Average Error: 17.5 → 0.0
Time: 2.3s
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 r538307 = x;
        double r538308 = y;
        double r538309 = r538307 * r538308;
        double r538310 = r538308 * r538308;
        double r538311 = r538309 + r538310;
        double r538312 = z;
        double r538313 = r538308 * r538312;
        double r538314 = r538311 - r538313;
        double r538315 = r538314 - r538310;
        return r538315;
}
↓
double f(double x, double y, double z) {
        double r538316 = y;
        double r538317 = x;
        double r538318 = z;
        double r538319 = r538317 - r538318;
        double r538320 = 0.0;
        double r538321 = fma(r538316, r538319, r538320);
        return r538321;
}

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 2020027 +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)))