Average Error: 17.5 → 0.0
Time: 3.2s
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 r510476 = x;
        double r510477 = y;
        double r510478 = r510476 * r510477;
        double r510479 = r510477 * r510477;
        double r510480 = r510478 + r510479;
        double r510481 = z;
        double r510482 = r510477 * r510481;
        double r510483 = r510480 - r510482;
        double r510484 = r510483 - r510479;
        return r510484;
}
↓
double f(double x, double y, double z) {
        double r510485 = y;
        double r510486 = x;
        double r510487 = z;
        double r510488 = r510486 - r510487;
        double r510489 = 0.0;
        double r510490 = fma(r510485, r510488, r510489);
        return r510490;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original17.5
Target0.0
Herbie0.0
\[\left(x - z\right) \cdot y\]

Derivation

  1. Initial program 17.5

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

  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 2020024 +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)))