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

Average Error: 0.0 → 0.0

Time: 2.0s

Precision: binary64

Cost: 6720

Math

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

↓

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

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 0.0

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

2. Simplified 0.0

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, 1 - y, y\right)} \]
   Proof
   (fma.f64 x (-.f64 1 y) y): 0 points increase in error, 0 points decrease in error
   (fma.f64 x (Rewrite<= unsub-neg_binary64 (+.f64 1 (neg.f64 y))) y): 0 points increase in error, 0 points decrease in error
   (fma.f64 x (Rewrite<= +-commutative_binary64 (+.f64 (neg.f64 y) 1)) y): 0 points increase in error, 0 points decrease in error
   (Rewrite<= fma-def_binary64 (+.f64 (*.f64 x (+.f64 (neg.f64 y) 1)) y)): 3 points increase in error, 0 points decrease in error
   (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (+.f64 (neg.f64 y) 1) x)) y): 0 points increase in error, 0 points decrease in error
   (+.f64 (Rewrite<= distribute-rgt1-in_binary64 (+.f64 x (*.f64 (neg.f64 y) x))) y): 0 points increase in error, 0 points decrease in error
   (+.f64 (+.f64 x (Rewrite<= distribute-lft-neg-in_binary64 (neg.f64 (*.f64 y x)))) y): 0 points increase in error, 0 points decrease in error
   (+.f64 (+.f64 x (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 x y)))) y): 0 points increase in error, 0 points decrease in error
   (+.f64 (+.f64 x (Rewrite<= distribute-lft-neg-out_binary64 (*.f64 (neg.f64 x) y))) y): 0 points increase in error, 0 points decrease in error
   (Rewrite=> associate-+l+_binary64 (+.f64 x (+.f64 (*.f64 (neg.f64 x) y) y))): 0 points increase in error, 0 points decrease in error
   (+.f64 x (Rewrite<= +-commutative_binary64 (+.f64 y (*.f64 (neg.f64 x) y)))): 0 points increase in error, 0 points decrease in error
   (+.f64 x (Rewrite<= cancel-sign-sub-inv_binary64 (-.f64 y (*.f64 x y)))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= associate--l+_binary64 (-.f64 (+.f64 x y) (*.f64 x y))): 0 points increase in error, 0 points decrease in error

3. Final simplification 0.0

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

Alternatives

Alternative 1
Error	23.8
Cost	1048

\[\begin{array}{l} t_0 := x \cdot \left(-y\right)\\ \mathbf{if}\;x \leq -2.15 \cdot 10^{+105}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -8 \cdot 10^{+61}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -6.4 \cdot 10^{-21}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -1.3 \cdot 10^{-65}:\\ \;\;\;\;y\\ \mathbf{elif}\;x \leq -1.5 \cdot 10^{-87}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	19.3
Cost	584

\[\begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{-92}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - x\right)\\ \end{array} \]

Alternative 3
Error	19.0
Cost	452

\[\begin{array}{l} \mathbf{if}\;y \leq 7 \cdot 10^{-93}:\\ \;\;\;\;x - x \cdot y\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - x\right)\\ \end{array} \]

Alternative 4
Error	0.0
Cost	448

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

Alternative 5
Error	27.8
Cost	196

\[\begin{array}{l} \mathbf{if}\;y \leq 1.75 \cdot 10^{-96}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

Alternative 6
Error	36.1
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022331 
(FPCore (x y)
  :name "Data.Colour.RGBSpace.HSL:hsl from colour-2.3.3, A"
  :precision binary64
  (- (+ x y) (* x y)))