Compound Interest

Average Error: 42.9 → 32.7

Time: 6.9m

Precision: 64

Math

\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]

↓

\[\begin{array}{l} \mathbf{if}\;n \le -8.97495351037423816126988860777680675991 \cdot 10^{177}:\\ \;\;\;\;\left(100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}\right) \cdot n\\ \mathbf{elif}\;n \le -36228113.1512196958065032958984375:\\ \;\;\;\;100 \cdot \frac{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{\frac{i}{n}}\\ \mathbf{elif}\;n \le -1.04145011309898066305251012619905231421 \cdot 10^{-294}:\\ \;\;\;\;100 \cdot \frac{\sqrt[3]{\frac{\frac{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot \left(2 \cdot n\right)\right)} + \left(-\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right)}{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + 1 \cdot 1}}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}} \cdot \sqrt[3]{\frac{\frac{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot \left(2 \cdot n\right)\right)} + \left(-\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right)}{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + 1 \cdot 1}}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}}{\frac{i}{\sqrt[3]{\frac{1}{\sqrt{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}}} \cdot \frac{\frac{1}{n}}{\sqrt[3]{\frac{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} - 1 \cdot 1}{\sqrt{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}}}}\\ \mathbf{elif}\;n \le 1.496434530671250065211050155721986673116 \cdot 10^{-224}:\\ \;\;\;\;100 \cdot \frac{\left(1 \cdot i + \left(\log 1 \cdot n + 1\right)\right) - 1}{\frac{i}{n}}\\ \mathbf{elif}\;n \le 7.653944076549357751092870318001482628066 \cdot 10^{-198}:\\ \;\;\;\;100 \cdot \left(\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right) \cdot \frac{n}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{\frac{i}{n}}\\ \end{array}\]

C

double f(double i, double n) {
        double r1661355 = 100.0;
        double r1661356 = 1.0;
        double r1661357 = i;
        double r1661358 = n;
        double r1661359 = r1661357 / r1661358;
        double r1661360 = r1661356 + r1661359;
        double r1661361 = pow(r1661360, r1661358);
        double r1661362 = r1661361 - r1661356;
        double r1661363 = r1661362 / r1661359;
        double r1661364 = r1661355 * r1661363;
        return r1661364;
}

↓

double f(double i, double n) {
        double r1661365 = n;
        double r1661366 = -8.974953510374238e+177;
        bool r1661367 = r1661365 <= r1661366;
        double r1661368 = 100.0;
        double r1661369 = 1.0;
        double r1661370 = i;
        double r1661371 = r1661370 / r1661365;
        double r1661372 = r1661369 + r1661371;
        double r1661373 = pow(r1661372, r1661365);
        double r1661374 = r1661373 - r1661369;
        double r1661375 = r1661374 / r1661370;
        double r1661376 = r1661368 * r1661375;
        double r1661377 = r1661376 * r1661365;
        double r1661378 = -36228113.151219696;
        bool r1661379 = r1661365 <= r1661378;
        double r1661380 = r1661369 * r1661370;
        double r1661381 = 0.5;
        double r1661382 = 2.0;
        double r1661383 = pow(r1661370, r1661382);
        double r1661384 = r1661381 * r1661383;
        double r1661385 = log(r1661369);
        double r1661386 = r1661385 * r1661365;
        double r1661387 = r1661384 + r1661386;
        double r1661388 = r1661380 + r1661387;
        double r1661389 = r1661383 * r1661385;
        double r1661390 = r1661381 * r1661389;
        double r1661391 = r1661388 - r1661390;
        double r1661392 = r1661391 / r1661371;
        double r1661393 = r1661368 * r1661392;
        double r1661394 = -1.0414501130989807e-294;
        bool r1661395 = r1661365 <= r1661394;
        double r1661396 = r1661382 * r1661365;
        double r1661397 = r1661382 * r1661396;
        double r1661398 = pow(r1661372, r1661397);
        double r1661399 = r1661369 * r1661369;
        double r1661400 = r1661399 * r1661399;
        double r1661401 = -r1661400;
        double r1661402 = r1661398 + r1661401;
        double r1661403 = pow(r1661372, r1661396);
        double r1661404 = r1661403 + r1661399;
        double r1661405 = r1661402 / r1661404;
        double r1661406 = r1661373 + r1661369;
        double r1661407 = r1661405 / r1661406;
        double r1661408 = cbrt(r1661407);
        double r1661409 = r1661408 * r1661408;
        double r1661410 = 1.0;
        double r1661411 = sqrt(r1661406);
        double r1661412 = r1661410 / r1661411;
        double r1661413 = cbrt(r1661412);
        double r1661414 = r1661370 / r1661413;
        double r1661415 = r1661410 / r1661365;
        double r1661416 = r1661403 - r1661399;
        double r1661417 = r1661416 / r1661411;
        double r1661418 = cbrt(r1661417);
        double r1661419 = r1661415 / r1661418;
        double r1661420 = r1661414 * r1661419;
        double r1661421 = r1661409 / r1661420;
        double r1661422 = r1661368 * r1661421;
        double r1661423 = 1.49643453067125e-224;
        bool r1661424 = r1661365 <= r1661423;
        double r1661425 = r1661386 + r1661410;
        double r1661426 = r1661380 + r1661425;
        double r1661427 = r1661426 - r1661369;
        double r1661428 = r1661427 / r1661371;
        double r1661429 = r1661368 * r1661428;
        double r1661430 = 7.653944076549358e-198;
        bool r1661431 = r1661365 <= r1661430;
        double r1661432 = r1661365 / r1661370;
        double r1661433 = r1661374 * r1661432;
        double r1661434 = r1661368 * r1661433;
        double r1661435 = r1661431 ? r1661434 : r1661393;
        double r1661436 = r1661424 ? r1661429 : r1661435;
        double r1661437 = r1661395 ? r1661422 : r1661436;
        double r1661438 = r1661379 ? r1661393 : r1661437;
        double r1661439 = r1661367 ? r1661377 : r1661438;
        return r1661439;
}

