Average Error: 12.9 → 0.0
Time: 2.1s
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 r512733 = x;
        double r512734 = y;
        double r512735 = r512733 * r512734;
        double r512736 = r512734 * r512734;
        double r512737 = r512735 - r512736;
        double r512738 = r512737 + r512736;
        double r512739 = z;
        double r512740 = r512734 * r512739;
        double r512741 = r512738 - r512740;
        return r512741;
}
↓
double f(double x, double y, double z) {
        double r512742 = y;
        double r512743 = x;
        double r512744 = z;
        double r512745 = -r512744;
        double r512746 = r512742 * r512745;
        double r512747 = fma(r512742, r512743, r512746);
        return r512747;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original12.9
Target0.0
Herbie0.0
\[\left(x - z\right) \cdot y\]

Derivation

  1. Initial program 12.9

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

  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 2020024 +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)))