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

Average Error: 0.3 → 0.2

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	0.3
Target	0.2
Herbie	0.2

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

Derivation

1. Initial program 0.3

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

2. 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)): 3 points increase in error, 1 points decrease in error
   (+.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (-.f64 y x) 6) z)) x): 30 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	24.7
Cost	848

\[\begin{array}{l} t_0 := 6 \cdot \left(y \cdot z\right)\\ \mathbf{if}\;x \leq -5.4 \cdot 10^{-88}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -1.8 \cdot 10^{-160}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-181}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{-131}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 2
Error	24.7
Cost	848

\[\begin{array}{l} t_0 := y \cdot \left(6 \cdot z\right)\\ \mathbf{if}\;x \leq -3.05 \cdot 10^{-89}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -1.95 \cdot 10^{-160}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-181}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{-130}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 3
Error	24.6
Cost	848

\[\begin{array}{l} \mathbf{if}\;x \leq -2.55 \cdot 10^{-92}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -1.8 \cdot 10^{-160}:\\ \;\;\;\;z \cdot \left(y \cdot 6\right)\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-181}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{-130}:\\ \;\;\;\;y \cdot \left(6 \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 4
Error	0.9
Cost	840

\[\begin{array}{l} \mathbf{if}\;z \leq -0.15:\\ \;\;\;\;\left(x - y\right) \cdot \left(z \cdot -6\right)\\ \mathbf{elif}\;z \leq 0.165:\\ \;\;\;\;x + 6 \cdot \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;6 \cdot \left(y \cdot z - x \cdot z\right)\\ \end{array} \]

Alternative 5
Error	12.2
Cost	712

\[\begin{array}{l} t_0 := 6 \cdot \left(\left(y - x\right) \cdot z\right)\\ \mathbf{if}\;z \leq -2.6 \cdot 10^{-6}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 3.4 \cdot 10^{-51}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	12.2
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -2.6 \cdot 10^{-6}:\\ \;\;\;\;\left(x - y\right) \cdot \left(z \cdot -6\right)\\ \mathbf{elif}\;z \leq 8.2 \cdot 10^{-51}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;6 \cdot \left(\left(y - x\right) \cdot z\right)\\ \end{array} \]

Alternative 7
Error	0.9
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -0.125:\\ \;\;\;\;\left(x - y\right) \cdot \left(z \cdot -6\right)\\ \mathbf{elif}\;z \leq 0.165:\\ \;\;\;\;x + 6 \cdot \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;6 \cdot \left(\left(y - x\right) \cdot z\right)\\ \end{array} \]

Alternative 8
Error	24.6
Cost	584

\[\begin{array}{l} t_0 := -6 \cdot \left(x \cdot z\right)\\ \mathbf{if}\;z \leq -0.165:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 2.7 \cdot 10^{+22}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 9
Error	0.3
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	34.0
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022329 
(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)))