Error

Bits error versus i

Bits error versus n

Target

Original	42.9
Target	43.1
Herbie	32.7

\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

1. Split input into 5 regimes

2. if n < -8.974953510374238e+177

   1. Initial program 53.2

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
   2. Using strategy rm

   3. Applied associate-/r/ 52.7

      \[\leadsto 100 \cdot \color{blue}{\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i} \cdot n\right)}\]

   4. Applied associate-*r* 52.7

      \[\leadsto \color{blue}{\left(100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}\right) \cdot n}\]

   if -8.974953510374238e+177 < n < -36228113.151219696 or 7.653944076549358e-198 < n

   1. Initial program 52.0

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]

   2. Taylor expanded around 0 36.1

      \[\leadsto 100 \cdot \frac{\color{blue}{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}}{\frac{i}{n}}\]

   if -36228113.151219696 < n < -1.0414501130989807e-294

   1. Initial program 16.9

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
   2. Using strategy rm

   3. Applied flip-- 16.9

      \[\leadsto 100 \cdot \frac{\color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} \cdot {\left(1 + \frac{i}{n}\right)}^{n} - 1 \cdot 1}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}}{\frac{i}{n}}\]

   4. Simplified 16.9

      \[\leadsto 100 \cdot \frac{\frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + \left(-1 \cdot 1\right)}}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{\frac{i}{n}}\]
   5. Using strategy rm

   6. Applied flip-+ 16.9

      \[\leadsto 100 \cdot \frac{\frac{\color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} \cdot {\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} - \left(-1 \cdot 1\right) \cdot \left(-1 \cdot 1\right)}{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} - \left(-1 \cdot 1\right)}}}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{\frac{i}{n}}\]

   7. Simplified 16.8

      \[\leadsto 100 \cdot \frac{\frac{\frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot \left(2 \cdot n\right)\right)} + \left(-\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right)}}{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} - \left(-1 \cdot 1\right)}}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{\frac{i}{n}}\]

   8. Simplified 16.8

      \[\leadsto 100 \cdot \frac{\frac{\frac{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot \left(2 \cdot n\right)\right)} + \left(-\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right)}{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + 1 \cdot 1}}}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{\frac{i}{n}}\]
   9. Using strategy rm

   10. Applied add-cube-cbrt 16.8

       \[\leadsto 100 \cdot \frac{\color{blue}{\left(\sqrt[3]{\frac{\frac{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot \left(2 \cdot n\right)\right)} + \left(-\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right)}{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + 1 \cdot 1}}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}} \cdot \sqrt[3]{\frac{\frac{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot \left(2 \cdot n\right)\right)} + \left(-\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right)}{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + 1 \cdot 1}}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}\right) \cdot \sqrt[3]{\frac{\frac{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot \left(2 \cdot n\right)\right)} + \left(-\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right)}{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + 1 \cdot 1}}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}}}{\frac{i}{n}}\]

   11. Applied associate-/l* 16.8

       \[\leadsto 100 \cdot \color{blue}{\frac{\sqrt[3]{\frac{\frac{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot \left(2 \cdot n\right)\right)} + \left(-\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right)}{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + 1 \cdot 1}}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}} \cdot \sqrt[3]{\frac{\frac{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot \left(2 \cdot n\right)\right)} + \left(-\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right)}{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + 1 \cdot 1}}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}}{\frac{\frac{i}{n}}{\sqrt[3]{\frac{\frac{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot \left(2 \cdot n\right)\right)} + \left(-\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right)}{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + 1 \cdot 1}}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}}}}\]
   12. Using strategy rm

