Average Error: 11.6 → 1.3
Time: 18.8s
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 r295980 = x;
        double r295981 = y;
        double r295982 = 2.0;
        double r295983 = r295981 * r295982;
        double r295984 = z;
        double r295985 = r295983 * r295984;
        double r295986 = r295984 * r295982;
        double r295987 = r295986 * r295984;
        double r295988 = t;
        double r295989 = r295981 * r295988;
        double r295990 = r295987 - r295989;
        double r295991 = r295985 / r295990;
        double r295992 = r295980 - r295991;
        return r295992;
}
↓
double f(double x, double y, double z, double t) {
        double r295993 = x;
        double r295994 = z;
        double r295995 = y;
        double r295996 = r295995 / r295994;
        double r295997 = r295994 / r295996;
        double r295998 = t;
        double r295999 = 2.0;
        double r296000 = r295998 / r295999;
        double r296001 = r295997 - r296000;
        double r296002 = r295994 / r296001;
        double r296003 = r295993 - r296002;
        return r296003;
}

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

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. Simplified3.5

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

  4. Applied associate-/l*1.3

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

  5. Final simplification1.3

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

Reproduce

herbie shell --seed 2019323 
(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)))))