Average Error: 17.6 → 0.0
Time: 24.4s
Precision: 64
\[\left(\left(x \cdot y - y \cdot z\right) - y \cdot y\right) + y \cdot y\]
↓
\[\mathsf{fma}\left(x, y, -z \cdot y\right) + 0 \cdot z\]
\left(\left(x \cdot y - y \cdot z\right) - y \cdot y\right) + y \cdot y
↓
\mathsf{fma}\left(x, y, -z \cdot y\right) + 0 \cdot z
double f(double x, double y, double z) {
        double r437847 = x;
        double r437848 = y;
        double r437849 = r437847 * r437848;
        double r437850 = z;
        double r437851 = r437848 * r437850;
        double r437852 = r437849 - r437851;
        double r437853 = r437848 * r437848;
        double r437854 = r437852 - r437853;
        double r437855 = r437854 + r437853;
        return r437855;
}
↓
double f(double x, double y, double z) {
        double r437856 = x;
        double r437857 = y;
        double r437858 = z;
        double r437859 = r437858 * r437857;
        double r437860 = -r437859;
        double r437861 = fma(r437856, r437857, r437860);
        double r437862 = 0.0;
        double r437863 = r437862 * r437858;
        double r437864 = r437861 + r437863;
        return r437864;
}

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

    \[\left(\left(x \cdot y - y \cdot z\right) - y \cdot y\right) + y \cdot y\]
  2. Using strategy rm

  3. Applied prod-diff17.6

    \[\leadsto \left(\color{blue}{\left(\mathsf{fma}\left(x, y, -z \cdot y\right) + \mathsf{fma}\left(-z, y, z \cdot y\right)\right)} - y \cdot y\right) + y \cdot y\]

  4. Applied associate--l+17.6

    \[\leadsto \color{blue}{\left(\mathsf{fma}\left(x, y, -z \cdot y\right) + \left(\mathsf{fma}\left(-z, y, z \cdot y\right) - y \cdot y\right)\right)} + y \cdot y\]

  5. Applied associate-+l+8.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, -z \cdot y\right) + \left(\left(\mathsf{fma}\left(-z, y, z \cdot y\right) - y \cdot y\right) + y \cdot y\right)}\]

  6. Simplified0.0

    \[\leadsto \mathsf{fma}\left(x, y, -z \cdot y\right) + \color{blue}{0 \cdot z}\]

  7. Final simplification0.0

    \[\leadsto \mathsf{fma}\left(x, y, -z \cdot y\right) + 0 \cdot z\]

Reproduce

herbie shell --seed 2019326 +o rules:numerics
(FPCore (x y z)
  :name "Linear.Quaternion:$c/ from linear-1.19.1.3, B"
  :precision binary64

  :herbie-target
  (* (- x z) y)

  (+ (- (- (* x y) (* y z)) (* y y)) (* y y)))