\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
Test:
Compound Interest
Bits:
128 bits
Bits error versus i
Bits error versus n
Time: 1.1 m
Input Error: 52.0
Output Error: 6.3
Log: ⚲
Profile: 🕒
\(\begin{cases} \frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}} & \text{when } i \le -3.777511286313303 \cdot 10^{+52} \\ \left({\left(e^{\frac{1}{24}}\right)}^{\left(i \cdot i\right)} \cdot 100\right) \cdot \left(n \cdot e^{\frac{1}{2} \cdot i}\right) & \text{when } i \le 6.840282922734168 \\ \left(\left(\frac{100 \cdot \left(n \cdot n\right)}{\frac{i}{\log i}} + \frac{{n}^3}{i} \cdot \frac{100}{i}\right) + \left(\left(\frac{50}{3} \cdot \left({\left(\log i\right)}^3 \cdot \frac{{n}^{4}}{i}\right) + \frac{{\left(\log n\right)}^2}{\frac{i}{{n}^3}} \cdot 50\right) + \frac{{n}^{4}}{\frac{i}{100}} \cdot \frac{\log i}{i}\right)\right) + \left(\left(\left(\left(\frac{50}{i} \cdot \left(\log i \cdot \log i\right)\right) \cdot {n}^3 + \frac{\left(50 \cdot {n}^{4}\right) \cdot {\left(\log n\right)}^2}{\frac{i}{\log i}}\right) - \left(\left({\left(\log n\right)}^3 \cdot \frac{50}{3}\right) \cdot \frac{{n}^{4}}{i} + \left(\frac{\frac{50}{i}}{i} \cdot \frac{{n}^{4}}{i} + \frac{{n}^{4}}{\frac{i}{100}} \cdot \frac{\log n}{i}\right)\right)\right) - \left(\frac{\left(50 \cdot {n}^{4}\right) \cdot \log n}{\frac{i}{\log i \cdot \log i}} + \left(\frac{\log n}{i} \cdot \left({n}^3 \cdot \log i\right) + \frac{\log n \cdot \left(n \cdot n\right)}{i}\right) \cdot 100\right)\right) & \text{when } i \le 4.127372531442322 \cdot 10^{+187} \\ \frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}} & \text{otherwise} \end{cases}\)

    if i < -3.777511286313303e+52 or 4.127372531442322e+187 < i

    1. Started with
      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
      25.4
    2. Using strategy rm
      25.4
    3. Applied associate-*r/ to get
      \[\color{red}{100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}} \leadsto \color{blue}{\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}}\]
      25.4

    if -3.777511286313303e+52 < i < 6.840282922734168

    1. Started with
      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
      61.7
    2. Using strategy rm
      61.7
    3. Applied add-exp-log to get
      \[100 \cdot \color{red}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}} \leadsto 100 \cdot \color{blue}{e^{\log \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\right)}}\]
      61.8
    4. Applied taylor to get
      \[100 \cdot e^{\log \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\right)} \leadsto 100 \cdot e^{\frac{1}{24} \cdot {i}^2 + \left(\log n + \frac{1}{2} \cdot i\right)}\]
      34.4
    5. Taylor expanded around 0 to get
      \[100 \cdot e^{\color{red}{\frac{1}{24} \cdot {i}^2 + \left(\log n + \frac{1}{2} \cdot i\right)}} \leadsto 100 \cdot e^{\color{blue}{\frac{1}{24} \cdot {i}^2 + \left(\log n + \frac{1}{2} \cdot i\right)}}\]
      34.4
    6. Applied simplify to get
      \[100 \cdot e^{\frac{1}{24} \cdot {i}^2 + \left(\log n + \frac{1}{2} \cdot i\right)} \leadsto \left({\left(e^{\frac{1}{24}}\right)}^{\left(i \cdot i\right)} \cdot 100\right) \cdot \left(n \cdot e^{\frac{1}{2} \cdot i}\right)\]
      0.1
    7. Applied final simplification

