Average Error: 17.5 → 0.0
Time: 2.4s
Precision: 64
\[\left(\left(x \cdot y + y \cdot y\right) - y \cdot z\right) - y \cdot y\]
↓
\[\mathsf{fma}\left(y, x - z, 0\right)\]
\left(\left(x \cdot y + y \cdot y\right) - y \cdot z\right) - y \cdot y
↓
\mathsf{fma}\left(y, x - z, 0\right)
double f(double x, double y, double z) {
        double r618660 = x;
        double r618661 = y;
        double r618662 = r618660 * r618661;
        double r618663 = r618661 * r618661;
        double r618664 = r618662 + r618663;
        double r618665 = z;
        double r618666 = r618661 * r618665;
        double r618667 = r618664 - r618666;
        double r618668 = r618667 - r618663;
        return r618668;
}
↓
double f(double x, double y, double z) {
        double r618669 = y;
        double r618670 = x;
        double r618671 = z;
        double r618672 = r618670 - r618671;
        double r618673 = 0.0;
        double r618674 = fma(r618669, r618672, r618673);
        return r618674;
}

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

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

  2. Simplified0.0

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2020083 +o rules:numerics
(FPCore (x y z)
  :name "Linear.Quaternion:$c/ from linear-1.19.1.3, C"
  :precision binary64

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

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