Compound Interest

Average Error: 42.5 → 29.4

Time: 45.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;n \le -2.8503749344901687 \cdot 10^{+128}:\\ \;\;\;\;\mathsf{fma}\left(e^{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n}, 100, -100\right) \cdot \frac{n}{i}\\ \mathbf{elif}\;n \le -6.741071980134238 \cdot 10^{+48}:\\ \;\;\;\;\frac{1}{i} \cdot \left(n \cdot \mathsf{fma}\left({\left(\frac{i}{n} + 1\right)}^{n}, 100, -100\right)\right)\\ \mathbf{elif}\;n \le -2.81251599798957 \cdot 10^{+26}:\\ \;\;\;\;\frac{\mathsf{fma}\left(50, i \cdot i, \left(100 + \left(i \cdot i\right) \cdot \frac{50}{3}\right) \cdot i\right)}{\frac{i}{n}}\\ \mathbf{elif}\;n \le -3.9175258711905035 \cdot 10^{-19}:\\ \;\;\;\;\frac{\frac{\sqrt[3]{\mathsf{fma}\left(e^{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n + \left(n + n\right) \cdot \mathsf{log1p}\left(\frac{i}{n}\right)}, 1000000, -1000000\right)}}{\sqrt[3]{\mathsf{fma}\left(e^{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n}, 10000 \cdot e^{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n}, 10000 \cdot e^{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n} + 10000\right)}}}{\frac{\sqrt[3]{i}}{\sqrt[3]{n}}} \cdot \frac{\frac{\sqrt[3]{\mathsf{fma}\left(e^{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n + \left(n + n\right) \cdot \mathsf{log1p}\left(\frac{i}{n}\right)}, 1000000, -1000000\right)} \cdot \sqrt[3]{\mathsf{fma}\left(e^{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n + \left(n + n\right) \cdot \mathsf{log1p}\left(\frac{i}{n}\right)}, 1000000, -1000000\right)}}{\sqrt[3]{\mathsf{fma}\left(e^{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n}, 10000 \cdot e^{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n}, 10000 \cdot e^{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n} + 10000\right)} \cdot \sqrt[3]{\mathsf{fma}\left(e^{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n}, 10000 \cdot e^{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n}, 10000 \cdot e^{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n} + 10000\right)}}}{\frac{\sqrt[3]{i} \cdot \sqrt[3]{i}}{\sqrt[3]{n} \cdot \sqrt[3]{n}}}\\ \mathbf{elif}\;n \le 1.8317810149458703 \cdot 10^{-135}:\\ \;\;\;\;0\\ \mathbf{elif}\;n \le 2.3783092808523403 \cdot 10^{+147}:\\ \;\;\;\;\frac{\mathsf{fma}\left(50, i \cdot i, \left(100 + \left(i \cdot i\right) \cdot \frac{50}{3}\right) \cdot i\right)}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(e^{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n + \left(n + n\right) \cdot \mathsf{log1p}\left(\frac{i}{n}\right)}, 1000000, -1000000\right)}{\mathsf{fma}\left(e^{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n}, 10000, 10000 \cdot e^{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n} + 10000\right)}}{\frac{i}{n}}\\ \end{array}\]

C

double f(double i, double n) {
        double r7002275 = 100.0;
        double r7002276 = 1.0;
        double r7002277 = i;
        double r7002278 = n;
        double r7002279 = r7002277 / r7002278;
        double r7002280 = r7002276 + r7002279;
        double r7002281 = pow(r7002280, r7002278);
        double r7002282 = r7002281 - r7002276;
        double r7002283 = r7002282 / r7002279;
        double r7002284 = r7002275 * r7002283;
        return r7002284;
}

