Average Error: 0.0 → 0.0
Time: 8.6s
Precision: 64
\[\frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}\]
↓
\[\frac{1 + \left(2 - \frac{2}{1 + t}\right) \cdot \left(2 - \frac{2}{1 + t}\right)}{\left(2 - \frac{2}{1 + t}\right) \cdot \log \left(\frac{e^{2}}{e^{\frac{2}{1 + t}}}\right) + 2}\]
\frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}
↓
\frac{1 + \left(2 - \frac{2}{1 + t}\right) \cdot \left(2 - \frac{2}{1 + t}\right)}{\left(2 - \frac{2}{1 + t}\right) \cdot \log \left(\frac{e^{2}}{e^{\frac{2}{1 + t}}}\right) + 2}
double f(double t) {
        double r973594 = 1.0;
        double r973595 = 2.0;
        double r973596 = t;
        double r973597 = r973595 / r973596;
        double r973598 = r973594 / r973596;
        double r973599 = r973594 + r973598;
        double r973600 = r973597 / r973599;
        double r973601 = r973595 - r973600;
        double r973602 = r973601 * r973601;
        double r973603 = r973594 + r973602;
        double r973604 = r973595 + r973602;
        double r973605 = r973603 / r973604;
        return r973605;
}
↓
double f(double t) {
        double r973606 = 1.0;
        double r973607 = 2.0;
        double r973608 = t;
        double r973609 = r973606 + r973608;
        double r973610 = r973607 / r973609;
        double r973611 = r973607 - r973610;
        double r973612 = r973611 * r973611;
        double r973613 = r973606 + r973612;
        double r973614 = exp(r973607);
        double r973615 = exp(r973610);
        double r973616 = r973614 / r973615;
        double r973617 = log(r973616);
        double r973618 = r973611 * r973617;
        double r973619 = r973618 + r973607;
        double r973620 = r973613 / r973619;
        return r973620;
}

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

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

  2. Simplified0.0

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

  4. Applied add-log-exp0.0

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

  5. Applied add-log-exp0.0

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

  6. Applied diff-log0.0

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

  7. Final simplification0.0

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

Reproduce

herbie shell --seed 2019132 
(FPCore (t)
  :name "Kahan p13 Example 2"
  (/ (+ 1 (* (- 2 (/ (/ 2 t) (+ 1 (/ 1 t)))) (- 2 (/ (/ 2 t) (+ 1 (/ 1 t)))))) (+ 2 (* (- 2 (/ (/ 2 t) (+ 1 (/ 1 t)))) (- 2 (/ (/ 2 t) (+ 1 (/ 1 t))))))))