Average Error: 11.3 → 1.1
Time: 18.1s
Precision: 64
\[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}\]
↓
\[x - \frac{z}{\frac{z}{\frac{y}{z}} - \frac{t}{2}}\]
x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}
↓
x - \frac{z}{\frac{z}{\frac{y}{z}} - \frac{t}{2}}
double f(double x, double y, double z, double t) {
        double r247967 = x;
        double r247968 = y;
        double r247969 = 2.0;
        double r247970 = r247968 * r247969;
        double r247971 = z;
        double r247972 = r247970 * r247971;
        double r247973 = r247971 * r247969;
        double r247974 = r247973 * r247971;
        double r247975 = t;
        double r247976 = r247968 * r247975;
        double r247977 = r247974 - r247976;
        double r247978 = r247972 / r247977;
        double r247979 = r247967 - r247978;
        return r247979;
}
↓
double f(double x, double y, double z, double t) {
        double r247980 = x;
        double r247981 = z;
        double r247982 = y;
        double r247983 = r247982 / r247981;
        double r247984 = r247981 / r247983;
        double r247985 = t;
        double r247986 = 2.0;
        double r247987 = r247985 / r247986;
        double r247988 = r247984 - r247987;
        double r247989 = r247981 / r247988;
        double r247990 = r247980 - r247989;
        return r247990;
}

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

Original11.3
Target0.1
Herbie1.1
\[x - \frac{1}{\frac{z}{y} - \frac{\frac{t}{2}}{z}}\]

Derivation

  1. Initial program 11.3

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

  2. Simplified3.1

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

  4. Applied associate-/l*1.1

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

  5. Final simplification1.1

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

Reproduce

herbie shell --seed 2019326 
(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)))))