Compound Interest

Average Error: 42.5 → 31.3

Time: 55.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;i \le -3.143171245412234223671663589649849512853 \cdot 10^{-8}:\\ \;\;\;\;\frac{100 \cdot \left({\left(\frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \le -6.202175580399479749610806763177730589725 \cdot 10^{-257}:\\ \;\;\;\;100 \cdot \frac{\mathsf{fma}\left(1, i, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 1.789886679054524215845529450016829481257 \cdot 10^{-195}:\\ \;\;\;\;100 \cdot \left(\frac{\frac{1}{\sqrt[3]{\mathsf{fma}\left(1, 1, {\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)}\right)} \cdot \sqrt[3]{\mathsf{fma}\left(1, 1, {\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)}\right)}}}{\sqrt[3]{i} \cdot \sqrt[3]{i}} \cdot \left(\frac{\frac{{\left(1 + \frac{i}{n}\right)}^{\left(4 \cdot n\right)} - 1 \cdot {1}^{3}}{\left({\left(1 + \frac{i}{n}\right)}^{n} + 1\right) \cdot \sqrt[3]{\mathsf{fma}\left(1, 1, {\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)}\right)}}}{\sqrt[3]{i}} \cdot n\right)\right)\\ \mathbf{elif}\;i \le 85896130569.3759002685546875:\\ \;\;\;\;100 \cdot \frac{\mathsf{fma}\left(1, i, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(\frac{\frac{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)}}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{\frac{i}{n}} - \frac{\frac{1 \cdot 1}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{\frac{i}{n}}\right)\\ \end{array}\]

C

double f(double i, double n) {
        double r240486 = 100.0;
        double r240487 = 1.0;
        double r240488 = i;
        double r240489 = n;
        double r240490 = r240488 / r240489;
        double r240491 = r240487 + r240490;
        double r240492 = pow(r240491, r240489);
        double r240493 = r240492 - r240487;
        double r240494 = r240493 / r240490;
        double r240495 = r240486 * r240494;
        return r240495;
}

↓

double f(double i, double n) {
        double r240496 = i;
        double r240497 = -3.143171245412234e-08;
        bool r240498 = r240496 <= r240497;
        double r240499 = 100.0;
        double r240500 = n;
        double r240501 = r240496 / r240500;
        double r240502 = pow(r240501, r240500);
        double r240503 = 1.0;
        double r240504 = r240502 - r240503;
        double r240505 = r240499 * r240504;
        double r240506 = r240505 / r240501;
        double r240507 = -6.20217558039948e-257;
        bool r240508 = r240496 <= r240507;
        double r240509 = 0.5;
        double r240510 = 2.0;
        double r240511 = pow(r240496, r240510);
        double r240512 = log(r240503);
        double r240513 = r240512 * r240500;
        double r240514 = fma(r240509, r240511, r240513);
        double r240515 = fma(r240503, r240496, r240514);
        double r240516 = r240511 * r240512;
        double r240517 = r240509 * r240516;
        double r240518 = r240515 - r240517;
        double r240519 = r240518 / r240501;
        double r240520 = r240499 * r240519;
        double r240521 = 1.7898866790545242e-195;
        bool r240522 = r240496 <= r240521;
        double r240523 = 1.0;
        double r240524 = r240503 + r240501;
        double r240525 = r240510 * r240500;
        double r240526 = pow(r240524, r240525);
        double r240527 = fma(r240503, r240503, r240526);
        double r240528 = cbrt(r240527);
        double r240529 = r240528 * r240528;
        double r240530 = r240523 / r240529;
        double r240531 = cbrt(r240496);
        double r240532 = r240531 * r240531;
        double r240533 = r240530 / r240532;
        double r240534 = 4.0;
        double r240535 = r240534 * r240500;
        double r240536 = pow(r240524, r240535);
        double r240537 = 3.0;
        double r240538 = pow(r240503, r240537);
        double r240539 = r240503 * r240538;
        double r240540 = r240536 - r240539;
        double r240541 = pow(r240524, r240500);
        double r240542 = r240541 + r240503;
        double r240543 = r240542 * r240528;
        double r240544 = r240540 / r240543;
        double r240545 = r240544 / r240531;
        double r240546 = r240545 * r240500;
        double r240547 = r240533 * r240546;
        double r240548 = r240499 * r240547;
        double r240549 = 85896130569.3759;
        bool r240550 = r240496 <= r240549;
        double r240551 = r240526 / r240542;
        double r240552 = r240551 / r240501;
        double r240553 = r240503 * r240503;
        double r240554 = r240553 / r240542;
        double r240555 = r240554 / r240501;
        double r240556 = r240552 - r240555;
        double r240557 = r240499 * r240556;
        double r240558 = r240550 ? r240520 : r240557;
        double r240559 = r240522 ? r240548 : r240558;
        double r240560 = r240508 ? r240520 : r240559;
        double r240561 = r240498 ? r240506 : r240560;
        return r240561;
}

