Compound Interest

Average Error: 42.3 → 20.1

Time: 1.6m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;i \le -4.890775921786639 \cdot 10^{-05}:\\ \;\;\;\;100 \cdot \frac{{\left({\left(\frac{i}{n} + 1\right)}^{n}\right)}^{3} - 1}{\frac{i}{n} \cdot \left({\left(\frac{i}{n} + 1\right)}^{n} \cdot {\left(\frac{i}{n} + 1\right)}^{n} + \left(1 + {\left(\frac{i}{n} + 1\right)}^{n}\right)\right)}\\ \mathbf{elif}\;i \le -1.4365268904089577 \cdot 10^{-258}:\\ \;\;\;\;\left(\frac{n \cdot i}{i} \cdot 100\right) \cdot \left(i \cdot \frac{1}{2} + \left(1 + \left(i \cdot i\right) \cdot \frac{1}{6}\right)\right)\\ \mathbf{elif}\;i \le 1.1743359908539163 \cdot 10^{-228}:\\ \;\;\;\;\left(\left(i \cdot i\right) \cdot n\right) \cdot \frac{50}{3} + \left(100 + i \cdot 50\right) \cdot n\\ \mathbf{elif}\;i \le 3.2268987471607203 \cdot 10^{-189}:\\ \;\;\;\;100 \cdot \left(\frac{1}{i} \cdot \sqrt[3]{\left(\left(\left(\left(i \cdot \frac{1}{2}\right) \cdot i + \left(\left(i \cdot i\right) \cdot \left(i \cdot \frac{1}{6}\right) + i\right)\right) \cdot n\right) \cdot \left(\left(\left(i \cdot \frac{1}{2}\right) \cdot i + \left(\left(i \cdot i\right) \cdot \left(i \cdot \frac{1}{6}\right) + i\right)\right) \cdot n\right)\right) \cdot \left(\left(\left(i \cdot \frac{1}{2}\right) \cdot i + \left(\left(i \cdot i\right) \cdot \left(i \cdot \frac{1}{6}\right) + i\right)\right) \cdot n\right)}\right)\\ \mathbf{elif}\;i \le 0.13449167275451146:\\ \;\;\;\;\left(\frac{n \cdot i}{i} \cdot 100\right) \cdot \left(i \cdot \frac{1}{2} + \left(1 + \left(i \cdot i\right) \cdot \frac{1}{6}\right)\right)\\ \mathbf{elif}\;i \le 7.218954252890614 \cdot 10^{+203}:\\ \;\;\;\;100 \cdot \frac{{\left({\left(\frac{i}{n} + 1\right)}^{n}\right)}^{3} - 1}{\frac{i}{n} \cdot \left({\left(\frac{i}{n} + 1\right)}^{n} \cdot {\left(\frac{i}{n} + 1\right)}^{n} + \left(1 + {\left(\frac{i}{n} + 1\right)}^{n}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

C

double f(double i, double n) {
        double r11875634 = 100.0;
        double r11875635 = 1.0;
        double r11875636 = i;
        double r11875637 = n;
        double r11875638 = r11875636 / r11875637;
        double r11875639 = r11875635 + r11875638;
        double r11875640 = pow(r11875639, r11875637);
        double r11875641 = r11875640 - r11875635;
        double r11875642 = r11875641 / r11875638;
        double r11875643 = r11875634 * r11875642;
        return r11875643;
}

