Average Error: 0.0 → 0.0
Time: 9.2s
Precision: 64
\[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}\]
↓
\[1 - \frac{1}{2 + \frac{\left({2}^{3} - {\left(\frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}^{3}\right) \cdot \left({2}^{3} - {\left(\frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}^{3}\right)}{\left(2 \cdot 2 + \frac{\frac{2}{t}}{1 + \frac{1}{t}} \cdot \left(2 + \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)\right) \cdot \left(2 \cdot 2 + \frac{\frac{2}{t}}{1 + \frac{1}{t}} \cdot \left(2 + \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)\right)}}\]
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 + \frac{\left({2}^{3} - {\left(\frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}^{3}\right) \cdot \left({2}^{3} - {\left(\frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}^{3}\right)}{\left(2 \cdot 2 + \frac{\frac{2}{t}}{1 + \frac{1}{t}} \cdot \left(2 + \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)\right) \cdot \left(2 \cdot 2 + \frac{\frac{2}{t}}{1 + \frac{1}{t}} \cdot \left(2 + \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)\right)}}
double f(double t) {
        double r66278 = 1.0;
        double r66279 = 2.0;
        double r66280 = t;
        double r66281 = r66279 / r66280;
        double r66282 = r66278 / r66280;
        double r66283 = r66278 + r66282;
        double r66284 = r66281 / r66283;
        double r66285 = r66279 - r66284;
        double r66286 = r66285 * r66285;
        double r66287 = r66279 + r66286;
        double r66288 = r66278 / r66287;
        double r66289 = r66278 - r66288;
        return r66289;
}
↓
double f(double t) {
        double r66290 = 1.0;
        double r66291 = 2.0;
        double r66292 = 3.0;
        double r66293 = pow(r66291, r66292);
        double r66294 = t;
        double r66295 = r66291 / r66294;
        double r66296 = r66290 / r66294;
        double r66297 = r66290 + r66296;
        double r66298 = r66295 / r66297;
        double r66299 = pow(r66298, r66292);
        double r66300 = r66293 - r66299;
        double r66301 = r66300 * r66300;
        double r66302 = r66291 * r66291;
        double r66303 = r66291 + r66298;
        double r66304 = r66298 * r66303;
        double r66305 = r66302 + r66304;
        double r66306 = r66305 * r66305;
        double r66307 = r66301 / r66306;
        double r66308 = r66291 + r66307;
        double r66309 = r66290 / r66308;
        double r66310 = r66290 - r66309;
        return r66310;
}

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 flip3--0.0

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

  4. Applied flip3--0.0

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

  5. Applied frac-times0.0

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

  6. Simplified0.0

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

  7. Final simplification0.0

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

Reproduce

herbie shell --seed 2020042 
(FPCore (t)
  :name "Kahan p13 Example 3"
  :precision binary64
  (- 1 (/ 1 (+ 2 (* (- 2 (/ (/ 2 t) (+ 1 (/ 1 t)))) (- 2 (/ (/ 2 t) (+ 1 (/ 1 t)))))))))