↓

double f(double i, double n) {
        double r7002285 = n;
        double r7002286 = -2.8503749344901687e+128;
        bool r7002287 = r7002285 <= r7002286;
        double r7002288 = i;
        double r7002289 = r7002288 / r7002285;
        double r7002290 = log1p(r7002289);
        double r7002291 = r7002290 * r7002285;
        double r7002292 = exp(r7002291);
        double r7002293 = 100.0;
        double r7002294 = -100.0;
        double r7002295 = fma(r7002292, r7002293, r7002294);
        double r7002296 = r7002285 / r7002288;
        double r7002297 = r7002295 * r7002296;
        double r7002298 = -6.741071980134238e+48;
        bool r7002299 = r7002285 <= r7002298;
        double r7002300 = 1.0;
        double r7002301 = r7002300 / r7002288;
        double r7002302 = r7002289 + r7002300;
        double r7002303 = pow(r7002302, r7002285);
        double r7002304 = fma(r7002303, r7002293, r7002294);
        double r7002305 = r7002285 * r7002304;
        double r7002306 = r7002301 * r7002305;
        double r7002307 = -2.81251599798957e+26;
        bool r7002308 = r7002285 <= r7002307;
        double r7002309 = 50.0;
        double r7002310 = r7002288 * r7002288;
        double r7002311 = 16.666666666666668;
        double r7002312 = r7002310 * r7002311;
        double r7002313 = r7002293 + r7002312;
        double r7002314 = r7002313 * r7002288;
        double r7002315 = fma(r7002309, r7002310, r7002314);
        double r7002316 = r7002315 / r7002289;
        double r7002317 = -3.9175258711905035e-19;
        bool r7002318 = r7002285 <= r7002317;
        double r7002319 = r7002285 + r7002285;
        double r7002320 = r7002319 * r7002290;
        double r7002321 = r7002291 + r7002320;
        double r7002322 = exp(r7002321);
        double r7002323 = 1000000.0;
        double r7002324 = -1000000.0;
        double r7002325 = fma(r7002322, r7002323, r7002324);
        double r7002326 = cbrt(r7002325);
        double r7002327 = 10000.0;
        double r7002328 = r7002327 * r7002292;
        double r7002329 = r7002328 + r7002327;
        double r7002330 = fma(r7002292, r7002328, r7002329);
        double r7002331 = cbrt(r7002330);
        double r7002332 = r7002326 / r7002331;
        double r7002333 = cbrt(r7002288);
        double r7002334 = cbrt(r7002285);
        double r7002335 = r7002333 / r7002334;
        double r7002336 = r7002332 / r7002335;
        double r7002337 = r7002326 * r7002326;
        double r7002338 = r7002331 * r7002331;
        double r7002339 = r7002337 / r7002338;
        double r7002340 = r7002333 * r7002333;
        double r7002341 = r7002334 * r7002334;
        double r7002342 = r7002340 / r7002341;
        double r7002343 = r7002339 / r7002342;
        double r7002344 = r7002336 * r7002343;
        double r7002345 = 1.8317810149458703e-135;
        bool r7002346 = r7002285 <= r7002345;
        double r7002347 = 0.0;
        double r7002348 = 2.3783092808523403e+147;
        bool r7002349 = r7002285 <= r7002348;
        double r7002350 = fma(r7002292, r7002327, r7002329);
        double r7002351 = r7002325 / r7002350;
        double r7002352 = r7002351 / r7002289;
        double r7002353 = r7002349 ? r7002316 : r7002352;
        double r7002354 = r7002346 ? r7002347 : r7002353;
        double r7002355 = r7002318 ? r7002344 : r7002354;
        double r7002356 = r7002308 ? r7002316 : r7002355;
        double r7002357 = r7002299 ? r7002306 : r7002356;
        double r7002358 = r7002287 ? r7002297 : r7002357;
        return r7002358;
}

Error

Bits error versus i

Bits error versus n

Target

Original	42.5
Target	41.8
Herbie	29.4

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

Derivation

1. Split input into 6 regimes

2. if n < -2.8503749344901687e+128

   1. Initial program 50.8

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

   2. Simplified 50.8

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

   4. Applied pow-to-exp 55.7

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

   5. Simplified 41.2

      \[\leadsto \frac{\mathsf{fma}\left(e^{\color{blue}{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n}}, 100, -100\right)}{\frac{i}{n}}\]
   6. Using strategy rm

