Average Error: 12.1 → 0.1
Time: 21.5s
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 r323259 = x;
        double r323260 = y;
        double r323261 = 2.0;
        double r323262 = r323260 * r323261;
        double r323263 = z;
        double r323264 = r323262 * r323263;
        double r323265 = r323263 * r323261;
        double r323266 = r323265 * r323263;
        double r323267 = t;
        double r323268 = r323260 * r323267;
        double r323269 = r323266 - r323268;
        double r323270 = r323264 / r323269;
        double r323271 = r323259 - r323270;
        return r323271;
}
↓
double f(double x, double y, double z, double t) {
        double r323272 = x;
        double r323273 = 1.0;
        double r323274 = z;
        double r323275 = y;
        double r323276 = r323274 / r323275;
        double r323277 = t;
        double r323278 = 2.0;
        double r323279 = r323277 / r323278;
        double r323280 = r323279 / r323274;
        double r323281 = r323276 - r323280;
        double r323282 = r323273 / r323281;
        double r323283 = r323272 - r323282;
        return r323283;
}

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

Original12.1
Target0.1
Herbie0.1
\[x - \frac{1}{\frac{z}{y} - \frac{\frac{t}{2}}{z}}\]

Derivation

  1. Initial program 12.1

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

  2. Simplified3.8

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

  4. Applied clear-num3.9

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

  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 2019322 
(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)))))