↓

double f(double i, double n) {
        double r11875644 = i;
        double r11875645 = -4.890775921786639e-05;
        bool r11875646 = r11875644 <= r11875645;
        double r11875647 = 100.0;
        double r11875648 = n;
        double r11875649 = r11875644 / r11875648;
        double r11875650 = 1.0;
        double r11875651 = r11875649 + r11875650;
        double r11875652 = pow(r11875651, r11875648);
        double r11875653 = 3.0;
        double r11875654 = pow(r11875652, r11875653);
        double r11875655 = r11875654 - r11875650;
        double r11875656 = r11875652 * r11875652;
        double r11875657 = r11875650 + r11875652;
        double r11875658 = r11875656 + r11875657;
        double r11875659 = r11875649 * r11875658;
        double r11875660 = r11875655 / r11875659;
        double r11875661 = r11875647 * r11875660;
        double r11875662 = -1.4365268904089577e-258;
        bool r11875663 = r11875644 <= r11875662;
        double r11875664 = r11875648 * r11875644;
        double r11875665 = r11875664 / r11875644;
        double r11875666 = r11875665 * r11875647;
        double r11875667 = 0.5;
        double r11875668 = r11875644 * r11875667;
        double r11875669 = r11875644 * r11875644;
        double r11875670 = 0.16666666666666666;
        double r11875671 = r11875669 * r11875670;
        double r11875672 = r11875650 + r11875671;
        double r11875673 = r11875668 + r11875672;
        double r11875674 = r11875666 * r11875673;
        double r11875675 = 1.1743359908539163e-228;
        bool r11875676 = r11875644 <= r11875675;
        double r11875677 = r11875669 * r11875648;
        double r11875678 = 16.666666666666668;
        double r11875679 = r11875677 * r11875678;
        double r11875680 = 50.0;
        double r11875681 = r11875644 * r11875680;
        double r11875682 = r11875647 + r11875681;
        double r11875683 = r11875682 * r11875648;
        double r11875684 = r11875679 + r11875683;
        double r11875685 = 3.2268987471607203e-189;
        bool r11875686 = r11875644 <= r11875685;
        double r11875687 = r11875650 / r11875644;
        double r11875688 = r11875668 * r11875644;
        double r11875689 = r11875644 * r11875670;
        double r11875690 = r11875669 * r11875689;
        double r11875691 = r11875690 + r11875644;
        double r11875692 = r11875688 + r11875691;
        double r11875693 = r11875692 * r11875648;
        double r11875694 = r11875693 * r11875693;
        double r11875695 = r11875694 * r11875693;
        double r11875696 = cbrt(r11875695);
        double r11875697 = r11875687 * r11875696;
        double r11875698 = r11875647 * r11875697;
        double r11875699 = 0.13449167275451146;
        bool r11875700 = r11875644 <= r11875699;
        double r11875701 = 7.218954252890614e+203;
        bool r11875702 = r11875644 <= r11875701;
        double r11875703 = 0.0;
        double r11875704 = r11875702 ? r11875661 : r11875703;
        double r11875705 = r11875700 ? r11875674 : r11875704;
        double r11875706 = r11875686 ? r11875698 : r11875705;
        double r11875707 = r11875676 ? r11875684 : r11875706;
        double r11875708 = r11875663 ? r11875674 : r11875707;
        double r11875709 = r11875646 ? r11875661 : r11875708;
        return r11875709;
}

Error

Bits error versus i

Bits error versus n

Target

Original	42.3
Target	42.4
Herbie	20.1

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

Derivation

1. Split input into 5 regimes

2. if i < -4.890775921786639e-05 or 0.13449167275451146 < i < 7.218954252890614e+203

   1. Initial program 27.8

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

   3. Applied flip3-- 27.8

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

   4. Applied associate-/l/ 27.8

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

   if -4.890775921786639e-05 < i < -1.4365268904089577e-258 or 3.2268987471607203e-189 < i < 0.13449167275451146

   1. Initial program 50.6

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

   2. Taylor expanded around 0 29.3

      \[\leadsto 100 \cdot \frac{\color{blue}{i + \left(\frac{1}{2} \cdot {i}^{2} + \frac{1}{6} \cdot {i}^{3}\right)}}{\frac{i}{n}}\]

   3. Simplified 29.3

      \[\leadsto 100 \cdot \frac{\color{blue}{\left(i + \frac{1}{6} \cdot \left(\left(i \cdot i\right) \cdot i\right)\right) + \left(\frac{1}{2} \cdot i\right) \cdot i}}{\frac{i}{n}}\]
   4. Using strategy rm

   5. Applied div-inv 29.3

      \[\leadsto 100 \cdot \frac{\left(i + \frac{1}{6} \cdot \left(\left(i \cdot i\right) \cdot i\right)\right) + \left(\frac{1}{2} \cdot i\right) \cdot i}{\color{blue}{i \cdot \frac{1}{n}}}\]

   6. Applied *-un-lft-identity 29.3

      \[\leadsto 100 \cdot \frac{\color{blue}{1 \cdot \left(\left(i + \frac{1}{6} \cdot \left(\left(i \cdot i\right) \cdot i\right)\right) + \left(\frac{1}{2} \cdot i\right) \cdot i\right)}}{i \cdot \frac{1}{n}}\]

