Average Error: 17.6 → 0.0
Time: 1.8s
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 r461042 = x;
        double r461043 = y;
        double r461044 = r461042 * r461043;
        double r461045 = r461043 * r461043;
        double r461046 = r461044 + r461045;
        double r461047 = z;
        double r461048 = r461043 * r461047;
        double r461049 = r461046 - r461048;
        double r461050 = r461049 - r461045;
        return r461050;
}
↓
double f(double x, double y, double z) {
        double r461051 = y;
        double r461052 = x;
        double r461053 = z;
        double r461054 = r461052 - r461053;
        double r461055 = 0.0;
        double r461056 = fma(r461051, r461054, r461055);
        return r461056;
}

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)))