Average Error: 11.6 → 0.1
Time: 19.0s
Precision: 64
\[x - \frac{\left(y \cdot 2.0\right) \cdot z}{\left(z \cdot 2.0\right) \cdot z - y \cdot t}\]
↓
\[x - \frac{1}{\frac{z}{y} - \frac{0.5}{\frac{z}{t}}}\]
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{0.5}{\frac{z}{t}}}
double f(double x, double y, double z, double t) {
        double r22174046 = x;
        double r22174047 = y;
        double r22174048 = 2.0;
        double r22174049 = r22174047 * r22174048;
        double r22174050 = z;
        double r22174051 = r22174049 * r22174050;
        double r22174052 = r22174050 * r22174048;
        double r22174053 = r22174052 * r22174050;
        double r22174054 = t;
        double r22174055 = r22174047 * r22174054;
        double r22174056 = r22174053 - r22174055;
        double r22174057 = r22174051 / r22174056;
        double r22174058 = r22174046 - r22174057;
        return r22174058;
}
↓
double f(double x, double y, double z, double t) {
        double r22174059 = x;
        double r22174060 = 1.0;
        double r22174061 = z;
        double r22174062 = y;
        double r22174063 = r22174061 / r22174062;
        double r22174064 = 0.5;
        double r22174065 = t;
        double r22174066 = r22174061 / r22174065;
        double r22174067 = r22174064 / r22174066;
        double r22174068 = r22174063 - r22174067;
        double r22174069 = r22174060 / r22174068;
        double r22174070 = r22174059 - r22174069;
        return r22174070;
}

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

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

  4. Applied clear-num1.0

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

  5. Taylor expanded around 0 0.1

    \[\leadsto x - \frac{1}{\color{blue}{\frac{z}{y} - 0.5 \cdot \frac{t}{z}}}\]
  6. Using strategy rm

  7. Applied clear-num0.1

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

  8. Simplified0.1

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

  9. Final simplification0.1

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

Reproduce

herbie shell --seed 2019163 +o rules:numerics
(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)))))