Average Error: 11.6 → 2.2
Time: 4.2s
Precision: 64
\[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}\]
↓
\[x - \frac{y}{\frac{2 \cdot z - t \cdot \frac{y}{z}}{2}}\]
x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}
↓
x - \frac{y}{\frac{2 \cdot z - t \cdot \frac{y}{z}}{2}}
double f(double x, double y, double z, double t) {
        double r452571 = x;
        double r452572 = y;
        double r452573 = 2.0;
        double r452574 = r452572 * r452573;
        double r452575 = z;
        double r452576 = r452574 * r452575;
        double r452577 = r452575 * r452573;
        double r452578 = r452577 * r452575;
        double r452579 = t;
        double r452580 = r452572 * r452579;
        double r452581 = r452578 - r452580;
        double r452582 = r452576 / r452581;
        double r452583 = r452571 - r452582;
        return r452583;
}
↓
double f(double x, double y, double z, double t) {
        double r452584 = x;
        double r452585 = y;
        double r452586 = 2.0;
        double r452587 = z;
        double r452588 = r452586 * r452587;
        double r452589 = t;
        double r452590 = r452585 / r452587;
        double r452591 = r452589 * r452590;
        double r452592 = r452588 - r452591;
        double r452593 = r452592 / r452586;
        double r452594 = r452585 / r452593;
        double r452595 = r452584 - r452594;
        return r452595;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original11.6
Target0.1
Herbie2.2
\[x - \frac{1}{\frac{z}{y} - \frac{\frac{t}{2}}{z}}\]

Derivation

  1. Initial program 11.6

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

  3. Applied associate-/l*6.4

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

  5. Applied associate-/l*6.4

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

  6. Simplified2.7

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

  8. Applied *-un-lft-identity2.7

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

  9. Applied times-frac2.2

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

  10. Simplified2.2

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

  11. Final simplification2.2

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

Reproduce

herbie shell --seed 2020100 
(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)))))