   13. Applied add-sqr-sqrt 16.8

       \[\leadsto 100 \cdot \frac{\sqrt[3]{\frac{\frac{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot \left(2 \cdot n\right)\right)} + \left(-\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right)}{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + 1 \cdot 1}}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}} \cdot \sqrt[3]{\frac{\frac{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot \left(2 \cdot n\right)\right)} + \left(-\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right)}{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + 1 \cdot 1}}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}}{\frac{\frac{i}{n}}{\sqrt[3]{\frac{\frac{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot \left(2 \cdot n\right)\right)} + \left(-\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right)}{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + 1 \cdot 1}}{\color{blue}{\sqrt{{\left(1 + \frac{i}{n}\right)}^{n} + 1} \cdot \sqrt{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}}}}}\]

   14. Applied flip-+ 50.8

       \[\leadsto 100 \cdot \frac{\sqrt[3]{\frac{\frac{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot \left(2 \cdot n\right)\right)} + \left(-\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right)}{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + 1 \cdot 1}}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}} \cdot \sqrt[3]{\frac{\frac{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot \left(2 \cdot n\right)\right)} + \left(-\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right)}{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + 1 \cdot 1}}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}}{\frac{\frac{i}{n}}{\sqrt[3]{\frac{\frac{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot \left(2 \cdot n\right)\right)} + \left(-\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right)}{\color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} \cdot {\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} - \left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)}{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} - 1 \cdot 1}}}}{\sqrt{{\left(1 + \frac{i}{n}\right)}^{n} + 1} \cdot \sqrt{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}}}}\]

   15. Applied associate-/r/ 50.8

       \[\leadsto 100 \cdot \frac{\sqrt[3]{\frac{\frac{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot \left(2 \cdot n\right)\right)} + \left(-\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right)}{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + 1 \cdot 1}}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}} \cdot \sqrt[3]{\frac{\frac{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot \left(2 \cdot n\right)\right)} + \left(-\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right)}{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + 1 \cdot 1}}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}}{\frac{\frac{i}{n}}{\sqrt[3]{\frac{\color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot \left(2 \cdot n\right)\right)} + \left(-\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right)}{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} \cdot {\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} - \left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)} \cdot \left({\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} - 1 \cdot 1\right)}}{\sqrt{{\left(1 + \frac{i}{n}\right)}^{n} + 1} \cdot \sqrt{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}}}}\]

   16. Applied times-frac 50.8

       \[\leadsto 100 \cdot \frac{\sqrt[3]{\frac{\frac{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot \left(2 \cdot n\right)\right)} + \left(-\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right)}{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + 1 \cdot 1}}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}} \cdot \sqrt[3]{\frac{\frac{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot \left(2 \cdot n\right)\right)} + \left(-\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right)}{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + 1 \cdot 1}}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}}{\frac{\frac{i}{n}}{\sqrt[3]{\color{blue}{\frac{\frac{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot \left(2 \cdot n\right)\right)} + \left(-\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right)}{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} \cdot {\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} - \left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)}}{\sqrt{{\left(1 + \frac{i}{n}\right)}^{n} + 1}} \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} - 1 \cdot 1}{\sqrt{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}}}}}\]

   17. Applied cbrt-prod 50.8

       \[\leadsto 100 \cdot \frac{\sqrt[3]{\frac{\frac{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot \left(2 \cdot n\right)\right)} + \left(-\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right)}{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + 1 \cdot 1}}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}} \cdot \sqrt[3]{\frac{\frac{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot \left(2 \cdot n\right)\right)} + \left(-\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right)}{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + 1 \cdot 1}}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}}{\frac{\frac{i}{n}}{\color{blue}{\sqrt[3]{\frac{\frac{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot \left(2 \cdot n\right)\right)} + \left(-\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right)}{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} \cdot {\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} - \left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)}}{\sqrt{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}} \cdot \sqrt[3]{\frac{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} - 1 \cdot 1}{\sqrt{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}}}}}\]

   18. Applied div-inv 50.8

       \[\leadsto 100 \cdot \frac{\sqrt[3]{\frac{\frac{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot \left(2 \cdot n\right)\right)} + \left(-\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right)}{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + 1 \cdot 1}}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}} \cdot \sqrt[3]{\frac{\frac{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot \left(2 \cdot n\right)\right)} + \left(-\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right)}{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + 1 \cdot 1}}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}}{\frac{\color{blue}{i \cdot \frac{1}{n}}}{\sqrt[3]{\frac{\frac{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot \left(2 \cdot n\right)\right)} + \left(-\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right)}{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} \cdot {\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} - \left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)}}{\sqrt{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}} \cdot \sqrt[3]{\frac{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} - 1 \cdot 1}{\sqrt{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}}}}\]

