Average Error: 13.4 → 0.0
Time: 7.9s
Precision: 64
\[\left(\left(x \cdot y - y \cdot y\right) + y \cdot y\right) - y \cdot z\]
↓
\[\mathsf{fma}\left(x, y, \left(-z\right) \cdot y\right)\]
\left(\left(x \cdot y - y \cdot y\right) + y \cdot y\right) - y \cdot z
↓
\mathsf{fma}\left(x, y, \left(-z\right) \cdot y\right)
double f(double x, double y, double z) {
        double r589620 = x;
        double r589621 = y;
        double r589622 = r589620 * r589621;
        double r589623 = r589621 * r589621;
        double r589624 = r589622 - r589623;
        double r589625 = r589624 + r589623;
        double r589626 = z;
        double r589627 = r589621 * r589626;
        double r589628 = r589625 - r589627;
        return r589628;
}
↓
double f(double x, double y, double z) {
        double r589629 = x;
        double r589630 = y;
        double r589631 = z;
        double r589632 = -r589631;
        double r589633 = r589632 * r589630;
        double r589634 = fma(r589629, r589630, r589633);
        return r589634;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original13.4
Target0.0
Herbie0.0
\[\left(x - z\right) \cdot y\]

Derivation

  1. Initial program 13.4

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

  2. Simplified0.0

    \[\leadsto \color{blue}{y \cdot \left(x - z\right)}\]
  3. Using strategy rm

  4. Applied sub-neg0.0

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

  5. Applied distribute-lft-in0.0

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

  6. Simplified0.0

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

  7. Simplified0.0

    \[\leadsto x \cdot y + \color{blue}{\left(-z\right) \cdot y}\]
  8. Using strategy rm

  9. Applied fma-def0.0

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

  10. Final simplification0.0

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

Reproduce

herbie shell --seed 2020046 +o rules:numerics
(FPCore (x y z)
  :name "Linear.Quaternion:$c/ from linear-1.19.1.3, D"
  :precision binary64

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

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