Average Error: 17.9 → 0.0
Time: 8.0s
Precision: 64
\[\left(\left(x \cdot y - y \cdot z\right) - y \cdot y\right) + y \cdot y\]
↓
\[\mathsf{fma}\left(x, y, \left(-z\right) \cdot y\right)\]
\left(\left(x \cdot y - y \cdot z\right) - y \cdot y\right) + y \cdot y
↓
\mathsf{fma}\left(x, y, \left(-z\right) \cdot y\right)
double f(double x, double y, double z) {
        double r526461 = x;
        double r526462 = y;
        double r526463 = r526461 * r526462;
        double r526464 = z;
        double r526465 = r526462 * r526464;
        double r526466 = r526463 - r526465;
        double r526467 = r526462 * r526462;
        double r526468 = r526466 - r526467;
        double r526469 = r526468 + r526467;
        return r526469;
}
↓
double f(double x, double y, double z) {
        double r526470 = x;
        double r526471 = y;
        double r526472 = z;
        double r526473 = -r526472;
        double r526474 = r526473 * r526471;
        double r526475 = fma(r526470, r526471, r526474);
        return r526475;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original17.9
Target0.0
Herbie0.0
\[\left(x - z\right) \cdot y\]

Derivation

  1. Initial program 17.9

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

  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. Simplified0.0

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

  7. Simplified0.0

    \[\leadsto x \cdot y + \color{blue}{\left(-z\right) \cdot y}\]
  8. Using strategy rm

  9. Applied fma-def0.0

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

  10. Final simplification0.0

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

Reproduce

herbie shell --seed 2020046 +o rules:numerics
(FPCore (x y z)
  :name "Linear.Quaternion:$c/ from linear-1.19.1.3, B"
  :precision binary64

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

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