Compound Interest

Average Error: 42.5 → 30.1

Time: 55.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;n \le -2.4372743425058105193615005673548969949 \cdot 10^{-310}:\\ \;\;\;\;100 \cdot \frac{\frac{\frac{1}{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + 1 \cdot 1} \cdot \sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + 1 \cdot 1}}}{\sqrt{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}}{\frac{i}{\frac{\frac{{\left(1 + \frac{i}{n}\right)}^{\left(4 \cdot n\right)} - 1 \cdot {1}^{3}}{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + 1 \cdot 1}}}{\sqrt{{\left(1 + \frac{i}{n}\right)}^{n} + 1}} \cdot n}}\\ \mathbf{elif}\;n \le 2.053192934454678974082105649395487884911 \cdot 10^{-111}:\\ \;\;\;\;100 \cdot \frac{\frac{1}{2} \cdot \left({\left(\log i\right)}^{2} \cdot {n}^{2} + {n}^{2} \cdot {\left(\log n\right)}^{2}\right) + \left(\left(\frac{1}{6} \cdot \left({\left(\log i\right)}^{3} \cdot {n}^{3}\right) + \left(\log i \cdot n + \frac{1}{2} \cdot \left(\log i \cdot \left({n}^{3} \cdot {\left(\log n\right)}^{2}\right)\right)\right)\right) - \left(\log n \cdot \left(\log i \cdot {n}^{2} + n\right) + \left(\frac{1}{2} \cdot \left({\left(\log i\right)}^{2} \cdot \left({n}^{3} \cdot \log n\right)\right) + \frac{1}{6} \cdot \left({n}^{3} \cdot {\left(\log n\right)}^{3}\right)\right)\right)\right)}{\frac{i}{n}}\\ \mathbf{elif}\;n \le 3.682516545739370661841848796102550693335 \cdot 10^{-44}:\\ \;\;\;\;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 9.089128901802147835523691243773490781497 \cdot 10^{-18}:\\ \;\;\;\;50 \cdot \left(\frac{{n}^{3} \cdot {\left(\log n\right)}^{2}}{i} + \frac{{\left(\log i\right)}^{2} \cdot {n}^{3}}{i}\right) + \left(\left(16.66666666666666429819088079966604709625 \cdot \frac{{\left(\log i\right)}^{3} \cdot {n}^{4}}{i} + \left(100 \cdot \frac{\log i \cdot {n}^{2}}{i} + 50 \cdot \frac{\log i \cdot \left({n}^{4} \cdot {\left(\log n\right)}^{2}\right)}{i}\right)\right) - \left(16.66666666666666429819088079966604709625 \cdot \frac{{n}^{4} \cdot {\left(\log n\right)}^{3}}{i} + \left(100 \cdot \left(\frac{\log i \cdot \left({n}^{3} \cdot \log n\right)}{i} + \frac{{n}^{2} \cdot \log n}{i}\right) + 50 \cdot \frac{{\left(\log i\right)}^{2} \cdot \left({n}^{4} \cdot \log n\right)}{i}\right)\right)\right)\\ \mathbf{elif}\;n \le 2.128047119799741370168835622013045914305 \cdot 10^{177}:\\ \;\;\;\;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{else}:\\ \;\;\;\;100 \cdot \frac{{\left(\frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\\ \end{array}\]

C

double f(double i, double n) {
        double r239372 = 100.0;
        double r239373 = 1.0;
        double r239374 = i;
        double r239375 = n;
        double r239376 = r239374 / r239375;
        double r239377 = r239373 + r239376;
        double r239378 = pow(r239377, r239375);
        double r239379 = r239378 - r239373;
        double r239380 = r239379 / r239376;
        double r239381 = r239372 * r239380;
        return r239381;
}

