Average Error: 11.4 → 0.9
Time: 16.1s
Precision: 64
\[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}\]
↓
\[x - \frac{y}{\mathsf{fma}\left(\frac{y}{2}, \frac{-t}{z}, z\right)}\]
x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}
↓
x - \frac{y}{\mathsf{fma}\left(\frac{y}{2}, \frac{-t}{z}, z\right)}
double f(double x, double y, double z, double t) {
        double r576398 = x;
        double r576399 = y;
        double r576400 = 2.0;
        double r576401 = r576399 * r576400;
        double r576402 = z;
        double r576403 = r576401 * r576402;
        double r576404 = r576402 * r576400;
        double r576405 = r576404 * r576402;
        double r576406 = t;
        double r576407 = r576399 * r576406;
        double r576408 = r576405 - r576407;
        double r576409 = r576403 / r576408;
        double r576410 = r576398 - r576409;
        return r576410;
}
↓
double f(double x, double y, double z, double t) {
        double r576411 = x;
        double r576412 = y;
        double r576413 = 2.0;
        double r576414 = r576412 / r576413;
        double r576415 = t;
        double r576416 = -r576415;
        double r576417 = z;
        double r576418 = r576416 / r576417;
        double r576419 = fma(r576414, r576418, r576417);
        double r576420 = r576412 / r576419;
        double r576421 = r576411 - r576420;
        return r576421;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original11.4
Target0.1
Herbie0.9
\[x - \frac{1}{\frac{z}{y} - \frac{\frac{t}{2}}{z}}\]

Derivation

  1. Initial program 11.4

    \[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}\]

  2. Simplified0.9

    \[\leadsto \color{blue}{x - \frac{y}{\mathsf{fma}\left(\frac{y}{2}, \frac{-t}{z}, z\right)}}\]
  3. Using strategy rm

  4. Applied *-un-lft-identity0.9

    \[\leadsto x - \frac{y}{\color{blue}{1 \cdot \mathsf{fma}\left(\frac{y}{2}, \frac{-t}{z}, z\right)}}\]

  5. Final simplification0.9

    \[\leadsto x - \frac{y}{\mathsf{fma}\left(\frac{y}{2}, \frac{-t}{z}, z\right)}\]

Reproduce

herbie shell --seed 2020047 +o rules:numerics
(FPCore (x y z t)
  :name "Numeric.AD.Rank1.Halley:findZero from ad-4.2.4"
  :precision binary64

  :herbie-target
  (- x (/ 1 (- (/ z y) (/ (/ t 2) z))))

  (- x (/ (* (* y 2) z) (- (* (* z 2) z) (* y t)))))