Average Error: 11.1 → 2.8
Time: 3.7s
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 r470451 = x;
        double r470452 = y;
        double r470453 = 2.0;
        double r470454 = r470452 * r470453;
        double r470455 = z;
        double r470456 = r470454 * r470455;
        double r470457 = r470455 * r470453;
        double r470458 = r470457 * r470455;
        double r470459 = t;
        double r470460 = r470452 * r470459;
        double r470461 = r470458 - r470460;
        double r470462 = r470456 / r470461;
        double r470463 = r470451 - r470462;
        return r470463;
}
↓
double f(double x, double y, double z, double t) {
        double r470464 = x;
        double r470465 = y;
        double r470466 = 2.0;
        double r470467 = r470465 * r470466;
        double r470468 = z;
        double r470469 = r470466 * r470468;
        double r470470 = t;
        double r470471 = r470470 * r470465;
        double r470472 = r470471 / r470468;
        double r470473 = r470469 - r470472;
        double r470474 = r470467 / r470473;
        double r470475 = r470464 - r470474;
        return r470475;
}

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 +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)))))