Average Error: 11.5 → 0.1
Time: 18.1s
Precision: 64
\[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}\]
↓
\[x - \frac{1}{\frac{z}{y} - 0.5 \cdot \frac{t}{z}}\]
x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}
↓
x - \frac{1}{\frac{z}{y} - 0.5 \cdot \frac{t}{z}}
double f(double x, double y, double z, double t) {
        double r279924 = x;
        double r279925 = y;
        double r279926 = 2.0;
        double r279927 = r279925 * r279926;
        double r279928 = z;
        double r279929 = r279927 * r279928;
        double r279930 = r279928 * r279926;
        double r279931 = r279930 * r279928;
        double r279932 = t;
        double r279933 = r279925 * r279932;
        double r279934 = r279931 - r279933;
        double r279935 = r279929 / r279934;
        double r279936 = r279924 - r279935;
        return r279936;
}
↓
double f(double x, double y, double z, double t) {
        double r279937 = x;
        double r279938 = 1.0;
        double r279939 = z;
        double r279940 = y;
        double r279941 = r279939 / r279940;
        double r279942 = 0.5;
        double r279943 = t;
        double r279944 = r279943 / r279939;
        double r279945 = r279942 * r279944;
        double r279946 = r279941 - r279945;
        double r279947 = r279938 / r279946;
        double r279948 = r279937 - r279947;
        return r279948;
}

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

Original11.5
Target0.1
Herbie0.1
\[x - \frac{1}{\frac{z}{y} - \frac{\frac{t}{2}}{z}}\]

Derivation

  1. Initial program 11.5

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

  2. Simplified0.9

    \[\leadsto \color{blue}{x - \frac{y}{\mathsf{fma}\left(\frac{t}{z}, -\frac{y}{2}, z\right)}}\]
  3. Using strategy rm

  4. Applied clear-num0.9

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

  5. Simplified2.9

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

  6. Taylor expanded around 0 0.1

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

  7. Final simplification0.1

    \[\leadsto x - \frac{1}{\frac{z}{y} - 0.5 \cdot \frac{t}{z}}\]

Reproduce

herbie shell --seed 2019325 +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)))))