Average Error: 0.0 → 0.0
Time: 31.9s
Precision: 64
\[\frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}\]
↓
\[\frac{1 + \sqrt[3]{{\left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}^{3}} \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{2 + \sqrt[3]{{\left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}^{3}} \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}\]
\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 + \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)}{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 r60919 = 1.0;
        double r60920 = 2.0;
        double r60921 = t;
        double r60922 = r60920 / r60921;
        double r60923 = r60919 / r60921;
        double r60924 = r60919 + r60923;
        double r60925 = r60922 / r60924;
        double r60926 = r60920 - r60925;
        double r60927 = r60926 * r60926;
        double r60928 = r60919 + r60927;
        double r60929 = r60920 + r60927;
        double r60930 = r60928 / r60929;
        return r60930;
}
↓
double f(double t) {
        double r60931 = 1.0;
        double r60932 = 2.0;
        double r60933 = t;
        double r60934 = r60932 / r60933;
        double r60935 = r60931 / r60933;
        double r60936 = r60931 + r60935;
        double r60937 = r60934 / r60936;
        double r60938 = r60932 - r60937;
        double r60939 = 3.0;
        double r60940 = pow(r60938, r60939);
        double r60941 = cbrt(r60940);
        double r60942 = r60941 * r60938;
        double r60943 = r60931 + r60942;
        double r60944 = r60932 + r60942;
        double r60945 = r60943 / r60944;
        return r60945;
}

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. Using strategy rm

  3. Applied add-cbrt-cube0.0

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

  4. Simplified0.0

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

  6. Applied add-cbrt-cube0.0

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

  7. Simplified0.0

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

  8. Final simplification0.0

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

Reproduce

herbie shell --seed 2019325 +o rules:numerics
(FPCore (t)
  :name "Kahan p13 Example 2"
  :precision binary64
  (/ (+ 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))))))))