Average Error: 11.7 → 0.1
Time: 20.9s
Precision: 64
\[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}\]
↓
\[x - \frac{1}{\frac{z}{y} - \frac{\frac{t}{2}}{z}}\]
x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}
↓
x - \frac{1}{\frac{z}{y} - \frac{\frac{t}{2}}{z}}
double f(double x, double y, double z, double t) {
        double r330127 = x;
        double r330128 = y;
        double r330129 = 2.0;
        double r330130 = r330128 * r330129;
        double r330131 = z;
        double r330132 = r330130 * r330131;
        double r330133 = r330131 * r330129;
        double r330134 = r330133 * r330131;
        double r330135 = t;
        double r330136 = r330128 * r330135;
        double r330137 = r330134 - r330136;
        double r330138 = r330132 / r330137;
        double r330139 = r330127 - r330138;
        return r330139;
}
↓
double f(double x, double y, double z, double t) {
        double r330140 = x;
        double r330141 = 1.0;
        double r330142 = z;
        double r330143 = y;
        double r330144 = r330142 / r330143;
        double r330145 = t;
        double r330146 = 2.0;
        double r330147 = r330145 / r330146;
        double r330148 = r330147 / r330142;
        double r330149 = r330144 - r330148;
        double r330150 = r330141 / r330149;
        double r330151 = r330140 - r330150;
        return r330151;
}

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

Original11.7
Target0.1
Herbie0.1
\[x - \frac{1}{\frac{z}{y} - \frac{\frac{t}{2}}{z}}\]

Derivation

  1. Initial program 11.7

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

  2. Simplified3.6

    \[\leadsto \color{blue}{x - \frac{z}{\frac{z \cdot z}{y} - \frac{t}{2}}}\]
  3. Using strategy rm

  4. Applied clear-num3.6

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

  5. Simplified0.1

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

  6. Final simplification0.1

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

Reproduce

herbie shell --seed 2019304 
(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)))))