   7. Applied times-frac 15.0

      \[\leadsto 100 \cdot \color{blue}{\left(\frac{1}{i} \cdot \frac{\left(i + \frac{1}{6} \cdot \left(\left(i \cdot i\right) \cdot i\right)\right) + \left(\frac{1}{2} \cdot i\right) \cdot i}{\frac{1}{n}}\right)}\]

   8. Simplified 15.0

      \[\leadsto 100 \cdot \left(\frac{1}{i} \cdot \color{blue}{\left(\left(\left(i + \frac{1}{6} \cdot \left(\left(i \cdot i\right) \cdot i\right)\right) + \left(i \cdot \frac{1}{2}\right) \cdot i\right) \cdot n\right)}\right)\]
   9. Using strategy rm

   10. Applied associate-*l* 15.0

       \[\leadsto 100 \cdot \left(\frac{1}{i} \cdot \left(\left(\left(i + \frac{1}{6} \cdot \left(\left(i \cdot i\right) \cdot i\right)\right) + \color{blue}{i \cdot \left(\frac{1}{2} \cdot i\right)}\right) \cdot n\right)\right)\]

   11. Applied associate-*r* 15.0

       \[\leadsto 100 \cdot \left(\frac{1}{i} \cdot \left(\left(\left(i + \color{blue}{\left(\frac{1}{6} \cdot \left(i \cdot i\right)\right) \cdot i}\right) + i \cdot \left(\frac{1}{2} \cdot i\right)\right) \cdot n\right)\right)\]

   12. Applied *-un-lft-identity 15.0

       \[\leadsto 100 \cdot \left(\frac{1}{i} \cdot \left(\left(\left(\color{blue}{1 \cdot i} + \left(\frac{1}{6} \cdot \left(i \cdot i\right)\right) \cdot i\right) + i \cdot \left(\frac{1}{2} \cdot i\right)\right) \cdot n\right)\right)\]

   13. Applied distribute-rgt-out 15.0

       \[\leadsto 100 \cdot \left(\frac{1}{i} \cdot \left(\left(\color{blue}{i \cdot \left(1 + \frac{1}{6} \cdot \left(i \cdot i\right)\right)} + i \cdot \left(\frac{1}{2} \cdot i\right)\right) \cdot n\right)\right)\]

   14. Applied distribute-lft-out 15.0

       \[\leadsto 100 \cdot \left(\frac{1}{i} \cdot \left(\color{blue}{\left(i \cdot \left(\left(1 + \frac{1}{6} \cdot \left(i \cdot i\right)\right) + \frac{1}{2} \cdot i\right)\right)} \cdot n\right)\right)\]

   15. Applied associate-*l* 15.0

       \[\leadsto 100 \cdot \left(\frac{1}{i} \cdot \color{blue}{\left(i \cdot \left(\left(\left(1 + \frac{1}{6} \cdot \left(i \cdot i\right)\right) + \frac{1}{2} \cdot i\right) \cdot n\right)\right)}\right)\]
   16. Using strategy rm

   17. Applied *-commutative 15.0

       \[\leadsto 100 \cdot \left(\frac{1}{i} \cdot \left(i \cdot \color{blue}{\left(n \cdot \left(\left(1 + \frac{1}{6} \cdot \left(i \cdot i\right)\right) + \frac{1}{2} \cdot i\right)\right)}\right)\right)\]

   18. Applied associate-*r* 15.0

       \[\leadsto 100 \cdot \left(\frac{1}{i} \cdot \color{blue}{\left(\left(i \cdot n\right) \cdot \left(\left(1 + \frac{1}{6} \cdot \left(i \cdot i\right)\right) + \frac{1}{2} \cdot i\right)\right)}\right)\]

   19. Applied associate-*r* 15.0

       \[\leadsto 100 \cdot \color{blue}{\left(\left(\frac{1}{i} \cdot \left(i \cdot n\right)\right) \cdot \left(\left(1 + \frac{1}{6} \cdot \left(i \cdot i\right)\right) + \frac{1}{2} \cdot i\right)\right)}\]

   20. Applied associate-*r* 15.0

       \[\leadsto \color{blue}{\left(100 \cdot \left(\frac{1}{i} \cdot \left(i \cdot n\right)\right)\right) \cdot \left(\left(1 + \frac{1}{6} \cdot \left(i \cdot i\right)\right) + \frac{1}{2} \cdot i\right)}\]