↓

double f(double i, double n) {
        double r239382 = n;
        double r239383 = -2.4372743425058e-310;
        bool r239384 = r239382 <= r239383;
        double r239385 = 100.0;
        double r239386 = 1.0;
        double r239387 = 1.0;
        double r239388 = i;
        double r239389 = r239388 / r239382;
        double r239390 = r239387 + r239389;
        double r239391 = 2.0;
        double r239392 = r239391 * r239382;
        double r239393 = pow(r239390, r239392);
        double r239394 = r239387 * r239387;
        double r239395 = r239393 + r239394;
        double r239396 = cbrt(r239395);
        double r239397 = r239396 * r239396;
        double r239398 = r239386 / r239397;
        double r239399 = pow(r239390, r239382);
        double r239400 = r239399 + r239387;
        double r239401 = sqrt(r239400);
        double r239402 = r239398 / r239401;
        double r239403 = 4.0;
        double r239404 = r239403 * r239382;
        double r239405 = pow(r239390, r239404);
        double r239406 = 3.0;
        double r239407 = pow(r239387, r239406);
        double r239408 = r239387 * r239407;
        double r239409 = r239405 - r239408;
        double r239410 = r239409 / r239396;
        double r239411 = r239410 / r239401;
        double r239412 = r239411 * r239382;
        double r239413 = r239388 / r239412;
        double r239414 = r239402 / r239413;
        double r239415 = r239385 * r239414;
        double r239416 = 2.053192934454679e-111;
        bool r239417 = r239382 <= r239416;
        double r239418 = 0.5;
        double r239419 = log(r239388);
        double r239420 = pow(r239419, r239391);
        double r239421 = pow(r239382, r239391);
        double r239422 = r239420 * r239421;
        double r239423 = log(r239382);
        double r239424 = pow(r239423, r239391);
        double r239425 = r239421 * r239424;
        double r239426 = r239422 + r239425;
        double r239427 = r239418 * r239426;
        double r239428 = 0.16666666666666666;
        double r239429 = pow(r239419, r239406);
        double r239430 = pow(r239382, r239406);
        double r239431 = r239429 * r239430;
        double r239432 = r239428 * r239431;
        double r239433 = r239419 * r239382;
        double r239434 = r239430 * r239424;
        double r239435 = r239419 * r239434;
        double r239436 = r239418 * r239435;
        double r239437 = r239433 + r239436;
        double r239438 = r239432 + r239437;
        double r239439 = r239419 * r239421;
        double r239440 = r239439 + r239382;
        double r239441 = r239423 * r239440;
        double r239442 = r239430 * r239423;
        double r239443 = r239420 * r239442;
        double r239444 = r239418 * r239443;
        double r239445 = pow(r239423, r239406);
        double r239446 = r239430 * r239445;
        double r239447 = r239428 * r239446;
        double r239448 = r239444 + r239447;
        double r239449 = r239441 + r239448;
        double r239450 = r239438 - r239449;
        double r239451 = r239427 + r239450;
        double r239452 = r239451 / r239389;
        double r239453 = r239385 * r239452;
        double r239454 = 3.6825165457393707e-44;
        bool r239455 = r239382 <= r239454;
        double r239456 = r239387 * r239388;
        double r239457 = 0.5;
        double r239458 = pow(r239388, r239391);
        double r239459 = r239457 * r239458;
        double r239460 = log(r239387);
        double r239461 = r239460 * r239382;
        double r239462 = r239459 + r239461;
        double r239463 = r239456 + r239462;
        double r239464 = r239458 * r239460;
        double r239465 = r239457 * r239464;
        double r239466 = r239463 - r239465;
        double r239467 = r239466 / r239389;
        double r239468 = r239385 * r239467;
        double r239469 = 9.089128901802148e-18;
        bool r239470 = r239382 <= r239469;
        double r239471 = 50.0;
        double r239472 = r239434 / r239388;
        double r239473 = r239420 * r239430;
        double r239474 = r239473 / r239388;
        double r239475 = r239472 + r239474;
        double r239476 = r239471 * r239475;
        double r239477 = 16.666666666666664;
        double r239478 = pow(r239382, r239403);
        double r239479 = r239429 * r239478;
        double r239480 = r239479 / r239388;
        double r239481 = r239477 * r239480;
        double r239482 = r239439 / r239388;
        double r239483 = r239385 * r239482;
        double r239484 = r239478 * r239424;
        double r239485 = r239419 * r239484;
        double r239486 = r239485 / r239388;
        double r239487 = r239471 * r239486;
        double r239488 = r239483 + r239487;
        double r239489 = r239481 + r239488;
        double r239490 = r239478 * r239445;
        double r239491 = r239490 / r239388;
        double r239492 = r239477 * r239491;
        double r239493 = r239419 * r239442;
        double r239494 = r239493 / r239388;
        double r239495 = r239421 * r239423;
        double r239496 = r239495 / r239388;
        double r239497 = r239494 + r239496;
        double r239498 = r239385 * r239497;
        double r239499 = r239478 * r239423;
        double r239500 = r239420 * r239499;
        double r239501 = r239500 / r239388;
        double r239502 = r239471 * r239501;
        double r239503 = r239498 + r239502;
        double r239504 = r239492 + r239503;
        double r239505 = r239489 - r239504;
        double r239506 = r239476 + r239505;
        double r239507 = 2.1280471197997414e+177;
        bool r239508 = r239382 <= r239507;
        double r239509 = pow(r239389, r239382);
        double r239510 = r239509 - r239387;
        double r239511 = r239510 / r239389;
        double r239512 = r239385 * r239511;
        double r239513 = r239508 ? r239468 : r239512;
        double r239514 = r239470 ? r239506 : r239513;
        double r239515 = r239455 ? r239468 : r239514;
        double r239516 = r239417 ? r239453 : r239515;
        double r239517 = r239384 ? r239415 : r239516;
        return r239517;
}

