Compound Interest

Average Error: 42.9 → 28.6

Time: 50.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;i \le -3.7966848740325338 \cdot 10^{-12}:\\ \;\;\;\;\frac{100 \cdot \left({\left(\frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \le -5.95710477349490714 \cdot 10^{-209}:\\ \;\;\;\;100 \cdot \frac{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 4.8800960479016703 \cdot 10^{-201}:\\ \;\;\;\;100 \cdot \left(n \cdot \left(0.50000000000000022 \cdot i + 1.00000000000000022\right) + \left(1 \cdot \frac{\log 1 \cdot {n}^{2}}{i} - 0.499999999999999944 \cdot \left(i \cdot \left(\log 1 \cdot n\right)\right)\right)\right)\\ \mathbf{elif}\;i \le 1.56110700502537925:\\ \;\;\;\;100 \cdot \frac{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(\frac{\frac{1}{\sqrt[3]{1 \cdot \left(1 + {\left(1 + \frac{i}{n}\right)}^{n}\right) + {\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)}} \cdot \sqrt[3]{1 \cdot \left(1 + {\left(1 + \frac{i}{n}\right)}^{n}\right) + {\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)}}}}{i} \cdot \left(n \cdot \left(\frac{{\left({\left(1 + \frac{i}{n}\right)}^{n}\right)}^{3}}{\sqrt[3]{1 \cdot \left(1 + {\left(1 + \frac{i}{n}\right)}^{n}\right) + {\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)}}} - \frac{{1}^{3}}{\sqrt[3]{1 \cdot \left(1 + {\left(1 + \frac{i}{n}\right)}^{n}\right) + {\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)}}}\right)\right)\right)\\ \end{array}\]

C

double f(double i, double n) {
        double r279307 = 100.0;
        double r279308 = 1.0;
        double r279309 = i;
        double r279310 = n;
        double r279311 = r279309 / r279310;
        double r279312 = r279308 + r279311;
        double r279313 = pow(r279312, r279310);
        double r279314 = r279313 - r279308;
        double r279315 = r279314 / r279311;
        double r279316 = r279307 * r279315;
        return r279316;
}

↓

double f(double i, double n) {
        double r279317 = i;
        double r279318 = -3.796684874032534e-12;
        bool r279319 = r279317 <= r279318;
        double r279320 = 100.0;
        double r279321 = n;
        double r279322 = r279317 / r279321;
        double r279323 = pow(r279322, r279321);
        double r279324 = 1.0;
        double r279325 = r279323 - r279324;
        double r279326 = r279320 * r279325;
        double r279327 = r279326 / r279322;
        double r279328 = -5.957104773494907e-209;
        bool r279329 = r279317 <= r279328;
        double r279330 = r279324 * r279317;
        double r279331 = 0.5;
        double r279332 = 2.0;
        double r279333 = pow(r279317, r279332);
        double r279334 = r279331 * r279333;
        double r279335 = log(r279324);
        double r279336 = r279335 * r279321;
        double r279337 = r279334 + r279336;
        double r279338 = r279330 + r279337;
        double r279339 = r279333 * r279335;
        double r279340 = r279331 * r279339;
        double r279341 = r279338 - r279340;
        double r279342 = r279341 / r279322;
        double r279343 = r279320 * r279342;
        double r279344 = 4.88009604790167e-201;
        bool r279345 = r279317 <= r279344;
        double r279346 = 0.5000000000000002;
        double r279347 = r279346 * r279317;
        double r279348 = 1.0000000000000002;
        double r279349 = r279347 + r279348;
        double r279350 = r279321 * r279349;
        double r279351 = pow(r279321, r279332);
        double r279352 = r279335 * r279351;
        double r279353 = r279352 / r279317;
        double r279354 = r279324 * r279353;
        double r279355 = 0.49999999999999994;
        double r279356 = r279317 * r279336;
        double r279357 = r279355 * r279356;
        double r279358 = r279354 - r279357;
        double r279359 = r279350 + r279358;
        double r279360 = r279320 * r279359;
        double r279361 = 1.5611070050253792;
        bool r279362 = r279317 <= r279361;
        double r279363 = 1.0;
        double r279364 = r279324 + r279322;
        double r279365 = pow(r279364, r279321);
        double r279366 = r279324 + r279365;
        double r279367 = r279324 * r279366;
        double r279368 = r279332 * r279321;
        double r279369 = pow(r279364, r279368);
        double r279370 = r279367 + r279369;
        double r279371 = cbrt(r279370);
        double r279372 = r279371 * r279371;
        double r279373 = r279363 / r279372;
        double r279374 = r279373 / r279317;
        double r279375 = 3.0;
        double r279376 = pow(r279365, r279375);
        double r279377 = r279376 / r279371;
        double r279378 = pow(r279324, r279375);
        double r279379 = r279378 / r279371;
        double r279380 = r279377 - r279379;
        double r279381 = r279321 * r279380;
        double r279382 = r279374 * r279381;
        double r279383 = r279320 * r279382;
        double r279384 = r279362 ? r279343 : r279383;
        double r279385 = r279345 ? r279360 : r279384;
        double r279386 = r279329 ? r279343 : r279385;
        double r279387 = r279319 ? r279327 : r279386;
        return r279387;
}

