Average Error: 11.4 → 0.9
Time: 6.7s
Precision: 64
\[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le 526596793.112873614:\\ \;\;\;\;x - \frac{z}{\frac{z}{\frac{y}{z}} - \frac{t}{2}}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{z}{z \cdot 2 - \frac{y}{z} \cdot t} \cdot \left(\frac{y}{z} \cdot 2\right)\\ \end{array}\]
x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}
↓
\begin{array}{l}
\mathbf{if}\;z \le 526596793.112873614:\\
\;\;\;\;x - \frac{z}{\frac{z}{\frac{y}{z}} - \frac{t}{2}}\\

\mathbf{else}:\\
\;\;\;\;x - \frac{z}{z \cdot 2 - \frac{y}{z} \cdot t} \cdot \left(\frac{y}{z} \cdot 2\right)\\

\end{array}
double f(double x, double y, double z, double t) {
        double r407272 = x;
        double r407273 = y;
        double r407274 = 2.0;
        double r407275 = r407273 * r407274;
        double r407276 = z;
        double r407277 = r407275 * r407276;
        double r407278 = r407276 * r407274;
        double r407279 = r407278 * r407276;
        double r407280 = t;
        double r407281 = r407273 * r407280;
        double r407282 = r407279 - r407281;
        double r407283 = r407277 / r407282;
        double r407284 = r407272 - r407283;
        return r407284;
}
↓
double f(double x, double y, double z, double t) {
        double r407285 = z;
        double r407286 = 526596793.1128736;
        bool r407287 = r407285 <= r407286;
        double r407288 = x;
        double r407289 = y;
        double r407290 = r407289 / r407285;
        double r407291 = r407285 / r407290;
        double r407292 = t;
        double r407293 = 2.0;
        double r407294 = r407292 / r407293;
        double r407295 = r407291 - r407294;
        double r407296 = r407285 / r407295;
        double r407297 = r407288 - r407296;
        double r407298 = r407285 * r407293;
        double r407299 = r407290 * r407292;
        double r407300 = r407298 - r407299;
        double r407301 = r407285 / r407300;
        double r407302 = r407290 * r407293;
        double r407303 = r407301 * r407302;
        double r407304 = r407288 - r407303;
        double r407305 = r407287 ? r407297 : r407304;
        return r407305;
}

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

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

Derivation

  1. Split input into 2 regimes

  2. if z < 526596793.1128736

    1. Initial program 9.3

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

    2. Simplified2.8

      \[\leadsto \color{blue}{x - \frac{z}{\frac{z \cdot z}{y} - \frac{t}{2}}}\]
    3. Using strategy rm

    4. Applied associate-/l*1.0

      \[\leadsto x - \frac{z}{\color{blue}{\frac{z}{\frac{y}{z}}} - \frac{t}{2}}\]

    if 526596793.1128736 < z

    1. Initial program 18.0

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

    2. Simplified6.5

      \[\leadsto \color{blue}{x - \frac{z}{\frac{z \cdot z}{y} - \frac{t}{2}}}\]
    3. Using strategy rm

    4. Applied associate-/l*3.1

      \[\leadsto x - \frac{z}{\color{blue}{\frac{z}{\frac{y}{z}}} - \frac{t}{2}}\]
    5. Using strategy rm

    6. Applied frac-sub3.8

      \[\leadsto x - \frac{z}{\color{blue}{\frac{z \cdot 2 - \frac{y}{z} \cdot t}{\frac{y}{z} \cdot 2}}}\]

    7. Applied associate-/r/0.7

      \[\leadsto x - \color{blue}{\frac{z}{z \cdot 2 - \frac{y}{z} \cdot t} \cdot \left(\frac{y}{z} \cdot 2\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le 526596793.112873614:\\ \;\;\;\;x - \frac{z}{\frac{z}{\frac{y}{z}} - \frac{t}{2}}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{z}{z \cdot 2 - \frac{y}{z} \cdot t} \cdot \left(\frac{y}{z} \cdot 2\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020045 
(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)))))