Average Error: 11.6 → 0.1
Time: 25.0s
Precision: 64
\[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}\]
↓
\[x - \frac{1}{\frac{z}{y} - 0.5 \cdot \frac{t}{z}}\]
x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}
↓
x - \frac{1}{\frac{z}{y} - 0.5 \cdot \frac{t}{z}}
double f(double x, double y, double z, double t) {
        double r400290 = x;
        double r400291 = y;
        double r400292 = 2.0;
        double r400293 = r400291 * r400292;
        double r400294 = z;
        double r400295 = r400293 * r400294;
        double r400296 = r400294 * r400292;
        double r400297 = r400296 * r400294;
        double r400298 = t;
        double r400299 = r400291 * r400298;
        double r400300 = r400297 - r400299;
        double r400301 = r400295 / r400300;
        double r400302 = r400290 - r400301;
        return r400302;
}
↓
double f(double x, double y, double z, double t) {
        double r400303 = x;
        double r400304 = 1.0;
        double r400305 = z;
        double r400306 = y;
        double r400307 = r400305 / r400306;
        double r400308 = 0.5;
        double r400309 = t;
        double r400310 = r400309 / r400305;
        double r400311 = r400308 * r400310;
        double r400312 = r400307 - r400311;
        double r400313 = r400304 / r400312;
        double r400314 = r400303 - r400313;
        return r400314;
}

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

Original11.6
Target0.1
Herbie0.1
\[x - \frac{1}{\frac{z}{y} - \frac{\frac{t}{2}}{z}}\]

Derivation

  1. Initial program 11.6

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

  2. Simplified2.7

    \[\leadsto \color{blue}{x - \frac{y}{\frac{z}{1} - \frac{\frac{y \cdot t}{z}}{2}}}\]
  3. Using strategy rm

  4. Applied clear-num2.7

    \[\leadsto x - \color{blue}{\frac{1}{\frac{\frac{z}{1} - \frac{\frac{y \cdot t}{z}}{2}}{y}}}\]

  5. Simplified2.7

    \[\leadsto x - \frac{1}{\color{blue}{\frac{z - \frac{\frac{y \cdot t}{z}}{2}}{y}}}\]

  6. Taylor expanded around 0 0.1

    \[\leadsto x - \frac{1}{\color{blue}{\frac{z}{y} - 0.5 \cdot \frac{t}{z}}}\]

  7. Final simplification0.1

    \[\leadsto x - \frac{1}{\frac{z}{y} - 0.5 \cdot \frac{t}{z}}\]

Reproduce

herbie shell --seed 2019199 
(FPCore (x y z t)
  :name "Numeric.AD.Rank1.Halley:findZero from ad-4.2.4"

  :herbie-target
  (- x (/ 1.0 (- (/ z y) (/ (/ t 2.0) z))))

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