   7. Applied div-inv 41.3

      \[\leadsto \color{blue}{\mathsf{fma}\left(e^{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n}, 100, -100\right) \cdot \frac{1}{\frac{i}{n}}}\]

   8. Simplified 41.3

      \[\leadsto \mathsf{fma}\left(e^{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n}, 100, -100\right) \cdot \color{blue}{\frac{n}{i}}\]

   if -2.8503749344901687e+128 < n < -6.741071980134238e+48

   1. Initial program 38.8

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

   2. Simplified 38.8

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

   4. Applied div-inv 38.8

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

   5. Applied *-un-lft-identity 38.8

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

   6. Applied times-frac 38.7

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

   7. Simplified 38.6

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

   if -6.741071980134238e+48 < n < -2.81251599798957e+26 or 1.8317810149458703e-135 < n < 2.3783092808523403e+147

   1. Initial program 55.4

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

   2. Simplified 55.4

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

   3. Taylor expanded around 0 25.0

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

   4. Simplified 25.0

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(50, i \cdot i, i \cdot \left(\left(i \cdot i\right) \cdot \frac{50}{3} + 100\right)\right)}}{\frac{i}{n}}\]

   if -2.81251599798957e+26 < n < -3.9175258711905035e-19

   1. Initial program 25.2

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

   2. Simplified 25.1

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

   4. Applied pow-to-exp 30.2

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

   5. Simplified 29.2

      \[\leadsto \frac{\mathsf{fma}\left(e^{\color{blue}{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n}}, 100, -100\right)}{\frac{i}{n}}\]
   6. Using strategy rm

   7. Applied fma-udef 29.3

      \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n} \cdot 100 + -100}}{\frac{i}{n}}\]
   8. Using strategy rm

   9. Applied flip3-+ 29.3

      \[\leadsto \frac{\color{blue}{\frac{{\left(e^{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n} \cdot 100\right)}^{3} + {-100}^{3}}{\left(e^{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n} \cdot 100\right) \cdot \left(e^{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n} \cdot 100\right) + \left(-100 \cdot -100 - \left(e^{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n} \cdot 100\right) \cdot -100\right)}}}{\frac{i}{n}}\]

   10. Simplified 29.0

       \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(e^{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n + \mathsf{log1p}\left(\frac{i}{n}\right) \cdot \left(n + n\right)}, 1000000, -1000000\right)}}{\left(e^{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n} \cdot 100\right) \cdot \left(e^{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n} \cdot 100\right) + \left(-100 \cdot -100 - \left(e^{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n} \cdot 100\right) \cdot -100\right)}}{\frac{i}{n}}\]

   11. Simplified 29.0

       \[\leadsto \frac{\frac{\mathsf{fma}\left(e^{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n + \mathsf{log1p}\left(\frac{i}{n}\right) \cdot \left(n + n\right)}, 1000000, -1000000\right)}{\color{blue}{\mathsf{fma}\left(e^{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n}, e^{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n} \cdot 10000, e^{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n} \cdot 10000 + 10000\right)}}}{\frac{i}{n}}\]
   12. Using strategy rm

   13. Applied add-cube-cbrt 29.2

       \[\leadsto \frac{\frac{\mathsf{fma}\left(e^{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n + \mathsf{log1p}\left(\frac{i}{n}\right) \cdot \left(n + n\right)}, 1000000, -1000000\right)}{\mathsf{fma}\left(e^{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n}, e^{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n} \cdot 10000, e^{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n} \cdot 10000 + 10000\right)}}{\frac{i}{\color{blue}{\left(\sqrt[3]{n} \cdot \sqrt[3]{n}\right) \cdot \sqrt[3]{n}}}}\]

   14. Applied add-cube-cbrt 29.3

       \[\leadsto \frac{\frac{\mathsf{fma}\left(e^{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n + \mathsf{log1p}\left(\frac{i}{n}\right) \cdot \left(n + n\right)}, 1000000, -1000000\right)}{\mathsf{fma}\left(e^{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n}, e^{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n} \cdot 10000, e^{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n} \cdot 10000 + 10000\right)}}{\frac{\color{blue}{\left(\sqrt[3]{i} \cdot \sqrt[3]{i}\right) \cdot \sqrt[3]{i}}}{\left(\sqrt[3]{n} \cdot \sqrt[3]{n}\right) \cdot \sqrt[3]{n}}}\]

