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

Average Error: 0.3 → 0.2

Time: 8.7s

Precision: binary64

Cost: 576

Math

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

↓

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

FPCore

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

↓

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

C

double code(double x, double y, double z) {
	return x + (((y - x) * 6.0) * z);
}

↓

double code(double x, double y, double z) {
	return x + ((y - x) * (6.0 * 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) * 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) * (6.0d0 * z))
end function

Java

public static double code(double x, double y, double z) {
	return x + (((y - x) * 6.0) * z);
}

↓

public static double code(double x, double y, double z) {
	return x + ((y - x) * (6.0 * z));
}

Python

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

↓

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

Julia

function code(x, y, z)
	return Float64(x + Float64(Float64(Float64(y - x) * 6.0) * z))
end

↓

function code(x, y, z)
	return Float64(x + Float64(Float64(y - x) * Float64(6.0 * z)))
end

MATLAB

function tmp = code(x, y, z)
	tmp = x + (((y - x) * 6.0) * z);
end

↓

function tmp = code(x, y, z)
	tmp = x + ((y - x) * (6.0 * z));
end

Wolfram

code[x_, y_, z_] := N[(x + N[(N[(N[(y - x), $MachinePrecision] * 6.0), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_] := N[(x + N[(N[(y - x), $MachinePrecision] * N[(6.0 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	0.3
Target	0.2
Herbie	0.2

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

Derivation

1. Initial program 0.3

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

2. Simplified 0.3

   \[\leadsto \color{blue}{\mathsf{fma}\left(\left(y - x\right) \cdot 6, z, x\right)} \]
   Proof
   (fma.f64 (*.f64 (-.f64 y x) 6) z x): 0 points increase in error, 0 points decrease in error
   (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (*.f64 (-.f64 y x) 6) z) x)): 5 points increase in error, 1 points decrease in error
   (Rewrite<= +-commutative_binary64 (+.f64 x (*.f64 (*.f64 (-.f64 y x) 6) z))): 0 points increase in error, 0 points decrease in error

3. Applied egg-rr 0.2

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

4. Final simplification 0.2

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

Alternatives

Alternative 1
Error	25.3
Cost	1244

\[\begin{array}{l} t_0 := x \cdot \left(z \cdot -6\right)\\ t_1 := y \cdot \left(6 \cdot z\right)\\ \mathbf{if}\;z \leq -4 \cdot 10^{+108}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -1.3911802600254683 \cdot 10^{-114}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 2.0208099600057898 \cdot 10^{-70}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 717315.7439188524:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 3.7 \cdot 10^{+79}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 8.5 \cdot 10^{+145}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.9 \cdot 10^{+245}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	25.3
Cost	1244

\[\begin{array}{l} t_0 := x \cdot \left(z \cdot -6\right)\\ t_1 := y \cdot \left(6 \cdot z\right)\\ \mathbf{if}\;z \leq -4 \cdot 10^{+108}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -1.3911802600254683 \cdot 10^{-114}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 2.0208099600057898 \cdot 10^{-70}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 717315.7439188524:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 3.7 \cdot 10^{+79}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 6 \cdot 10^{+152}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.9 \cdot 10^{+245}:\\ \;\;\;\;-6 \cdot \left(x \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	14.0
Cost	712

\[\begin{array}{l} t_0 := z \cdot \frac{y - x}{0.16666666666666666}\\ \mathbf{if}\;z \leq -1.3911802600254683 \cdot 10^{-114}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 2.0208099600057898 \cdot 10^{-70}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	13.9
Cost	712

\[\begin{array}{l} t_0 := \left(y - x\right) \cdot \left(6 \cdot z\right)\\ \mathbf{if}\;z \leq -1.3911802600254683 \cdot 10^{-114}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 2.0208099600057898 \cdot 10^{-70}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	13.1
Cost	712

\[\begin{array}{l} t_0 := x \cdot \left(1 + z \cdot -6\right)\\ \mathbf{if}\;x \leq -5.872105785230448 \cdot 10^{-74}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 2.1562139141002143 \cdot 10^{+20}:\\ \;\;\;\;\left(y - x\right) \cdot \left(6 \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	1.2
Cost	712

\[\begin{array}{l} t_0 := \left(y - x\right) \cdot \left(6 \cdot z\right)\\ \mathbf{if}\;z \leq -22009204.077657487:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 0.011881861933520465:\\ \;\;\;\;x + z \cdot \left(y \cdot 6\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	1.3
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -4339070671785.9253:\\ \;\;\;\;\left(y - x\right) \cdot \left(6 \cdot z\right)\\ \mathbf{elif}\;z \leq 717315.7439188524:\\ \;\;\;\;x + 6 \cdot \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{y - x}{0.16666666666666666}\\ \end{array} \]

Alternative 8
Error	24.5
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -5.872105785230448 \cdot 10^{-74}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2.0105680278433627 \cdot 10^{-88}:\\ \;\;\;\;y \cdot \left(6 \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 9
Error	35.2
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022300 
(FPCore (x y z)
  :name "Data.Colour.RGBSpace.HSL:hsl from colour-2.3.3, E"
  :precision binary64

  :herbie-target
  (- x (* (* 6.0 z) (- x y)))

  (+ x (* (* (- y x) 6.0) z)))