Average Error: 17.8 → 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 r516196 = x;
        double r516197 = y;
        double r516198 = r516196 * r516197;
        double r516199 = r516197 * r516197;
        double r516200 = r516198 + r516199;
        double r516201 = z;
        double r516202 = r516197 * r516201;
        double r516203 = r516200 - r516202;
        double r516204 = r516203 - r516199;
        return r516204;
}
↓
double f(double x, double y, double z) {
        double r516205 = y;
        double r516206 = x;
        double r516207 = z;
        double r516208 = r516206 - r516207;
        double r516209 = 0.0;
        double r516210 = fma(r516205, r516208, r516209);
        return r516210;
}

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 2020062 +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)))