Compound Interest

Average Error: 40.9 → 32.8

Time: 35.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;i \le -0.67216965178613575:\\ \;\;\;\;100 \cdot \left(\frac{\frac{\frac{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot \left(2 \cdot n\right)\right)} + \left(-\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right)} \cdot \sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot \left(2 \cdot n\right)\right)} + \left(-\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right)}}{\sqrt{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + 1 \cdot 1}}}{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} + 1} \cdot \sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}}{i} \cdot \frac{\frac{\frac{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot \left(2 \cdot n\right)\right)} + \left(-\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right)}}{\sqrt{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + 1 \cdot 1}}}{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}}{\frac{1}{n}}\right)\\ \mathbf{elif}\;i \le 5.4848294551835708 \cdot 10^{145}:\\ \;\;\;\;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}\;i \le 3.45170537215142098 \cdot 10^{222}:\\ \;\;\;\;100 \cdot \left(\frac{\frac{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + \left(-1 \cdot 1\right)} \cdot \sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + \left(-1 \cdot 1\right)}}{\sqrt{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}}{i} \cdot \left(\frac{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + \left(-1 \cdot 1\right)}}{\sqrt{{\left(1 + \frac{i}{n}\right)}^{n} + 1}} \cdot n\right)\right)\\ \mathbf{elif}\;i \le 2.74199199550633987 \cdot 10^{289}:\\ \;\;\;\;100 \cdot \frac{\frac{\frac{\left(4 \cdot i + \left(4 \cdot \left(\log 1 \cdot n\right) + 1\right)\right) + \left(-\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\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}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(\frac{\frac{\frac{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot \left(2 \cdot n\right)\right)} + \left(-\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right)} \cdot \sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot \left(2 \cdot n\right)\right)} + \left(-\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right)}}{\sqrt{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + 1 \cdot 1}}}{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} + 1} \cdot \sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}}{i} \cdot \frac{\frac{\frac{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot \left(2 \cdot n\right)\right)} + \left(-\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right)}}{\sqrt{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + 1 \cdot 1}}}{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}}{\frac{1}{n}}\right)\\ \end{array}\]

C

double f(double i, double n) {
        double r281649 = 100.0;
        double r281650 = 1.0;
        double r281651 = i;
        double r281652 = n;
        double r281653 = r281651 / r281652;
        double r281654 = r281650 + r281653;
        double r281655 = pow(r281654, r281652);
        double r281656 = r281655 - r281650;
        double r281657 = r281656 / r281653;
        double r281658 = r281649 * r281657;
        return r281658;
}

↓

double f(double i, double n) {
        double r281659 = i;
        double r281660 = -0.6721696517861357;
        bool r281661 = r281659 <= r281660;
        double r281662 = 100.0;
        double r281663 = 1.0;
        double r281664 = n;
        double r281665 = r281659 / r281664;
        double r281666 = r281663 + r281665;
        double r281667 = 2.0;
        double r281668 = r281667 * r281664;
        double r281669 = r281667 * r281668;
        double r281670 = pow(r281666, r281669);
        double r281671 = r281663 * r281663;
        double r281672 = r281671 * r281671;
        double r281673 = -r281672;
        double r281674 = r281670 + r281673;
        double r281675 = cbrt(r281674);
        double r281676 = r281675 * r281675;
        double r281677 = pow(r281666, r281668);
        double r281678 = r281677 + r281671;
        double r281679 = sqrt(r281678);
        double r281680 = r281676 / r281679;
        double r281681 = pow(r281666, r281664);
        double r281682 = r281681 + r281663;
        double r281683 = cbrt(r281682);
        double r281684 = r281683 * r281683;
        double r281685 = r281680 / r281684;
        double r281686 = r281685 / r281659;
        double r281687 = r281675 / r281679;
        double r281688 = r281687 / r281683;
        double r281689 = 1.0;
        double r281690 = r281689 / r281664;
        double r281691 = r281688 / r281690;
        double r281692 = r281686 * r281691;
        double r281693 = r281662 * r281692;
        double r281694 = 5.484829455183571e+145;
        bool r281695 = r281659 <= r281694;
        double r281696 = r281663 * r281659;
        double r281697 = 0.5;
        double r281698 = pow(r281659, r281667);
        double r281699 = r281697 * r281698;
        double r281700 = log(r281663);
        double r281701 = r281700 * r281664;
        double r281702 = r281699 + r281701;
        double r281703 = r281696 + r281702;
        double r281704 = r281698 * r281700;
        double r281705 = r281697 * r281704;
        double r281706 = r281703 - r281705;
        double r281707 = r281706 / r281665;
        double r281708 = r281662 * r281707;
        double r281709 = 3.451705372151421e+222;
        bool r281710 = r281659 <= r281709;
        double r281711 = -r281671;
        double r281712 = r281677 + r281711;
        double r281713 = cbrt(r281712);
        double r281714 = r281713 * r281713;
        double r281715 = sqrt(r281682);
        double r281716 = r281714 / r281715;
        double r281717 = r281716 / r281659;
        double r281718 = r281713 / r281715;
        double r281719 = r281718 * r281664;
        double r281720 = r281717 * r281719;
        double r281721 = r281662 * r281720;
        double r281722 = 2.74199199550634e+289;
        bool r281723 = r281659 <= r281722;
        double r281724 = 4.0;
        double r281725 = r281724 * r281659;
        double r281726 = 4.0;
        double r281727 = r281726 * r281701;
        double r281728 = r281727 + r281689;
        double r281729 = r281725 + r281728;
        double r281730 = r281729 + r281673;
        double r281731 = r281730 / r281678;
        double r281732 = r281731 / r281682;
        double r281733 = r281732 / r281665;
        double r281734 = r281662 * r281733;
        double r281735 = r281723 ? r281734 : r281693;
        double r281736 = r281710 ? r281721 : r281735;
        double r281737 = r281695 ? r281708 : r281736;
        double r281738 = r281661 ? r281693 : r281737;
        return r281738;
}

