Average Error: 13.1 → 0.0
Time: 1.0m
Precision: 64
\[\left(\left(x \cdot y - y \cdot y\right) + y \cdot y\right) - y \cdot z\]
↓
\[\mathsf{fma}\left(y, x, \left(-y\right) \cdot z\right)\]
\left(\left(x \cdot y - y \cdot y\right) + y \cdot y\right) - y \cdot z
↓
\mathsf{fma}\left(y, x, \left(-y\right) \cdot z\right)
double f(double x, double y, double z) {
        double r24580626 = x;
        double r24580627 = y;
        double r24580628 = r24580626 * r24580627;
        double r24580629 = r24580627 * r24580627;
        double r24580630 = r24580628 - r24580629;
        double r24580631 = r24580630 + r24580629;
        double r24580632 = z;
        double r24580633 = r24580627 * r24580632;
        double r24580634 = r24580631 - r24580633;
        return r24580634;
}
↓
double f(double x, double y, double z) {
        double r24580635 = y;
        double r24580636 = x;
        double r24580637 = -r24580635;
        double r24580638 = z;
        double r24580639 = r24580637 * r24580638;
        double r24580640 = fma(r24580635, r24580636, r24580639);
        return r24580640;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original13.1
Target0.0
Herbie0.0
\[\left(x - z\right) \cdot y\]

Derivation

  1. Initial program 13.1

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

  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

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

Reproduce

herbie shell --seed 2019200 +o rules:numerics
(FPCore (x y z)
  :name "Linear.Quaternion:$c/ from linear-1.19.1.3, D"

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

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