Average Error: 11.1 → 2.8
Time: 3.6s
Precision: 64
\[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}\]
↓
\[x - \frac{y \cdot 2}{2 \cdot z - \frac{t \cdot y}{z}}\]
x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}
↓
x - \frac{y \cdot 2}{2 \cdot z - \frac{t \cdot y}{z}}
double f(double x, double y, double z, double t) {
        double r496508 = x;
        double r496509 = y;
        double r496510 = 2.0;
        double r496511 = r496509 * r496510;
        double r496512 = z;
        double r496513 = r496511 * r496512;
        double r496514 = r496512 * r496510;
        double r496515 = r496514 * r496512;
        double r496516 = t;
        double r496517 = r496509 * r496516;
        double r496518 = r496515 - r496517;
        double r496519 = r496513 / r496518;
        double r496520 = r496508 - r496519;
        return r496520;
}
↓
double f(double x, double y, double z, double t) {
        double r496521 = x;
        double r496522 = y;
        double r496523 = 2.0;
        double r496524 = r496522 * r496523;
        double r496525 = z;
        double r496526 = r496523 * r496525;
        double r496527 = t;
        double r496528 = r496527 * r496522;
        double r496529 = r496528 / r496525;
        double r496530 = r496526 - r496529;
        double r496531 = r496524 / r496530;
        double r496532 = r496521 - r496531;
        return r496532;
}

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

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

Derivation

  1. Initial program 11.1

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

  3. Applied associate-/l*6.3

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

  4. Taylor expanded around 0 2.8

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

  5. Final simplification2.8

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

Reproduce

herbie shell --seed 2020018 
(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)))))