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

Average Error: 0.3 → 0.2

Time: 8.6s

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)): 1 points increase in error, 0 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	23.6
Cost	980

\[\begin{array}{l} t_0 := z \cdot \left(y \cdot 6\right)\\ \mathbf{if}\;z \leq -3.8 \cdot 10^{+212}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -2.5279591138776153 \cdot 10^{+30}:\\ \;\;\;\;-6 \cdot \left(x \cdot z\right)\\ \mathbf{elif}\;z \leq -8.203906431019795 \cdot 10^{-36}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 8.640531797436794 \cdot 10^{-8}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 6.4 \cdot 10^{+176}:\\ \;\;\;\;x \cdot \left(z \cdot -6\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	23.6
Cost	980

\[\begin{array}{l} t_0 := 6 \cdot \left(y \cdot z\right)\\ \mathbf{if}\;z \leq -3.8 \cdot 10^{+212}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -2.5279591138776153 \cdot 10^{+30}:\\ \;\;\;\;-6 \cdot \left(x \cdot z\right)\\ \mathbf{elif}\;z \leq -8.203906431019795 \cdot 10^{-36}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 8.640531797436794 \cdot 10^{-8}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 6.4 \cdot 10^{+176}:\\ \;\;\;\;x \cdot \left(z \cdot -6\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(y \cdot 6\right)\\ \end{array} \]

Alternative 3
Error	11.9
Cost	712

\[\begin{array}{l} t_0 := \left(y - x\right) \cdot \left(6 \cdot z\right)\\ \mathbf{if}\;z \leq -3.447659526510953 \cdot 10^{-48}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 6.771056242562191 \cdot 10^{-27}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	1.8
Cost	712

\[\begin{array}{l} t_0 := \left(y - x\right) \cdot \left(6 \cdot z\right)\\ \mathbf{if}\;z \leq -16577504821.054987:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 6.771056242562191 \cdot 10^{-27}:\\ \;\;\;\;x + z \cdot \left(y \cdot 6\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	1.8
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -19505355390682.14:\\ \;\;\;\;z \cdot \left(\left(y - x\right) \cdot 6\right)\\ \mathbf{elif}\;z \leq 6.771056242562191 \cdot 10^{-27}:\\ \;\;\;\;x + z \cdot \left(y \cdot 6\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y - x\right) \cdot \left(6 \cdot z\right)\\ \end{array} \]

Alternative 6
Error	1.8
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -16577504821.054987:\\ \;\;\;\;6 \cdot \left(\left(y - x\right) \cdot z\right)\\ \mathbf{elif}\;z \leq 6.771056242562191 \cdot 10^{-27}:\\ \;\;\;\;x + z \cdot \left(y \cdot 6\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y - x\right) \cdot \left(6 \cdot z\right)\\ \end{array} \]

Alternative 7
Error	23.7
Cost	584

\[\begin{array}{l} t_0 := z \cdot \left(x \cdot -6\right)\\ \mathbf{if}\;z \leq -932729590.0480419:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 8.640531797436794 \cdot 10^{-8}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	23.7
Cost	584

\[\begin{array}{l} \mathbf{if}\;z \leq -932729590.0480419:\\ \;\;\;\;-6 \cdot \left(x \cdot z\right)\\ \mathbf{elif}\;z \leq 8.640531797436794 \cdot 10^{-8}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(x \cdot -6\right)\\ \end{array} \]

Alternative 9
Error	23.7
Cost	584

\[\begin{array}{l} \mathbf{if}\;z \leq -932729590.0480419:\\ \;\;\;\;-6 \cdot \left(x \cdot z\right)\\ \mathbf{elif}\;z \leq 8.640531797436794 \cdot 10^{-8}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(z \cdot -6\right)\\ \end{array} \]

Alternative 10
Error	34.4
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022316 
(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)))