Average Error: 11.2 → 1.0
Time: 4.5s
Precision: 64
\[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}\]
↓
\[x - \frac{y \cdot 2}{2 \cdot z - \frac{\frac{t}{z}}{\frac{1}{y}}}\]
x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}
↓
x - \frac{y \cdot 2}{2 \cdot z - \frac{\frac{t}{z}}{\frac{1}{y}}}
double f(double x, double y, double z, double t) {
        double r432043 = x;
        double r432044 = y;
        double r432045 = 2.0;
        double r432046 = r432044 * r432045;
        double r432047 = z;
        double r432048 = r432046 * r432047;
        double r432049 = r432047 * r432045;
        double r432050 = r432049 * r432047;
        double r432051 = t;
        double r432052 = r432044 * r432051;
        double r432053 = r432050 - r432052;
        double r432054 = r432048 / r432053;
        double r432055 = r432043 - r432054;
        return r432055;
}
↓
double f(double x, double y, double z, double t) {
        double r432056 = x;
        double r432057 = y;
        double r432058 = 2.0;
        double r432059 = r432057 * r432058;
        double r432060 = z;
        double r432061 = r432058 * r432060;
        double r432062 = t;
        double r432063 = r432062 / r432060;
        double r432064 = 1.0;
        double r432065 = r432064 / r432057;
        double r432066 = r432063 / r432065;
        double r432067 = r432061 - r432066;
        double r432068 = r432059 / r432067;
        double r432069 = r432056 - r432068;
        return r432069;
}

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

Original11.2
Target0.1
Herbie1.0
\[x - \frac{1}{\frac{z}{y} - \frac{\frac{t}{2}}{z}}\]

Derivation

  1. Initial program 11.2

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

  3. Applied associate-/l*6.3

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

  4. Taylor expanded around 0 2.3

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

  6. Applied associate-/l*2.2

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

  8. Applied div-inv2.3

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

  9. Applied associate-/r*1.0

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

  10. Final simplification1.0

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

Reproduce

herbie shell --seed 2019353 +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)))))