Average Error: 17.4 → 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 r490933 = x;
        double r490934 = y;
        double r490935 = r490933 * r490934;
        double r490936 = r490934 * r490934;
        double r490937 = r490935 + r490936;
        double r490938 = z;
        double r490939 = r490934 * r490938;
        double r490940 = r490937 - r490939;
        double r490941 = r490940 - r490936;
        return r490941;
}
↓
double f(double x, double y, double z) {
        double r490942 = y;
        double r490943 = x;
        double r490944 = z;
        double r490945 = r490943 - r490944;
        double r490946 = 0.0;
        double r490947 = fma(r490942, r490945, r490946);
        return r490947;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original17.4
Target0.0
Herbie0.0
\[\left(x - z\right) \cdot y\]

Derivation

  1. Initial program 17.4

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

  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 2020056 +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)))