Compound Interest

Average Error: 43.1 → 22.3

Time: 20.9s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;n \le -1.00297232240918336 \cdot 10^{174}:\\ \;\;\;\;\left(100 \cdot n\right) \cdot \frac{\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)}{i}\\ \mathbf{elif}\;n \le -4.03699116408299863 \cdot 10^{85}:\\ \;\;\;\;\frac{100 \cdot \left(n \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)\right)}{i}\\ \mathbf{elif}\;n \le -3.31747608845659503 \cdot 10^{63}:\\ \;\;\;\;\left(100 \cdot n\right) \cdot \frac{\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)}{i}\\ \mathbf{elif}\;n \le -1.0912719332224593 \cdot 10^{26}:\\ \;\;\;\;\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}\\ \mathbf{elif}\;n \le -934893.903368213796:\\ \;\;\;\;\left(100 \cdot n\right) \cdot \frac{\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)}{i}\\ \mathbf{elif}\;n \le 2.44249498211844666 \cdot 10^{-299}:\\ \;\;\;\;100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \frac{1}{\frac{i}{n}}\right)\\ \mathbf{elif}\;n \le 6.74497016401884472 \cdot 10^{-102}:\\ \;\;\;\;\mathsf{fma}\left(50, \frac{{n}^{3} \cdot {\left(\log n\right)}^{2}}{i}, \mathsf{fma}\left(50, \frac{{\left(\log i\right)}^{2} \cdot {n}^{3}}{i}, \mathsf{fma}\left(16.666666666666664, \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)\right) - \mathsf{fma}\left(16.666666666666664, \frac{{n}^{4} \cdot {\left(\log n\right)}^{3}}{i}, \mathsf{fma}\left(100, \frac{\log i \cdot \left({n}^{3} \cdot \log n\right)}{i}, \mathsf{fma}\left(100, \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)\\ \mathbf{else}:\\ \;\;\;\;\left(100 \cdot n\right) \cdot \frac{\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)}{i}\\ \end{array}\]

C

double f(double i, double n) {
        double r122358 = 100.0;
        double r122359 = 1.0;
        double r122360 = i;
        double r122361 = n;
        double r122362 = r122360 / r122361;
        double r122363 = r122359 + r122362;
        double r122364 = pow(r122363, r122361);
        double r122365 = r122364 - r122359;
        double r122366 = r122365 / r122362;
        double r122367 = r122358 * r122366;
        return r122367;
}

