Average Error: 17.8 → 0.0
Time: 2.0s
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 r476134 = x;
        double r476135 = y;
        double r476136 = r476134 * r476135;
        double r476137 = r476135 * r476135;
        double r476138 = r476136 + r476137;
        double r476139 = z;
        double r476140 = r476135 * r476139;
        double r476141 = r476138 - r476140;
        double r476142 = r476141 - r476137;
        return r476142;
}
↓
double f(double x, double y, double z) {
        double r476143 = y;
        double r476144 = x;
        double r476145 = z;
        double r476146 = r476144 - r476145;
        double r476147 = 0.0;
        double r476148 = fma(r476143, r476146, r476147);
        return r476148;
}

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)))