Average Error: 11.5 → 1.1
Time: 12.7s
Precision: 64
\[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}\]
↓
\[x - \frac{y}{\frac{2 \cdot z - \frac{t}{z} \cdot y}{2}}\]
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}{z} \cdot y}{2}}
double f(double x, double y, double z, double t) {
        double r453429 = x;
        double r453430 = y;
        double r453431 = 2.0;
        double r453432 = r453430 * r453431;
        double r453433 = z;
        double r453434 = r453432 * r453433;
        double r453435 = r453433 * r453431;
        double r453436 = r453435 * r453433;
        double r453437 = t;
        double r453438 = r453430 * r453437;
        double r453439 = r453436 - r453438;
        double r453440 = r453434 / r453439;
        double r453441 = r453429 - r453440;
        return r453441;
}
↓
double f(double x, double y, double z, double t) {
        double r453442 = x;
        double r453443 = y;
        double r453444 = 2.0;
        double r453445 = z;
        double r453446 = r453444 * r453445;
        double r453447 = t;
        double r453448 = r453447 / r453445;
        double r453449 = r453448 * r453443;
        double r453450 = r453446 - r453449;
        double r453451 = r453450 / r453444;
        double r453452 = r453443 / r453451;
        double r453453 = r453442 - r453452;
        return r453453;
}

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

Original11.5
Target0.1
Herbie1.1
\[x - \frac{1}{\frac{z}{y} - \frac{\frac{t}{2}}{z}}\]

Derivation

  1. Initial program 11.5

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

  3. Applied associate-/l*6.9

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

  5. Applied associate-/l*6.9

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

  6. Simplified2.8

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

  8. Applied associate-/l*2.3

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

  10. Applied associate-/r/1.1

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

  11. Final simplification1.1

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

Reproduce

herbie shell --seed 2019298 
(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)))))