   21. Simplified 14.9

       \[\leadsto \color{blue}{\left(100 \cdot \frac{i \cdot n}{i}\right)} \cdot \left(\left(1 + \frac{1}{6} \cdot \left(i \cdot i\right)\right) + \frac{1}{2} \cdot i\right)\]

   if -1.4365268904089577e-258 < i < 1.1743359908539163e-228

   1. Initial program 47.7

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

   2. Taylor expanded around 0 41.5

      \[\leadsto 100 \cdot \frac{\color{blue}{i + \left(\frac{1}{2} \cdot {i}^{2} + \frac{1}{6} \cdot {i}^{3}\right)}}{\frac{i}{n}}\]

   3. Simplified 41.5

      \[\leadsto 100 \cdot \frac{\color{blue}{\left(i + \frac{1}{6} \cdot \left(\left(i \cdot i\right) \cdot i\right)\right) + \left(\frac{1}{2} \cdot i\right) \cdot i}}{\frac{i}{n}}\]
   4. Using strategy rm

   5. Applied div-inv 41.5

      \[\leadsto 100 \cdot \frac{\left(i + \frac{1}{6} \cdot \left(\left(i \cdot i\right) \cdot i\right)\right) + \left(\frac{1}{2} \cdot i\right) \cdot i}{\color{blue}{i \cdot \frac{1}{n}}}\]

   6. Applied *-un-lft-identity 41.5

      \[\leadsto 100 \cdot \frac{\color{blue}{1 \cdot \left(\left(i + \frac{1}{6} \cdot \left(\left(i \cdot i\right) \cdot i\right)\right) + \left(\frac{1}{2} \cdot i\right) \cdot i\right)}}{i \cdot \frac{1}{n}}\]

   7. Applied times-frac 16.9

      \[\leadsto 100 \cdot \color{blue}{\left(\frac{1}{i} \cdot \frac{\left(i + \frac{1}{6} \cdot \left(\left(i \cdot i\right) \cdot i\right)\right) + \left(\frac{1}{2} \cdot i\right) \cdot i}{\frac{1}{n}}\right)}\]

   8. Simplified 16.9

      \[\leadsto 100 \cdot \left(\frac{1}{i} \cdot \color{blue}{\left(\left(\left(i + \frac{1}{6} \cdot \left(\left(i \cdot i\right) \cdot i\right)\right) + \left(i \cdot \frac{1}{2}\right) \cdot i\right) \cdot n\right)}\right)\]
   9. Using strategy rm

   10. Applied associate-*l* 16.9

       \[\leadsto 100 \cdot \left(\frac{1}{i} \cdot \left(\left(\left(i + \frac{1}{6} \cdot \left(\left(i \cdot i\right) \cdot i\right)\right) + \color{blue}{i \cdot \left(\frac{1}{2} \cdot i\right)}\right) \cdot n\right)\right)\]

   11. Applied associate-*r* 16.9

       \[\leadsto 100 \cdot \left(\frac{1}{i} \cdot \left(\left(\left(i + \color{blue}{\left(\frac{1}{6} \cdot \left(i \cdot i\right)\right) \cdot i}\right) + i \cdot \left(\frac{1}{2} \cdot i\right)\right) \cdot n\right)\right)\]

   12. Applied *-un-lft-identity 16.9

       \[\leadsto 100 \cdot \left(\frac{1}{i} \cdot \left(\left(\left(\color{blue}{1 \cdot i} + \left(\frac{1}{6} \cdot \left(i \cdot i\right)\right) \cdot i\right) + i \cdot \left(\frac{1}{2} \cdot i\right)\right) \cdot n\right)\right)\]

   13. Applied distribute-rgt-out 16.9

       \[\leadsto 100 \cdot \left(\frac{1}{i} \cdot \left(\left(\color{blue}{i \cdot \left(1 + \frac{1}{6} \cdot \left(i \cdot i\right)\right)} + i \cdot \left(\frac{1}{2} \cdot i\right)\right) \cdot n\right)\right)\]

   14. Applied distribute-lft-out 16.9

       \[\leadsto 100 \cdot \left(\frac{1}{i} \cdot \left(\color{blue}{\left(i \cdot \left(\left(1 + \frac{1}{6} \cdot \left(i \cdot i\right)\right) + \frac{1}{2} \cdot i\right)\right)} \cdot n\right)\right)\]

