Average Error: 17.9 → 0.0
Time: 2.7s
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 r448044 = x;
        double r448045 = y;
        double r448046 = r448044 * r448045;
        double r448047 = r448045 * r448045;
        double r448048 = r448046 + r448047;
        double r448049 = z;
        double r448050 = r448045 * r448049;
        double r448051 = r448048 - r448050;
        double r448052 = r448051 - r448047;
        return r448052;
}
↓
double f(double x, double y, double z) {
        double r448053 = y;
        double r448054 = x;
        double r448055 = z;
        double r448056 = r448054 - r448055;
        double r448057 = 0.0;
        double r448058 = fma(r448053, r448056, r448057);
        return r448058;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original17.9
Target0.0
Herbie0.0
\[\left(x - z\right) \cdot y\]

Derivation

  1. Initial program 17.9

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

  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 2020001 +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)))