Data.Colour.CIE:cieLABView from colour-2.3.3, B

Average Error: 0.03% → 0.01%

Time: 1.9s

Precision: binary64

Cost: 6720

Math

\[500 \cdot \left(x - y\right) \]

↓

\[\mathsf{fma}\left(500, x, -500 \cdot y\right) \]

FPCore

(FPCore (x y) :precision binary64 (* 500.0 (- x y)))

↓

(FPCore (x y) :precision binary64 (fma 500.0 x (* -500.0 y)))

C

double code(double x, double y) {
	return 500.0 * (x - y);
}

↓

double code(double x, double y) {
	return fma(500.0, x, (-500.0 * y));
}

Julia

function code(x, y)
	return Float64(500.0 * Float64(x - y))
end

↓

function code(x, y)
	return fma(500.0, x, Float64(-500.0 * y))
end

Wolfram

code[x_, y_] := N[(500.0 * N[(x - y), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_] := N[(500.0 * x + N[(-500.0 * y), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 0.03

   \[500 \cdot \left(x - y\right) \]

2. Taylor expanded in x around 0 0.03

   \[\leadsto \color{blue}{500 \cdot x + -500 \cdot y} \]

3. Simplified 0.01

   \[\leadsto \color{blue}{\mathsf{fma}\left(500, x, -500 \cdot y\right)} \]
   Proof

   [Start] 0.03	\[ 500 \cdot x + -500 \cdot y \]
   fma-def [=>] 0.01	\[ \color{blue}{\mathsf{fma}\left(500, x, -500 \cdot y\right)} \]

4. Final simplification 0.01

   \[\leadsto \mathsf{fma}\left(500, x, -500 \cdot y\right) \]

Alternatives

Alternative 1
Error	26.35%
Cost	721

\[\begin{array}{l} \mathbf{if}\;x \leq -30000000:\\ \;\;\;\;500 \cdot x\\ \mathbf{elif}\;x \leq 1.12 \cdot 10^{-76} \lor \neg \left(x \leq 2.6 \cdot 10^{+116}\right) \land x \leq 7 \cdot 10^{+137}:\\ \;\;\;\;-500 \cdot y\\ \mathbf{else}:\\ \;\;\;\;500 \cdot x\\ \end{array} \]

Alternative 2
Error	0.03%
Cost	448

\[-500 \cdot y + 500 \cdot x \]

Alternative 3
Error	0.03%
Cost	320

\[500 \cdot \left(x - y\right) \]

Alternative 4
Error	49.3%
Cost	192

\[-500 \cdot y \]

Reproduce

herbie shell --seed 2023090 
(FPCore (x y)
  :name "Data.Colour.CIE:cieLABView from colour-2.3.3, B"
  :precision binary64
  (* 500.0 (- x y)))