Average Error: 17.5 → 0.0
Time: 1.6s
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 r455561 = x;
        double r455562 = y;
        double r455563 = r455561 * r455562;
        double r455564 = r455562 * r455562;
        double r455565 = r455563 + r455564;
        double r455566 = z;
        double r455567 = r455562 * r455566;
        double r455568 = r455565 - r455567;
        double r455569 = r455568 - r455564;
        return r455569;
}
↓
double f(double x, double y, double z) {
        double r455570 = y;
        double r455571 = x;
        double r455572 = z;
        double r455573 = r455571 - r455572;
        double r455574 = 0.0;
        double r455575 = fma(r455570, r455573, r455574);
        return r455575;
}

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 2020100 +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)))