Average Error: 13.4 → 0.0
Time: 1.5s
Precision: 64
\[\left(\left(x \cdot y - y \cdot y\right) + y \cdot y\right) - y \cdot z\]
↓
\[\mathsf{fma}\left(y, x, y \cdot \left(-z\right)\right)\]
\left(\left(x \cdot y - y \cdot y\right) + y \cdot y\right) - y \cdot z
↓
\mathsf{fma}\left(y, x, y \cdot \left(-z\right)\right)
double f(double x, double y, double z) {
        double r539614 = x;
        double r539615 = y;
        double r539616 = r539614 * r539615;
        double r539617 = r539615 * r539615;
        double r539618 = r539616 - r539617;
        double r539619 = r539618 + r539617;
        double r539620 = z;
        double r539621 = r539615 * r539620;
        double r539622 = r539619 - r539621;
        return r539622;
}
↓
double f(double x, double y, double z) {
        double r539623 = y;
        double r539624 = x;
        double r539625 = z;
        double r539626 = -r539625;
        double r539627 = r539623 * r539626;
        double r539628 = fma(r539623, r539624, r539627);
        return r539628;
}

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. Using strategy rm

  7. Applied fma-def0.0

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

  8. Final simplification0.0

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

Reproduce

herbie shell --seed 2020001 +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)))