   19. Applied times-frac 50.8

       \[\leadsto 100 \cdot \frac{\sqrt[3]{\frac{\frac{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot \left(2 \cdot n\right)\right)} + \left(-\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right)}{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + 1 \cdot 1}}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}} \cdot \sqrt[3]{\frac{\frac{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot \left(2 \cdot n\right)\right)} + \left(-\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right)}{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + 1 \cdot 1}}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}}{\color{blue}{\frac{i}{\sqrt[3]{\frac{\frac{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot \left(2 \cdot n\right)\right)} + \left(-\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right)}{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} \cdot {\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} - \left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)}}{\sqrt{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}}} \cdot \frac{\frac{1}{n}}{\sqrt[3]{\frac{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} - 1 \cdot 1}{\sqrt{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}}}}}\]

   20. Simplified 16.8

       \[\leadsto 100 \cdot \frac{\sqrt[3]{\frac{\frac{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot \left(2 \cdot n\right)\right)} + \left(-\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right)}{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + 1 \cdot 1}}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}} \cdot \sqrt[3]{\frac{\frac{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot \left(2 \cdot n\right)\right)} + \left(-\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right)}{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + 1 \cdot 1}}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}}{\color{blue}{\frac{i}{\sqrt[3]{\frac{1}{\sqrt{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}}}} \cdot \frac{\frac{1}{n}}{\sqrt[3]{\frac{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} - 1 \cdot 1}{\sqrt{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}}}}\]

   if -1.0414501130989807e-294 < n < 1.49643453067125e-224

   1. Initial program 33.7

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]

   2. Taylor expanded around 0 14.9

      \[\leadsto 100 \cdot \frac{\color{blue}{\left(1 \cdot i + \left(\log 1 \cdot n + 1\right)\right)} - 1}{\frac{i}{n}}\]

   if 1.49643453067125e-224 < n < 7.653944076549358e-198

   1. Initial program 47.4

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
   2. Using strategy rm

   3. Applied div-inv 47.4

      \[\leadsto 100 \cdot \color{blue}{\left(\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right) \cdot \frac{1}{\frac{i}{n}}\right)}\]

   4. Simplified 47.4

      \[\leadsto 100 \cdot \left(\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right) \cdot \color{blue}{\frac{n}{i}}\right)\]
3. Recombined 5 regimes into one program.

4. Final simplification 32.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -8.97495351037423816126988860777680675991 \cdot 10^{177}:\\ \;\;\;\;\left(100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}\right) \cdot n\\ \mathbf{elif}\;n \le -36228113.1512196958065032958984375:\\ \;\;\;\;100 \cdot \frac{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{\frac{i}{n}}\\ \mathbf{elif}\;n \le -1.04145011309898066305251012619905231421 \cdot 10^{-294}:\\ \;\;\;\;100 \cdot \frac{\sqrt[3]{\frac{\frac{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot \left(2 \cdot n\right)\right)} + \left(-\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right)}{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + 1 \cdot 1}}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}} \cdot \sqrt[3]{\frac{\frac{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot \left(2 \cdot n\right)\right)} + \left(-\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right)}{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + 1 \cdot 1}}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}}{\frac{i}{\sqrt[3]{\frac{1}{\sqrt{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}}} \cdot \frac{\frac{1}{n}}{\sqrt[3]{\frac{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} - 1 \cdot 1}{\sqrt{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}}}}\\ \mathbf{elif}\;n \le 1.496434530671250065211050155721986673116 \cdot 10^{-224}:\\ \;\;\;\;100 \cdot \frac{\left(1 \cdot i + \left(\log 1 \cdot n + 1\right)\right) - 1}{\frac{i}{n}}\\ \mathbf{elif}\;n \le 7.653944076549357751092870318001482628066 \cdot 10^{-198}:\\ \;\;\;\;100 \cdot \left(\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right) \cdot \frac{n}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{\frac{i}{n}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 
(FPCore (i n)
  :name "Compound Interest"
  :precision binary64

  :herbie-target
  (* 100 (/ (- (exp (* n (if (== (+ 1 (/ i n)) 1) (/ i n) (/ (* (/ i n) (log (+ 1 (/ i n)))) (- (+ (/ i n) 1) 1))))) 1) (/ i n)))

  (* 100 (/ (- (pow (+ 1 (/ i n)) n) 1) (/ i n))))