Error

Bits error versus i

Bits error versus n

Target

Original	42.5
Target	42.4
Herbie	30.1

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

Derivation

1. Split input into 5 regimes

2. if n < -2.4372743425058e-310

   1. Initial program 31.8

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

   3. Applied flip-- 31.8

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

   4. Simplified 31.8

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

   6. Applied flip-- 31.8

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

   7. Simplified 31.8

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

   9. Applied add-sqr-sqrt 31.8

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

   10. Applied add-cube-cbrt 31.8

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

   11. Applied *-un-lft-identity 31.8

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

   12. Applied times-frac 31.8

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

   13. Applied times-frac 31.8

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

   14. Applied associate-/l* 31.8

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

   15. Simplified 31.6

       \[\leadsto 100 \cdot \frac{\frac{\frac{1}{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + 1 \cdot 1} \cdot \sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + 1 \cdot 1}}}{\sqrt{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}}{\color{blue}{\frac{i}{\frac{\frac{{\left(1 + \frac{i}{n}\right)}^{\left(4 \cdot n\right)} - 1 \cdot {1}^{3}}{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + 1 \cdot 1}}}{\sqrt{{\left(1 + \frac{i}{n}\right)}^{n} + 1}} \cdot n}}}\]

   if -2.4372743425058e-310 < n < 2.053192934454679e-111

   1. Initial program 43.9

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

   2. Taylor expanded around inf 26.9

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

   3. Simplified 44.3

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

   4. Taylor expanded around 0 17.7

      \[\leadsto 100 \cdot \frac{\color{blue}{\left(\frac{1}{2} \cdot \left({\left(\log i\right)}^{2} \cdot {n}^{2}\right) + \left(\frac{1}{2} \cdot \left({n}^{2} \cdot {\left(\log n\right)}^{2}\right) + \left(\frac{1}{6} \cdot \left({\left(\log i\right)}^{3} \cdot {n}^{3}\right) + \left(\log i \cdot n + \frac{1}{2} \cdot \left(\log i \cdot \left({n}^{3} \cdot {\left(\log n\right)}^{2}\right)\right)\right)\right)\right)\right) - \left(n \cdot \log n + \left(\log i \cdot \left({n}^{2} \cdot \log n\right) + \left(\frac{1}{2} \cdot \left({\left(\log i\right)}^{2} \cdot \left({n}^{3} \cdot \log n\right)\right) + \frac{1}{6} \cdot \left({n}^{3} \cdot {\left(\log n\right)}^{3}\right)\right)\right)\right)}}{\frac{i}{n}}\]

