Average Error: 11.9 → 0.1
Time: 14.3s
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 r519532 = x;
        double r519533 = y;
        double r519534 = 2.0;
        double r519535 = r519533 * r519534;
        double r519536 = z;
        double r519537 = r519535 * r519536;
        double r519538 = r519536 * r519534;
        double r519539 = r519538 * r519536;
        double r519540 = t;
        double r519541 = r519533 * r519540;
        double r519542 = r519539 - r519541;
        double r519543 = r519537 / r519542;
        double r519544 = r519532 - r519543;
        return r519544;
}
↓
double f(double x, double y, double z, double t) {
        double r519545 = x;
        double r519546 = 1.0;
        double r519547 = z;
        double r519548 = y;
        double r519549 = r519547 / r519548;
        double r519550 = t;
        double r519551 = 2.0;
        double r519552 = r519550 / r519551;
        double r519553 = r519552 / r519547;
        double r519554 = r519549 - r519553;
        double r519555 = r519546 / r519554;
        double r519556 = r519545 - r519555;
        return r519556;
}

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

Original11.9
Target0.1
Herbie0.1
\[x - \frac{1}{\frac{z}{y} - \frac{\frac{t}{2}}{z}}\]

Derivation

  1. Initial program 11.9

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

  2. Simplified3.5

    \[\leadsto \color{blue}{x - \frac{z}{\frac{z \cdot z}{y} - \frac{t}{2}}}\]
  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 2019350 
(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)))))