Average Error: 11.2 → 2.7
Time: 5.0s
Precision: 64
\[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}\]
↓
\[x - \frac{y \cdot 2}{2 \cdot z - \frac{t \cdot y}{z}}\]
x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}
↓
x - \frac{y \cdot 2}{2 \cdot z - \frac{t \cdot y}{z}}
double f(double x, double y, double z, double t) {
        double r387857 = x;
        double r387858 = y;
        double r387859 = 2.0;
        double r387860 = r387858 * r387859;
        double r387861 = z;
        double r387862 = r387860 * r387861;
        double r387863 = r387861 * r387859;
        double r387864 = r387863 * r387861;
        double r387865 = t;
        double r387866 = r387858 * r387865;
        double r387867 = r387864 - r387866;
        double r387868 = r387862 / r387867;
        double r387869 = r387857 - r387868;
        return r387869;
}
↓
double f(double x, double y, double z, double t) {
        double r387870 = x;
        double r387871 = y;
        double r387872 = 2.0;
        double r387873 = r387871 * r387872;
        double r387874 = z;
        double r387875 = r387872 * r387874;
        double r387876 = t;
        double r387877 = r387876 * r387871;
        double r387878 = r387877 / r387874;
        double r387879 = r387875 - r387878;
        double r387880 = r387873 / r387879;
        double r387881 = r387870 - r387880;
        return r387881;
}

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

Original11.2
Target0.1
Herbie2.7
\[x - \frac{1}{\frac{z}{y} - \frac{\frac{t}{2}}{z}}\]

Derivation

  1. Initial program 11.2

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

  3. Applied associate-/l*6.6

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

  4. Taylor expanded around 0 2.7

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

  5. Final simplification2.7

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

Reproduce

herbie shell --seed 2020035 
(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)))))