   5. Simplified 17.7

      \[\leadsto 100 \cdot \frac{\color{blue}{\frac{1}{2} \cdot \left({\left(\log i\right)}^{2} \cdot {n}^{2} + {n}^{2} \cdot {\left(\log n\right)}^{2}\right) + \left(\left(\frac{1}{6} \cdot \left({\left(\log i\right)}^{3} \cdot {n}^{3}\right) + \left(\log i \cdot n + \frac{1}{2} \cdot \left(\log i \cdot \left({n}^{3} \cdot {\left(\log n\right)}^{2}\right)\right)\right)\right) - \left(\log n \cdot \left(\log i \cdot {n}^{2} + n\right) + \left(\frac{1}{2} \cdot \left({\left(\log i\right)}^{2} \cdot \left({n}^{3} \cdot \log n\right)\right) + \frac{1}{6} \cdot \left({n}^{3} \cdot {\left(\log n\right)}^{3}\right)\right)\right)\right)}}{\frac{i}{n}}\]

   if 2.053192934454679e-111 < n < 3.6825165457393707e-44 or 9.089128901802148e-18 < n < 2.1280471197997414e+177

   1. Initial program 58.8

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

   2. Taylor expanded around 0 25.0

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

   if 3.6825165457393707e-44 < n < 9.089128901802148e-18

   1. Initial program 60.2

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

   2. Taylor expanded around inf 57.9

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

   3. Simplified 60.8

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

   4. Taylor expanded around 0 36.4

      \[\leadsto \color{blue}{\left(50 \cdot \frac{{n}^{3} \cdot {\left(\log n\right)}^{2}}{i} + \left(50 \cdot \frac{{\left(\log i\right)}^{2} \cdot {n}^{3}}{i} + \left(16.66666666666666429819088079966604709625 \cdot \frac{{\left(\log i\right)}^{3} \cdot {n}^{4}}{i} + \left(100 \cdot \frac{\log i \cdot {n}^{2}}{i} + 50 \cdot \frac{\log i \cdot \left({n}^{4} \cdot {\left(\log n\right)}^{2}\right)}{i}\right)\right)\right)\right) - \left(16.66666666666666429819088079966604709625 \cdot \frac{{n}^{4} \cdot {\left(\log n\right)}^{3}}{i} + \left(100 \cdot \frac{\log i \cdot \left({n}^{3} \cdot \log n\right)}{i} + \left(100 \cdot \frac{{n}^{2} \cdot \log n}{i} + 50 \cdot \frac{{\left(\log i\right)}^{2} \cdot \left({n}^{4} \cdot \log n\right)}{i}\right)\right)\right)}\]

   5. Simplified 36.4

      \[\leadsto \color{blue}{50 \cdot \left(\frac{{n}^{3} \cdot {\left(\log n\right)}^{2}}{i} + \frac{{\left(\log i\right)}^{2} \cdot {n}^{3}}{i}\right) + \left(\left(16.66666666666666429819088079966604709625 \cdot \frac{{\left(\log i\right)}^{3} \cdot {n}^{4}}{i} + \left(100 \cdot \frac{\log i \cdot {n}^{2}}{i} + 50 \cdot \frac{\log i \cdot \left({n}^{4} \cdot {\left(\log n\right)}^{2}\right)}{i}\right)\right) - \left(16.66666666666666429819088079966604709625 \cdot \frac{{n}^{4} \cdot {\left(\log n\right)}^{3}}{i} + \left(100 \cdot \left(\frac{\log i \cdot \left({n}^{3} \cdot \log n\right)}{i} + \frac{{n}^{2} \cdot \log n}{i}\right) + 50 \cdot \frac{{\left(\log i\right)}^{2} \cdot \left({n}^{4} \cdot \log n\right)}{i}\right)\right)\right)}\]

