Average Error: 11.7 → 0.1
Time: 13.5s
Precision: 64
\[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}\]
↓
\[x - \frac{1}{\frac{z}{y} - 0.5 \cdot \frac{t}{z}}\]
x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}
↓
x - \frac{1}{\frac{z}{y} - 0.5 \cdot \frac{t}{z}}
double f(double x, double y, double z, double t) {
        double r648515 = x;
        double r648516 = y;
        double r648517 = 2.0;
        double r648518 = r648516 * r648517;
        double r648519 = z;
        double r648520 = r648518 * r648519;
        double r648521 = r648519 * r648517;
        double r648522 = r648521 * r648519;
        double r648523 = t;
        double r648524 = r648516 * r648523;
        double r648525 = r648522 - r648524;
        double r648526 = r648520 / r648525;
        double r648527 = r648515 - r648526;
        return r648527;
}
↓
double f(double x, double y, double z, double t) {
        double r648528 = x;
        double r648529 = 1.0;
        double r648530 = z;
        double r648531 = y;
        double r648532 = r648530 / r648531;
        double r648533 = 0.5;
        double r648534 = t;
        double r648535 = r648534 / r648530;
        double r648536 = r648533 * r648535;
        double r648537 = r648532 - r648536;
        double r648538 = r648529 / r648537;
        double r648539 = r648528 - r648538;
        return r648539;
}

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
Herbie0.1
\[x - \frac{1}{\frac{z}{y} - \frac{\frac{t}{2}}{z}}\]

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. Simplified1.2

    \[\leadsto \color{blue}{x - \frac{y}{\mathsf{fma}\left(\frac{y}{2}, \frac{-t}{z}, z\right)}}\]
  3. Using strategy rm

  4. Applied clear-num1.2

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

  5. Taylor expanded around 0 0.1

    \[\leadsto x - \frac{1}{\color{blue}{\frac{z}{y} - 0.5 \cdot \frac{t}{z}}}\]

  6. Final simplification0.1

    \[\leadsto x - \frac{1}{\frac{z}{y} - 0.5 \cdot \frac{t}{z}}\]

Reproduce

herbie shell --seed 2020042 +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)))))