   15. Applied times-frac 29.3

       \[\leadsto \frac{\frac{\mathsf{fma}\left(e^{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n + \mathsf{log1p}\left(\frac{i}{n}\right) \cdot \left(n + n\right)}, 1000000, -1000000\right)}{\mathsf{fma}\left(e^{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n}, e^{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n} \cdot 10000, e^{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n} \cdot 10000 + 10000\right)}}{\color{blue}{\frac{\sqrt[3]{i} \cdot \sqrt[3]{i}}{\sqrt[3]{n} \cdot \sqrt[3]{n}} \cdot \frac{\sqrt[3]{i}}{\sqrt[3]{n}}}}\]

   16. Applied add-cube-cbrt 29.3

       \[\leadsto \frac{\frac{\mathsf{fma}\left(e^{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n + \mathsf{log1p}\left(\frac{i}{n}\right) \cdot \left(n + n\right)}, 1000000, -1000000\right)}{\color{blue}{\left(\sqrt[3]{\mathsf{fma}\left(e^{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n}, e^{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n} \cdot 10000, e^{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n} \cdot 10000 + 10000\right)} \cdot \sqrt[3]{\mathsf{fma}\left(e^{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n}, e^{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n} \cdot 10000, e^{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n} \cdot 10000 + 10000\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(e^{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n}, e^{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n} \cdot 10000, e^{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n} \cdot 10000 + 10000\right)}}}}{\frac{\sqrt[3]{i} \cdot \sqrt[3]{i}}{\sqrt[3]{n} \cdot \sqrt[3]{n}} \cdot \frac{\sqrt[3]{i}}{\sqrt[3]{n}}}\]

   17. Applied add-cube-cbrt 29.3

       \[\leadsto \frac{\frac{\color{blue}{\left(\sqrt[3]{\mathsf{fma}\left(e^{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n + \mathsf{log1p}\left(\frac{i}{n}\right) \cdot \left(n + n\right)}, 1000000, -1000000\right)} \cdot \sqrt[3]{\mathsf{fma}\left(e^{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n + \mathsf{log1p}\left(\frac{i}{n}\right) \cdot \left(n + n\right)}, 1000000, -1000000\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(e^{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n + \mathsf{log1p}\left(\frac{i}{n}\right) \cdot \left(n + n\right)}, 1000000, -1000000\right)}}}{\left(\sqrt[3]{\mathsf{fma}\left(e^{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n}, e^{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n} \cdot 10000, e^{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n} \cdot 10000 + 10000\right)} \cdot \sqrt[3]{\mathsf{fma}\left(e^{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n}, e^{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n} \cdot 10000, e^{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n} \cdot 10000 + 10000\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(e^{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n}, e^{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n} \cdot 10000, e^{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n} \cdot 10000 + 10000\right)}}}{\frac{\sqrt[3]{i} \cdot \sqrt[3]{i}}{\sqrt[3]{n} \cdot \sqrt[3]{n}} \cdot \frac{\sqrt[3]{i}}{\sqrt[3]{n}}}\]

