Average Error: 0.2 → 0.2
Time: 2.5s
Precision: binary64
\[\left(\left(x - \frac{16}{116}\right) \cdot 3\right) \cdot y \]
↓
\[\mathsf{fma}\left(3, y \cdot x, y \cdot -0.41379310344827586\right) \]
\left(\left(x - \frac{16}{116}\right) \cdot 3\right) \cdot y
↓
\mathsf{fma}\left(3, y \cdot x, y \cdot -0.41379310344827586\right)

(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)))
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));
}

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. Taylor expanded in x around 0 0.2

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

  3. Applied fma-neg_binary640.2

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

  4. Simplified0.2

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

  5. Final simplification0.2

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

Reproduce

herbie shell --seed 2022077 
(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))