Average Error: 12.7 → 0.0
Time: 22.2s
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 r344187 = x;
        double r344188 = y;
        double r344189 = r344187 * r344188;
        double r344190 = r344188 * r344188;
        double r344191 = r344189 - r344190;
        double r344192 = r344191 + r344190;
        double r344193 = z;
        double r344194 = r344188 * r344193;
        double r344195 = r344192 - r344194;
        return r344195;
}
↓
double f(double x, double y, double z) {
        double r344196 = x;
        double r344197 = y;
        double r344198 = z;
        double r344199 = -r344198;
        double r344200 = r344197 * r344199;
        double r344201 = fma(r344196, r344197, r344200);
        return r344201;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original12.7
Target0.0
Herbie0.0
\[\left(x - z\right) \cdot y\]

Derivation

  1. Initial program 12.7

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

  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 2019198 +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)))