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

Average Error: 0.4 → 0.2

Time: 11.6s

Precision: binary64

Cost: 960

Math

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

↓

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

FPCore

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

↓

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

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 (4.0 * (y - x)) + (x + (-6.0 * ((y - x) * z)));
}

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 = (4.0d0 * (y - x)) + (x + ((-6.0d0) * ((y - x) * z)))
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 (4.0 * (y - x)) + (x + (-6.0 * ((y - x) * z)));
}

Python

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

↓

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

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(4.0 * Float64(y - x)) + Float64(x + Float64(-6.0 * Float64(Float64(y - x) * z))))
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 = (4.0 * (y - x)) + (x + (-6.0 * ((y - x) * z)));
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[(4.0 * N[(y - x), $MachinePrecision]), $MachinePrecision] + N[(x + N[(-6.0 * N[(N[(y - x), $MachinePrecision] * z), $MachinePrecision]), $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
   (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): 5 points increase in error, 6 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): 2 points increase in error, 1 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)): 3 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): 68 points increase in error, 17 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. Taylor expanded in z around 0 0.2

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

4. Final simplification 0.2

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

Alternatives

Alternative 1
Error	33.0
Cost	1112

\[\begin{array}{l} t_0 := y \cdot \left(-6 \cdot z\right)\\ \mathbf{if}\;z \leq -2.4774148561027497 \cdot 10^{+21}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -3.4389908808816314 \cdot 10^{-66}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq -1.4764790011037582 \cdot 10^{-216}:\\ \;\;\;\;4 \cdot y\\ \mathbf{elif}\;z \leq -4.267252325917222 \cdot 10^{-230}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq -2.4314014463301733 \cdot 10^{-297}:\\ \;\;\;\;4 \cdot y\\ \mathbf{elif}\;z \leq 0.0036699405484121803:\\ \;\;\;\;x \cdot -3\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	32.3
Cost	1112

\[\begin{array}{l} t_0 := 6 \cdot \left(x \cdot z\right)\\ \mathbf{if}\;z \leq -55867.329838425474:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -3.4389908808816314 \cdot 10^{-66}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq -1.4764790011037582 \cdot 10^{-216}:\\ \;\;\;\;4 \cdot y\\ \mathbf{elif}\;z \leq -4.267252325917222 \cdot 10^{-230}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq -2.4314014463301733 \cdot 10^{-297}:\\ \;\;\;\;4 \cdot y\\ \mathbf{elif}\;z \leq 0.0036699405484121803:\\ \;\;\;\;x \cdot -3\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	32.3
Cost	1112

\[\begin{array}{l} \mathbf{if}\;z \leq -55867.329838425474:\\ \;\;\;\;x \cdot \left(z \cdot 6\right)\\ \mathbf{elif}\;z \leq -3.4389908808816314 \cdot 10^{-66}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq -1.4764790011037582 \cdot 10^{-216}:\\ \;\;\;\;4 \cdot y\\ \mathbf{elif}\;z \leq -4.267252325917222 \cdot 10^{-230}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq -2.4314014463301733 \cdot 10^{-297}:\\ \;\;\;\;4 \cdot y\\ \mathbf{elif}\;z \leq 0.0036699405484121803:\\ \;\;\;\;x \cdot -3\\ \mathbf{else}:\\ \;\;\;\;6 \cdot \left(x \cdot z\right)\\ \end{array} \]

Alternative 4
Error	1.3
Cost	1096

\[\begin{array}{l} \mathbf{if}\;z \leq -55867.329838425474:\\ \;\;\;\;z \cdot \left(\left(y - x\right) \cdot -6\right)\\ \mathbf{elif}\;z \leq 10.180784999286313:\\ \;\;\;\;x \cdot -4 + \left(x + 6 \cdot \left(y \cdot \left(0.6666666666666666 - z\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \left(y \cdot -6 + x \cdot 6\right)\\ \end{array} \]

Alternative 5
Error	1.8
Cost	968

\[\begin{array}{l} \mathbf{if}\;z \leq -55867.329838425474:\\ \;\;\;\;z \cdot \left(\left(y - x\right) \cdot -6\right)\\ \mathbf{elif}\;z \leq 0.0036699405484121803:\\ \;\;\;\;4 \cdot y + x \cdot -3\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \left(y \cdot -6 + x \cdot 6\right)\\ \end{array} \]

Alternative 6
Error	12.9
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -55867.329838425474:\\ \;\;\;\;x \cdot \left(z \cdot 6\right)\\ \mathbf{elif}\;z \leq 0.0036699405484121803:\\ \;\;\;\;4 \cdot y + x \cdot -3\\ \mathbf{else}:\\ \;\;\;\;6 \cdot \left(x \cdot z\right)\\ \end{array} \]

Alternative 7
Error	1.8
Cost	712

\[\begin{array}{l} t_0 := z \cdot \left(\left(y - x\right) \cdot -6\right)\\ \mathbf{if}\;z \leq -55867.329838425474:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 0.0036699405484121803:\\ \;\;\;\;4 \cdot y + x \cdot -3\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	0.4
Cost	704

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

Alternative 9
Error	33.8
Cost	456

\[\begin{array}{l} \mathbf{if}\;y \leq -1.417898523361797 \cdot 10^{-32}:\\ \;\;\;\;4 \cdot y\\ \mathbf{elif}\;y \leq 1.0475370519041326 \cdot 10^{+63}:\\ \;\;\;\;x \cdot -3\\ \mathbf{else}:\\ \;\;\;\;4 \cdot y\\ \end{array} \]

Alternative 10
Error	43.2
Cost	192

\[4 \cdot y \]

Alternative 11
Error	62.4
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022295 
(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))))