Compound Interest

Average Error: 42.9 → 33.3

Time: 1.1m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;i \le -1.21329936542595784:\\ \;\;\;\;100 \cdot \frac{\frac{{\left({\left(1 + \frac{i}{n}\right)}^{n}\right)}^{3} - {1}^{3}}{\mathsf{fma}\left(1, {\left(1 + \frac{i}{n}\right)}^{n} + 1, {\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)}\right)}}{\frac{i}{n}}\\ \mathbf{elif}\;i \le -1.13133054522964566 \cdot 10^{-274}:\\ \;\;\;\;100 \cdot \frac{\mathsf{fma}\left(i, 1, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 1.64797021054671768 \cdot 10^{-275}:\\ \;\;\;\;100 \cdot \left(\left(\frac{\frac{\sqrt[3]{{\left({\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)}\right)}^{3} + {\left(-1 \cdot 1\right)}^{3}} \cdot \sqrt[3]{{\left({\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)}\right)}^{3} + {\left(-1 \cdot 1\right)}^{3}}}{\sqrt[3]{\mathsf{fma}\left(-1 \cdot 1, \left(-1 \cdot 1\right) - {\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)}, {\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot \left(2 \cdot n\right)\right)}\right)} \cdot \sqrt[3]{\mathsf{fma}\left(-1 \cdot 1, \left(-1 \cdot 1\right) - {\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)}, {\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot \left(2 \cdot n\right)\right)}\right)}}}{\sqrt[3]{i} \cdot \sqrt[3]{i}} \cdot \left(\sqrt[3]{n} \cdot \sqrt[3]{n}\right)\right) \cdot \frac{\frac{\frac{\sqrt[3]{{\left({\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)}\right)}^{3} + {\left(-1 \cdot 1\right)}^{3}}}{\sqrt[3]{\mathsf{fma}\left(-1 \cdot 1, \left(-1 \cdot 1\right) - {\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)}, {\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot \left(2 \cdot n\right)\right)}\right)}}}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{\frac{\sqrt[3]{i}}{\sqrt[3]{n}}}\right)\\ \mathbf{elif}\;i \le 8.0570023823781865 \cdot 10^{-197}:\\ \;\;\;\;100 \cdot \frac{\mathsf{fma}\left(i, 1, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 2.87388849271851564 \cdot 10^{-134}:\\ \;\;\;\;100 \cdot \left(\frac{\frac{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + \left(-1 \cdot 1\right)} \cdot \sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + \left(-1 \cdot 1\right)}}{\sqrt{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}}{\frac{\sqrt{i}}{\sqrt[3]{n} \cdot \sqrt[3]{n}}} \cdot \frac{\frac{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + \left(-1 \cdot 1\right)}}{\sqrt{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}}{\frac{\sqrt{i}}{\sqrt[3]{n}}}\right)\\ \mathbf{elif}\;i \le 7.1470850675780769 \cdot 10^{-9}:\\ \;\;\;\;100 \cdot \frac{\mathsf{fma}\left(i, 1, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right)}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(\frac{\frac{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + \left(-1 \cdot 1\right)} \cdot \sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + \left(-1 \cdot 1\right)}}{\sqrt{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}}{\frac{\sqrt{i}}{\sqrt[3]{n} \cdot \sqrt[3]{n}}} \cdot \frac{\frac{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + \left(-1 \cdot 1\right)}}{\sqrt{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}}{\frac{\sqrt{i}}{\sqrt[3]{n}}}\right)\\ \end{array}\]

C

double f(double i, double n) {
        double r417447 = 100.0;
        double r417448 = 1.0;
        double r417449 = i;
        double r417450 = n;
        double r417451 = r417449 / r417450;
        double r417452 = r417448 + r417451;
        double r417453 = pow(r417452, r417450);
        double r417454 = r417453 - r417448;
        double r417455 = r417454 / r417451;
        double r417456 = r417447 * r417455;
        return r417456;
}

