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

Average Error: 0.4 → 0.2

Time: 15.3s

Precision: binary64

Cost: 960

Math

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

↓

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

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (+ (+ x (* (- y x) (* z -6.0))) (* (- 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 (x + ((y - x) * (z * -6.0))) + ((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 = (x + ((y - x) * (z * (-6.0d0)))) + ((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 (x + ((y - x) * (z * -6.0))) + ((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 (x + ((y - x) * (z * -6.0))) + ((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(x + Float64(Float64(y - x) * Float64(z * -6.0))) + 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 = (x + ((y - x) * (z * -6.0))) + ((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[(x + N[(N[(y - x), $MachinePrecision] * N[(z * -6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(y - x), $MachinePrecision] * 4.0), $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
   (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, 12 points decrease in error
   (fma.f64 (-.f64 y x) (fma.f64 z (*.f64 6 -1) (Rewrite<= metadata-eval (*.f64 6 2/3))) x): 0 points increase in error, 0 points decrease in error
   (fma.f64 (-.f64 y x) (fma.f64 z (*.f64 6 -1) (*.f64 6 (Rewrite<= metadata-eval (/.f64 2 3)))) x): 14 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 6 (/.f64 2 3)))) x): 0 points increase in error, 12 points decrease in error
   (fma.f64 (-.f64 y x) (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 6 -1) z)) (*.f64 6 (/.f64 2 3))) x): 0 points increase in error, 2 points decrease in error
   (fma.f64 (-.f64 y x) (+.f64 (Rewrite<= associate-*r*_binary64 (*.f64 6 (*.f64 -1 z))) (*.f64 6 (/.f64 2 3))) x): 14 points increase in error, 0 points decrease in error
   (fma.f64 (-.f64 y x) (+.f64 (*.f64 6 (Rewrite<= neg-mul-1_binary64 (neg.f64 z))) (*.f64 6 (/.f64 2 3))) x): 0 points increase in error, 14 points decrease in error
   (fma.f64 (-.f64 y x) (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 6 (/.f64 2 3)) (*.f64 6 (neg.f64 z)))) x): 12 points increase in error, 0 points decrease in error
   (fma.f64 (-.f64 y x) (Rewrite<= distribute-lft-in_binary64 (*.f64 6 (+.f64 (/.f64 2 3) (neg.f64 z)))) x): 0 points increase in error, 12 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)): 14 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): 0 points increase in error, 6 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, 8 points decrease in error

3. Applied egg-rr 0.2

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

4. Final simplification 0.2

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

Alternatives

Alternative 1
Error	25.0
Cost	1373

\[\begin{array}{l} t_0 := 6 \cdot \left(y \cdot \left(0.6666666666666666 - z\right)\right)\\ \mathbf{if}\;x \leq -2.1 \cdot 10^{+132}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;x \leq -1.05 \cdot 10^{+61}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -3.7 \cdot 10^{+20}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;x \leq -2.15 \cdot 10^{-45}:\\ \;\;\;\;6 \cdot \left(x \cdot z\right)\\ \mathbf{elif}\;x \leq 280000000 \lor \neg \left(x \leq 6.1 \cdot 10^{+85}\right) \land x \leq 8.2 \cdot 10^{+129}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x \cdot -3\\ \end{array} \]

Alternative 2
Error	21.2
Cost	1372

\[\begin{array}{l} t_0 := x \cdot \left(-3 + z \cdot 6\right)\\ t_1 := z \cdot \left(\left(y - x\right) \cdot -6\right)\\ \mathbf{if}\;z \leq -0.032:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -3.5 \cdot 10^{-307}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq 3.7 \cdot 10^{-214}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 1.16 \cdot 10^{-150}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq 1.35 \cdot 10^{-111}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq 4 \cdot 10^{-46}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq 170000:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	21.0
Cost	1372

