Average Error: 17.8 → 0.0
Time: 1.5s
Precision: 64
\[\left(\left(x \cdot y + y \cdot y\right) - y \cdot z\right) - y \cdot y\]
↓
\[\mathsf{fma}\left(y, x - z, 0\right)\]
\left(\left(x \cdot y + y \cdot y\right) - y \cdot z\right) - y \cdot y
↓
\mathsf{fma}\left(y, x - z, 0\right)
double f(double x, double y, double z) {
        double r569853 = x;
        double r569854 = y;
        double r569855 = r569853 * r569854;
        double r569856 = r569854 * r569854;
        double r569857 = r569855 + r569856;
        double r569858 = z;
        double r569859 = r569854 * r569858;
        double r569860 = r569857 - r569859;
        double r569861 = r569860 - r569856;
        return r569861;
}
↓
double f(double x, double y, double z) {
        double r569862 = y;
        double r569863 = x;
        double r569864 = z;
        double r569865 = r569863 - r569864;
        double r569866 = 0.0;
        double r569867 = fma(r569862, r569865, r569866);
        return r569867;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original17.8
Target0.0
Herbie0.0
\[\left(x - z\right) \cdot y\]

Derivation

  1. Initial program 17.8

    \[\left(\left(x \cdot y + y \cdot y\right) - y \cdot z\right) - y \cdot y\]

  2. Simplified0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(y, x - z, 0\right)}\]

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2020059 +o rules:numerics
(FPCore (x y z)
  :name "Linear.Quaternion:$c/ from linear-1.19.1.3, C"
  :precision binary64

  :herbie-target
  (* (- x z) y)

  (- (- (+ (* x y) (* y y)) (* y z)) (* y y)))