   18. Applied times-frac 29.3

       \[\leadsto \frac{\color{blue}{\frac{\sqrt[3]{\mathsf{fma}\left(e^{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n + \mathsf{log1p}\left(\frac{i}{n}\right) \cdot \left(n + n\right)}, 1000000, -1000000\right)} \cdot \sqrt[3]{\mathsf{fma}\left(e^{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n + \mathsf{log1p}\left(\frac{i}{n}\right) \cdot \left(n + n\right)}, 1000000, -1000000\right)}}{\sqrt[3]{\mathsf{fma}\left(e^{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n}, e^{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n} \cdot 10000, e^{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n} \cdot 10000 + 10000\right)} \cdot \sqrt[3]{\mathsf{fma}\left(e^{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n}, e^{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n} \cdot 10000, e^{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n} \cdot 10000 + 10000\right)}} \cdot \frac{\sqrt[3]{\mathsf{fma}\left(e^{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n + \mathsf{log1p}\left(\frac{i}{n}\right) \cdot \left(n + n\right)}, 1000000, -1000000\right)}}{\sqrt[3]{\mathsf{fma}\left(e^{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n}, e^{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n} \cdot 10000, e^{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n} \cdot 10000 + 10000\right)}}}}{\frac{\sqrt[3]{i} \cdot \sqrt[3]{i}}{\sqrt[3]{n} \cdot \sqrt[3]{n}} \cdot \frac{\sqrt[3]{i}}{\sqrt[3]{n}}}\]

   19. Applied times-frac 29.4

       \[\leadsto \color{blue}{\frac{\frac{\sqrt[3]{\mathsf{fma}\left(e^{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n + \mathsf{log1p}\left(\frac{i}{n}\right) \cdot \left(n + n\right)}, 1000000, -1000000\right)} \cdot \sqrt[3]{\mathsf{fma}\left(e^{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n + \mathsf{log1p}\left(\frac{i}{n}\right) \cdot \left(n + n\right)}, 1000000, -1000000\right)}}{\sqrt[3]{\mathsf{fma}\left(e^{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n}, e^{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n} \cdot 10000, e^{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n} \cdot 10000 + 10000\right)} \cdot \sqrt[3]{\mathsf{fma}\left(e^{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n}, e^{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n} \cdot 10000, e^{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n} \cdot 10000 + 10000\right)}}}{\frac{\sqrt[3]{i} \cdot \sqrt[3]{i}}{\sqrt[3]{n} \cdot \sqrt[3]{n}}} \cdot \frac{\frac{\sqrt[3]{\mathsf{fma}\left(e^{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n + \mathsf{log1p}\left(\frac{i}{n}\right) \cdot \left(n + n\right)}, 1000000, -1000000\right)}}{\sqrt[3]{\mathsf{fma}\left(e^{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n}, e^{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n} \cdot 10000, e^{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n} \cdot 10000 + 10000\right)}}}{\frac{\sqrt[3]{i}}{\sqrt[3]{n}}}}\]

   if -3.9175258711905035e-19 < n < 1.8317810149458703e-135

   1. Initial program 25.5

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

   2. Simplified 25.5

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

   3. Taylor expanded around 0 18.1

      \[\leadsto \color{blue}{0}\]

   if 2.3783092808523403e+147 < n

   1. Initial program 60.2

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

   2. Simplified 60.2

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

   4. Applied pow-to-exp 60.2

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

   5. Simplified 42.4

      \[\leadsto \frac{\mathsf{fma}\left(e^{\color{blue}{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n}}, 100, -100\right)}{\frac{i}{n}}\]
   6. Using strategy rm

   7. Applied fma-udef 42.4

      \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n} \cdot 100 + -100}}{\frac{i}{n}}\]
   8. Using strategy rm

   9. Applied flip3-+ 42.4

      \[\leadsto \frac{\color{blue}{\frac{{\left(e^{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n} \cdot 100\right)}^{3} + {-100}^{3}}{\left(e^{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n} \cdot 100\right) \cdot \left(e^{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n} \cdot 100\right) + \left(-100 \cdot -100 - \left(e^{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n} \cdot 100\right) \cdot -100\right)}}}{\frac{i}{n}}\]

   10. Simplified 42.4

       \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(e^{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n + \mathsf{log1p}\left(\frac{i}{n}\right) \cdot \left(n + n\right)}, 1000000, -1000000\right)}}{\left(e^{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n} \cdot 100\right) \cdot \left(e^{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n} \cdot 100\right) + \left(-100 \cdot -100 - \left(e^{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n} \cdot 100\right) \cdot -100\right)}}{\frac{i}{n}}\]

