Average Error: 18.0 → 0.0
Time: 13.5s
Precision: 64
\[\left(\left(x \cdot y + y \cdot y\right) - y \cdot z\right) - y \cdot y\]
↓
\[\mathsf{fma}\left(y, x, y \cdot \left(-z\right)\right)\]
\left(\left(x \cdot y + y \cdot y\right) - y \cdot z\right) - y \cdot y
↓
\mathsf{fma}\left(y, x, y \cdot \left(-z\right)\right)
double f(double x, double y, double z) {
        double r746373 = x;
        double r746374 = y;
        double r746375 = r746373 * r746374;
        double r746376 = r746374 * r746374;
        double r746377 = r746375 + r746376;
        double r746378 = z;
        double r746379 = r746374 * r746378;
        double r746380 = r746377 - r746379;
        double r746381 = r746380 - r746376;
        return r746381;
}
↓
double f(double x, double y, double z) {
        double r746382 = y;
        double r746383 = x;
        double r746384 = z;
        double r746385 = -r746384;
        double r746386 = r746382 * r746385;
        double r746387 = fma(r746382, r746383, r746386);
        return r746387;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original18.0
Target0.0
Herbie0.0
\[\left(x - z\right) \cdot y\]

Derivation

  1. Initial program 18.0

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

  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 2019350 +o rules:numerics
(FPCore (x y z)
  :name "Linear.Quaternion:$c/ from linear-1.19.1.3, C"
  :precision binary64

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

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