Average Error: 0.2 → 0.2
Time: 2.0s
Precision: binary64
\[\left(\left(x - \frac{16}{116}\right) \cdot 3\right) \cdot y \]
↓
\[y \cdot \mathsf{fma}\left(3, x, -0.41379310344827586\right) \]
\left(\left(x - \frac{16}{116}\right) \cdot 3\right) \cdot y
↓
y \cdot \mathsf{fma}\left(3, x, -0.41379310344827586\right)

(FPCore (x y) :precision binary64 (* (* (- x (/ 16.0 116.0)) 3.0) y))
↓
(FPCore (x y) :precision binary64 (* y (fma 3.0 x -0.41379310344827586)))
double code(double x, double y) {
	return ((x - (16.0 / 116.0)) * 3.0) * y;
}
↓
double code(double x, double y) {
	return y * fma(3.0, x, -0.41379310344827586);
}

Error

Bits error versus x

Bits error versus y

Target

Original0.2
Target0.2
Herbie0.2
\[y \cdot \left(x \cdot 3 - 0.41379310344827586\right) \]

Derivation

  1. Initial program 0.2

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

  2. Simplified0.2

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

  3. Taylor expanded in y around 0 0.2

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

  4. Applied *-un-lft-identity_binary640.2

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

  5. Applied associate-*l*_binary640.2

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

  6. Simplified0.2

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

  7. Final simplification0.2

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

Reproduce

herbie shell --seed 2022088 
(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))