Average Error: 12.1 → 2.9
Time: 11.4s
Precision: 64
\[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}\]
↓
\[x - \frac{y \cdot 2}{1 \cdot \left(2 \cdot z - \frac{t \cdot y}{z}\right)}\]
x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}
↓
x - \frac{y \cdot 2}{1 \cdot \left(2 \cdot z - \frac{t \cdot y}{z}\right)}
double f(double x, double y, double z, double t) {
        double r341476 = x;
        double r341477 = y;
        double r341478 = 2.0;
        double r341479 = r341477 * r341478;
        double r341480 = z;
        double r341481 = r341479 * r341480;
        double r341482 = r341480 * r341478;
        double r341483 = r341482 * r341480;
        double r341484 = t;
        double r341485 = r341477 * r341484;
        double r341486 = r341483 - r341485;
        double r341487 = r341481 / r341486;
        double r341488 = r341476 - r341487;
        return r341488;
}
↓
double f(double x, double y, double z, double t) {
        double r341489 = x;
        double r341490 = y;
        double r341491 = 2.0;
        double r341492 = r341490 * r341491;
        double r341493 = 1.0;
        double r341494 = z;
        double r341495 = r341491 * r341494;
        double r341496 = t;
        double r341497 = r341496 * r341490;
        double r341498 = r341497 / r341494;
        double r341499 = r341495 - r341498;
        double r341500 = r341493 * r341499;
        double r341501 = r341492 / r341500;
        double r341502 = r341489 - r341501;
        return r341502;
}

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

Original12.1
Target0.1
Herbie2.9
\[x - \frac{1}{\frac{z}{y} - \frac{\frac{t}{2}}{z}}\]

Derivation

  1. Initial program 12.1

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

  3. Applied associate-/l*6.9

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

  5. Applied *-un-lft-identity6.9

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

  6. Applied *-un-lft-identity6.9

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

  7. Applied times-frac6.9

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

  8. Simplified6.9

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

  9. Simplified2.9

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

  10. Final simplification2.9

    \[\leadsto x - \frac{y \cdot 2}{1 \cdot \left(2 \cdot z - \frac{t \cdot y}{z}\right)}\]

Reproduce

herbie shell --seed 2019297 
(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)))))