Average Error: 11.2 → 0.1
Time: 13.7s
Precision: 64
\[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}\]
↓
\[x - \frac{1}{\frac{z \cdot 2}{y} - \frac{t}{z}} \cdot 2\]
x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}
↓
x - \frac{1}{\frac{z \cdot 2}{y} - \frac{t}{z}} \cdot 2
double f(double x, double y, double z, double t) {
        double r410512 = x;
        double r410513 = y;
        double r410514 = 2.0;
        double r410515 = r410513 * r410514;
        double r410516 = z;
        double r410517 = r410515 * r410516;
        double r410518 = r410516 * r410514;
        double r410519 = r410518 * r410516;
        double r410520 = t;
        double r410521 = r410513 * r410520;
        double r410522 = r410519 - r410521;
        double r410523 = r410517 / r410522;
        double r410524 = r410512 - r410523;
        return r410524;
}
↓
double f(double x, double y, double z, double t) {
        double r410525 = x;
        double r410526 = 1.0;
        double r410527 = z;
        double r410528 = 2.0;
        double r410529 = r410527 * r410528;
        double r410530 = y;
        double r410531 = r410529 / r410530;
        double r410532 = t;
        double r410533 = r410532 / r410527;
        double r410534 = r410531 - r410533;
        double r410535 = r410526 / r410534;
        double r410536 = r410535 * r410528;
        double r410537 = r410525 - r410536;
        return r410537;
}

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

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. Simplified2.5

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

  4. Applied clear-num2.5

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

  5. Simplified2.1

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

  6. Taylor expanded around 0 0.1

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

  7. Simplified0.1

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

  8. Final simplification0.1

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

Reproduce

herbie shell --seed 2019194 
(FPCore (x y z t)
  :name "Numeric.AD.Rank1.Halley:findZero from ad-4.2.4"

  :herbie-target
  (- x (/ 1.0 (- (/ z y) (/ (/ t 2.0) z))))

  (- x (/ (* (* y 2.0) z) (- (* (* z 2.0) z) (* y t)))))