Average Error: 11.6 → 0.1
Time: 15.2s
Precision: 64
\[x - \frac{\left(y \cdot 2.0\right) \cdot z}{\left(z \cdot 2.0\right) \cdot z - y \cdot t}\]
↓
\[x - \frac{1}{\frac{z}{y} - \frac{t}{z} \cdot 0.5}\]
x - \frac{\left(y \cdot 2.0\right) \cdot z}{\left(z \cdot 2.0\right) \cdot z - y \cdot t}
↓
x - \frac{1}{\frac{z}{y} - \frac{t}{z} \cdot 0.5}
double f(double x, double y, double z, double t) {
        double r22364998 = x;
        double r22364999 = y;
        double r22365000 = 2.0;
        double r22365001 = r22364999 * r22365000;
        double r22365002 = z;
        double r22365003 = r22365001 * r22365002;
        double r22365004 = r22365002 * r22365000;
        double r22365005 = r22365004 * r22365002;
        double r22365006 = t;
        double r22365007 = r22364999 * r22365006;
        double r22365008 = r22365005 - r22365007;
        double r22365009 = r22365003 / r22365008;
        double r22365010 = r22364998 - r22365009;
        return r22365010;
}
↓
double f(double x, double y, double z, double t) {
        double r22365011 = x;
        double r22365012 = 1.0;
        double r22365013 = z;
        double r22365014 = y;
        double r22365015 = r22365013 / r22365014;
        double r22365016 = t;
        double r22365017 = r22365016 / r22365013;
        double r22365018 = 0.5;
        double r22365019 = r22365017 * r22365018;
        double r22365020 = r22365015 - r22365019;
        double r22365021 = r22365012 / r22365020;
        double r22365022 = r22365011 - r22365021;
        return r22365022;
}

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
Herbie0.1
\[x - \frac{1}{\frac{z}{y} - \frac{\frac{t}{2.0}}{z}}\]

Derivation

  1. Initial program 11.6

    \[x - \frac{\left(y \cdot 2.0\right) \cdot z}{\left(z \cdot 2.0\right) \cdot z - y \cdot t}\]

  2. Simplified1.0

    \[\leadsto \color{blue}{x - \frac{y}{z - \frac{\frac{t}{z} \cdot y}{2.0}}}\]
  3. Using strategy rm

  4. Applied clear-num1.0

    \[\leadsto x - \color{blue}{\frac{1}{\frac{z - \frac{\frac{t}{z} \cdot y}{2.0}}{y}}}\]

  5. Taylor expanded around 0 0.1

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

  6. Final simplification0.1

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

Reproduce

herbie shell --seed 2019163 
(FPCore (x y z t)
  :name "Numeric.AD.Rank1.Halley:findZero from ad-4.2.4"

  :herbie-target
  (- x (/ 1 (- (/ z y) (/ (/ t 2.0) z))))

  (- x (/ (* (* y 2.0) z) (- (* (* z 2.0) z) (* y t)))))