Data.Colour.CIE:cieLAB from colour-2.3.3, A

Average Error: 0.2 → 0.2

Time: 2.2s

Precision: binary64

Math

\[\left(\left(x - \frac{16}{116}\right) \cdot 3\right) \cdot y \]

↓

\[\mathsf{fma}\left(3 \cdot y, x, y \cdot -0.41379310344827586\right) \]

FPCore

(FPCore (x y) :precision binary64 (* (* (- x (/ 16.0 116.0)) 3.0) y))

↓

(FPCore (x y) :precision binary64 (fma (* 3.0 y) x (* y -0.41379310344827586)))

C

double code(double x, double y) {
	return ((x - (16.0 / 116.0)) * 3.0) * y;
}

↓

double code(double x, double y) {
	return fma((3.0 * y), x, (y * -0.41379310344827586));
}

Julia

function code(x, y)
	return Float64(Float64(Float64(x - Float64(16.0 / 116.0)) * 3.0) * y)
end

↓

function code(x, y)
	return fma(Float64(3.0 * y), x, Float64(y * -0.41379310344827586))
end

Wolfram

code[x_, y_] := N[(N[(N[(x - N[(16.0 / 116.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision] * y), $MachinePrecision]

↓

code[x_, y_] := N[(N[(3.0 * y), $MachinePrecision] * x + N[(y * -0.41379310344827586), $MachinePrecision]), $MachinePrecision]

Error

Bits error versus x

Bits error versus y

Target

Original	0.2
Target	0.2
Herbie	0.2

\[y \cdot \left(x \cdot 3 - 0.41379310344827586\right) \]

Derivation

1. Initial program 0.2

   \[\left(\left(x - \frac{16}{116}\right) \cdot 3\right) \cdot y \]

2. Simplified 0.2

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, 3, -0.41379310344827586\right) \cdot y} \]

3. Taylor expanded in x around 0 0.2

   \[\leadsto \color{blue}{3 \cdot \left(y \cdot x\right) - 0.41379310344827586 \cdot y} \]

4. Applied egg-rr 0.2

   \[\leadsto \color{blue}{\mathsf{fma}\left(3 \cdot y, x, y \cdot -0.41379310344827586\right)} \]

5. Final simplification 0.2

   \[\leadsto \mathsf{fma}\left(3 \cdot y, x, y \cdot -0.41379310344827586\right) \]

Reproduce

herbie shell --seed 2022153 
(FPCore (x y)
  :name "Data.Colour.CIE:cieLAB from colour-2.3.3, A"
  :precision binary64

  :herbie-target
  (* y (- (* x 3.0) 0.41379310344827586))

  (* (* (- x (/ 16.0 116.0)) 3.0) y))