Average Error: 13.2 → 0.0
Time: 21.9s
Precision: 64
\[\left(\left(x \cdot y - y \cdot y\right) + y \cdot y\right) - y \cdot z\]
↓
\[\mathsf{fma}\left(x, y, y \cdot \left(-z\right)\right)\]
\left(\left(x \cdot y - y \cdot y\right) + y \cdot y\right) - y \cdot z
↓
\mathsf{fma}\left(x, y, y \cdot \left(-z\right)\right)
double f(double x, double y, double z) {
        double r391165 = x;
        double r391166 = y;
        double r391167 = r391165 * r391166;
        double r391168 = r391166 * r391166;
        double r391169 = r391167 - r391168;
        double r391170 = r391169 + r391168;
        double r391171 = z;
        double r391172 = r391166 * r391171;
        double r391173 = r391170 - r391172;
        return r391173;
}
↓
double f(double x, double y, double z) {
        double r391174 = x;
        double r391175 = y;
        double r391176 = z;
        double r391177 = -r391176;
        double r391178 = r391175 * r391177;
        double r391179 = fma(r391174, r391175, r391178);
        return r391179;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original13.2
Target0.0
Herbie0.0
\[\left(x - z\right) \cdot y\]

Derivation

  1. Initial program 13.2

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

  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-rgt-in0.0

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

  6. Simplified0.0

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

  8. Applied fma-def0.0

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

  9. Final simplification0.0

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

Reproduce

herbie shell --seed 2019212 +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)))