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

Average Error: 0.2 → 0.2

Time: 4.3s

Precision: binary64

Math

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

↓

\[\mathsf{fma}\left(y \cdot x, 3, \mathsf{fma}\left(y, -0.41379310344827586, y \cdot 0\right)\right) \]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (fma (* y x) 3.0 (fma y -0.41379310344827586 (* y 0.0))))

C

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

↓

double code(double x, double y) {
	return fma((y * x), 3.0, fma(y, -0.41379310344827586, (y * 0.0)));
}

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 0.4

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

5. Applied egg-rr 0.2

   \[\leadsto \mathsf{fma}\left(y \cdot x, 3, \color{blue}{\mathsf{fma}\left(y, -0.41379310344827586, y \cdot 0\right) \cdot 1}\right) \]

6. Final simplification 0.2

   \[\leadsto \mathsf{fma}\left(y \cdot x, 3, \mathsf{fma}\left(y, -0.41379310344827586, y \cdot 0\right)\right) \]

Reproduce

herbie shell --seed 2022130 
(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))