Average Error: 18.2 → 0.0
Time: 25.9s
Precision: 64
\[\left(\left(x \cdot y - y \cdot z\right) - y \cdot y\right) + y \cdot y\]
↓
\[\mathsf{fma}\left(x, y, -z \cdot y\right) + 0 \cdot z\]
\left(\left(x \cdot y - y \cdot z\right) - y \cdot y\right) + y \cdot y
↓
\mathsf{fma}\left(x, y, -z \cdot y\right) + 0 \cdot z
double f(double x, double y, double z) {
        double r292013 = x;
        double r292014 = y;
        double r292015 = r292013 * r292014;
        double r292016 = z;
        double r292017 = r292014 * r292016;
        double r292018 = r292015 - r292017;
        double r292019 = r292014 * r292014;
        double r292020 = r292018 - r292019;
        double r292021 = r292020 + r292019;
        return r292021;
}
↓
double f(double x, double y, double z) {
        double r292022 = x;
        double r292023 = y;
        double r292024 = z;
        double r292025 = r292024 * r292023;
        double r292026 = -r292025;
        double r292027 = fma(r292022, r292023, r292026);
        double r292028 = 0.0;
        double r292029 = r292028 * r292024;
        double r292030 = r292027 + r292029;
        return r292030;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original18.2
Target0.0
Herbie0.0
\[\left(x - z\right) \cdot y\]

Derivation

  1. Initial program 18.2

    \[\left(\left(x \cdot y - y \cdot z\right) - y \cdot y\right) + y \cdot y\]
  2. Using strategy rm

  3. Applied prod-diff18.2

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

  4. Applied associate--l+18.2

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

  5. Applied associate-+l+8.3

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

  6. Simplified0.0

    \[\leadsto \mathsf{fma}\left(x, y, -z \cdot y\right) + \color{blue}{0 \cdot z}\]

  7. Final simplification0.0

    \[\leadsto \mathsf{fma}\left(x, y, -z \cdot y\right) + 0 \cdot z\]

Reproduce

herbie shell --seed 2019325 +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)))