Average Error: 13.1 → 0.0
Time: 1.4s
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 r398487 = x;
        double r398488 = y;
        double r398489 = r398487 * r398488;
        double r398490 = r398488 * r398488;
        double r398491 = r398489 - r398490;
        double r398492 = r398491 + r398490;
        double r398493 = z;
        double r398494 = r398488 * r398493;
        double r398495 = r398492 - r398494;
        return r398495;
}
↓
double f(double x, double y, double z) {
        double r398496 = y;
        double r398497 = x;
        double r398498 = z;
        double r398499 = -r398498;
        double r398500 = r398496 * r398499;
        double r398501 = fma(r398496, r398497, r398500);
        return r398501;
}

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

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

Reproduce

herbie shell --seed 2020062 +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)))