   if 2.1280471197997414e+177 < n

   1. Initial program 61.9

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

   2. Taylor expanded around inf 64.0

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

   3. Simplified 45.7

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

4. Final simplification 30.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -2.4372743425058105193615005673548969949 \cdot 10^{-310}:\\ \;\;\;\;100 \cdot \frac{\frac{\frac{1}{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + 1 \cdot 1} \cdot \sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + 1 \cdot 1}}}{\sqrt{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}}{\frac{i}{\frac{\frac{{\left(1 + \frac{i}{n}\right)}^{\left(4 \cdot n\right)} - 1 \cdot {1}^{3}}{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + 1 \cdot 1}}}{\sqrt{{\left(1 + \frac{i}{n}\right)}^{n} + 1}} \cdot n}}\\ \mathbf{elif}\;n \le 2.053192934454678974082105649395487884911 \cdot 10^{-111}:\\ \;\;\;\;100 \cdot \frac{\frac{1}{2} \cdot \left({\left(\log i\right)}^{2} \cdot {n}^{2} + {n}^{2} \cdot {\left(\log n\right)}^{2}\right) + \left(\left(\frac{1}{6} \cdot \left({\left(\log i\right)}^{3} \cdot {n}^{3}\right) + \left(\log i \cdot n + \frac{1}{2} \cdot \left(\log i \cdot \left({n}^{3} \cdot {\left(\log n\right)}^{2}\right)\right)\right)\right) - \left(\log n \cdot \left(\log i \cdot {n}^{2} + n\right) + \left(\frac{1}{2} \cdot \left({\left(\log i\right)}^{2} \cdot \left({n}^{3} \cdot \log n\right)\right) + \frac{1}{6} \cdot \left({n}^{3} \cdot {\left(\log n\right)}^{3}\right)\right)\right)\right)}{\frac{i}{n}}\\ \mathbf{elif}\;n \le 3.682516545739370661841848796102550693335 \cdot 10^{-44}:\\ \;\;\;\;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 9.089128901802147835523691243773490781497 \cdot 10^{-18}:\\ \;\;\;\;50 \cdot \left(\frac{{n}^{3} \cdot {\left(\log n\right)}^{2}}{i} + \frac{{\left(\log i\right)}^{2} \cdot {n}^{3}}{i}\right) + \left(\left(16.66666666666666429819088079966604709625 \cdot \frac{{\left(\log i\right)}^{3} \cdot {n}^{4}}{i} + \left(100 \cdot \frac{\log i \cdot {n}^{2}}{i} + 50 \cdot \frac{\log i \cdot \left({n}^{4} \cdot {\left(\log n\right)}^{2}\right)}{i}\right)\right) - \left(16.66666666666666429819088079966604709625 \cdot \frac{{n}^{4} \cdot {\left(\log n\right)}^{3}}{i} + \left(100 \cdot \left(\frac{\log i \cdot \left({n}^{3} \cdot \log n\right)}{i} + \frac{{n}^{2} \cdot \log n}{i}\right) + 50 \cdot \frac{{\left(\log i\right)}^{2} \cdot \left({n}^{4} \cdot \log n\right)}{i}\right)\right)\right)\\ \mathbf{elif}\;n \le 2.128047119799741370168835622013045914305 \cdot 10^{177}:\\ \;\;\;\;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{else}:\\ \;\;\;\;100 \cdot \frac{{\left(\frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019303 
(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))))