Average Error: 0.0 → 0.0
Time: 8.2s
Precision: 64
\[\frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}\]
↓
\[\frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(\left(2 - \frac{\frac{\sqrt{2}}{\sqrt[3]{t}}}{\sqrt[3]{1 + \frac{1}{t}}} \cdot \frac{\frac{\sqrt{2}}{\sqrt[3]{t} \cdot \sqrt[3]{t}}}{\sqrt[3]{1 + \frac{1}{t}} \cdot \sqrt[3]{1 + \frac{1}{t}}}\right) + \frac{\frac{\sqrt{2}}{\sqrt[3]{t} \cdot \sqrt[3]{t}}}{\sqrt[3]{1 + \frac{1}{t}} \cdot \sqrt[3]{1 + \frac{1}{t}}} \cdot \left(\left(-\frac{\frac{\sqrt{2}}{\sqrt[3]{t}}}{\sqrt[3]{1 + \frac{1}{t}}}\right) + \frac{\frac{\sqrt{2}}{\sqrt[3]{t}}}{\sqrt[3]{1 + \frac{1}{t}}}\right)\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}\]
\frac{1 + \left(2 - \frac{\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{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(\left(2 - \frac{\frac{\sqrt{2}}{\sqrt[3]{t}}}{\sqrt[3]{1 + \frac{1}{t}}} \cdot \frac{\frac{\sqrt{2}}{\sqrt[3]{t} \cdot \sqrt[3]{t}}}{\sqrt[3]{1 + \frac{1}{t}} \cdot \sqrt[3]{1 + \frac{1}{t}}}\right) + \frac{\frac{\sqrt{2}}{\sqrt[3]{t} \cdot \sqrt[3]{t}}}{\sqrt[3]{1 + \frac{1}{t}} \cdot \sqrt[3]{1 + \frac{1}{t}}} \cdot \left(\left(-\frac{\frac{\sqrt{2}}{\sqrt[3]{t}}}{\sqrt[3]{1 + \frac{1}{t}}}\right) + \frac{\frac{\sqrt{2}}{\sqrt[3]{t}}}{\sqrt[3]{1 + \frac{1}{t}}}\right)\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)}
double f(double t) {
        double r76279 = 1.0;
        double r76280 = 2.0;
        double r76281 = t;
        double r76282 = r76280 / r76281;
        double r76283 = r76279 / r76281;
        double r76284 = r76279 + r76283;
        double r76285 = r76282 / r76284;
        double r76286 = r76280 - r76285;
        double r76287 = r76286 * r76286;
        double r76288 = r76279 + r76287;
        double r76289 = r76280 + r76287;
        double r76290 = r76288 / r76289;
        return r76290;
}
↓
double f(double t) {
        double r76291 = 1.0;
        double r76292 = 2.0;
        double r76293 = t;
        double r76294 = r76292 / r76293;
        double r76295 = r76291 / r76293;
        double r76296 = r76291 + r76295;
        double r76297 = r76294 / r76296;
        double r76298 = r76292 - r76297;
        double r76299 = sqrt(r76292);
        double r76300 = cbrt(r76293);
        double r76301 = r76299 / r76300;
        double r76302 = cbrt(r76296);
        double r76303 = r76301 / r76302;
        double r76304 = r76300 * r76300;
        double r76305 = r76299 / r76304;
        double r76306 = r76302 * r76302;
        double r76307 = r76305 / r76306;
        double r76308 = r76303 * r76307;
        double r76309 = r76292 - r76308;
        double r76310 = -r76303;
        double r76311 = r76310 + r76303;
        double r76312 = r76307 * r76311;
        double r76313 = r76309 + r76312;
        double r76314 = r76298 * r76313;
        double r76315 = r76291 + r76314;
        double r76316 = r76298 * r76298;
        double r76317 = r76292 + r76316;
        double r76318 = r76315 / r76317;
        return r76318;
}

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-cube-cbrt0.0

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

  4. Applied add-cube-cbrt0.0

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

  5. Applied add-sqr-sqrt0.0

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

  6. Applied times-frac0.0

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

  7. Applied times-frac0.0

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

  8. Applied add-sqr-sqrt0.5

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

  9. Applied prod-diff0.5

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

  10. Simplified0.0

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

  11. Simplified0.0

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

  12. Final simplification0.0

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

Reproduce

herbie shell --seed 2020001 +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))))))))