Average Error: 0.0 → 0.0
Time: 16.8s
Precision: 64
\[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}\]
↓
\[1 - \frac{1}{\frac{{2}^{3} - {\left(\frac{2}{t \cdot 1 + 1}\right)}^{3}}{2 \cdot 2 + \frac{2}{t \cdot 1 + 1} \cdot \left(2 + \frac{2}{t \cdot 1 + 1}\right)} \cdot \left(2 - \frac{2}{t \cdot 1 + 1}\right) + 2}\]
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}{\frac{{2}^{3} - {\left(\frac{2}{t \cdot 1 + 1}\right)}^{3}}{2 \cdot 2 + \frac{2}{t \cdot 1 + 1} \cdot \left(2 + \frac{2}{t \cdot 1 + 1}\right)} \cdot \left(2 - \frac{2}{t \cdot 1 + 1}\right) + 2}
double f(double t) {
        double r35530 = 1.0;
        double r35531 = 2.0;
        double r35532 = t;
        double r35533 = r35531 / r35532;
        double r35534 = r35530 / r35532;
        double r35535 = r35530 + r35534;
        double r35536 = r35533 / r35535;
        double r35537 = r35531 - r35536;
        double r35538 = r35537 * r35537;
        double r35539 = r35531 + r35538;
        double r35540 = r35530 / r35539;
        double r35541 = r35530 - r35540;
        return r35541;
}
↓
double f(double t) {
        double r35542 = 1.0;
        double r35543 = 2.0;
        double r35544 = 3.0;
        double r35545 = pow(r35543, r35544);
        double r35546 = t;
        double r35547 = r35546 * r35542;
        double r35548 = r35547 + r35542;
        double r35549 = r35543 / r35548;
        double r35550 = pow(r35549, r35544);
        double r35551 = r35545 - r35550;
        double r35552 = r35543 * r35543;
        double r35553 = r35543 + r35549;
        double r35554 = r35549 * r35553;
        double r35555 = r35552 + r35554;
        double r35556 = r35551 / r35555;
        double r35557 = r35543 - r35549;
        double r35558 = r35556 * r35557;
        double r35559 = r35558 + r35543;
        double r35560 = r35542 / r35559;
        double r35561 = r35542 - r35560;
        return r35561;
}

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. Simplified0.0

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

  4. Applied flip3--0.0

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

  5. Simplified0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2019347 
(FPCore (t)
  :name "Kahan p13 Example 3"
  :precision binary64
  (- 1 (/ 1 (+ 2 (* (- 2 (/ (/ 2 t) (+ 1 (/ 1 t)))) (- 2 (/ (/ 2 t) (+ 1 (/ 1 t)))))))))