   15. Applied associate-*l* 16.9

       \[\leadsto 100 \cdot \left(\frac{1}{i} \cdot \color{blue}{\left(i \cdot \left(\left(\left(1 + \frac{1}{6} \cdot \left(i \cdot i\right)\right) + \frac{1}{2} \cdot i\right) \cdot n\right)\right)}\right)\]

   16. Taylor expanded around 0 15.5

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

   17. Simplified 15.5

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

   if 1.1743359908539163e-228 < i < 3.2268987471607203e-189

   1. Initial program 48.7

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

   2. Taylor expanded around 0 37.0

      \[\leadsto 100 \cdot \frac{\color{blue}{i + \left(\frac{1}{2} \cdot {i}^{2} + \frac{1}{6} \cdot {i}^{3}\right)}}{\frac{i}{n}}\]

   3. Simplified 37.0

      \[\leadsto 100 \cdot \frac{\color{blue}{\left(i + \frac{1}{6} \cdot \left(\left(i \cdot i\right) \cdot i\right)\right) + \left(\frac{1}{2} \cdot i\right) \cdot i}}{\frac{i}{n}}\]
   4. Using strategy rm

   5. Applied div-inv 37.0

      \[\leadsto 100 \cdot \frac{\left(i + \frac{1}{6} \cdot \left(\left(i \cdot i\right) \cdot i\right)\right) + \left(\frac{1}{2} \cdot i\right) \cdot i}{\color{blue}{i \cdot \frac{1}{n}}}\]

   6. Applied *-un-lft-identity 37.0

      \[\leadsto 100 \cdot \frac{\color{blue}{1 \cdot \left(\left(i + \frac{1}{6} \cdot \left(\left(i \cdot i\right) \cdot i\right)\right) + \left(\frac{1}{2} \cdot i\right) \cdot i\right)}}{i \cdot \frac{1}{n}}\]

   7. Applied times-frac 17.6

      \[\leadsto 100 \cdot \color{blue}{\left(\frac{1}{i} \cdot \frac{\left(i + \frac{1}{6} \cdot \left(\left(i \cdot i\right) \cdot i\right)\right) + \left(\frac{1}{2} \cdot i\right) \cdot i}{\frac{1}{n}}\right)}\]

   8. Simplified 17.6

      \[\leadsto 100 \cdot \left(\frac{1}{i} \cdot \color{blue}{\left(\left(\left(i + \frac{1}{6} \cdot \left(\left(i \cdot i\right) \cdot i\right)\right) + \left(i \cdot \frac{1}{2}\right) \cdot i\right) \cdot n\right)}\right)\]
   9. Using strategy rm

   10. Applied add-cbrt-cube 37.7

       \[\leadsto 100 \cdot \left(\frac{1}{i} \cdot \left(\left(\left(i + \frac{1}{6} \cdot \left(\left(i \cdot i\right) \cdot i\right)\right) + \left(i \cdot \frac{1}{2}\right) \cdot i\right) \cdot \color{blue}{\sqrt[3]{\left(n \cdot n\right) \cdot n}}\right)\right)\]

   11. Applied add-cbrt-cube 48.7

       \[\leadsto 100 \cdot \left(\frac{1}{i} \cdot \left(\color{blue}{\sqrt[3]{\left(\left(\left(i + \frac{1}{6} \cdot \left(\left(i \cdot i\right) \cdot i\right)\right) + \left(i \cdot \frac{1}{2}\right) \cdot i\right) \cdot \left(\left(i + \frac{1}{6} \cdot \left(\left(i \cdot i\right) \cdot i\right)\right) + \left(i \cdot \frac{1}{2}\right) \cdot i\right)\right) \cdot \left(\left(i + \frac{1}{6} \cdot \left(\left(i \cdot i\right) \cdot i\right)\right) + \left(i \cdot \frac{1}{2}\right) \cdot i\right)}} \cdot \sqrt[3]{\left(n \cdot n\right) \cdot n}\right)\right)\]

   12. Applied cbrt-unprod 48.7

       \[\leadsto 100 \cdot \left(\frac{1}{i} \cdot \color{blue}{\sqrt[3]{\left(\left(\left(\left(i + \frac{1}{6} \cdot \left(\left(i \cdot i\right) \cdot i\right)\right) + \left(i \cdot \frac{1}{2}\right) \cdot i\right) \cdot \left(\left(i + \frac{1}{6} \cdot \left(\left(i \cdot i\right) \cdot i\right)\right) + \left(i \cdot \frac{1}{2}\right) \cdot i\right)\right) \cdot \left(\left(i + \frac{1}{6} \cdot \left(\left(i \cdot i\right) \cdot i\right)\right) + \left(i \cdot \frac{1}{2}\right) \cdot i\right)\right) \cdot \left(\left(n \cdot n\right) \cdot n\right)}}\right)\]

