Compound Interest

Average Error: 43.0 → 21.2

Time: 28.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;n \le -1.221351164035121487084940040764531680711 \cdot 10^{167}:\\ \;\;\;\;n \cdot \frac{100 \cdot \left(\mathsf{fma}\left(1, i, \mathsf{fma}\left(0.5, i \cdot i, \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right)}{i}\\ \mathbf{elif}\;n \le -2.480881740905884775675974535607866708188 \cdot 10^{155}:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{{\left(\frac{i}{n}\right)}^{n} - 1}}\\ \mathbf{elif}\;n \le -2.725703542154751437414591467328010381395 \cdot 10^{65}:\\ \;\;\;\;n \cdot \frac{100 \cdot \left(\mathsf{fma}\left(1, i, \mathsf{fma}\left(0.5, i \cdot i, \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right)}{i}\\ \mathbf{elif}\;n \le 2.651124307599290160581024077772320494899 \cdot 10^{-310}:\\ \;\;\;\;\frac{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} \cdot 100 + \left(-1\right) \cdot 100} \cdot \sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} \cdot 100 + \left(-1\right) \cdot 100}}{\frac{\frac{i}{n}}{\sqrt[3]{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}}}\\ \mathbf{elif}\;n \le 1.803619210409391642244204445393223551086 \cdot 10^{-92}:\\ \;\;\;\;\mathsf{fma}\left(50, \mathsf{fma}\left(\frac{{n}^{3}}{i}, {\left(\log n\right)}^{2}, \frac{{\left(\log i\right)}^{2} \cdot {n}^{3}}{i}\right), \mathsf{fma}\left(16.66666666666666429819088079966604709625, \frac{{\left(\log i\right)}^{3} \cdot {n}^{4}}{i}, \mathsf{fma}\left(100, \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) - \mathsf{fma}\left(\frac{{n}^{4} \cdot {\left(\log n\right)}^{3}}{i}, 16.66666666666666429819088079966604709625, \mathsf{fma}\left(100, \mathsf{fma}\left(\frac{\log i}{i}, {n}^{3} \cdot \log n, \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)\\ \mathbf{else}:\\ \;\;\;\;n \cdot \frac{100 \cdot \left(\mathsf{fma}\left(1, i, \mathsf{fma}\left(0.5, i \cdot i, \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right)}{i}\\ \end{array}\]

C

double f(double i, double n) {
        double r117243 = 100.0;
        double r117244 = 1.0;
        double r117245 = i;
        double r117246 = n;
        double r117247 = r117245 / r117246;
        double r117248 = r117244 + r117247;
        double r117249 = pow(r117248, r117246);
        double r117250 = r117249 - r117244;
        double r117251 = r117250 / r117247;
        double r117252 = r117243 * r117251;
        return r117252;
}

