Average Error: 0.0 → 0.0
Time: 19.3s
Precision: 64
\[\frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}\]
↓
\[\frac{1 + \sqrt[3]{{\left(2 - \frac{2}{1 + t \cdot 1}\right)}^{3}} \cdot \left(2 - \frac{2}{1 + t \cdot 1}\right)}{2 + \left(2 - \frac{2}{1 + t \cdot 1}\right) \cdot \sqrt[3]{{\left(2 - \frac{2}{1 + t \cdot 1}\right)}^{3}}}\]
\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{2}{1 + t \cdot 1}\right)}^{3}} \cdot \left(2 - \frac{2}{1 + t \cdot 1}\right)}{2 + \left(2 - \frac{2}{1 + t \cdot 1}\right) \cdot \sqrt[3]{{\left(2 - \frac{2}{1 + t \cdot 1}\right)}^{3}}}
double f(double t) {
        double r36214 = 1.0;
        double r36215 = 2.0;
        double r36216 = t;
        double r36217 = r36215 / r36216;
        double r36218 = r36214 / r36216;
        double r36219 = r36214 + r36218;
        double r36220 = r36217 / r36219;
        double r36221 = r36215 - r36220;
        double r36222 = r36221 * r36221;
        double r36223 = r36214 + r36222;
        double r36224 = r36215 + r36222;
        double r36225 = r36223 / r36224;
        return r36225;
}
↓
double f(double t) {
        double r36226 = 1.0;
        double r36227 = 2.0;
        double r36228 = t;
        double r36229 = r36228 * r36226;
        double r36230 = r36226 + r36229;
        double r36231 = r36227 / r36230;
        double r36232 = r36227 - r36231;
        double r36233 = 3.0;
        double r36234 = pow(r36232, r36233);
        double r36235 = cbrt(r36234);
        double r36236 = r36235 * r36232;
        double r36237 = r36226 + r36236;
        double r36238 = r36232 * r36235;
        double r36239 = r36227 + r36238;
        double r36240 = r36237 / r36239;
        return r36240;
}

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

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

  4. Applied add-cbrt-cube0.0

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

  5. Simplified0.0

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

  7. Applied add-cbrt-cube0.0

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

  8. Simplified0.0

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

  9. Final simplification0.0

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

Reproduce

herbie shell --seed 2019325 
(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))))))))