Average Error: 17.7 → 0.0
Time: 1.7s
Precision: 64
\[\left(\left(x \cdot y - y \cdot z\right) - y \cdot y\right) + y \cdot y\]
↓
\[\mathsf{fma}\left(y, x, y \cdot \left(-z\right)\right)\]
\left(\left(x \cdot y - y \cdot z\right) - y \cdot y\right) + y \cdot y
↓
\mathsf{fma}\left(y, x, y \cdot \left(-z\right)\right)
double f(double x, double y, double z) {
        double r515018 = x;
        double r515019 = y;
        double r515020 = r515018 * r515019;
        double r515021 = z;
        double r515022 = r515019 * r515021;
        double r515023 = r515020 - r515022;
        double r515024 = r515019 * r515019;
        double r515025 = r515023 - r515024;
        double r515026 = r515025 + r515024;
        return r515026;
}
↓
double f(double x, double y, double z) {
        double r515027 = y;
        double r515028 = x;
        double r515029 = z;
        double r515030 = -r515029;
        double r515031 = r515027 * r515030;
        double r515032 = fma(r515027, r515028, r515031);
        return r515032;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original17.7
Target0.0
Herbie0.0
\[\left(x - z\right) \cdot y\]

Derivation

  1. Initial program 17.7

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

  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

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

Reproduce

herbie shell --seed 2020033 +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)))