Average Error: 11.6 → 1.2
Time: 22.9s
Precision: 64
\[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}\]
↓
\[x - y \cdot \frac{-1}{\frac{t}{z} \cdot \frac{y}{2} - z}\]
x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}
↓
x - y \cdot \frac{-1}{\frac{t}{z} \cdot \frac{y}{2} - z}
double f(double x, double y, double z, double t) {
        double r298858 = x;
        double r298859 = y;
        double r298860 = 2.0;
        double r298861 = r298859 * r298860;
        double r298862 = z;
        double r298863 = r298861 * r298862;
        double r298864 = r298862 * r298860;
        double r298865 = r298864 * r298862;
        double r298866 = t;
        double r298867 = r298859 * r298866;
        double r298868 = r298865 - r298867;
        double r298869 = r298863 / r298868;
        double r298870 = r298858 - r298869;
        return r298870;
}
↓
double f(double x, double y, double z, double t) {
        double r298871 = x;
        double r298872 = y;
        double r298873 = -1.0;
        double r298874 = t;
        double r298875 = z;
        double r298876 = r298874 / r298875;
        double r298877 = 2.0;
        double r298878 = r298872 / r298877;
        double r298879 = r298876 * r298878;
        double r298880 = r298879 - r298875;
        double r298881 = r298873 / r298880;
        double r298882 = r298872 * r298881;
        double r298883 = r298871 - r298882;
        return r298883;
}

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

Original11.6
Target0.1
Herbie1.2
\[x - \frac{1}{\frac{z}{y} - \frac{\frac{t}{2}}{z}}\]

Derivation

  1. Initial program 11.6

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

  2. Simplified1.1

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

  4. Applied div-inv1.2

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

  6. Applied frac-2neg1.2

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

  7. Simplified1.2

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

  8. Simplified1.2

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

  9. Final simplification1.2

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

Reproduce

herbie shell --seed 2019323 +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)))))