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

Average Error: 0.34% → 0.28%

Time: 11.5s

Precision: binary64

Cost: 6848

Math

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

↓

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

FPCore

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

↓

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

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

Julia

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

↓

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

Target

Original	0.34%
Target	0.28%
Herbie	0.28%

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

Derivation

1. Initial program 0.34

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

2. Simplified 0.28

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

   [Start] 0.34	\[ x + \left(\left(y - x\right) \cdot 6\right) \cdot z \]
   +-commutative [=>] 0.34	\[ \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot z + x} \]
   associate-*l* [=>] 0.28	\[ \color{blue}{\left(y - x\right) \cdot \left(6 \cdot z\right)} + x \]
   fma-def [=>] 0.28	\[ \color{blue}{\mathsf{fma}\left(y - x, 6 \cdot z, x\right)} \]

3. Final simplification 0.28

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

Alternatives

Alternative 1
Error	37.29%
Cost	848

\[\begin{array}{l} t_0 := 6 \cdot \left(y \cdot z\right)\\ t_1 := -6 \cdot \left(x \cdot z\right)\\ \mathbf{if}\;z \leq -1.4 \cdot 10^{+98}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -8.2 \cdot 10^{+27}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -7 \cdot 10^{-57}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 0.165:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	37.32%
Cost	848

\[\begin{array}{l} t_0 := 6 \cdot \left(y \cdot z\right)\\ \mathbf{if}\;z \leq -1.5 \cdot 10^{+99}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -8.5 \cdot 10^{+27}:\\ \;\;\;\;-6 \cdot \left(x \cdot z\right)\\ \mathbf{elif}\;z \leq -2.5 \cdot 10^{-57}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 0.165:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(z \cdot -6\right)\\ \end{array} \]

Alternative 3
Error	37.27%
Cost	848

\[\begin{array}{l} t_0 := y \cdot \left(6 \cdot z\right)\\ \mathbf{if}\;z \leq -5.5 \cdot 10^{+100}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -2.4 \cdot 10^{+27}:\\ \;\;\;\;-6 \cdot \left(x \cdot z\right)\\ \mathbf{elif}\;z \leq -2.35 \cdot 10^{-57}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 0.165:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(z \cdot -6\right)\\ \end{array} \]

Alternative 4
Error	37.29%
Cost	848

\[\begin{array}{l} \mathbf{if}\;z \leq -1.7 \cdot 10^{+98}:\\ \;\;\;\;z \cdot \left(y \cdot 6\right)\\ \mathbf{elif}\;z \leq -2.02 \cdot 10^{+27}:\\ \;\;\;\;-6 \cdot \left(x \cdot z\right)\\ \mathbf{elif}\;z \leq -8 \cdot 10^{-59}:\\ \;\;\;\;y \cdot \left(6 \cdot z\right)\\ \mathbf{elif}\;z \leq 0.165:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(z \cdot -6\right)\\ \end{array} \]

Alternative 5
Error	18.93%
Cost	713

\[\begin{array}{l} \mathbf{if}\;z \leq -2.1 \cdot 10^{-65} \lor \neg \left(z \leq 1.7 \cdot 10^{-39}\right):\\ \;\;\;\;6 \cdot \left(\left(y - x\right) \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 6
Error	18.91%
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -1.25 \cdot 10^{-65}:\\ \;\;\;\;6 \cdot \left(\left(y - x\right) \cdot z\right)\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{-40}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\left(y - x\right) \cdot \left(6 \cdot z\right)\\ \end{array} \]

Alternative 7
Error	1.68%
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -0.185:\\ \;\;\;\;6 \cdot \left(\left(y - x\right) \cdot z\right)\\ \mathbf{elif}\;z \leq 0.165:\\ \;\;\;\;x + z \cdot \left(y \cdot 6\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y - x\right) \cdot \left(6 \cdot z\right)\\ \end{array} \]

Alternative 8
Error	37.98%
Cost	585

\[\begin{array}{l} \mathbf{if}\;z \leq -0.165 \lor \neg \left(z \leq 0.165\right):\\ \;\;\;\;-6 \cdot \left(x \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 9
Error	0.34%
Cost	576

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

Alternative 10
Error	0.28%
Cost	576

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

Alternative 11
Error	54.7%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023103 
(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)))