Error

Bits error versus i

Bits error versus n

Target

Original	42.9
Target	42.4
Herbie	28.6

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

Derivation

1. Split input into 4 regimes

2. if i < -3.796684874032534e-12

   1. Initial program 29.1

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

   3. Applied associate-*r/ 29.1

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

   4. Taylor expanded around inf 64.0

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

   5. Simplified 19.3

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

   if -3.796684874032534e-12 < i < -5.957104773494907e-209 or 4.88009604790167e-201 < i < 1.5611070050253792

   1. Initial program 50.8

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

   2. Taylor expanded around 0 30.0

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

   if -5.957104773494907e-209 < i < 4.88009604790167e-201

   1. Initial program 49.1

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

   3. Applied flip3-- 49.1

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

   4. Simplified 49.1

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

   6. Applied div-sub 49.1

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

   7. Applied div-sub 49.2

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

   8. Taylor expanded around 0 32.7

      \[\leadsto 100 \cdot \color{blue}{\left(\left(0.50000000000000022 \cdot \left(i \cdot n\right) + \left(1.00000000000000022 \cdot n + 1 \cdot \frac{\log 1 \cdot {n}^{2}}{i}\right)\right) - 0.499999999999999944 \cdot \left(i \cdot \left(\log 1 \cdot n\right)\right)\right)}\]

   9. Simplified 32.7

      \[\leadsto 100 \cdot \color{blue}{\left(n \cdot \left(0.50000000000000022 \cdot i + 1.00000000000000022\right) + \left(1 \cdot \frac{\log 1 \cdot {n}^{2}}{i} - 0.499999999999999944 \cdot \left(i \cdot \left(\log 1 \cdot n\right)\right)\right)\right)}\]

   if 1.5611070050253792 < i

   1. Initial program 33.3

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

   3. Applied flip3-- 33.3

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

   4. Simplified 33.3

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

   6. Applied div-sub 33.3

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

   7. Applied div-sub 33.3

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

   9. Applied div-inv 35.8

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

   10. Applied add-cube-cbrt 35.9

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

   11. Applied *-un-lft-identity 35.9

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

   12. Applied times-frac 41.4

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

   13. Applied times-frac 40.3

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

   14. Applied div-inv 40.1

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

   15. Applied add-cube-cbrt 40.1

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

   16. Applied *-un-lft-identity 40.1

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

   17. Applied times-frac 37.3

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

   18. Applied times-frac 33.4

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

   19. Applied distribute-lft-out-- 33.4

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

   20. Simplified 33.3

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

4. Final simplification 28.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -3.7966848740325338 \cdot 10^{-12}:\\ \;\;\;\;\frac{100 \cdot \left({\left(\frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \le -5.95710477349490714 \cdot 10^{-209}:\\ \;\;\;\;100 \cdot \frac{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 4.8800960479016703 \cdot 10^{-201}:\\ \;\;\;\;100 \cdot \left(n \cdot \left(0.50000000000000022 \cdot i + 1.00000000000000022\right) + \left(1 \cdot \frac{\log 1 \cdot {n}^{2}}{i} - 0.499999999999999944 \cdot \left(i \cdot \left(\log 1 \cdot n\right)\right)\right)\right)\\ \mathbf{elif}\;i \le 1.56110700502537925:\\ \;\;\;\;100 \cdot \frac{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(\frac{\frac{1}{\sqrt[3]{1 \cdot \left(1 + {\left(1 + \frac{i}{n}\right)}^{n}\right) + {\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)}} \cdot \sqrt[3]{1 \cdot \left(1 + {\left(1 + \frac{i}{n}\right)}^{n}\right) + {\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)}}}}{i} \cdot \left(n \cdot \left(\frac{{\left({\left(1 + \frac{i}{n}\right)}^{n}\right)}^{3}}{\sqrt[3]{1 \cdot \left(1 + {\left(1 + \frac{i}{n}\right)}^{n}\right) + {\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)}}} - \frac{{1}^{3}}{\sqrt[3]{1 \cdot \left(1 + {\left(1 + \frac{i}{n}\right)}^{n}\right) + {\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)}}}\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020003 
(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))))