Average Error: 17.4 → 0.0
Time: 2.1s
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 r524032 = x;
        double r524033 = y;
        double r524034 = r524032 * r524033;
        double r524035 = r524033 * r524033;
        double r524036 = r524034 + r524035;
        double r524037 = z;
        double r524038 = r524033 * r524037;
        double r524039 = r524036 - r524038;
        double r524040 = r524039 - r524035;
        return r524040;
}
↓
double f(double x, double y, double z) {
        double r524041 = y;
        double r524042 = x;
        double r524043 = z;
        double r524044 = r524042 - r524043;
        double r524045 = 0.0;
        double r524046 = fma(r524041, r524044, r524045);
        return r524046;
}

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 2020081 +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)))