↓

double f(double i, double n) {
        double r117253 = n;
        double r117254 = -1.2213511640351215e+167;
        bool r117255 = r117253 <= r117254;
        double r117256 = 100.0;
        double r117257 = 1.0;
        double r117258 = i;
        double r117259 = 0.5;
        double r117260 = r117258 * r117258;
        double r117261 = log(r117257);
        double r117262 = r117261 * r117253;
        double r117263 = fma(r117259, r117260, r117262);
        double r117264 = fma(r117257, r117258, r117263);
        double r117265 = 2.0;
        double r117266 = pow(r117258, r117265);
        double r117267 = r117266 * r117261;
        double r117268 = r117259 * r117267;
        double r117269 = r117264 - r117268;
        double r117270 = r117256 * r117269;
        double r117271 = r117270 / r117258;
        double r117272 = r117253 * r117271;
        double r117273 = -2.480881740905885e+155;
        bool r117274 = r117253 <= r117273;
        double r117275 = r117258 / r117253;
        double r117276 = pow(r117275, r117253);
        double r117277 = r117276 - r117257;
        double r117278 = r117258 / r117277;
        double r117279 = r117253 / r117278;
        double r117280 = r117256 * r117279;
        double r117281 = -2.7257035421547514e+65;
        bool r117282 = r117253 <= r117281;
        double r117283 = 2.6511243075993e-310;
        bool r117284 = r117253 <= r117283;
        double r117285 = r117257 + r117275;
        double r117286 = pow(r117285, r117253);
        double r117287 = r117286 * r117256;
        double r117288 = -r117257;
        double r117289 = r117288 * r117256;
        double r117290 = r117287 + r117289;
        double r117291 = cbrt(r117290);
        double r117292 = r117291 * r117291;
        double r117293 = r117286 - r117257;
        double r117294 = r117256 * r117293;
        double r117295 = cbrt(r117294);
        double r117296 = r117275 / r117295;
        double r117297 = r117292 / r117296;
        double r117298 = 1.8036192104093916e-92;
        bool r117299 = r117253 <= r117298;
        double r117300 = 50.0;
        double r117301 = 3.0;
        double r117302 = pow(r117253, r117301);
        double r117303 = r117302 / r117258;
        double r117304 = log(r117253);
        double r117305 = pow(r117304, r117265);
        double r117306 = log(r117258);
        double r117307 = pow(r117306, r117265);
        double r117308 = r117307 * r117302;
        double r117309 = r117308 / r117258;
        double r117310 = fma(r117303, r117305, r117309);
        double r117311 = 16.666666666666664;
        double r117312 = pow(r117306, r117301);
        double r117313 = 4.0;
        double r117314 = pow(r117253, r117313);
        double r117315 = r117312 * r117314;
        double r117316 = r117315 / r117258;
        double r117317 = pow(r117253, r117265);
        double r117318 = r117306 * r117317;
        double r117319 = r117318 / r117258;
        double r117320 = r117314 * r117305;
        double r117321 = r117306 * r117320;
        double r117322 = r117321 / r117258;
        double r117323 = r117300 * r117322;
        double r117324 = fma(r117256, r117319, r117323);
        double r117325 = fma(r117311, r117316, r117324);
        double r117326 = fma(r117300, r117310, r117325);
        double r117327 = pow(r117304, r117301);
        double r117328 = r117314 * r117327;
        double r117329 = r117328 / r117258;
        double r117330 = r117306 / r117258;
        double r117331 = r117302 * r117304;
        double r117332 = r117317 * r117304;
        double r117333 = r117332 / r117258;
        double r117334 = fma(r117330, r117331, r117333);
        double r117335 = r117314 * r117304;
        double r117336 = r117307 * r117335;
        double r117337 = r117336 / r117258;
        double r117338 = r117300 * r117337;
        double r117339 = fma(r117256, r117334, r117338);
        double r117340 = fma(r117329, r117311, r117339);
        double r117341 = r117326 - r117340;
        double r117342 = r117299 ? r117341 : r117272;
        double r117343 = r117284 ? r117297 : r117342;
        double r117344 = r117282 ? r117272 : r117343;
        double r117345 = r117274 ? r117280 : r117344;
        double r117346 = r117255 ? r117272 : r117345;
        return r117346;
}

Error

Bits error versus i

Bits error versus n

Target

Original	43.0
Target	42.6
Herbie	21.2

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

Derivation

1. Split input into 4 regimes

2. if n < -1.2213511640351215e+167 or -2.480881740905885e+155 < n < -2.7257035421547514e+65 or 1.8036192104093916e-92 < n

