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

Average Error: 0.0 → 0.0

Time: 2.5s

Precision: binary64

Cost: 6720

\[ \begin{array}{c}[x, y] = \mathsf{sort}([x, y])\\ \end{array} \]

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

Julia

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

↓

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

Wolfram

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

↓

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

Derivation

1. Initial program 0.0

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

2. Simplified 0.0

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

   [Start] 0.0	\[ \left(x + y\right) - x \cdot y \]
   associate--l+ [=>] 0.0	\[ \color{blue}{x + \left(y - x \cdot y\right)} \]
   +-commutative [=>] 0.0	\[ \color{blue}{\left(y - x \cdot y\right) + x} \]
   cancel-sign-sub-inv [=>] 0.0	\[ \color{blue}{\left(y + \left(-x\right) \cdot y\right)} + x \]
   distribute-rgt1-in [=>] 0.0	\[ \color{blue}{\left(\left(-x\right) + 1\right) \cdot y} + x \]
   *-commutative [=>] 0.0	\[ \color{blue}{y \cdot \left(\left(-x\right) + 1\right)} + x \]
   fma-def [=>] 0.0	\[ \color{blue}{\mathsf{fma}\left(y, \left(-x\right) + 1, x\right)} \]
   +-commutative [=>] 0.0	\[ \mathsf{fma}\left(y, \color{blue}{1 + \left(-x\right)}, x\right) \]
   unsub-neg [=>] 0.0	\[ \mathsf{fma}\left(y, \color{blue}{1 - x}, x\right) \]

3. Final simplification 0.0

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

Alternatives

Alternative 1
Error	17.1
Cost	784

\[\begin{array}{l} t_0 := y \cdot \left(-x\right)\\ \mathbf{if}\;y \leq -1:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.65 \cdot 10^{-37}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{+110}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 1.06 \cdot 10^{+170}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

Alternative 2
Error	8.6
Cost	584

\[\begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;y \leq 4 \cdot 10^{-38}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - x\right)\\ \end{array} \]

Alternative 3
Error	8.2
Cost	452

\[\begin{array}{l} \mathbf{if}\;y \leq 1.9 \cdot 10^{-37}:\\ \;\;\;\;x \cdot \left(1 - y\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - x\right)\\ \end{array} \]

Alternative 4
Error	8.1
Cost	452

\[\begin{array}{l} \mathbf{if}\;y \leq 3.1 \cdot 10^{-38}:\\ \;\;\;\;x - y \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - x\right)\\ \end{array} \]

Alternative 5
Error	0.0
Cost	448

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

Alternative 6
Error	17.7
Cost	196

\[\begin{array}{l} \mathbf{if}\;y \leq 5.6 \cdot 10^{-38}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

Alternative 7
Error	36.3
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023060 
(FPCore (x y)
  :name "Data.Colour.RGBSpace.HSL:hsl from colour-2.3.3, A"
  :precision binary64
  (- (+ x y) (* x y)))