Average Error: 17.6 → 0.0
Time: 24.8s
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 r347701 = x;
        double r347702 = y;
        double r347703 = r347701 * r347702;
        double r347704 = z;
        double r347705 = r347702 * r347704;
        double r347706 = r347703 - r347705;
        double r347707 = r347702 * r347702;
        double r347708 = r347706 - r347707;
        double r347709 = r347708 + r347707;
        return r347709;
}
↓
double f(double x, double y, double z) {
        double r347710 = x;
        double r347711 = y;
        double r347712 = z;
        double r347713 = r347712 * r347711;
        double r347714 = -r347713;
        double r347715 = fma(r347710, r347711, r347714);
        double r347716 = 0.0;
        double r347717 = r347716 * r347712;
        double r347718 = r347715 + r347717;
        return r347718;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original17.6
Target0.0
Herbie0.0
\[\left(x - z\right) \cdot y\]

Derivation

  1. Initial program 17.6

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

  3. Applied prod-diff17.6

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

  4. Applied associate--l+17.6

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

  5. Applied associate-+l+8.0

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

  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 2019326 +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)))