Average Error: 17.4 → 0.0
Time: 2.2s
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 r788683 = x;
        double r788684 = y;
        double r788685 = r788683 * r788684;
        double r788686 = r788684 * r788684;
        double r788687 = r788685 + r788686;
        double r788688 = z;
        double r788689 = r788684 * r788688;
        double r788690 = r788687 - r788689;
        double r788691 = r788690 - r788686;
        return r788691;
}
↓
double f(double x, double y, double z) {
        double r788692 = y;
        double r788693 = x;
        double r788694 = z;
        double r788695 = r788693 - r788694;
        double r788696 = 0.0;
        double r788697 = fma(r788692, r788695, r788696);
        return r788697;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original17.4
Target0.0
Herbie0.0
\[\left(x - z\right) \cdot y\]

Derivation

  1. Initial program 17.4

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

  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 2020034 +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)))