Data.Colour.SRGB:transferFunction from colour-2.3.3

Average Accuracy: 100.0% → 100.0%

Time: 3.2s

Precision: binary64

Cost: 6720

Math

\[\left(x + 1\right) \cdot y - x \]

↓

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

FPCore

(FPCore (x y) :precision binary64 (- (* (+ x 1.0) y) x))

↓

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

C

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

↓

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

Julia

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

↓

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

Wolfram

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

↓

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

Derivation

1. Initial program 100.0%

   \[\left(x + 1\right) \cdot y - x \]

2. Simplified 100.0%

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

   [Start] 100.0	\[ \left(x + 1\right) \cdot y - x \]
   *-commutative [=>] 100.0	\[ \color{blue}{y \cdot \left(x + 1\right)} - x \]
   distribute-rgt-in [=>] 100.0	\[ \color{blue}{\left(x \cdot y + 1 \cdot y\right)} - x \]
   fma-def [=>] 100.0	\[ \color{blue}{\mathsf{fma}\left(x, y, 1 \cdot y\right)} - x \]
   *-lft-identity [=>] 100.0	\[ \mathsf{fma}\left(x, y, \color{blue}{y}\right) - x \]

3. Final simplification 100.0%

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

Alternatives

Alternative 1
Accuracy	71.1%
Cost	656

\[\begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{+65}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq -3.1 \cdot 10^{+21}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq -4.6 \cdot 10^{-13}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 2.4 \cdot 10^{-27}:\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

Alternative 2
Accuracy	97.9%
Cost	585

\[\begin{array}{l} \mathbf{if}\;y \leq -1.06 \cdot 10^{+21} \lor \neg \left(y \leq 3.1 \cdot 10^{-6}\right):\\ \;\;\;\;y \cdot \left(x + 1\right)\\ \mathbf{else}:\\ \;\;\;\;y - x\\ \end{array} \]

Alternative 3
Accuracy	97.9%
Cost	584

\[\begin{array}{l} \mathbf{if}\;y \leq -1.06 \cdot 10^{+21}:\\ \;\;\;\;y + x \cdot y\\ \mathbf{elif}\;y \leq 3.1 \cdot 10^{-6}:\\ \;\;\;\;y - x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x + 1\right)\\ \end{array} \]

Alternative 4
Accuracy	84.4%
Cost	456

\[\begin{array}{l} \mathbf{if}\;y \leq -1.85 \cdot 10^{+54}:\\ \;\;\;\;y - x\\ \mathbf{elif}\;y \leq -7.5 \cdot 10^{+22}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;y - x\\ \end{array} \]

Alternative 5
Accuracy	100.0%
Cost	448

\[y \cdot \left(x + 1\right) - x \]

Alternative 6
Accuracy	71.3%
Cost	392

\[\begin{array}{l} \mathbf{if}\;y \leq -1.55 \cdot 10^{-13}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 1.52 \cdot 10^{-27}:\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

Alternative 7
Accuracy	43.9%
Cost	64

\[y \]

Reproduce

herbie shell --seed 2023151 
(FPCore (x y)
  :name "Data.Colour.SRGB:transferFunction from colour-2.3.3"
  :precision binary64
  (- (* (+ x 1.0) y) x))