\[\begin{array}{l} t_0 := z \cdot \left(\left(y - x\right) \cdot -6\right)\\ t_1 := x \cdot \left(-3 + z \cdot 6\right)\\ \mathbf{if}\;z \leq -1600:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -7.4 \cdot 10^{-307}:\\ \;\;\;\;y \cdot \left(4 + z \cdot -6\right)\\ \mathbf{elif}\;z \leq 5.3 \cdot 10^{-214}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{-147}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{-111}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq 1.55 \cdot 10^{-45}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq 205000:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	33.0
Cost	1244

\[\begin{array}{l} t_0 := 6 \cdot \left(x \cdot z\right)\\ \mathbf{if}\;z \leq -1.6:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -3.25 \cdot 10^{-307}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{-214}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq 1.15 \cdot 10^{-149}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq 2.05 \cdot 10^{-112}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq 4.5 \cdot 10^{-46}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq 0.96:\\ \;\;\;\;x \cdot -3\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	33.1
Cost	1244

\[\begin{array}{l} \mathbf{if}\;z \leq -1.4:\\ \;\;\;\;x \cdot \left(z \cdot 6\right)\\ \mathbf{elif}\;z \leq -5.1 \cdot 10^{-307}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq 8.6 \cdot 10^{-214}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{-148}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{-111}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq 3 \cdot 10^{-45}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq 0.96:\\ \;\;\;\;x \cdot -3\\ \mathbf{else}:\\ \;\;\;\;6 \cdot \left(x \cdot z\right)\\ \end{array} \]

Alternative 6
Error	33.1
Cost	1244

\[\begin{array}{l} \mathbf{if}\;z \leq -3300:\\ \;\;\;\;y \cdot \left(z \cdot -6\right)\\ \mathbf{elif}\;z \leq -8.5 \cdot 10^{-308}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq 3.25 \cdot 10^{-214}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq 3.3 \cdot 10^{-144}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq 2 \cdot 10^{-112}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq 7.5 \cdot 10^{-46}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq 0.96:\\ \;\;\;\;x \cdot -3\\ \mathbf{else}:\\ \;\;\;\;6 \cdot \left(x \cdot z\right)\\ \end{array} \]

Alternative 7
Error	16.6
Cost	978

\[\begin{array}{l} \mathbf{if}\;x \leq -2.8 \cdot 10^{+80} \lor \neg \left(x \leq -1.16 \cdot 10^{+61} \lor \neg \left(x \leq -2.1 \cdot 10^{-45}\right) \land x \leq 1.55 \cdot 10^{-44}\right):\\ \;\;\;\;x \cdot \left(-3 + z \cdot 6\right)\\ \mathbf{else}:\\ \;\;\;\;6 \cdot \left(y \cdot \left(0.6666666666666666 - z\right)\right)\\ \end{array} \]

Alternative 8
Error	34.5
Cost	721

\[\begin{array}{l} \mathbf{if}\;x \leq -1.85 \cdot 10^{+129}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;x \leq -7 \cdot 10^{+61} \lor \neg \left(x \leq -1.95 \cdot 10^{-120}\right) \land x \leq 1.72 \cdot 10^{-9}:\\ \;\;\;\;y \cdot 4\\ \mathbf{else}:\\ \;\;\;\;x \cdot -3\\ \end{array} \]

Alternative 9
Error	1.9
Cost	713

\[\begin{array}{l} \mathbf{if}\;z \leq -0.6 \lor \neg \left(z \leq 0.66\right):\\ \;\;\;\;z \cdot \left(\left(y - x\right) \cdot -6\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot 4 + x \cdot -3\\ \end{array} \]

Alternative 10
Error	0.4
Cost	704

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

Alternative 11
Error	0.3
Cost	704

\[x + \left(z + -0.6666666666666666\right) \cdot \frac{y - x}{-0.16666666666666666} \]

Alternative 12
Error	0.3
Cost	704

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

Alternative 13
Error	0.2
Cost	704

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

Alternative 14
Error	42.7
Cost	192

\[x \cdot -3 \]

Reproduce

herbie shell --seed 2022340 
(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))))