Error

Bits error versus i

Bits error versus n

Target

Original	40.9
Target	41.1
Herbie	32.8

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

Derivation

1. Split input into 4 regimes

2. if i < -0.6721696517861357 or 2.74199199550634e+289 < i

   1. Initial program 27.7

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

   3. Applied flip-- 27.7

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

   4. Simplified 27.7

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

   6. Applied flip-+ 27.7

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

   7. Simplified 27.6

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

   8. Simplified 27.6

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

   10. Applied div-inv 27.7

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

   11. Applied add-cube-cbrt 27.7

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

   12. Applied add-sqr-sqrt 27.7

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

   13. Applied add-cube-cbrt 27.7

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

   14. Applied times-frac 27.7

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

   15. Applied times-frac 27.7

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

   16. Applied times-frac 28.1

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

   if -0.6721696517861357 < i < 5.484829455183571e+145

   1. Initial program 47.1

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

   2. Taylor expanded around 0 34.6

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

   if 5.484829455183571e+145 < i < 3.451705372151421e+222

   1. Initial program 32.8

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

   3. Applied flip-- 32.8

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

   4. Simplified 32.7

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

   6. Applied div-inv 32.8

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

   7. Applied add-sqr-sqrt 32.8

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

   8. Applied add-cube-cbrt 32.8

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

   9. Applied times-frac 32.8

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

   10. Applied times-frac 32.8

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

   11. Simplified 32.7

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

   if 3.451705372151421e+222 < i < 2.74199199550634e+289

   1. Initial program 31.9

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

   3. Applied flip-- 31.9

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

   4. Simplified 31.9

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

   6. Applied flip-+ 31.9

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

   7. Simplified 31.8

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

   8. Simplified 31.8

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

   9. Taylor expanded around 0 33.8

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

4. Final simplification 32.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -0.67216965178613575:\\ \;\;\;\;100 \cdot \left(\frac{\frac{\frac{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot \left(2 \cdot n\right)\right)} + \left(-\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right)} \cdot \sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot \left(2 \cdot n\right)\right)} + \left(-\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right)}}{\sqrt{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + 1 \cdot 1}}}{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} + 1} \cdot \sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}}{i} \cdot \frac{\frac{\frac{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot \left(2 \cdot n\right)\right)} + \left(-\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right)}}{\sqrt{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + 1 \cdot 1}}}{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}}{\frac{1}{n}}\right)\\ \mathbf{elif}\;i \le 5.4848294551835708 \cdot 10^{145}:\\ \;\;\;\;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}\;i \le 3.45170537215142098 \cdot 10^{222}:\\ \;\;\;\;100 \cdot \left(\frac{\frac{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + \left(-1 \cdot 1\right)} \cdot \sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + \left(-1 \cdot 1\right)}}{\sqrt{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}}{i} \cdot \left(\frac{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + \left(-1 \cdot 1\right)}}{\sqrt{{\left(1 + \frac{i}{n}\right)}^{n} + 1}} \cdot n\right)\right)\\ \mathbf{elif}\;i \le 2.74199199550633987 \cdot 10^{289}:\\ \;\;\;\;100 \cdot \frac{\frac{\frac{\left(4 \cdot i + \left(4 \cdot \left(\log 1 \cdot n\right) + 1\right)\right) + \left(-\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\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}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(\frac{\frac{\frac{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot \left(2 \cdot n\right)\right)} + \left(-\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right)} \cdot \sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot \left(2 \cdot n\right)\right)} + \left(-\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right)}}{\sqrt{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + 1 \cdot 1}}}{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} + 1} \cdot \sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}}{i} \cdot \frac{\frac{\frac{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot \left(2 \cdot n\right)\right)} + \left(-\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right)}}{\sqrt{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + 1 \cdot 1}}}{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}}{\frac{1}{n}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020024 
(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))))