Average Error: 17.9 → 0.0
Time: 1.1m
Precision: 64
\[\left(\left(x \cdot y - y \cdot z\right) - y \cdot y\right) + y \cdot y\]
↓
\[\mathsf{fma}\left(y, x, \left(-y\right) \cdot z\right)\]
\left(\left(x \cdot y - y \cdot z\right) - y \cdot y\right) + y \cdot y
↓
\mathsf{fma}\left(y, x, \left(-y\right) \cdot z\right)
double f(double x, double y, double z) {
        double r21407886 = x;
        double r21407887 = y;
        double r21407888 = r21407886 * r21407887;
        double r21407889 = z;
        double r21407890 = r21407887 * r21407889;
        double r21407891 = r21407888 - r21407890;
        double r21407892 = r21407887 * r21407887;
        double r21407893 = r21407891 - r21407892;
        double r21407894 = r21407893 + r21407892;
        return r21407894;
}
↓
double f(double x, double y, double z) {
        double r21407895 = y;
        double r21407896 = x;
        double r21407897 = -r21407895;
        double r21407898 = z;
        double r21407899 = r21407897 * r21407898;
        double r21407900 = fma(r21407895, r21407896, r21407899);
        return r21407900;
}

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. 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

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

Reproduce

herbie shell --seed 2019200 +o rules:numerics
(FPCore (x y z)
  :name "Linear.Quaternion:$c/ from linear-1.19.1.3, B"

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

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