↓

double f(double i, double n) {
        double r417457 = i;
        double r417458 = -1.2132993654259578;
        bool r417459 = r417457 <= r417458;
        double r417460 = 100.0;
        double r417461 = 1.0;
        double r417462 = n;
        double r417463 = r417457 / r417462;
        double r417464 = r417461 + r417463;
        double r417465 = pow(r417464, r417462);
        double r417466 = 3.0;
        double r417467 = pow(r417465, r417466);
        double r417468 = pow(r417461, r417466);
        double r417469 = r417467 - r417468;
        double r417470 = r417465 + r417461;
        double r417471 = 2.0;
        double r417472 = r417471 * r417462;
        double r417473 = pow(r417464, r417472);
        double r417474 = fma(r417461, r417470, r417473);
        double r417475 = r417469 / r417474;
        double r417476 = r417475 / r417463;
        double r417477 = r417460 * r417476;
        double r417478 = -1.1313305452296457e-274;
        bool r417479 = r417457 <= r417478;
        double r417480 = 0.5;
        double r417481 = pow(r417457, r417471);
        double r417482 = log(r417461);
        double r417483 = r417482 * r417462;
        double r417484 = fma(r417480, r417481, r417483);
        double r417485 = r417481 * r417482;
        double r417486 = r417480 * r417485;
        double r417487 = r417484 - r417486;
        double r417488 = fma(r417457, r417461, r417487);
        double r417489 = r417488 / r417463;
        double r417490 = r417460 * r417489;
        double r417491 = 1.6479702105467177e-275;
        bool r417492 = r417457 <= r417491;
        double r417493 = pow(r417473, r417466);
        double r417494 = r417461 * r417461;
        double r417495 = -r417494;
        double r417496 = pow(r417495, r417466);
        double r417497 = r417493 + r417496;
        double r417498 = cbrt(r417497);
        double r417499 = r417498 * r417498;
        double r417500 = r417495 - r417473;
        double r417501 = r417471 * r417472;
        double r417502 = pow(r417464, r417501);
        double r417503 = fma(r417495, r417500, r417502);
        double r417504 = cbrt(r417503);
        double r417505 = r417504 * r417504;
        double r417506 = r417499 / r417505;
        double r417507 = cbrt(r417457);
        double r417508 = r417507 * r417507;
        double r417509 = r417506 / r417508;
        double r417510 = cbrt(r417462);
        double r417511 = r417510 * r417510;
        double r417512 = r417509 * r417511;
        double r417513 = r417498 / r417504;
        double r417514 = r417513 / r417470;
        double r417515 = r417507 / r417510;
        double r417516 = r417514 / r417515;
        double r417517 = r417512 * r417516;
        double r417518 = r417460 * r417517;
        double r417519 = 8.057002382378186e-197;
        bool r417520 = r417457 <= r417519;
        double r417521 = 2.8738884927185156e-134;
        bool r417522 = r417457 <= r417521;
        double r417523 = r417473 + r417495;
        double r417524 = cbrt(r417523);
        double r417525 = r417524 * r417524;
        double r417526 = sqrt(r417470);
        double r417527 = r417525 / r417526;
        double r417528 = sqrt(r417457);
        double r417529 = r417528 / r417511;
        double r417530 = r417527 / r417529;
        double r417531 = r417524 / r417526;
        double r417532 = r417528 / r417510;
        double r417533 = r417531 / r417532;
        double r417534 = r417530 * r417533;
        double r417535 = r417460 * r417534;
        double r417536 = 7.147085067578077e-09;
        bool r417537 = r417457 <= r417536;
        double r417538 = r417537 ? r417490 : r417535;
        double r417539 = r417522 ? r417535 : r417538;
        double r417540 = r417520 ? r417490 : r417539;
        double r417541 = r417492 ? r417518 : r417540;
        double r417542 = r417479 ? r417490 : r417541;
        double r417543 = r417459 ? r417477 : r417542;
        return r417543;
}

Error

Bits error versus i

Bits error versus n

Target

Original	42.9
Target	42.8
Herbie	33.3

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

Derivation

1. Split input into 4 regimes

2. if i < -1.2132993654259578

   1. Initial program 28.1

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

   3. Applied flip3-- 28.2

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

   4. Simplified 28.2

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

   if -1.2132993654259578 < i < -1.1313305452296457e-274 or 1.6479702105467177e-275 < i < 8.057002382378186e-197 or 2.8738884927185156e-134 < i < 7.147085067578077e-09

