Average Error: 11.9 → 0.1
Time: 17.5s
Precision: 64
\[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}\]
↓
\[x - \frac{1}{\frac{z}{y} - \left(0.5 \cdot t\right) \cdot \frac{1}{z}}\]
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} - \left(0.5 \cdot t\right) \cdot \frac{1}{z}}
double f(double x, double y, double z, double t) {
        double r28670607 = x;
        double r28670608 = y;
        double r28670609 = 2.0;
        double r28670610 = r28670608 * r28670609;
        double r28670611 = z;
        double r28670612 = r28670610 * r28670611;
        double r28670613 = r28670611 * r28670609;
        double r28670614 = r28670613 * r28670611;
        double r28670615 = t;
        double r28670616 = r28670608 * r28670615;
        double r28670617 = r28670614 - r28670616;
        double r28670618 = r28670612 / r28670617;
        double r28670619 = r28670607 - r28670618;
        return r28670619;
}
↓
double f(double x, double y, double z, double t) {
        double r28670620 = x;
        double r28670621 = 1.0;
        double r28670622 = z;
        double r28670623 = y;
        double r28670624 = r28670622 / r28670623;
        double r28670625 = 0.5;
        double r28670626 = t;
        double r28670627 = r28670625 * r28670626;
        double r28670628 = r28670621 / r28670622;
        double r28670629 = r28670627 * r28670628;
        double r28670630 = r28670624 - r28670629;
        double r28670631 = r28670621 / r28670630;
        double r28670632 = r28670620 - r28670631;
        return r28670632;
}

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

Original11.9
Target0.1
Herbie0.1
\[x - \frac{1}{\frac{z}{y} - \frac{\frac{t}{2}}{z}}\]

Derivation

  1. Initial program 11.9

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

  2. Simplified0.9

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

  4. Applied clear-num1.0

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

  5. Taylor expanded around 0 0.1

    \[\leadsto x - \frac{1}{\color{blue}{\frac{z}{y} - 0.5 \cdot \frac{t}{z}}}\]
  6. Using strategy rm

  7. Applied div-inv0.1

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

  8. Applied associate-*r*0.1

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

  9. Final simplification0.1

    \[\leadsto x - \frac{1}{\frac{z}{y} - \left(0.5 \cdot t\right) \cdot \frac{1}{z}}\]

Reproduce

herbie shell --seed 2019169 
(FPCore (x y z t)
  :name "Numeric.AD.Rank1.Halley:findZero from ad-4.2.4"

  :herbie-target
  (- x (/ 1.0 (- (/ z y) (/ (/ t 2.0) z))))

  (- x (/ (* (* y 2.0) z) (- (* (* z 2.0) z) (* y t)))))