Compound Interest

Average Error: 42.5 → 31.3

Time: 53.6s

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 r229320 = 100.0;
        double r229321 = 1.0;
        double r229322 = i;
        double r229323 = n;
        double r229324 = r229322 / r229323;
        double r229325 = r229321 + r229324;
        double r229326 = pow(r229325, r229323);
        double r229327 = r229326 - r229321;
        double r229328 = r229327 / r229324;
        double r229329 = r229320 * r229328;
        return r229329;
}

↓

double f(double i, double n) {
        double r229330 = i;
        double r229331 = -3.143171245412234e-08;
        bool r229332 = r229330 <= r229331;
        double r229333 = 100.0;
        double r229334 = n;
        double r229335 = r229330 / r229334;
        double r229336 = pow(r229335, r229334);
        double r229337 = 1.0;
        double r229338 = r229336 - r229337;
        double r229339 = r229333 * r229338;
        double r229340 = r229339 / r229335;
        double r229341 = -6.20217558039948e-257;
        bool r229342 = r229330 <= r229341;
        double r229343 = 0.5;
        double r229344 = 2.0;
        double r229345 = pow(r229330, r229344);
        double r229346 = log(r229337);
        double r229347 = r229346 * r229334;
        double r229348 = fma(r229343, r229345, r229347);
        double r229349 = fma(r229337, r229330, r229348);
        double r229350 = r229345 * r229346;
        double r229351 = r229343 * r229350;
        double r229352 = r229349 - r229351;
        double r229353 = r229352 / r229335;
        double r229354 = r229333 * r229353;
        double r229355 = 1.7898866790545242e-195;
        bool r229356 = r229330 <= r229355;
        double r229357 = 1.0;
        double r229358 = r229337 + r229335;
        double r229359 = r229344 * r229334;
        double r229360 = pow(r229358, r229359);
        double r229361 = fma(r229337, r229337, r229360);
        double r229362 = cbrt(r229361);
        double r229363 = r229362 * r229362;
        double r229364 = r229357 / r229363;
        double r229365 = cbrt(r229330);
        double r229366 = r229365 * r229365;
        double r229367 = r229364 / r229366;
        double r229368 = 4.0;
        double r229369 = r229368 * r229334;
        double r229370 = pow(r229358, r229369);
        double r229371 = 3.0;
        double r229372 = pow(r229337, r229371);
        double r229373 = r229337 * r229372;
        double r229374 = r229370 - r229373;
        double r229375 = pow(r229358, r229334);
        double r229376 = r229375 + r229337;
        double r229377 = r229376 * r229362;
        double r229378 = r229374 / r229377;
        double r229379 = r229378 / r229365;
        double r229380 = r229379 * r229334;
        double r229381 = r229367 * r229380;
        double r229382 = r229333 * r229381;
        double r229383 = 85896130569.3759;
        bool r229384 = r229330 <= r229383;
        double r229385 = r229360 / r229376;
        double r229386 = r229385 / r229335;
        double r229387 = r229337 * r229337;
        double r229388 = r229387 / r229376;
        double r229389 = r229388 / r229335;
        double r229390 = r229386 - r229389;
        double r229391 = r229333 * r229390;
        double r229392 = r229384 ? r229354 : r229391;
        double r229393 = r229356 ? r229382 : r229392;
        double r229394 = r229342 ? r229354 : r229393;
        double r229395 = r229332 ? r229340 : r229394;
        return r229395;
}

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))))