↓

double f(double i, double n) {
        double r122368 = n;
        double r122369 = -1.0029723224091834e+174;
        bool r122370 = r122368 <= r122369;
        double r122371 = 100.0;
        double r122372 = r122371 * r122368;
        double r122373 = 1.0;
        double r122374 = i;
        double r122375 = 0.5;
        double r122376 = r122374 * r122374;
        double r122377 = log(r122373);
        double r122378 = r122377 * r122368;
        double r122379 = fma(r122375, r122376, r122378);
        double r122380 = fma(r122373, r122374, r122379);
        double r122381 = 2.0;
        double r122382 = pow(r122374, r122381);
        double r122383 = r122382 * r122377;
        double r122384 = r122375 * r122383;
        double r122385 = r122380 - r122384;
        double r122386 = r122385 / r122374;
        double r122387 = r122372 * r122386;
        double r122388 = -4.0369911640829986e+85;
        bool r122389 = r122368 <= r122388;
        double r122390 = r122374 / r122368;
        double r122391 = r122373 + r122390;
        double r122392 = pow(r122391, r122368);
        double r122393 = r122392 - r122373;
        double r122394 = r122368 * r122393;
        double r122395 = r122371 * r122394;
        double r122396 = r122395 / r122374;
        double r122397 = -3.317476088456595e+63;
        bool r122398 = r122368 <= r122397;
        double r122399 = -1.0912719332224593e+26;
        bool r122400 = r122368 <= r122399;
        double r122401 = r122371 * r122393;
        double r122402 = r122401 / r122390;
        double r122403 = -934893.9033682138;
        bool r122404 = r122368 <= r122403;
        double r122405 = 2.4424949821184467e-299;
        bool r122406 = r122368 <= r122405;
        double r122407 = r122392 / r122390;
        double r122408 = r122373 / r122390;
        double r122409 = r122407 - r122408;
        double r122410 = r122371 * r122409;
        double r122411 = 6.744970164018845e-102;
        bool r122412 = r122368 <= r122411;
        double r122413 = 50.0;
        double r122414 = 3.0;
        double r122415 = pow(r122368, r122414);
        double r122416 = log(r122368);
        double r122417 = pow(r122416, r122381);
        double r122418 = r122415 * r122417;
        double r122419 = r122418 / r122374;
        double r122420 = log(r122374);
        double r122421 = pow(r122420, r122381);
        double r122422 = r122421 * r122415;
        double r122423 = r122422 / r122374;
        double r122424 = 16.666666666666664;
        double r122425 = pow(r122420, r122414);
        double r122426 = 4.0;
        double r122427 = pow(r122368, r122426);
        double r122428 = r122425 * r122427;
        double r122429 = r122428 / r122374;
        double r122430 = pow(r122368, r122381);
        double r122431 = r122420 * r122430;
        double r122432 = r122431 / r122374;
        double r122433 = r122427 * r122417;
        double r122434 = r122420 * r122433;
        double r122435 = r122434 / r122374;
        double r122436 = r122413 * r122435;
        double r122437 = fma(r122371, r122432, r122436);
        double r122438 = fma(r122424, r122429, r122437);
        double r122439 = fma(r122413, r122423, r122438);
        double r122440 = fma(r122413, r122419, r122439);
        double r122441 = pow(r122416, r122414);
        double r122442 = r122427 * r122441;
        double r122443 = r122442 / r122374;
        double r122444 = r122415 * r122416;
        double r122445 = r122420 * r122444;
        double r122446 = r122445 / r122374;
        double r122447 = r122430 * r122416;
        double r122448 = r122447 / r122374;
        double r122449 = r122427 * r122416;
        double r122450 = r122421 * r122449;
        double r122451 = r122450 / r122374;
        double r122452 = r122413 * r122451;
        double r122453 = fma(r122371, r122448, r122452);
        double r122454 = fma(r122371, r122446, r122453);
        double r122455 = fma(r122424, r122443, r122454);
        double r122456 = r122440 - r122455;
        double r122457 = r122412 ? r122456 : r122387;
        double r122458 = r122406 ? r122410 : r122457;
        double r122459 = r122404 ? r122387 : r122458;
        double r122460 = r122400 ? r122402 : r122459;
        double r122461 = r122398 ? r122387 : r122460;
        double r122462 = r122389 ? r122396 : r122461;
        double r122463 = r122370 ? r122387 : r122462;
        return r122463;
}

Error

Bits error versus i

Bits error versus n

Target

Original	43.1
Target	43.0
Herbie	22.3

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

Derivation

1. Split input into 5 regimes

2. if n < -1.0029723224091834e+174 or -4.0369911640829986e+85 < n < -3.317476088456595e+63 or -1.0912719332224593e+26 < n < -934893.9033682138 or 6.744970164018845e-102 < n

