Average Error: 11.4 → 1.1
Time: 7.3s
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 r563924 = x;
        double r563925 = y;
        double r563926 = 2.0;
        double r563927 = r563925 * r563926;
        double r563928 = z;
        double r563929 = r563927 * r563928;
        double r563930 = r563928 * r563926;
        double r563931 = r563930 * r563928;
        double r563932 = t;
        double r563933 = r563925 * r563932;
        double r563934 = r563931 - r563933;
        double r563935 = r563929 / r563934;
        double r563936 = r563924 - r563935;
        return r563936;
}
↓
double f(double x, double y, double z, double t) {
        double r563937 = x;
        double r563938 = z;
        double r563939 = y;
        double r563940 = r563939 / r563938;
        double r563941 = r563938 / r563940;
        double r563942 = t;
        double r563943 = 2.0;
        double r563944 = r563942 / r563943;
        double r563945 = r563941 - r563944;
        double r563946 = r563938 / r563945;
        double r563947 = r563937 - r563946;
        return r563947;
}

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

Original11.4
Target0.1
Herbie1.1
\[x - \frac{1}{\frac{z}{y} - \frac{\frac{t}{2}}{z}}\]

Derivation

  1. Initial program 11.4

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

  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*1.1

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

  5. Final simplification1.1

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

Reproduce

herbie shell --seed 2020047 +o rules:numerics
(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)))))