   1. Initial program 55.1

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

   3. Applied associate-*r/ 55.1

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

   5. Applied sub-neg 55.1

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

   6. Applied distribute-lft-in 55.1

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

   7. Simplified 55.1

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

   8. Simplified 55.1

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

   10. Applied *-un-lft-identity 55.1

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

   11. Applied *-un-lft-identity 55.1

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

   12. Applied times-frac 55.1

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

   13. Applied *-un-lft-identity 55.1

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

   14. Applied times-frac 55.1

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

   15. Simplified 55.1

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

   16. Simplified 54.7

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

   17. Taylor expanded around 0 20.6

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

   18. Simplified 20.6

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

   if -1.2213511640351215e+167 < n < -2.480881740905885e+155

   1. Initial program 41.5

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

   2. Taylor expanded around inf 64.0

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

   3. Simplified 42.0

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

   if -2.7257035421547514e+65 < n < 2.6511243075993e-310

   1. Initial program 19.8

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

   3. Applied associate-*r/ 19.7

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

   5. Applied sub-neg 19.7

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

   6. Applied distribute-lft-in 19.7

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

   7. Simplified 19.7

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

   8. Simplified 19.7

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

   10. Applied add-cube-cbrt 19.9

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

   11. Applied associate-/l* 19.9

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

   12. Simplified 19.9

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

   if 2.6511243075993e-310 < n < 1.8036192104093916e-92

   1. Initial program 45.7

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

   3. Applied associate-*r/ 45.7

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

   5. Applied sub-neg 45.7

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

   6. Applied distribute-lft-in 45.7

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

   7. Simplified 45.7

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

   8. Simplified 45.7

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

   10. Applied *-un-lft-identity 45.7

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

   11. Applied *-un-lft-identity 45.7

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

   12. Applied times-frac 45.7

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

   13. Applied *-un-lft-identity 45.7

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

   14. Applied times-frac 45.7

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

   15. Simplified 45.7

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

   16. Simplified 45.7

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

   17. Taylor expanded around 0 24.2

       \[\leadsto 1 \cdot \color{blue}{\left(\left(50 \cdot \frac{{n}^{3} \cdot {\left(\log n\right)}^{2}}{i} + \left(50 \cdot \frac{{\left(\log i\right)}^{2} \cdot {n}^{3}}{i} + \left(16.66666666666666429819088079966604709625 \cdot \frac{{\left(\log i\right)}^{3} \cdot {n}^{4}}{i} + \left(100 \cdot \frac{\log i \cdot {n}^{2}}{i} + 50 \cdot \frac{\log i \cdot \left({n}^{4} \cdot {\left(\log n\right)}^{2}\right)}{i}\right)\right)\right)\right) - \left(16.66666666666666429819088079966604709625 \cdot \frac{{n}^{4} \cdot {\left(\log n\right)}^{3}}{i} + \left(100 \cdot \frac{\log i \cdot \left({n}^{3} \cdot \log n\right)}{i} + \left(100 \cdot \frac{{n}^{2} \cdot \log n}{i} + 50 \cdot \frac{{\left(\log i\right)}^{2} \cdot \left({n}^{4} \cdot \log n\right)}{i}\right)\right)\right)\right)}\]

   18. Simplified 24.3

       \[\leadsto 1 \cdot \color{blue}{\left(\mathsf{fma}\left(50, \mathsf{fma}\left(\frac{{n}^{3}}{i}, {\left(\log n\right)}^{2}, \frac{{\left(\log i\right)}^{2} \cdot {n}^{3}}{i}\right), \mathsf{fma}\left(16.66666666666666429819088079966604709625, \frac{{\left(\log i\right)}^{3} \cdot {n}^{4}}{i}, \mathsf{fma}\left(100, \frac{\log i \cdot {n}^{2}}{i}, 50 \cdot \frac{\log i \cdot \left({n}^{4} \cdot {\left(\log n\right)}^{2}\right)}{i}\right)\right)\right) - \mathsf{fma}\left(\frac{{n}^{4} \cdot {\left(\log n\right)}^{3}}{i}, 16.66666666666666429819088079966604709625, \mathsf{fma}\left(100, \mathsf{fma}\left(\frac{\log i}{i}, {n}^{3} \cdot \log n, \frac{{n}^{2} \cdot \log n}{i}\right), 50 \cdot \frac{{\left(\log i\right)}^{2} \cdot \left({n}^{4} \cdot \log n\right)}{i}\right)\right)\right)}\]
3. Recombined 4 regimes into one program.

4. Final simplification 21.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -1.221351164035121487084940040764531680711 \cdot 10^{167}:\\ \;\;\;\;n \cdot \frac{100 \cdot \left(\mathsf{fma}\left(1, i, \mathsf{fma}\left(0.5, i \cdot i, \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right)}{i}\\ \mathbf{elif}\;n \le -2.480881740905884775675974535607866708188 \cdot 10^{155}:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{{\left(\frac{i}{n}\right)}^{n} - 1}}\\ \mathbf{elif}\;n \le -2.725703542154751437414591467328010381395 \cdot 10^{65}:\\ \;\;\;\;n \cdot \frac{100 \cdot \left(\mathsf{fma}\left(1, i, \mathsf{fma}\left(0.5, i \cdot i, \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right)}{i}\\ \mathbf{elif}\;n \le 2.651124307599290160581024077772320494899 \cdot 10^{-310}:\\ \;\;\;\;\frac{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} \cdot 100 + \left(-1\right) \cdot 100} \cdot \sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} \cdot 100 + \left(-1\right) \cdot 100}}{\frac{\frac{i}{n}}{\sqrt[3]{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}}}\\ \mathbf{elif}\;n \le 1.803619210409391642244204445393223551086 \cdot 10^{-92}:\\ \;\;\;\;\mathsf{fma}\left(50, \mathsf{fma}\left(\frac{{n}^{3}}{i}, {\left(\log n\right)}^{2}, \frac{{\left(\log i\right)}^{2} \cdot {n}^{3}}{i}\right), \mathsf{fma}\left(16.66666666666666429819088079966604709625, \frac{{\left(\log i\right)}^{3} \cdot {n}^{4}}{i}, \mathsf{fma}\left(100, \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) - \mathsf{fma}\left(\frac{{n}^{4} \cdot {\left(\log n\right)}^{3}}{i}, 16.66666666666666429819088079966604709625, \mathsf{fma}\left(100, \mathsf{fma}\left(\frac{\log i}{i}, {n}^{3} \cdot \log n, \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)\\ \mathbf{else}:\\ \;\;\;\;n \cdot \frac{100 \cdot \left(\mathsf{fma}\left(1, i, \mathsf{fma}\left(0.5, i \cdot i, \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right)}{i}\\ \end{array}\]

Reproduce

herbie shell --seed 2019347 +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))))