Average Error: 11.6 → 1.2
Time: 22.7s
Precision: 64
\[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}\]
↓
\[x - y \cdot \frac{1}{\frac{t}{z} \cdot \left(-\frac{y}{2}\right) + z}\]
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 \left(-\frac{y}{2}\right) + z}
double f(double x, double y, double z, double t) {
        double r323844 = x;
        double r323845 = y;
        double r323846 = 2.0;
        double r323847 = r323845 * r323846;
        double r323848 = z;
        double r323849 = r323847 * r323848;
        double r323850 = r323848 * r323846;
        double r323851 = r323850 * r323848;
        double r323852 = t;
        double r323853 = r323845 * r323852;
        double r323854 = r323851 - r323853;
        double r323855 = r323849 / r323854;
        double r323856 = r323844 - r323855;
        return r323856;
}
↓
double f(double x, double y, double z, double t) {
        double r323857 = x;
        double r323858 = y;
        double r323859 = 1.0;
        double r323860 = t;
        double r323861 = z;
        double r323862 = r323860 / r323861;
        double r323863 = 2.0;
        double r323864 = r323858 / r323863;
        double r323865 = -r323864;
        double r323866 = r323862 * r323865;
        double r323867 = r323866 + r323861;
        double r323868 = r323859 / r323867;
        double r323869 = r323858 * r323868;
        double r323870 = r323857 - r323869;
        return r323870;
}

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 fma-udef1.2

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

  7. Final simplification1.2

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

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)))))