Data.Colour.SRGB:transferFunction from colour-2.3.3

Average Error: 0.0 → 0.0

Time: 2.3s

Precision: binary64

Cost: 6720

Math

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

↓

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

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 fma(x, y, Float64(y - x))
end

Wolfram

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

↓

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

Derivation

1. Initial program 0.0

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

2. Simplified 0.0

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

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

3. Final simplification 0.0

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

Alternatives

Alternative 1
Error	9.8
Cost	848

\[\begin{array}{l} t_0 := y \cdot \left(x + 1\right)\\ \mathbf{if}\;y \leq -3.2 \cdot 10^{-23}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 9 \cdot 10^{-64}:\\ \;\;\;\;-x\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{-25}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 4.1 \cdot 10^{-9}:\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	9.7
Cost	848

\[\begin{array}{l} t_0 := y \cdot \left(x + 1\right)\\ t_1 := x \cdot \left(y + -1\right)\\ \mathbf{if}\;y \leq -3.85 \cdot 10^{-25}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 8.2 \cdot 10^{-66}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-25}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 2800000:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	18.8
Cost	656

\[\begin{array}{l} \mathbf{if}\;y \leq -2.1 \cdot 10^{-23}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 6.6 \cdot 10^{-62}:\\ \;\;\;\;-x\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{-24}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 2 \cdot 10^{-8}:\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

Alternative 4
Error	0.0
Cost	448

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

Alternative 5
Error	36.6
Cost	64

\[y \]

Reproduce

herbie shell --seed 2023040 
(FPCore (x y)
  :name "Data.Colour.SRGB:transferFunction from colour-2.3.3"
  :precision binary64
  (- (* (+ x 1.0) y) x))