Average Error: 11.7 → 2.3
Time: 6.7s
Precision: 64
\[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}\]
↓
\[x - \frac{y \cdot 2}{1 \cdot \left(2 \cdot z - \left(t \cdot \left(\sqrt[3]{\frac{y}{z}} \cdot \sqrt[3]{\frac{y}{z}}\right)\right) \cdot \sqrt[3]{\frac{y}{z}}\right)}\]
x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}
↓
x - \frac{y \cdot 2}{1 \cdot \left(2 \cdot z - \left(t \cdot \left(\sqrt[3]{\frac{y}{z}} \cdot \sqrt[3]{\frac{y}{z}}\right)\right) \cdot \sqrt[3]{\frac{y}{z}}\right)}
double f(double x, double y, double z, double t) {
        double r2142 = x;
        double r2143 = y;
        double r2144 = 2.0;
        double r2145 = r2143 * r2144;
        double r2146 = z;
        double r2147 = r2145 * r2146;
        double r2148 = r2146 * r2144;
        double r2149 = r2148 * r2146;
        double r2150 = t;
        double r2151 = r2143 * r2150;
        double r2152 = r2149 - r2151;
        double r2153 = r2147 / r2152;
        double r2154 = r2142 - r2153;
        return r2154;
}
↓
double f(double x, double y, double z, double t) {
        double r2155 = x;
        double r2156 = y;
        double r2157 = 2.0;
        double r2158 = r2156 * r2157;
        double r2159 = 1.0;
        double r2160 = z;
        double r2161 = r2157 * r2160;
        double r2162 = t;
        double r2163 = r2156 / r2160;
        double r2164 = cbrt(r2163);
        double r2165 = r2164 * r2164;
        double r2166 = r2162 * r2165;
        double r2167 = r2166 * r2164;
        double r2168 = r2161 - r2167;
        double r2169 = r2159 * r2168;
        double r2170 = r2158 / r2169;
        double r2171 = r2155 - r2170;
        return r2171;
}

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

Original11.7
Target0.1
Herbie2.3
\[x - \frac{1}{\frac{z}{y} - \frac{\frac{t}{2}}{z}}\]

Derivation

  1. Initial program 11.7

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

  5. Applied *-un-lft-identity6.6

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

  6. Applied *-un-lft-identity6.6

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

  7. Applied times-frac6.6

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

  8. Simplified6.6

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

  9. Simplified2.7

    \[\leadsto x - \frac{y \cdot 2}{1 \cdot \color{blue}{\left(2 \cdot z - \frac{t \cdot y}{z}\right)}}\]
  10. Using strategy rm

  11. Applied *-un-lft-identity2.7

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

  12. Applied times-frac2.2

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

  13. Simplified2.2

    \[\leadsto x - \frac{y \cdot 2}{1 \cdot \left(2 \cdot z - \color{blue}{t} \cdot \frac{y}{z}\right)}\]
  14. Using strategy rm

  15. Applied add-cube-cbrt2.3

    \[\leadsto x - \frac{y \cdot 2}{1 \cdot \left(2 \cdot z - t \cdot \color{blue}{\left(\left(\sqrt[3]{\frac{y}{z}} \cdot \sqrt[3]{\frac{y}{z}}\right) \cdot \sqrt[3]{\frac{y}{z}}\right)}\right)}\]

  16. Applied associate-*r*2.3

    \[\leadsto x - \frac{y \cdot 2}{1 \cdot \left(2 \cdot z - \color{blue}{\left(t \cdot \left(\sqrt[3]{\frac{y}{z}} \cdot \sqrt[3]{\frac{y}{z}}\right)\right) \cdot \sqrt[3]{\frac{y}{z}}}\right)}\]

  17. Final simplification2.3

    \[\leadsto x - \frac{y \cdot 2}{1 \cdot \left(2 \cdot z - \left(t \cdot \left(\sqrt[3]{\frac{y}{z}} \cdot \sqrt[3]{\frac{y}{z}}\right)\right) \cdot \sqrt[3]{\frac{y}{z}}\right)}\]

Reproduce

herbie shell --seed 2020025 
(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)))))