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

Average Error: 0.3 → 0.3

Time: 8.4s

Precision: binary64

Cost: 6976

Math

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

↓

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

FPCore

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

↓

(FPCore (x y z) :precision binary64 (+ x (* (fma 6.0 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 + (fma(6.0, 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(fma(6.0, y, Float64(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[(6.0 * y + N[(x * -6.0), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]

Target

Original	0.3
Target	0.2
Herbie	0.3

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

Derivation

1. Initial program 0.3

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

2. Taylor expanded in y around 0 0.3

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

3. Simplified 0.3

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

   [Start] 0.3	\[ x + \left(6 \cdot y + -6 \cdot x\right) \cdot z \]
   fma-def [=>] 0.3	\[ x + \color{blue}{\mathsf{fma}\left(6, y, -6 \cdot x\right)} \cdot z \]

4. Final simplification 0.3

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

Alternatives

Alternative 1
Error	23.9
Cost	849

\[\begin{array}{l} t_0 := 6 \cdot \left(y \cdot z\right)\\ \mathbf{if}\;z \leq -6 \cdot 10^{-70}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 0.00065:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 1.36 \cdot 10^{+124} \lor \neg \left(z \leq 2 \cdot 10^{+178}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-6 \cdot \left(x \cdot z\right)\\ \end{array} \]

Alternative 2
Error	24.0
Cost	849

\[\begin{array}{l} \mathbf{if}\;z \leq -6.5 \cdot 10^{-70}:\\ \;\;\;\;y \cdot \left(6 \cdot z\right)\\ \mathbf{elif}\;z \leq 0.00086:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{+134} \lor \neg \left(z \leq 5.6 \cdot 10^{+177}\right):\\ \;\;\;\;6 \cdot \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;-6 \cdot \left(x \cdot z\right)\\ \end{array} \]

Alternative 3
Error	24.0
Cost	849

\[\begin{array}{l} \mathbf{if}\;z \leq -6.5 \cdot 10^{-70}:\\ \;\;\;\;z \cdot \left(6 \cdot y\right)\\ \mathbf{elif}\;z \leq 0.0065:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 2.7 \cdot 10^{+125} \lor \neg \left(z \leq 1.25 \cdot 10^{+178}\right):\\ \;\;\;\;6 \cdot \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;-6 \cdot \left(x \cdot z\right)\\ \end{array} \]

Alternative 4
Error	12.6
Cost	713

\[\begin{array}{l} \mathbf{if}\;z \leq -6.5 \cdot 10^{-70} \lor \neg \left(z \leq 0.00065\right):\\ \;\;\;\;6 \cdot \left(z \cdot \left(y - x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5
Error	12.6
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -5.2 \cdot 10^{-70}:\\ \;\;\;\;z \cdot \left(6 \cdot \left(y - x\right)\right)\\ \mathbf{elif}\;z \leq 0.00065:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;6 \cdot \left(z \cdot \left(y - x\right)\right)\\ \end{array} \]

Alternative 6
Error	12.7
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -6.5 \cdot 10^{-70}:\\ \;\;\;\;z \cdot \left(6 \cdot \left(y - x\right)\right)\\ \mathbf{elif}\;z \leq 86000000000000:\\ \;\;\;\;x \cdot \left(1 + -6 \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;6 \cdot \left(z \cdot \left(y - x\right)\right)\\ \end{array} \]

Alternative 7
Error	1.0
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -24500:\\ \;\;\;\;z \cdot \left(6 \cdot \left(y - x\right)\right)\\ \mathbf{elif}\;z \leq 0.165:\\ \;\;\;\;x + 6 \cdot \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;6 \cdot \left(z \cdot \left(y - x\right)\right)\\ \end{array} \]

Alternative 8
Error	24.2
Cost	585

\[\begin{array}{l} \mathbf{if}\;z \leq -0.17 \lor \neg \left(z \leq 0.165\right):\\ \;\;\;\;-6 \cdot \left(x \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 9
Error	0.3
Cost	576

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

Alternative 10
Error	34.8
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023016 
(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)))