Average Error: 17.5 → 0.0
Time: 13.0s
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 r493039 = x;
        double r493040 = y;
        double r493041 = r493039 * r493040;
        double r493042 = z;
        double r493043 = r493040 * r493042;
        double r493044 = r493041 - r493043;
        double r493045 = r493040 * r493040;
        double r493046 = r493044 - r493045;
        double r493047 = r493046 + r493045;
        return r493047;
}
↓
double f(double x, double y, double z) {
        double r493048 = x;
        double r493049 = y;
        double r493050 = z;
        double r493051 = -r493050;
        double r493052 = r493051 * r493049;
        double r493053 = fma(r493048, r493049, r493052);
        return r493053;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original17.5
Target0.0
Herbie0.0
\[\left(x - z\right) \cdot y\]

Derivation

  1. Initial program 17.5

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

  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 2020045 +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)))