points increase in error, 2 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): 3 points increase in error, 0 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, 0 points decrease in error
   (+.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (-.f64 y x) 6) (-.f64 (/.f64 2 3) z))) x): 57 points increase in error, 20 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	32.4
Cost	1640

\[\begin{array}{l} t_0 := z \cdot \left(x \cdot 6\right)\\ \mathbf{if}\;z \leq -7.4 \cdot 10^{+124}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -2.9 \cdot 10^{+92}:\\ \;\;\;\;-6 \cdot \left(y \cdot z\right)\\ \mathbf{elif}\;z \leq -160033704.54018337:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 2.9853275654269707 \cdot 10^{-255}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq 2.384464029021005 \cdot 10^{-234}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq 3.1809576474284375 \cdot 10^{-180}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq 1.1378691758781069 \cdot 10^{-151}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq 4.1095201108998124 \cdot 10^{-117}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq 1.1447581398977339 \cdot 10^{-34}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq 0.31326431225840107:\\ \;\;\;\;x \cdot -3\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	32.4
Cost	1640

\[\begin{array}{l} t_0 := 6 \cdot \left(x \cdot z\right)\\ \mathbf{if}\;z \leq -7.4 \cdot 10^{+124}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -2.9 \cdot 10^{+92}:\\ \;\;\;\;-6 \cdot \left(y \cdot z\right)\\ \mathbf{elif}\;z \leq -160033704.54018337:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 2.9853275654269707 \cdot 10^{-255}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq 2.384464029021005 \cdot 10^{-234}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq 3.1809576474284375 \cdot 10^{-180}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq 1.1378691758781069 \cdot 10^{-151}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq 4.1095201108998124 \cdot 10^{-117}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq 1.1447581398977339 \cdot 10^{-34}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq 0.31326431225840107:\\ \;\;\;\;x \cdot -3\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	21.0
Cost	1504

\[\begin{array}{l} t_0 := x \cdot \left(-3 + z \cdot 6\right)\\ t_1 := \left(y - x\right) \cdot \left(z \cdot -6\right)\\ \mathbf{if}\;z \leq -0.002926451270112743:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 2.9853275654269707 \cdot 10^{-255}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 2.384464029021005 \cdot 10^{-234}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq 3.1809576474284375 \cdot 10^{-180}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq 1.1378691758781069 \cdot 10^{-151}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq 4.1095201108998124 \cdot 10^{-117}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq 1.1447581398977339 \cdot 10^{-34}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq 30284.525102377243:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	32.7
Cost	1376

\[\begin{array}{l} t_0 := z \cdot \left(y \cdot -6\right)\\ \mathbf{if}\;z \leq -0.002926451270112743:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 2.9853275654269707 \cdot 10^{-255}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq 2.384464029021005 \cdot 10^{-234}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq 3.1809576474284375 \cdot 10^{-180}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq 1.1378691758781069 \cdot 10^{-151}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq 4.1095201108998124 \cdot 10^{-117}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq 1.1447581398977339 \cdot 10^{-34}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq 0.31326431225840107:\\ \;\;\;\;x \cdot -3\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	32.7
Cost	1376

\[\begin{array}{l} \mathbf{if}\;z \leq -0.002926451270112743:\\ \;\;\;\;y \cdot \left(z \cdot -6\right)\\ \mathbf{elif}\;z \leq 2.9853275654269707 \cdot 10^{-255}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq 2.384464029021005 \cdot 10^{-234}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq 3.1809576474284375 \cdot 10^{-180}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq 1.1378691758781069 \cdot 10^{-151}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq 4.1095201108998124 \cdot 10^{-117}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq 1.1447581398977339 \cdot 10^{-34}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq 0.31326431225840107:\\ \;\;\;\;x \cdot -3\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(y \cdot -6\right)\\ \end{array} \]

Alternative 6
Error	32.7
Cost	1376

\[\begin{array}{l} \mathbf{if}\;z \leq -0.002926451270112743:\\ \;\;\;\;-6 \cdot \left(y \cdot z\right)\\ \mathbf{elif}\;z \leq 2.9853275654269707 \cdot 10^{-255}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq 2.384464029021005 \cdot 10^{-234}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq 3.1809576474284375 \cdot 10^{-180}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq 1.1378691758781069 \cdot 10^{-151}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq 4.1095201108998124 \cdot 10^{-117}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq 1.1447581398977339 \cdot 10^{-34}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq 0.31326431225840107:\\ \;\;\;\;x \cdot -3\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(y \cdot -6\right)\\ \end{array} \]

Alternative 7
Error	0.2
Cost	1088

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

Alternative 8
Error	23.9
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq -6.8 \cdot 10^{+170}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;y \leq 1.0444233137369338 \cdot 10^{+49}:\\ \;\;\;\;x \cdot \left(-3 + z \cdot 6\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot 4\\ \end{array} \]

Alternative 9
Error	1.9
Cost	712

\[\begin{array}{l} t_0 := \left(y - x\right) \cdot \left(z \cdot -6\right)\\ \mathbf{if}\;z \leq -11746.95108953019:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 0.31326431225840107:\\ \;\;\;\;y \cdot 4 + x \cdot -3\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 10
Error	1.9
Cost	712

\[\begin{array}{l} t_0 := -6 \cdot \left(\left(y - x\right) \cdot z\right)\\ \mathbf{if}\;z \leq -11746.95108953019:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 0.31326431225840107:\\ \;\;\;\;y \cdot 4 + x \cdot -3\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 11
Error	0.4
Cost	704

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

Alternative 12
Error	0.4
Cost	704

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

Alternative 13
Error	33.4
Cost	456

\[\begin{array}{l} \mathbf{if}\;y \leq -173733.08394468194:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;y \leq 2.6810942204944528 \cdot 10^{+25}:\\ \;\;\;\;x \cdot -3\\ \mathbf{else}:\\ \;\;\;\;y \cdot 4\\ \end{array} \]

Alternative 14
Error	42.7
Cost	192

\[y \cdot 4 \]

Reproduce

herbie shell --seed 2022297 
(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))))