Average Error: 0.2 → 0.2
Time: 15.4s
Precision: 64
\[\left(\left(x - \frac{16.0}{116.0}\right) \cdot 3.0\right) \cdot y\]
↓
\[\mathsf{fma}\left(3.0, x, -0.41379310344827586\right) \cdot y\]
\left(\left(x - \frac{16.0}{116.0}\right) \cdot 3.0\right) \cdot y
↓
\mathsf{fma}\left(3.0, x, -0.41379310344827586\right) \cdot y
double f(double x, double y) {
        double r36320868 = x;
        double r36320869 = 16.0;
        double r36320870 = 116.0;
        double r36320871 = r36320869 / r36320870;
        double r36320872 = r36320868 - r36320871;
        double r36320873 = 3.0;
        double r36320874 = r36320872 * r36320873;
        double r36320875 = y;
        double r36320876 = r36320874 * r36320875;
        return r36320876;
}
↓
double f(double x, double y) {
        double r36320877 = 3.0;
        double r36320878 = x;
        double r36320879 = 0.41379310344827586;
        double r36320880 = -r36320879;
        double r36320881 = fma(r36320877, r36320878, r36320880);
        double r36320882 = y;
        double r36320883 = r36320881 * r36320882;
        return r36320883;
}

Error

Bits error versus x

Bits error versus y

Target

Original0.2
Target0.2
Herbie0.2
\[y \cdot \left(x \cdot 3.0 - 0.41379310344827586\right)\]

Derivation

  1. Initial program 0.2

    \[\left(\left(x - \frac{16.0}{116.0}\right) \cdot 3.0\right) \cdot y\]

  2. Taylor expanded around 0 0.2

    \[\leadsto \color{blue}{\left(3.0 \cdot x - 0.41379310344827586\right)} \cdot y\]
  3. Using strategy rm

  4. Applied fma-neg0.2

    \[\leadsto \color{blue}{\mathsf{fma}\left(3.0, x, -0.41379310344827586\right)} \cdot y\]

  5. Final simplification0.2

    \[\leadsto \mathsf{fma}\left(3.0, x, -0.41379310344827586\right) \cdot y\]

Reproduce

herbie shell --seed 2019164 +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))