Average Error: 17.7 → 0.0
Time: 1.7s
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 r519903 = x;
        double r519904 = y;
        double r519905 = r519903 * r519904;
        double r519906 = r519904 * r519904;
        double r519907 = r519905 + r519906;
        double r519908 = z;
        double r519909 = r519904 * r519908;
        double r519910 = r519907 - r519909;
        double r519911 = r519910 - r519906;
        return r519911;
}
↓
double f(double x, double y, double z) {
        double r519912 = y;
        double r519913 = x;
        double r519914 = z;
        double r519915 = r519913 - r519914;
        double r519916 = 0.0;
        double r519917 = fma(r519912, r519915, r519916);
        return r519917;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original17.7
Target0.0
Herbie0.0
\[\left(x - z\right) \cdot y\]

Derivation

  1. Initial program 17.7

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

  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 2019353 +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)))