Average Error: 11.3 → 0.1
Time: 16.0s
Precision: 64
\[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}\]
↓
\[x - \frac{1}{\mathsf{fma}\left(\frac{z}{y}, 1, -\frac{t}{z \cdot 2}\right)}\]
x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}
↓
x - \frac{1}{\mathsf{fma}\left(\frac{z}{y}, 1, -\frac{t}{z \cdot 2}\right)}
double f(double x, double y, double z, double t) {
        double r321913 = x;
        double r321914 = y;
        double r321915 = 2.0;
        double r321916 = r321914 * r321915;
        double r321917 = z;
        double r321918 = r321916 * r321917;
        double r321919 = r321917 * r321915;
        double r321920 = r321919 * r321917;
        double r321921 = t;
        double r321922 = r321914 * r321921;
        double r321923 = r321920 - r321922;
        double r321924 = r321918 / r321923;
        double r321925 = r321913 - r321924;
        return r321925;
}
↓
double f(double x, double y, double z, double t) {
        double r321926 = x;
        double r321927 = 1.0;
        double r321928 = z;
        double r321929 = y;
        double r321930 = r321928 / r321929;
        double r321931 = t;
        double r321932 = 2.0;
        double r321933 = r321928 * r321932;
        double r321934 = r321931 / r321933;
        double r321935 = -r321934;
        double r321936 = fma(r321930, r321927, r321935);
        double r321937 = r321927 / r321936;
        double r321938 = r321926 - r321937;
        return r321938;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original11.3
Target0.1
Herbie0.1
\[x - \frac{1}{\frac{z}{y} - \frac{\frac{t}{2}}{z}}\]

Derivation

  1. Initial program 11.3

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

  3. Applied clear-num11.4

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

  4. Simplified0.1

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

  5. Final simplification0.1

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

Reproduce

herbie shell --seed 2019326 +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)))))