Error

Bits error versus i

Bits error versus n

Target

Original	42.5
Target	42.4
Herbie	31.3

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

Derivation

1. Split input into 4 regimes

2. if i < -3.143171245412234e-08

   1. Initial program 27.6

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

   2. Taylor expanded around inf 64.0

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

   3. Simplified 19.0

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

   if -3.143171245412234e-08 < i < -6.20217558039948e-257 or 1.7898866790545242e-195 < i < 85896130569.3759

   1. Initial program 51.1

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

   2. Taylor expanded around 0 31.2

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

   3. Simplified 31.2

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

   if -6.20217558039948e-257 < i < 1.7898866790545242e-195

   1. Initial program 48.7

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

   3. Applied flip-- 48.7

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

   4. Simplified 48.7

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

   6. Applied flip-- 48.7

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

   7. Simplified 48.7

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

   8. Simplified 48.7

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

   10. Applied *-un-lft-identity 48.7

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

   11. Applied add-cube-cbrt 48.7

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

   12. Applied times-frac 48.7

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

   13. Applied *-un-lft-identity 48.7

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

   14. Applied add-cube-cbrt 48.7

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

   15. Applied *-un-lft-identity 48.7

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

   16. Applied times-frac 48.7

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

   17. Applied times-frac 48.7

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

   18. Applied times-frac 48.3

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

   19. Simplified 48.3

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

   20. Simplified 48.1

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

   if 85896130569.3759 < i

   1. Initial program 30.6

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

   3. Applied flip-- 30.6

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

   4. Simplified 30.6

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

   6. Applied div-sub 30.6

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

   7. Applied div-sub 30.6

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

4. Final simplification 31.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -3.143171245412234223671663589649849512853 \cdot 10^{-8}:\\ \;\;\;\;\frac{100 \cdot \left({\left(\frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \le -6.202175580399479749610806763177730589725 \cdot 10^{-257}:\\ \;\;\;\;100 \cdot \frac{\mathsf{fma}\left(1, i, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 1.789886679054524215845529450016829481257 \cdot 10^{-195}:\\ \;\;\;\;100 \cdot \left(\frac{\frac{1}{\sqrt[3]{\mathsf{fma}\left(1, 1, {\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)}\right)} \cdot \sqrt[3]{\mathsf{fma}\left(1, 1, {\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)}\right)}}}{\sqrt[3]{i} \cdot \sqrt[3]{i}} \cdot \left(\frac{\frac{{\left(1 + \frac{i}{n}\right)}^{\left(4 \cdot n\right)} - 1 \cdot {1}^{3}}{\left({\left(1 + \frac{i}{n}\right)}^{n} + 1\right) \cdot \sqrt[3]{\mathsf{fma}\left(1, 1, {\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)}\right)}}}{\sqrt[3]{i}} \cdot n\right)\right)\\ \mathbf{elif}\;i \le 85896130569.3759002685546875:\\ \;\;\;\;100 \cdot \frac{\mathsf{fma}\left(1, i, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(\frac{\frac{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)}}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{\frac{i}{n}} - \frac{\frac{1 \cdot 1}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{\frac{i}{n}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019303 +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))))