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

Average Error: 0.2 → 0.2

Time: 5.7s

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	0.2
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. Simplified 0.2

   \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, 6 \cdot z, x\right)} \]
   Proof
   (fma.f64 (-.f64 y x) (*.f64 6 z) x): 0 points increase in error, 0 points decrease in error
   (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (-.f64 y x) (*.f64 6 z)) x)): 2 points increase in error, 0 points decrease in error
   (+.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (-.f64 y x) 6) z)) x): 27 points increase in error, 22 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. Final simplification 0.2

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

Alternatives

Alternative 1
Error	12.6
Cost	712

\[\begin{array}{l} t_0 := 6 \cdot \left(\left(y - x\right) \cdot z\right)\\ \mathbf{if}\;z \leq -3 \cdot 10^{-69}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 6.2 \cdot 10^{-15}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	12.7
Cost	712

\[\begin{array}{l} t_0 := z \cdot \left(\left(y - x\right) \cdot 6\right)\\ \mathbf{if}\;z \leq -4.15 \cdot 10^{-72}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 9.8 \cdot 10^{-15}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	12.6
Cost	712

\[\begin{array}{l} t_0 := z \cdot \left(\left(y - x\right) \cdot 6\right)\\ \mathbf{if}\;z \leq -2.3 \cdot 10^{-71}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 6.6 \cdot 10^{-15}:\\ \;\;\;\;x \cdot \left(z \cdot -6 + 1\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	12.7
Cost	712

\[\begin{array}{l} t_0 := z \cdot \left(\left(y - x\right) \cdot 6\right)\\ \mathbf{if}\;z \leq -4.1 \cdot 10^{-72}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 7.8 \cdot 10^{-15}:\\ \;\;\;\;x + x \cdot \left(z \cdot -6\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	1.1
Cost	712

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

Alternative 6
Error	24.2
Cost	584

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

Alternative 7
Error	24.2
Cost	584

\[\begin{array}{l} \mathbf{if}\;z \leq -0.165:\\ \;\;\;\;x \cdot \left(z \cdot -6\right)\\ \mathbf{elif}\;z \leq 2.4 \cdot 10^{-6}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;-6 \cdot \left(x \cdot z\right)\\ \end{array} \]

Alternative 8
Error	24.2
Cost	584

\[\begin{array}{l} \mathbf{if}\;z \leq -0.165:\\ \;\;\;\;x \cdot \left(z \cdot -6\right)\\ \mathbf{elif}\;z \leq 2.4 \cdot 10^{-6}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(x \cdot -6\right)\\ \end{array} \]

Alternative 9
Error	0.2
Cost	576

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

Alternative 10
Error	0.2
Cost	576

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

Alternative 11
Error	35.1
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022325 
(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)))