Average Error: 0.0 → 0.0
Time: 2.8s
Precision: 64
\[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}\]
↓
\[1 - \frac{1}{2 + \sqrt[3]{{\left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}^{3}} \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}\]
1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}
↓
1 - \frac{1}{2 + \sqrt[3]{{\left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}^{3}} \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}
double f(double t) {
        double r56335 = 1.0;
        double r56336 = 2.0;
        double r56337 = t;
        double r56338 = r56336 / r56337;
        double r56339 = r56335 / r56337;
        double r56340 = r56335 + r56339;
        double r56341 = r56338 / r56340;
        double r56342 = r56336 - r56341;
        double r56343 = r56342 * r56342;
        double r56344 = r56336 + r56343;
        double r56345 = r56335 / r56344;
        double r56346 = r56335 - r56345;
        return r56346;
}
↓
double f(double t) {
        double r56347 = 1.0;
        double r56348 = 2.0;
        double r56349 = t;
        double r56350 = r56348 / r56349;
        double r56351 = r56347 / r56349;
        double r56352 = r56347 + r56351;
        double r56353 = r56350 / r56352;
        double r56354 = r56348 - r56353;
        double r56355 = 3.0;
        double r56356 = pow(r56354, r56355);
        double r56357 = cbrt(r56356);
        double r56358 = r56357 * r56354;
        double r56359 = r56348 + r56358;
        double r56360 = r56347 / r56359;
        double r56361 = r56347 - r56360;
        return r56361;
}

Error

Bits error versus t

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.0

    \[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}\]
  2. Using strategy rm

  3. Applied add-cbrt-cube0.0

    \[\leadsto 1 - \frac{1}{2 + \color{blue}{\sqrt[3]{\left(\left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}} \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}\]

  4. Simplified0.0

    \[\leadsto 1 - \frac{1}{2 + \sqrt[3]{\color{blue}{{\left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}^{3}}} \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}\]

  5. Final simplification0.0

    \[\leadsto 1 - \frac{1}{2 + \sqrt[3]{{\left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}^{3}} \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}\]

Reproduce

herbie shell --seed 2020057 
(FPCore (t)
  :name "Kahan p13 Example 3"
  :precision binary64
  (- 1 (/ 1 (+ 2 (* (- 2 (/ (/ 2 t) (+ 1 (/ 1 t)))) (- 2 (/ (/ 2 t) (+ 1 (/ 1 t)))))))))