   13. Simplified 27.2

       \[\leadsto 100 \cdot \left(\frac{1}{i} \cdot \sqrt[3]{\color{blue}{\left(\left(\left(\left(\frac{1}{6} \cdot i\right) \cdot \left(i \cdot i\right) + i\right) + i \cdot \left(\frac{1}{2} \cdot i\right)\right) \cdot n\right) \cdot \left(\left(\left(\left(\left(\frac{1}{6} \cdot i\right) \cdot \left(i \cdot i\right) + i\right) + i \cdot \left(\frac{1}{2} \cdot i\right)\right) \cdot n\right) \cdot \left(\left(\left(\left(\frac{1}{6} \cdot i\right) \cdot \left(i \cdot i\right) + i\right) + i \cdot \left(\frac{1}{2} \cdot i\right)\right) \cdot n\right)\right)}}\right)\]

   if 7.218954252890614e+203 < i

   1. Initial program 30.8

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

   2. Taylor expanded around 0 30.5

      \[\leadsto \color{blue}{0}\]
3. Recombined 5 regimes into one program.

4. Final simplification 20.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -4.890775921786639 \cdot 10^{-05}:\\ \;\;\;\;100 \cdot \frac{{\left({\left(\frac{i}{n} + 1\right)}^{n}\right)}^{3} - 1}{\frac{i}{n} \cdot \left({\left(\frac{i}{n} + 1\right)}^{n} \cdot {\left(\frac{i}{n} + 1\right)}^{n} + \left(1 + {\left(\frac{i}{n} + 1\right)}^{n}\right)\right)}\\ \mathbf{elif}\;i \le -1.4365268904089577 \cdot 10^{-258}:\\ \;\;\;\;\left(\frac{n \cdot i}{i} \cdot 100\right) \cdot \left(i \cdot \frac{1}{2} + \left(1 + \left(i \cdot i\right) \cdot \frac{1}{6}\right)\right)\\ \mathbf{elif}\;i \le 1.1743359908539163 \cdot 10^{-228}:\\ \;\;\;\;\left(\left(i \cdot i\right) \cdot n\right) \cdot \frac{50}{3} + \left(100 + i \cdot 50\right) \cdot n\\ \mathbf{elif}\;i \le 3.2268987471607203 \cdot 10^{-189}:\\ \;\;\;\;100 \cdot \left(\frac{1}{i} \cdot \sqrt[3]{\left(\left(\left(\left(i \cdot \frac{1}{2}\right) \cdot i + \left(\left(i \cdot i\right) \cdot \left(i \cdot \frac{1}{6}\right) + i\right)\right) \cdot n\right) \cdot \left(\left(\left(i \cdot \frac{1}{2}\right) \cdot i + \left(\left(i \cdot i\right) \cdot \left(i \cdot \frac{1}{6}\right) + i\right)\right) \cdot n\right)\right) \cdot \left(\left(\left(i \cdot \frac{1}{2}\right) \cdot i + \left(\left(i \cdot i\right) \cdot \left(i \cdot \frac{1}{6}\right) + i\right)\right) \cdot n\right)}\right)\\ \mathbf{elif}\;i \le 0.13449167275451146:\\ \;\;\;\;\left(\frac{n \cdot i}{i} \cdot 100\right) \cdot \left(i \cdot \frac{1}{2} + \left(1 + \left(i \cdot i\right) \cdot \frac{1}{6}\right)\right)\\ \mathbf{elif}\;i \le 7.218954252890614 \cdot 10^{+203}:\\ \;\;\;\;100 \cdot \frac{{\left({\left(\frac{i}{n} + 1\right)}^{n}\right)}^{3} - 1}{\frac{i}{n} \cdot \left({\left(\frac{i}{n} + 1\right)}^{n} \cdot {\left(\frac{i}{n} + 1\right)}^{n} + \left(1 + {\left(\frac{i}{n} + 1\right)}^{n}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Reproduce

herbie shell --seed 2019158 
(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))))