   1. Initial program 56.4

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

   3. Applied div-inv 56.4

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

   4. Applied add-sqr-sqrt 56.4

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

   5. Applied add-sqr-sqrt 56.4

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

   6. Applied difference-of-squares 56.4

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

   7. Applied times-frac 56.1

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

   8. Simplified 56.1

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

   10. Applied associate-*l/ 56.1

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

   11. Simplified 56.0

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

   12. Taylor expanded around 0 21.3

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

   13. Simplified 21.3

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

   15. Applied *-un-lft-identity 21.3

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

   16. Applied times-frac 20.1

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

   17. Applied associate-*r* 20.1

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

   18. Simplified 20.1

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

   if -1.0029723224091834e+174 < n < -4.0369911640829986e+85

   1. Initial program 40.6

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

   3. Applied div-inv 40.6

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

   4. Applied add-sqr-sqrt 40.6

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

   5. Applied add-sqr-sqrt 40.6

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

   6. Applied difference-of-squares 40.6

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

   7. Applied times-frac 40.3

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

   8. Simplified 40.3

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

   10. Applied associate-*l/ 40.2

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

   11. Simplified 40.2

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

   13. Applied associate-*r/ 40.2

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

   if -3.317476088456595e+63 < n < -1.0912719332224593e+26

   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 associate-*r/ 30.5

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

   if -934893.9033682138 < n < 2.4424949821184467e-299

   1. Initial program 17.7

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

   3. Applied div-sub 17.7

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

   if 2.4424949821184467e-299 < n < 6.744970164018845e-102

   1. Initial program 45.4

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

   3. Applied div-inv 45.4

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

   4. Applied add-sqr-sqrt 45.4

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

   5. Applied add-sqr-sqrt 45.4

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

   6. Applied difference-of-squares 45.4

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

   7. Applied times-frac 45.4

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

   8. Simplified 45.4

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

   10. Applied associate-*l/ 45.4

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

   11. Simplified 45.4

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

   12. Taylor expanded around 0 24.1

       \[\leadsto \color{blue}{\left(50 \cdot \frac{{n}^{3} \cdot {\left(\log n\right)}^{2}}{i} + \left(50 \cdot \frac{{\left(\log i\right)}^{2} \cdot {n}^{3}}{i} + \left(16.666666666666664 \cdot \frac{{\left(\log i\right)}^{3} \cdot {n}^{4}}{i} + \left(100 \cdot \frac{\log i \cdot {n}^{2}}{i} + 50 \cdot \frac{\log i \cdot \left({n}^{4} \cdot {\left(\log n\right)}^{2}\right)}{i}\right)\right)\right)\right) - \left(16.666666666666664 \cdot \frac{{n}^{4} \cdot {\left(\log n\right)}^{3}}{i} + \left(100 \cdot \frac{\log i \cdot \left({n}^{3} \cdot \log n\right)}{i} + \left(100 \cdot \frac{{n}^{2} \cdot \log n}{i} + 50 \cdot \frac{{\left(\log i\right)}^{2} \cdot \left({n}^{4} \cdot \log n\right)}{i}\right)\right)\right)}\]

   13. Simplified 24.1

       \[\leadsto \color{blue}{\mathsf{fma}\left(50, \frac{{n}^{3} \cdot {\left(\log n\right)}^{2}}{i}, \mathsf{fma}\left(50, \frac{{\left(\log i\right)}^{2} \cdot {n}^{3}}{i}, \mathsf{fma}\left(16.666666666666664, \frac{{\left(\log i\right)}^{3} \cdot {n}^{4}}{i}, \mathsf{fma}\left(100, \frac{\log i \cdot {n}^{2}}{i}, 50 \cdot \frac{\log i \cdot \left({n}^{4} \cdot {\left(\log n\right)}^{2}\right)}{i}\right)\right)\right)\right) - \mathsf{fma}\left(16.666666666666664, \frac{{n}^{4} \cdot {\left(\log n\right)}^{3}}{i}, \mathsf{fma}\left(100, \frac{\log i \cdot \left({n}^{3} \cdot \log n\right)}{i}, \mathsf{fma}\left(100, \frac{{n}^{2} \cdot \log n}{i}, 50 \cdot \frac{{\left(\log i\right)}^{2} \cdot \left({n}^{4} \cdot \log n\right)}{i}\right)\right)\right)}\]
3. Recombined 5 regimes into one program.

4. Final simplification 22.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -1.00297232240918336 \cdot 10^{174}:\\ \;\;\;\;\left(100 \cdot n\right) \cdot \frac{\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)}{i}\\ \mathbf{elif}\;n \le -4.03699116408299863 \cdot 10^{85}:\\ \;\;\;\;\frac{100 \cdot \left(n \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)\right)}{i}\\ \mathbf{elif}\;n \le -3.31747608845659503 \cdot 10^{63}:\\ \;\;\;\;\left(100 \cdot n\right) \cdot \frac{\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)}{i}\\ \mathbf{elif}\;n \le -1.0912719332224593 \cdot 10^{26}:\\ \;\;\;\;\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}\\ \mathbf{elif}\;n \le -934893.903368213796:\\ \;\;\;\;\left(100 \cdot n\right) \cdot \frac{\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)}{i}\\ \mathbf{elif}\;n \le 2.44249498211844666 \cdot 10^{-299}:\\ \;\;\;\;100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \frac{1}{\frac{i}{n}}\right)\\ \mathbf{elif}\;n \le 6.74497016401884472 \cdot 10^{-102}:\\ \;\;\;\;\mathsf{fma}\left(50, \frac{{n}^{3} \cdot {\left(\log n\right)}^{2}}{i}, \mathsf{fma}\left(50, \frac{{\left(\log i\right)}^{2} \cdot {n}^{3}}{i}, \mathsf{fma}\left(16.666666666666664, \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)\right) - \mathsf{fma}\left(16.666666666666664, \frac{{n}^{4} \cdot {\left(\log n\right)}^{3}}{i}, \mathsf{fma}\left(100, \frac{\log i \cdot \left({n}^{3} \cdot \log n\right)}{i}, \mathsf{fma}\left(100, \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)\\ \mathbf{else}:\\ \;\;\;\;\left(100 \cdot n\right) \cdot \frac{\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)}{i}\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 +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))))