Average Error: 17.6 → 0.0
Time: 2.5s
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 r471057 = x;
        double r471058 = y;
        double r471059 = r471057 * r471058;
        double r471060 = r471058 * r471058;
        double r471061 = r471059 + r471060;
        double r471062 = z;
        double r471063 = r471058 * r471062;
        double r471064 = r471061 - r471063;
        double r471065 = r471064 - r471060;
        return r471065;
}
↓
double f(double x, double y, double z) {
        double r471066 = y;
        double r471067 = x;
        double r471068 = z;
        double r471069 = r471067 - r471068;
        double r471070 = 0.0;
        double r471071 = fma(r471066, r471069, r471070);
        return r471071;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original17.6
Target0.0
Herbie0.0
\[\left(x - z\right) \cdot y\]

Derivation

  1. Initial program 17.6

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

  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 2020036 +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)))