Average Error: 11.2 → 0.1
Time: 16.6s
Precision: 64
\[x - \frac{\left(y \cdot 2.0\right) \cdot z}{\left(z \cdot 2.0\right) \cdot z - y \cdot t}\]
↓
\[x - \frac{1}{\frac{z}{y} - \left(0.5 \cdot t\right) \cdot \frac{1}{z}}\]
x - \frac{\left(y \cdot 2.0\right) \cdot z}{\left(z \cdot 2.0\right) \cdot z - y \cdot t}
↓
x - \frac{1}{\frac{z}{y} - \left(0.5 \cdot t\right) \cdot \frac{1}{z}}
double f(double x, double y, double z, double t) {
        double r15592845 = x;
        double r15592846 = y;
        double r15592847 = 2.0;
        double r15592848 = r15592846 * r15592847;
        double r15592849 = z;
        double r15592850 = r15592848 * r15592849;
        double r15592851 = r15592849 * r15592847;
        double r15592852 = r15592851 * r15592849;
        double r15592853 = t;
        double r15592854 = r15592846 * r15592853;
        double r15592855 = r15592852 - r15592854;
        double r15592856 = r15592850 / r15592855;
        double r15592857 = r15592845 - r15592856;
        return r15592857;
}
↓
double f(double x, double y, double z, double t) {
        double r15592858 = x;
        double r15592859 = 1.0;
        double r15592860 = z;
        double r15592861 = y;
        double r15592862 = r15592860 / r15592861;
        double r15592863 = 0.5;
        double r15592864 = t;
        double r15592865 = r15592863 * r15592864;
        double r15592866 = r15592859 / r15592860;
        double r15592867 = r15592865 * r15592866;
        double r15592868 = r15592862 - r15592867;
        double r15592869 = r15592859 / r15592868;
        double r15592870 = r15592858 - r15592869;
        return r15592870;
}

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
Herbie0.1
\[x - \frac{1}{\frac{z}{y} - \frac{\frac{t}{2.0}}{z}}\]

Derivation

  1. Initial program 11.2

    \[x - \frac{\left(y \cdot 2.0\right) \cdot z}{\left(z \cdot 2.0\right) \cdot z - y \cdot t}\]

  2. Simplified1.0

    \[\leadsto \color{blue}{x - \frac{y}{z - \frac{y}{2.0} \cdot \frac{t}{z}}}\]
  3. Using strategy rm

  4. Applied clear-num1.0

    \[\leadsto x - \color{blue}{\frac{1}{\frac{z - \frac{y}{2.0} \cdot \frac{t}{z}}{y}}}\]

  5. Taylor expanded around 0 0.1

    \[\leadsto x - \frac{1}{\color{blue}{\frac{z}{y} - 0.5 \cdot \frac{t}{z}}}\]
  6. Using strategy rm

  7. Applied div-inv0.1

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

  8. Applied associate-*r*0.1

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

  9. Final simplification0.1

    \[\leadsto x - \frac{1}{\frac{z}{y} - \left(0.5 \cdot t\right) \cdot \frac{1}{z}}\]

Reproduce

herbie shell --seed 2019162 +o rules:numerics
(FPCore (x y z t)
  :name "Numeric.AD.Rank1.Halley:findZero from ad-4.2.4"

  :herbie-target
  (- x (/ 1 (- (/ z y) (/ (/ t 2.0) z))))

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