Average Error: 11.6 → 1.3
Time: 18.7s
Precision: 64
\[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}\]
↓
\[x - \frac{z}{\frac{z}{\frac{y}{z}} - \frac{t}{2}}\]
x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}
↓
x - \frac{z}{\frac{z}{\frac{y}{z}} - \frac{t}{2}}
double f(double x, double y, double z, double t) {
        double r290569 = x;
        double r290570 = y;
        double r290571 = 2.0;
        double r290572 = r290570 * r290571;
        double r290573 = z;
        double r290574 = r290572 * r290573;
        double r290575 = r290573 * r290571;
        double r290576 = r290575 * r290573;
        double r290577 = t;
        double r290578 = r290570 * r290577;
        double r290579 = r290576 - r290578;
        double r290580 = r290574 / r290579;
        double r290581 = r290569 - r290580;
        return r290581;
}
↓
double f(double x, double y, double z, double t) {
        double r290582 = x;
        double r290583 = z;
        double r290584 = y;
        double r290585 = r290584 / r290583;
        double r290586 = r290583 / r290585;
        double r290587 = t;
        double r290588 = 2.0;
        double r290589 = r290587 / r290588;
        double r290590 = r290586 - r290589;
        double r290591 = r290583 / r290590;
        double r290592 = r290582 - r290591;
        return r290592;
}

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

Original11.6
Target0.1
Herbie1.3
\[x - \frac{1}{\frac{z}{y} - \frac{\frac{t}{2}}{z}}\]

Derivation

  1. Initial program 11.6

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

  2. Simplified3.5

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

  4. Applied associate-/l*1.3

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

  5. Final simplification1.3

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

Reproduce

herbie shell --seed 2019323 
(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)))))