Average Error: 17.8 → 0.0
Time: 1.4s
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 r501748 = x;
        double r501749 = y;
        double r501750 = r501748 * r501749;
        double r501751 = r501749 * r501749;
        double r501752 = r501750 + r501751;
        double r501753 = z;
        double r501754 = r501749 * r501753;
        double r501755 = r501752 - r501754;
        double r501756 = r501755 - r501751;
        return r501756;
}
↓
double f(double x, double y, double z) {
        double r501757 = y;
        double r501758 = x;
        double r501759 = z;
        double r501760 = r501758 - r501759;
        double r501761 = 0.0;
        double r501762 = fma(r501757, r501760, r501761);
        return r501762;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original17.8
Target0.0
Herbie0.0
\[\left(x - z\right) \cdot y\]

Derivation

  1. Initial program 17.8

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

  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 2020064 +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)))