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

Average Error: 0.2 → 0.2

Time: 7.9s

Precision: binary64

Cost: 7104

Math

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

↓

\[x + \mathsf{fma}\left(-6, x \cdot z, z \cdot \left(6 \cdot y\right)\right) \]

FPCore

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

↓

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

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 + fma(-6.0, (x * z), (z * (6.0 * y)));
}

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 + fma(-6.0, Float64(x * z), Float64(z * Float64(6.0 * y))))
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[(-6.0 * N[(x * z), $MachinePrecision] + N[(z * N[(6.0 * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	0.2
Target	0.2
Herbie	0.2

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

Derivation

1. Initial program 0.2

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

2. Taylor expanded in y around 0 0.2

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

3. Simplified 0.2

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

   [Start] 0.2	\[ x + \left(-6 \cdot \left(z \cdot x\right) + 6 \cdot \left(y \cdot z\right)\right) \]
   *-commutative [=>] 0.2	\[ x + \left(-6 \cdot \left(z \cdot x\right) + 6 \cdot \color{blue}{\left(z \cdot y\right)}\right) \]
   associate-*l* [<=] 0.2	\[ x + \left(-6 \cdot \left(z \cdot x\right) + \color{blue}{\left(6 \cdot z\right) \cdot y}\right) \]
   fma-def [=>] 0.2	\[ x + \color{blue}{\mathsf{fma}\left(-6, z \cdot x, \left(6 \cdot z\right) \cdot y\right)} \]
   *-commutative [=>] 0.2	\[ x + \mathsf{fma}\left(-6, z \cdot x, \color{blue}{\left(z \cdot 6\right)} \cdot y\right) \]
   associate-*r* [<=] 0.2	\[ x + \mathsf{fma}\left(-6, z \cdot x, \color{blue}{z \cdot \left(6 \cdot y\right)}\right) \]

4. Final simplification 0.2

   \[\leadsto x + \mathsf{fma}\left(-6, x \cdot z, z \cdot \left(6 \cdot y\right)\right) \]

Alternatives

Alternative 1
Error	0.2
Cost	6976

\[x + z \cdot \mathsf{fma}\left(6, y, x \cdot -6\right) \]

Alternative 2
Error	8.7
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -1.25 \cdot 10^{+28} \lor \neg \left(y \leq 2.3 \cdot 10^{-167}\right):\\ \;\;\;\;x + 6 \cdot \left(z \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x + -6 \cdot \left(x \cdot z\right)\\ \end{array} \]

Alternative 3
Error	8.6
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq -1.3 \cdot 10^{+23}:\\ \;\;\;\;x + y \cdot \left(z \cdot 6\right)\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{-165}:\\ \;\;\;\;x + -6 \cdot \left(x \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x + 6 \cdot \left(z \cdot y\right)\\ \end{array} \]

Alternative 4
Error	8.6
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq -1.3 \cdot 10^{+23}:\\ \;\;\;\;x + y \cdot \left(z \cdot 6\right)\\ \mathbf{elif}\;y \leq 9 \cdot 10^{-167}:\\ \;\;\;\;x + x \cdot \left(-6 \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x + 6 \cdot \left(z \cdot y\right)\\ \end{array} \]

Alternative 5
Error	8.6
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq -1.3 \cdot 10^{+23}:\\ \;\;\;\;x + y \cdot \left(z \cdot 6\right)\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{-164}:\\ \;\;\;\;x + x \cdot \left(-6 \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \left(6 \cdot y\right)\\ \end{array} \]

Alternative 6
Error	0.2
Cost	576

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

Alternative 7
Error	23.7
Cost	448

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

Alternative 8
Error	35.4
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023010 
(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)))