Average Error: 0.2 → 0.2
Time: 10.3s
Precision: 64
\[\left(\left(x - \frac{16}{116}\right) \cdot 3\right) \cdot y\]
↓
\[\mathsf{fma}\left(x, 3, -0.4137931034482758563264326312491903081536\right) \cdot y\]
\left(\left(x - \frac{16}{116}\right) \cdot 3\right) \cdot y
↓
\mathsf{fma}\left(x, 3, -0.4137931034482758563264326312491903081536\right) \cdot y
double f(double x, double y) {
        double r632540 = x;
        double r632541 = 16.0;
        double r632542 = 116.0;
        double r632543 = r632541 / r632542;
        double r632544 = r632540 - r632543;
        double r632545 = 3.0;
        double r632546 = r632544 * r632545;
        double r632547 = y;
        double r632548 = r632546 * r632547;
        return r632548;
}
↓
double f(double x, double y) {
        double r632549 = x;
        double r632550 = 3.0;
        double r632551 = 0.41379310344827586;
        double r632552 = -r632551;
        double r632553 = fma(r632549, r632550, r632552);
        double r632554 = y;
        double r632555 = r632553 * r632554;
        return r632555;
}

Error

Bits error versus x

Bits error versus y

Target

Original0.2
Target0.2
Herbie0.2
\[y \cdot \left(x \cdot 3 - 0.4137931034482758563264326312491903081536\right)\]

Derivation

  1. Initial program 0.2

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

  2. Taylor expanded around 0 0.2

    \[\leadsto \color{blue}{3 \cdot \left(x \cdot y\right) - 0.4137931034482758563264326312491903081536 \cdot y}\]

  3. Simplified0.2

    \[\leadsto \color{blue}{y \cdot \mathsf{fma}\left(x, 3, -0.4137931034482758563264326312491903081536\right)}\]

  4. Final simplification0.2

    \[\leadsto \mathsf{fma}\left(x, 3, -0.4137931034482758563264326312491903081536\right) \cdot y\]

Reproduce

herbie shell --seed 2019194 +o rules:numerics
(FPCore (x y)
  :name "Data.Colour.CIE:cieLAB from colour-2.3.3, A"

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

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