Average Error: 11.6 → 2.9
Time: 4.4s
Precision: 64
\[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}\]
↓
\[x - \left(y \cdot 2\right) \cdot \frac{1}{2 \cdot z - \frac{t \cdot y}{z}}\]
x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}
↓
x - \left(y \cdot 2\right) \cdot \frac{1}{2 \cdot z - \frac{t \cdot y}{z}}
double f(double x, double y, double z, double t) {
        double r495732 = x;
        double r495733 = y;
        double r495734 = 2.0;
        double r495735 = r495733 * r495734;
        double r495736 = z;
        double r495737 = r495735 * r495736;
        double r495738 = r495736 * r495734;
        double r495739 = r495738 * r495736;
        double r495740 = t;
        double r495741 = r495733 * r495740;
        double r495742 = r495739 - r495741;
        double r495743 = r495737 / r495742;
        double r495744 = r495732 - r495743;
        return r495744;
}
↓
double f(double x, double y, double z, double t) {
        double r495745 = x;
        double r495746 = y;
        double r495747 = 2.0;
        double r495748 = r495746 * r495747;
        double r495749 = 1.0;
        double r495750 = z;
        double r495751 = r495747 * r495750;
        double r495752 = t;
        double r495753 = r495752 * r495746;
        double r495754 = r495753 / r495750;
        double r495755 = r495751 - r495754;
        double r495756 = r495749 / r495755;
        double r495757 = r495748 * r495756;
        double r495758 = r495745 - r495757;
        return r495758;
}

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

Original11.6
Target0.1
Herbie2.9
\[x - \frac{1}{\frac{z}{y} - \frac{\frac{t}{2}}{z}}\]

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*7.0

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

  4. Taylor expanded around 0 2.9

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

  6. Applied div-inv2.9

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

  7. Final simplification2.9

    \[\leadsto x - \left(y \cdot 2\right) \cdot \frac{1}{2 \cdot z - \frac{t \cdot y}{z}}\]

Reproduce

herbie shell --seed 2020089 +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)))))