Average Error: 11.7 → 1.0
Time: 20.2s
Precision: 64
\[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}\]
↓
\[x - \frac{y}{\mathsf{fma}\left(\frac{t}{z}, -\frac{y}{2}, z\right)}\]
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{t}{z}, -\frac{y}{2}, z\right)}
double f(double x, double y, double z, double t) {
        double r280440 = x;
        double r280441 = y;
        double r280442 = 2.0;
        double r280443 = r280441 * r280442;
        double r280444 = z;
        double r280445 = r280443 * r280444;
        double r280446 = r280444 * r280442;
        double r280447 = r280446 * r280444;
        double r280448 = t;
        double r280449 = r280441 * r280448;
        double r280450 = r280447 - r280449;
        double r280451 = r280445 / r280450;
        double r280452 = r280440 - r280451;
        return r280452;
}
↓
double f(double x, double y, double z, double t) {
        double r280453 = x;
        double r280454 = y;
        double r280455 = t;
        double r280456 = z;
        double r280457 = r280455 / r280456;
        double r280458 = 2.0;
        double r280459 = r280454 / r280458;
        double r280460 = -r280459;
        double r280461 = fma(r280457, r280460, r280456);
        double r280462 = r280454 / r280461;
        double r280463 = r280453 - r280462;
        return r280463;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original11.7
Target0.1
Herbie1.0
\[x - \frac{1}{\frac{z}{y} - \frac{\frac{t}{2}}{z}}\]

Derivation

  1. Initial program 11.7

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

  2. Simplified1.0

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

  3. Final simplification1.0

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

Reproduce

herbie shell --seed 2019304 +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)))))