Average Error: 11.2 → 2.8
Time: 12.2s
Precision: 64
\[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}\]
↓
\[x - \frac{y}{\frac{2 \cdot z - \frac{t \cdot y}{z}}{2}}\]
x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}
↓
x - \frac{y}{\frac{2 \cdot z - \frac{t \cdot y}{z}}{2}}
double f(double x, double y, double z, double t) {
        double r440388 = x;
        double r440389 = y;
        double r440390 = 2.0;
        double r440391 = r440389 * r440390;
        double r440392 = z;
        double r440393 = r440391 * r440392;
        double r440394 = r440392 * r440390;
        double r440395 = r440394 * r440392;
        double r440396 = t;
        double r440397 = r440389 * r440396;
        double r440398 = r440395 - r440397;
        double r440399 = r440393 / r440398;
        double r440400 = r440388 - r440399;
        return r440400;
}
↓
double f(double x, double y, double z, double t) {
        double r440401 = x;
        double r440402 = y;
        double r440403 = 2.0;
        double r440404 = z;
        double r440405 = r440403 * r440404;
        double r440406 = t;
        double r440407 = r440406 * r440402;
        double r440408 = r440407 / r440404;
        double r440409 = r440405 - r440408;
        double r440410 = r440409 / r440403;
        double r440411 = r440402 / r440410;
        double r440412 = r440401 - r440411;
        return r440412;
}

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

Original11.2
Target0.1
Herbie2.8
\[x - \frac{1}{\frac{z}{y} - \frac{\frac{t}{2}}{z}}\]

Derivation

  1. Initial program 11.2

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

  3. Applied associate-/l*6.6

    \[\leadsto x - \color{blue}{\frac{y \cdot 2}{\frac{\left(z \cdot 2\right) \cdot z - y \cdot t}{z}}}\]
  4. Using strategy rm

  5. Applied associate-/l*6.6

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

  6. Simplified2.8

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

  7. Final simplification2.8

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

Reproduce

herbie shell --seed 2019294 
(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)))))