Average Error: 11.5 → 1.1
Time: 18.9s
Precision: 64
\[x - \frac{\left(y \cdot 2.0\right) \cdot z}{\left(z \cdot 2.0\right) \cdot z - y \cdot t}\]
↓
\[x - \frac{y}{z - \frac{t}{z} \cdot \frac{y}{2.0}}\]
x - \frac{\left(y \cdot 2.0\right) \cdot z}{\left(z \cdot 2.0\right) \cdot z - y \cdot t}
↓
x - \frac{y}{z - \frac{t}{z} \cdot \frac{y}{2.0}}
double f(double x, double y, double z, double t) {
        double r21853260 = x;
        double r21853261 = y;
        double r21853262 = 2.0;
        double r21853263 = r21853261 * r21853262;
        double r21853264 = z;
        double r21853265 = r21853263 * r21853264;
        double r21853266 = r21853264 * r21853262;
        double r21853267 = r21853266 * r21853264;
        double r21853268 = t;
        double r21853269 = r21853261 * r21853268;
        double r21853270 = r21853267 - r21853269;
        double r21853271 = r21853265 / r21853270;
        double r21853272 = r21853260 - r21853271;
        return r21853272;
}
↓
double f(double x, double y, double z, double t) {
        double r21853273 = x;
        double r21853274 = y;
        double r21853275 = z;
        double r21853276 = t;
        double r21853277 = r21853276 / r21853275;
        double r21853278 = 2.0;
        double r21853279 = r21853274 / r21853278;
        double r21853280 = r21853277 * r21853279;
        double r21853281 = r21853275 - r21853280;
        double r21853282 = r21853274 / r21853281;
        double r21853283 = r21853273 - r21853282;
        return r21853283;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original11.5
Target0.1
Herbie1.1
\[x - \frac{1}{\frac{z}{y} - \frac{\frac{t}{2.0}}{z}}\]

Derivation

  1. Initial program 11.5

    \[x - \frac{\left(y \cdot 2.0\right) \cdot z}{\left(z \cdot 2.0\right) \cdot z - y \cdot t}\]

  2. Simplified1.1

    \[\leadsto \color{blue}{x - \frac{y}{z - \frac{y}{2.0} \cdot \frac{t}{z}}}\]

  3. Final simplification1.1

    \[\leadsto x - \frac{y}{z - \frac{t}{z} \cdot \frac{y}{2.0}}\]

Reproduce

herbie shell --seed 2019168 +o rules:numerics
(FPCore (x y z t)
  :name "Numeric.AD.Rank1.Halley:findZero from ad-4.2.4"

  :herbie-target
  (- x (/ 1 (- (/ z y) (/ (/ t 2.0) z))))

  (- x (/ (* (* y 2.0) z) (- (* (* z 2.0) z) (* y t)))))