Average Error: 12.1 → 2.4
Time: 4.1s
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 r497638 = x;
        double r497639 = y;
        double r497640 = 2.0;
        double r497641 = r497639 * r497640;
        double r497642 = z;
        double r497643 = r497641 * r497642;
        double r497644 = r497642 * r497640;
        double r497645 = r497644 * r497642;
        double r497646 = t;
        double r497647 = r497639 * r497646;
        double r497648 = r497645 - r497647;
        double r497649 = r497643 / r497648;
        double r497650 = r497638 - r497649;
        return r497650;
}
↓
double f(double x, double y, double z, double t) {
        double r497651 = x;
        double r497652 = y;
        double r497653 = 2.0;
        double r497654 = z;
        double r497655 = r497653 * r497654;
        double r497656 = t;
        double r497657 = r497652 / r497654;
        double r497658 = r497656 * r497657;
        double r497659 = r497655 - r497658;
        double r497660 = r497659 / r497653;
        double r497661 = r497652 / r497660;
        double r497662 = r497651 - r497661;
        return r497662;
}

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

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Target

Original12.1
Target0.1
Herbie2.4
\[x - \frac{1}{\frac{z}{y} - \frac{\frac{t}{2}}{z}}\]

Derivation

  1. Initial program 12.1

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

  3. Applied associate-/l*7.1

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

  5. Applied associate-/l*7.1

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

  6. Simplified2.9

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

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

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

  9. Applied times-frac2.4

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

  10. Simplified2.4

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

  11. Final simplification2.4

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

Reproduce

herbie shell --seed 2020021 
(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)))))