   1. Initial program 51.4

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

   2. Taylor expanded around 0 31.5

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

   3. Simplified 31.5

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

   if -1.1313305452296457e-274 < i < 1.6479702105467177e-275

   1. Initial program 47.2

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

   3. Applied flip-- 47.2

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

   4. Simplified 47.2

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

   6. Applied flip3-+ 47.2

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

   7. Simplified 47.2

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

   9. Applied add-cube-cbrt 47.2

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

   10. Applied add-cube-cbrt 47.2

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

   11. Applied times-frac 47.2

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

   12. Applied *-un-lft-identity 47.2

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

   13. Applied add-cube-cbrt 47.2

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

   14. Applied add-cube-cbrt 47.2

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

   15. Applied times-frac 47.2

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

   16. Applied times-frac 47.2

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

   17. Applied times-frac 46.9

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

   18. Simplified 46.7

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

   if 8.057002382378186e-197 < i < 2.8738884927185156e-134 or 7.147085067578077e-09 < i

   1. Initial program 37.9

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

   3. Applied flip-- 37.9

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

   4. Simplified 37.9

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

   6. Applied add-cube-cbrt 38.1

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

   7. Applied add-sqr-sqrt 38.1

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

   8. Applied times-frac 38.1

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

   9. Applied add-sqr-sqrt 38.1

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

   10. Applied add-cube-cbrt 38.1

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

   11. Applied times-frac 38.1

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

   12. Applied times-frac 38.0

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

4. Final simplification 33.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -1.21329936542595784:\\ \;\;\;\;100 \cdot \frac{\frac{{\left({\left(1 + \frac{i}{n}\right)}^{n}\right)}^{3} - {1}^{3}}{\mathsf{fma}\left(1, {\left(1 + \frac{i}{n}\right)}^{n} + 1, {\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)}\right)}}{\frac{i}{n}}\\ \mathbf{elif}\;i \le -1.13133054522964566 \cdot 10^{-274}:\\ \;\;\;\;100 \cdot \frac{\mathsf{fma}\left(i, 1, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 1.64797021054671768 \cdot 10^{-275}:\\ \;\;\;\;100 \cdot \left(\left(\frac{\frac{\sqrt[3]{{\left({\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)}\right)}^{3} + {\left(-1 \cdot 1\right)}^{3}} \cdot \sqrt[3]{{\left({\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)}\right)}^{3} + {\left(-1 \cdot 1\right)}^{3}}}{\sqrt[3]{\mathsf{fma}\left(-1 \cdot 1, \left(-1 \cdot 1\right) - {\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)}, {\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot \left(2 \cdot n\right)\right)}\right)} \cdot \sqrt[3]{\mathsf{fma}\left(-1 \cdot 1, \left(-1 \cdot 1\right) - {\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)}, {\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot \left(2 \cdot n\right)\right)}\right)}}}{\sqrt[3]{i} \cdot \sqrt[3]{i}} \cdot \left(\sqrt[3]{n} \cdot \sqrt[3]{n}\right)\right) \cdot \frac{\frac{\frac{\sqrt[3]{{\left({\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)}\right)}^{3} + {\left(-1 \cdot 1\right)}^{3}}}{\sqrt[3]{\mathsf{fma}\left(-1 \cdot 1, \left(-1 \cdot 1\right) - {\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)}, {\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot \left(2 \cdot n\right)\right)}\right)}}}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{\frac{\sqrt[3]{i}}{\sqrt[3]{n}}}\right)\\ \mathbf{elif}\;i \le 8.0570023823781865 \cdot 10^{-197}:\\ \;\;\;\;100 \cdot \frac{\mathsf{fma}\left(i, 1, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 2.87388849271851564 \cdot 10^{-134}:\\ \;\;\;\;100 \cdot \left(\frac{\frac{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + \left(-1 \cdot 1\right)} \cdot \sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + \left(-1 \cdot 1\right)}}{\sqrt{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}}{\frac{\sqrt{i}}{\sqrt[3]{n} \cdot \sqrt[3]{n}}} \cdot \frac{\frac{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + \left(-1 \cdot 1\right)}}{\sqrt{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}}{\frac{\sqrt{i}}{\sqrt[3]{n}}}\right)\\ \mathbf{elif}\;i \le 7.1470850675780769 \cdot 10^{-9}:\\ \;\;\;\;100 \cdot \frac{\mathsf{fma}\left(i, 1, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right)}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(\frac{\frac{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + \left(-1 \cdot 1\right)} \cdot \sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + \left(-1 \cdot 1\right)}}{\sqrt{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}}{\frac{\sqrt{i}}{\sqrt[3]{n} \cdot \sqrt[3]{n}}} \cdot \frac{\frac{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + \left(-1 \cdot 1\right)}}{\sqrt{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}}{\frac{\sqrt{i}}{\sqrt[3]{n}}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020018 +o rules:numerics
(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))))