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

Average Error: 0.33% → 0.26%

Time: 7.3s

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	0.33%
Target	0.26%
Herbie	0.26%

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

Derivation

1. Initial program 0.33

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

2. Simplified 0.26

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

   [Start] 0.33	\[ x + \left(\left(y - x\right) \cdot 6\right) \cdot z \]
   +-commutative [=>] 0.33	\[ \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot z + x} \]
   associate-*l* [=>] 0.26	\[ \color{blue}{\left(y - x\right) \cdot \left(6 \cdot z\right)} + x \]
   fma-def [=>] 0.26	\[ \color{blue}{\mathsf{fma}\left(y - x, 6 \cdot z, x\right)} \]

3. Final simplification 0.26

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

Alternatives

Alternative 1
Error	39.37%
Cost	980

\[\begin{array}{l} t_0 := 6 \cdot \left(y \cdot z\right)\\ \mathbf{if}\;z \leq -5.9 \cdot 10^{-49}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -5.4 \cdot 10^{-67}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -2.8 \cdot 10^{-80}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 8.5 \cdot 10^{-83}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 9.2 \cdot 10^{+179}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-6 \cdot \left(x \cdot z\right)\\ \end{array} \]

Alternative 2
Error	39.6%
Cost	980

\[\begin{array}{l} t_0 := y \cdot \left(6 \cdot z\right)\\ \mathbf{if}\;z \leq -1.9 \cdot 10^{-48}:\\ \;\;\;\;6 \cdot \left(y \cdot z\right)\\ \mathbf{elif}\;z \leq -1.4 \cdot 10^{-64}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -1.55 \cdot 10^{-85}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 1.22 \cdot 10^{-82}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 8.5 \cdot 10^{+179}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-6 \cdot \left(x \cdot z\right)\\ \end{array} \]

Alternative 3
Error	21.81%
Cost	978

\[\begin{array}{l} \mathbf{if}\;z \leq -5 \cdot 10^{-51} \lor \neg \left(z \leq -2.25 \cdot 10^{-63}\right) \land \left(z \leq -1.5 \cdot 10^{-84} \lor \neg \left(z \leq 7.2 \cdot 10^{-83}\right)\right):\\ \;\;\;\;\left(y - x\right) \cdot \left(6 \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 4
Error	21.74%
Cost	977

\[\begin{array}{l} t_0 := \left(y - x\right) \cdot \left(6 \cdot z\right)\\ \mathbf{if}\;z \leq -5.5 \cdot 10^{-50}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -1.45 \cdot 10^{-63}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -3.2 \cdot 10^{-84} \lor \neg \left(z \leq 9.5 \cdot 10^{-83}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + z \cdot -6\right)\\ \end{array} \]

Alternative 5
Error	2.17%
Cost	713

\[\begin{array}{l} \mathbf{if}\;z \leq -0.142 \lor \neg \left(z \leq 8.6 \cdot 10^{-13}\right):\\ \;\;\;\;\left(y - x\right) \cdot \left(6 \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \left(y \cdot 6\right)\\ \end{array} \]

Alternative 6
Error	38.85%
Cost	585

\[\begin{array}{l} \mathbf{if}\;z \leq -4.5 \cdot 10^{-5} \lor \neg \left(z \leq 8.6 \cdot 10^{-13}\right):\\ \;\;\;\;-6 \cdot \left(x \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 7
Error	0.33%
Cost	576

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

Alternative 8
Error	0.26%
Cost	576

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

Alternative 9
Error	55.52%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023121 
(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)))