Average Error: 11.7 → 2.3
Time: 3.9s
Precision: 64
\[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}\]
↓
\[x - \frac{y \cdot 2}{z \cdot 2 - t \cdot \frac{y}{z}}\]
x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}
↓
x - \frac{y \cdot 2}{z \cdot 2 - t \cdot \frac{y}{z}}
double f(double x, double y, double z, double t) {
        double r506383 = x;
        double r506384 = y;
        double r506385 = 2.0;
        double r506386 = r506384 * r506385;
        double r506387 = z;
        double r506388 = r506386 * r506387;
        double r506389 = r506387 * r506385;
        double r506390 = r506389 * r506387;
        double r506391 = t;
        double r506392 = r506384 * r506391;
        double r506393 = r506390 - r506392;
        double r506394 = r506388 / r506393;
        double r506395 = r506383 - r506394;
        return r506395;
}
↓
double f(double x, double y, double z, double t) {
        double r506396 = x;
        double r506397 = y;
        double r506398 = 2.0;
        double r506399 = r506397 * r506398;
        double r506400 = z;
        double r506401 = r506400 * r506398;
        double r506402 = t;
        double r506403 = r506397 / r506400;
        double r506404 = r506402 * r506403;
        double r506405 = r506401 - r506404;
        double r506406 = r506399 / r506405;
        double r506407 = r506396 - r506406;
        return r506407;
}

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

Original11.7
Target0.1
Herbie2.3
\[x - \frac{1}{\frac{z}{y} - \frac{\frac{t}{2}}{z}}\]

Derivation

  1. Initial program 11.7

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

  3. Applied associate-/l*6.7

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

  5. Applied div-sub6.7

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

  6. Simplified2.9

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

  7. Simplified2.9

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

  9. Applied *-un-lft-identity2.9

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

  10. Applied times-frac2.3

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

  11. Simplified2.3

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

  12. Final simplification2.3

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

Reproduce

herbie shell --seed 2020001 +o rules:numerics
(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)))))