Compound Interest

Average Error: 42.5 → 30.2

Time: 55.7s

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}:\\ \;\;\;\;\left(100 \cdot \sqrt{\frac{{\left(\frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}}\right) \cdot \sqrt{\frac{{\left(\frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}}\\ \end{array}\]

C

double f(double i, double n) {
        double r239186 = 100.0;
        double r239187 = 1.0;
        double r239188 = i;
        double r239189 = n;
        double r239190 = r239188 / r239189;
        double r239191 = r239187 + r239190;
        double r239192 = pow(r239191, r239189);
        double r239193 = r239192 - r239187;
        double r239194 = r239193 / r239190;
        double r239195 = r239186 * r239194;
        return r239195;
}

↓

double f(double i, double n) {
        double r239196 = n;
        double r239197 = -2.4372743425058e-310;
        bool r239198 = r239196 <= r239197;
        double r239199 = 100.0;
        double r239200 = 1.0;
        double r239201 = 1.0;
        double r239202 = i;
        double r239203 = r239202 / r239196;
        double r239204 = r239201 + r239203;
        double r239205 = 2.0;
        double r239206 = r239205 * r239196;
        double r239207 = pow(r239204, r239206);
        double r239208 = r239201 * r239201;
        double r239209 = r239207 + r239208;
        double r239210 = cbrt(r239209);
        double r239211 = r239210 * r239210;
        double r239212 = r239200 / r239211;
        double r239213 = pow(r239204, r239196);
        double r239214 = r239213 + r239201;
        double r239215 = sqrt(r239214);
        double r239216 = r239212 / r239215;
        double r239217 = 4.0;
        double r239218 = r239217 * r239196;
        double r239219 = pow(r239204, r239218);
        double r239220 = 3.0;
        double r239221 = pow(r239201, r239220);
        double r239222 = r239201 * r239221;
        double r239223 = r239219 - r239222;
        double r239224 = r239223 / r239210;
        double r239225 = r239224 / r239215;
        double r239226 = r239225 * r239196;
        double r239227 = r239202 / r239226;
        double r239228 = r239216 / r239227;
        double r239229 = r239199 * r239228;
        double r239230 = 2.053192934454679e-111;
        bool r239231 = r239196 <= r239230;
        double r239232 = 0.5;
        double r239233 = log(r239202);
        double r239234 = pow(r239233, r239205);
        double r239235 = pow(r239196, r239205);
        double r239236 = r239234 * r239235;
        double r239237 = log(r239196);
        double r239238 = pow(r239237, r239205);
        double r239239 = r239235 * r239238;
        double r239240 = r239236 + r239239;
        double r239241 = r239232 * r239240;
        double r239242 = 0.16666666666666666;
        double r239243 = pow(r239233, r239220);
        double r239244 = pow(r239196, r239220);
        double r239245 = r239243 * r239244;
        double r239246 = r239242 * r239245;
        double r239247 = r239233 * r239196;
        double r239248 = r239244 * r239238;
        double r239249 = r239233 * r239248;
        double r239250 = r239232 * r239249;
        double r239251 = r239247 + r239250;
        double r239252 = r239246 + r239251;
        double r239253 = r239233 * r239235;
        double r239254 = r239253 + r239196;
        double r239255 = r239237 * r239254;
        double r239256 = r239244 * r239237;
        double r239257 = r239234 * r239256;
        double r239258 = r239232 * r239257;
        double r239259 = pow(r239237, r239220);
        double r239260 = r239244 * r239259;
        double r239261 = r239242 * r239260;
        double r239262 = r239258 + r239261;
        double r239263 = r239255 + r239262;
        double r239264 = r239252 - r239263;
        double r239265 = r239241 + r239264;
        double r239266 = r239265 / r239203;
        double r239267 = r239199 * r239266;
        double r239268 = 3.6825165457393707e-44;
        bool r239269 = r239196 <= r239268;
        double r239270 = r239201 * r239202;
        double r239271 = 0.5;
        double r239272 = pow(r239202, r239205);
        double r239273 = r239271 * r239272;
        double r239274 = log(r239201);
        double r239275 = r239274 * r239196;
        double r239276 = r239273 + r239275;
        double r239277 = r239270 + r239276;
        double r239278 = r239272 * r239274;
        double r239279 = r239271 * r239278;
        double r239280 = r239277 - r239279;
        double r239281 = r239280 / r239203;
        double r239282 = r239199 * r239281;
        double r239283 = 9.089128901802148e-18;
        bool r239284 = r239196 <= r239283;
        double r239285 = 50.0;
        double r239286 = r239248 / r239202;
        double r239287 = r239234 * r239244;
        double r239288 = r239287 / r239202;
        double r239289 = r239286 + r239288;
        double r239290 = r239285 * r239289;
        double r239291 = 16.666666666666664;
        double r239292 = pow(r239196, r239217);
        double r239293 = r239243 * r239292;
        double r239294 = r239293 / r239202;
        double r239295 = r239291 * r239294;
        double r239296 = r239253 / r239202;
        double r239297 = r239199 * r239296;
        double r239298 = r239292 * r239238;
        double r239299 = r239233 * r239298;
        double r239300 = r239299 / r239202;
        double r239301 = r239285 * r239300;
        double r239302 = r239297 + r239301;
        double r239303 = r239295 + r239302;
        double r239304 = r239292 * r239259;
        double r239305 = r239304 / r239202;
        double r239306 = r239291 * r239305;
        double r239307 = r239233 * r239256;
        double r239308 = r239307 / r239202;
        double r239309 = r239235 * r239237;
        double r239310 = r239309 / r239202;
        double r239311 = r239308 + r239310;
        double r239312 = r239199 * r239311;
        double r239313 = r239292 * r239237;
        double r239314 = r239234 * r239313;
        double r239315 = r239314 / r239202;
        double r239316 = r239285 * r239315;
        double r239317 = r239312 + r239316;
        double r239318 = r239306 + r239317;
        double r239319 = r239303 - r239318;
        double r239320 = r239290 + r239319;
        double r239321 = 2.1280471197997414e+177;
        bool r239322 = r239196 <= r239321;
        double r239323 = pow(r239203, r239196);
        double r239324 = r239323 - r239201;
        double r239325 = r239324 / r239203;
        double r239326 = sqrt(r239325);
        double r239327 = r239199 * r239326;
        double r239328 = r239327 * r239326;
        double r239329 = r239322 ? r239282 : r239328;
        double r239330 = r239284 ? r239320 : r239329;
        double r239331 = r239269 ? r239282 : r239330;
        double r239332 = r239231 ? r239267 : r239331;
        double r239333 = r239198 ? r239229 : r239332;
        return r239333;
}

Error

Bits error versus i

Bits error versus n

Target

Original	42.5
Target	42.4
Herbie	30.2

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

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

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

   5. Applied add-sqr-sqrt 45.8

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

   6. Applied associate-*r* 45.8

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

4. Final simplification 30.2

   \[\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}:\\ \;\;\;\;\left(100 \cdot \sqrt{\frac{{\left(\frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}}\right) \cdot \sqrt{\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))))