   11. Simplified 42.4

       \[\leadsto \frac{\frac{\mathsf{fma}\left(e^{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n + \mathsf{log1p}\left(\frac{i}{n}\right) \cdot \left(n + n\right)}, 1000000, -1000000\right)}{\color{blue}{\mathsf{fma}\left(e^{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n}, e^{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n} \cdot 10000, e^{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n} \cdot 10000 + 10000\right)}}}{\frac{i}{n}}\]

   12. Taylor expanded around inf 43.0

       \[\leadsto \frac{\frac{\mathsf{fma}\left(e^{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n + \mathsf{log1p}\left(\frac{i}{n}\right) \cdot \left(n + n\right)}, 1000000, -1000000\right)}{\mathsf{fma}\left(e^{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n}, e^{\color{blue}{0}} \cdot 10000, e^{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n} \cdot 10000 + 10000\right)}}{\frac{i}{n}}\]
3. Recombined 6 regimes into one program.

4. Final simplification 29.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -2.8503749344901687 \cdot 10^{+128}:\\ \;\;\;\;\mathsf{fma}\left(e^{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n}, 100, -100\right) \cdot \frac{n}{i}\\ \mathbf{elif}\;n \le -6.741071980134238 \cdot 10^{+48}:\\ \;\;\;\;\frac{1}{i} \cdot \left(n \cdot \mathsf{fma}\left({\left(\frac{i}{n} + 1\right)}^{n}, 100, -100\right)\right)\\ \mathbf{elif}\;n \le -2.81251599798957 \cdot 10^{+26}:\\ \;\;\;\;\frac{\mathsf{fma}\left(50, i \cdot i, \left(100 + \left(i \cdot i\right) \cdot \frac{50}{3}\right) \cdot i\right)}{\frac{i}{n}}\\ \mathbf{elif}\;n \le -3.9175258711905035 \cdot 10^{-19}:\\ \;\;\;\;\frac{\frac{\sqrt[3]{\mathsf{fma}\left(e^{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n + \left(n + n\right) \cdot \mathsf{log1p}\left(\frac{i}{n}\right)}, 1000000, -1000000\right)}}{\sqrt[3]{\mathsf{fma}\left(e^{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n}, 10000 \cdot e^{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n}, 10000 \cdot e^{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n} + 10000\right)}}}{\frac{\sqrt[3]{i}}{\sqrt[3]{n}}} \cdot \frac{\frac{\sqrt[3]{\mathsf{fma}\left(e^{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n + \left(n + n\right) \cdot \mathsf{log1p}\left(\frac{i}{n}\right)}, 1000000, -1000000\right)} \cdot \sqrt[3]{\mathsf{fma}\left(e^{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n + \left(n + n\right) \cdot \mathsf{log1p}\left(\frac{i}{n}\right)}, 1000000, -1000000\right)}}{\sqrt[3]{\mathsf{fma}\left(e^{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n}, 10000 \cdot e^{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n}, 10000 \cdot e^{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n} + 10000\right)} \cdot \sqrt[3]{\mathsf{fma}\left(e^{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n}, 10000 \cdot e^{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n}, 10000 \cdot e^{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n} + 10000\right)}}}{\frac{\sqrt[3]{i} \cdot \sqrt[3]{i}}{\sqrt[3]{n} \cdot \sqrt[3]{n}}}\\ \mathbf{elif}\;n \le 1.8317810149458703 \cdot 10^{-135}:\\ \;\;\;\;0\\ \mathbf{elif}\;n \le 2.3783092808523403 \cdot 10^{+147}:\\ \;\;\;\;\frac{\mathsf{fma}\left(50, i \cdot i, \left(100 + \left(i \cdot i\right) \cdot \frac{50}{3}\right) \cdot i\right)}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(e^{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n + \left(n + n\right) \cdot \mathsf{log1p}\left(\frac{i}{n}\right)}, 1000000, -1000000\right)}{\mathsf{fma}\left(e^{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n}, 10000, 10000 \cdot e^{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n} + 10000\right)}}{\frac{i}{n}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019143 +o rules:numerics
(FPCore (i n)
  :name "Compound Interest"

  :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))))