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

Percentage Accurate: 100.0% → 100.0%

Time: 2.0s

Precision: binary64

Cost: 6720

• Unsound rule application detected in e-graph. Results may not be sound.

Math

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

↓

\[\begin{array}{l} \\ \mathsf{fma}\left(x, 1 - y, y\right) \end{array} \]

FPCore

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

↓

(FPCore (x y) :precision binary64 (fma x (- 1.0 y) y))

C

double code(double x, double y) {
	return (x + y) - (x * y);
}

↓

double code(double x, double y) {
	return fma(x, (1.0 - y), y);
}

Julia

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

↓

function code(x, y)
	return fma(x, Float64(1.0 - y), y)
end

Wolfram

code[x_, y_] := N[(N[(x + y), $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_] := N[(x * N[(1.0 - y), $MachinePrecision] + y), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

2. Simplified 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, 1 - y, y\right)} \]
   Step-by-step derivation

   [Start] 100.0%	\[ \left(x + y\right) - x \cdot y \]
   +-commutative [=>] 100.0%	\[ \color{blue}{\left(y + x\right)} - x \cdot y \]
   associate--l+ [=>] 100.0%	\[ \color{blue}{y + \left(x - x \cdot y\right)} \]
   +-commutative [=>] 100.0%	\[ \color{blue}{\left(x - x \cdot y\right) + y} \]
   *-rgt-identity [<=] 100.0%	\[ \left(\color{blue}{x \cdot 1} - x \cdot y\right) + y \]
   distribute-lft-out-- [=>] 100.0%	\[ \color{blue}{x \cdot \left(1 - y\right)} + y \]
   unsub-neg [<=] 100.0%	\[ x \cdot \color{blue}{\left(1 + \left(-y\right)\right)} + y \]
   +-commutative [<=] 100.0%	\[ x \cdot \color{blue}{\left(\left(-y\right) + 1\right)} + y \]
   fma-def [=>] 100.0%	\[ \color{blue}{\mathsf{fma}\left(x, \left(-y\right) + 1, y\right)} \]
   +-commutative [=>] 100.0%	\[ \mathsf{fma}\left(x, \color{blue}{1 + \left(-y\right)}, y\right) \]
   unsub-neg [=>] 100.0%	\[ \mathsf{fma}\left(x, \color{blue}{1 - y}, y\right) \]

3. Final simplification 100.0%

   \[\leadsto \mathsf{fma}\left(x, 1 - y, y\right) \]

Alternatives

Alternative 1
Accuracy	100.0%
Cost	6720

\[\mathsf{fma}\left(x, 1 - y, y\right) \]

Alternative 2
Accuracy	62.0%
Cost	1180

\[\begin{array}{l} t_0 := y \cdot \left(-x\right)\\ \mathbf{if}\;x \leq -7 \cdot 10^{+230}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -1.1 \cdot 10^{+182}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -3.1 \cdot 10^{+155}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -3.4 \cdot 10^{-65}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -3 \cdot 10^{-102}:\\ \;\;\;\;y\\ \mathbf{elif}\;x \leq -1.95 \cdot 10^{-125}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Accuracy	71.2%
Cost	584

\[\begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;y \leq 4.4 \cdot 10^{-153}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - x\right)\\ \end{array} \]

Alternative 4
Accuracy	71.6%
Cost	452

\[\begin{array}{l} \mathbf{if}\;y \leq 4.4 \cdot 10^{-153}:\\ \;\;\;\;x - x \cdot y\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - x\right)\\ \end{array} \]

Alternative 5
Accuracy	100.0%
Cost	448

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

Alternative 6
Accuracy	47.4%
Cost	196

\[\begin{array}{l} \mathbf{if}\;y \leq 4.4 \cdot 10^{-153}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

Alternative 7
Accuracy	37.5%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023167 
(FPCore (x y)
  :name "Data.Colour.RGBSpace.HSL:hsl from colour-2.3.3, A"
  :precision binary64
  (- (+ x y) (* x y)))