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

Average Error: 0.01% → 0.01%

Time: 2.8s

Precision: binary64

Cost: 6720

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 0.01

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

2. Simplified 0.01

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

   [Start] 0.01	\[ \left(x + y\right) - x \cdot y \]
   associate--l+ [=>] 0.01	\[ \color{blue}{x + \left(y - x \cdot y\right)} \]
   +-commutative [=>] 0.01	\[ \color{blue}{\left(y - x \cdot y\right) + x} \]
   cancel-sign-sub-inv [=>] 0.01	\[ \color{blue}{\left(y + \left(-x\right) \cdot y\right)} + x \]
   distribute-rgt1-in [=>] 0.01	\[ \color{blue}{\left(\left(-x\right) + 1\right) \cdot y} + x \]
   *-commutative [=>] 0.01	\[ \color{blue}{y \cdot \left(\left(-x\right) + 1\right)} + x \]
   fma-def [=>] 0.01	\[ \color{blue}{\mathsf{fma}\left(y, \left(-x\right) + 1, x\right)} \]
   +-commutative [=>] 0.01	\[ \mathsf{fma}\left(y, \color{blue}{1 + \left(-x\right)}, x\right) \]
   unsub-neg [=>] 0.01	\[ \mathsf{fma}\left(y, \color{blue}{1 - x}, x\right) \]

3. Final simplification 0.01

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

Alternatives

Alternative 1
Error	37.75%
Cost	1048

\[\begin{array}{l} t_0 := y \cdot \left(-x\right)\\ \mathbf{if}\;x \leq -7.1 \cdot 10^{+168}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -7.2 \cdot 10^{+128}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-28}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -9.2 \cdot 10^{-73}:\\ \;\;\;\;y\\ \mathbf{elif}\;x \leq -2.8 \cdot 10^{-111}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	15.95%
Cost	850

\[\begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-28} \lor \neg \left(x \leq -3.2 \cdot 10^{-74} \lor \neg \left(x \leq -2 \cdot 10^{-111}\right) \land x \leq 8 \cdot 10^{-5}\right):\\ \;\;\;\;x \cdot \left(1 - y\right)\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

Alternative 3
Error	28.98%
Cost	716

\[\begin{array}{l} t_0 := x \cdot \left(1 - y\right)\\ \mathbf{if}\;x \leq -1.7 \cdot 10^{-28}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -6.5 \cdot 10^{-74}:\\ \;\;\;\;y\\ \mathbf{elif}\;x \leq -2.8 \cdot 10^{-111}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;y - y \cdot x\\ \end{array} \]

Alternative 4
Error	43.7%
Cost	460

\[\begin{array}{l} \mathbf{if}\;x \leq -3.6 \cdot 10^{-28}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -1.36 \cdot 10^{-70}:\\ \;\;\;\;y\\ \mathbf{elif}\;x \leq -2.7 \cdot 10^{-111}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

Alternative 5
Error	0.01%
Cost	448

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

Alternative 6
Error	57.15%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023090 
(FPCore (x y)
  :name "Data.Colour.RGBSpace.HSL:hsl from colour-2.3.3, A"
  :precision binary64
  (- (+ x y) (* x y)))