Average Error: 17.5 → 0.0
Time: 1.9s
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 r1619 = x;
        double r1620 = y;
        double r1621 = r1619 * r1620;
        double r1622 = r1620 * r1620;
        double r1623 = r1621 + r1622;
        double r1624 = z;
        double r1625 = r1620 * r1624;
        double r1626 = r1623 - r1625;
        double r1627 = r1626 - r1622;
        return r1627;
}
↓
double f(double x, double y, double z) {
        double r1628 = y;
        double r1629 = x;
        double r1630 = z;
        double r1631 = r1629 - r1630;
        double r1632 = 0.0;
        double r1633 = fma(r1628, r1631, r1632);
        return r1633;
}

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 2020025 +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)))