    if 6.840282922734168 < i < 4.127372531442322e+187

    1. Started with
      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
      45.6
    2. Using strategy rm
      45.6
    3. Applied add-cbrt-cube to get
      \[\color{red}{100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}} \leadsto \color{blue}{\sqrt[3]{{\left(100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\right)}^3}}\]
      46.2
    4. Applied taylor to get
      \[\sqrt[3]{{\left(100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\right)}^3} \leadsto \left(100 \cdot \frac{{n}^{4} \cdot \log i}{{i}^2} + \left(\frac{50}{3} \cdot \frac{{n}^{4} \cdot {\left(\log i\right)}^{3}}{i} + \left(50 \cdot \frac{{\left(\log n\right)}^2 \cdot {n}^{3}}{i} + \left(100 \cdot \frac{{n}^2 \cdot \log i}{i} + \left(100 \cdot \frac{{n}^{3}}{{i}^2} + \left(50 \cdot \frac{{n}^{3} \cdot {\left(\log i\right)}^2}{i} + 50 \cdot \frac{{n}^{4} \cdot \left({\left(\log n\right)}^2 \cdot \log i\right)}{i}\right)\right)\right)\right)\right)\right) - \left(50 \cdot \frac{{n}^{4}}{{i}^{3}} + \left(100 \cdot \frac{{n}^{4} \cdot \log n}{{i}^2} + \left(\frac{50}{3} \cdot \frac{{n}^{4} \cdot {\left(\log n\right)}^{3}}{i} + \left(100 \cdot \frac{{n}^2 \cdot \log n}{i} + \left(100 \cdot \frac{\log n \cdot \left({n}^{3} \cdot \log i\right)}{i} + 50 \cdot \frac{{n}^{4} \cdot \left(\log n \cdot {\left(\log i\right)}^2\right)}{i}\right)\right)\right)\right)\right)\]
      3.8
    5. Taylor expanded around 0 to get
      \[\color{red}{\left(100 \cdot \frac{{n}^{4} \cdot \log i}{{i}^2} + \left(\frac{50}{3} \cdot \frac{{n}^{4} \cdot {\left(\log i\right)}^{3}}{i} + \left(50 \cdot \frac{{\left(\log n\right)}^2 \cdot {n}^{3}}{i} + \left(100 \cdot \frac{{n}^2 \cdot \log i}{i} + \left(100 \cdot \frac{{n}^{3}}{{i}^2} + \left(50 \cdot \frac{{n}^{3} \cdot {\left(\log i\right)}^2}{i} + 50 \cdot \frac{{n}^{4} \cdot \left({\left(\log n\right)}^2 \cdot \log i\right)}{i}\right)\right)\right)\right)\right)\right) - \left(50 \cdot \frac{{n}^{4}}{{i}^{3}} + \left(100 \cdot \frac{{n}^{4} \cdot \log n}{{i}^2} + \left(\frac{50}{3} \cdot \frac{{n}^{4} \cdot {\left(\log n\right)}^{3}}{i} + \left(100 \cdot \frac{{n}^2 \cdot \log n}{i} + \left(100 \cdot \frac{\log n \cdot \left({n}^{3} \cdot \log i\right)}{i} + 50 \cdot \frac{{n}^{4} \cdot \left(\log n \cdot {\left(\log i\right)}^2\right)}{i}\right)\right)\right)\right)\right)} \leadsto \color{blue}{\left(100 \cdot \frac{{n}^{4} \cdot \log i}{{i}^2} + \left(\frac{50}{3} \cdot \frac{{n}^{4} \cdot {\left(\log i\right)}^{3}}{i} + \left(50 \cdot \frac{{\left(\log n\right)}^2 \cdot {n}^{3}}{i} + \left(100 \cdot \frac{{n}^2 \cdot \log i}{i} + \left(100 \cdot \frac{{n}^{3}}{{i}^2} + \left(50 \cdot \frac{{n}^{3} \cdot {\left(\log i\right)}^2}{i} + 50 \cdot \frac{{n}^{4} \cdot \left({\left(\log n\right)}^2 \cdot \log i\right)}{i}\right)\right)\right)\right)\right)\right) - \left(50 \cdot \frac{{n}^{4}}{{i}^{3}} + \left(100 \cdot \frac{{n}^{4} \cdot \log n}{{i}^2} + \left(\frac{50}{3} \cdot \frac{{n}^{4} \cdot {\left(\log n\right)}^{3}}{i} + \left(100 \cdot \frac{{n}^2 \cdot \log n}{i} + \left(100 \cdot \frac{\log n \cdot \left({n}^{3} \cdot \log i\right)}{i} + 50 \cdot \frac{{n}^{4} \cdot \left(\log n \cdot {\left(\log i\right)}^2\right)}{i}\right)\right)\right)\right)\right)}\]
      3.8
    6. Applied simplify to get
      \[\left(100 \cdot \frac{{n}^{4} \cdot \log i}{{i}^2} + \left(\frac{50}{3} \cdot \frac{{n}^{4} \cdot {\left(\log i\right)}^{3}}{i} + \left(50 \cdot \frac{{\left(\log n\right)}^2 \cdot {n}^{3}}{i} + \left(100 \cdot \frac{{n}^2 \cdot \log i}{i} + \left(100 \cdot \frac{{n}^{3}}{{i}^2} + \left(50 \cdot \frac{{n}^{3} \cdot {\left(\log i\right)}^2}{i} + 50 \cdot \frac{{n}^{4} \cdot \left({\left(\log n\right)}^2 \cdot \log i\right)}{i}\right)\right)\right)\right)\right)\right) - \left(50 \cdot \frac{{n}^{4}}{{i}^{3}} + \left(100 \cdot \frac{{n}^{4} \cdot \log n}{{i}^2} + \left(\frac{50}{3} \cdot \frac{{n}^{4} \cdot {\left(\log n\right)}^{3}}{i} + \left(100 \cdot \frac{{n}^2 \cdot \log n}{i} + \left(100 \cdot \frac{\log n \cdot \left({n}^{3} \cdot \log i\right)}{i} + 50 \cdot \frac{{n}^{4} \cdot \left(\log n \cdot {\left(\log i\right)}^2\right)}{i}\right)\right)\right)\right)\right) \leadsto \left(\left(\left(\frac{50}{3} \cdot \left({\left(\log i\right)}^3 \cdot \frac{{n}^{4}}{i}\right) + \frac{100 \cdot {n}^{4}}{\frac{i \cdot i}{\log i}}\right) + \frac{\log n \cdot \log n}{\frac{i}{{n}^3}} \cdot 50\right) + \left(\left(\frac{{n}^3 \cdot 100}{i \cdot i} + \frac{100 \cdot \left(n \cdot n\right)}{\frac{i}{\log i}}\right) + \left(\frac{50 \cdot {n}^3}{\frac{i}{\log i \cdot \log i}} + \frac{\left({n}^{4} \cdot 50\right) \cdot \left(\log n \cdot \log n\right)}{\frac{i}{\log i}}\right)\right)\right) - \left(\left(\left(\frac{\log n \cdot \left(n \cdot n\right)}{i} + \frac{\log n}{i} \cdot \left({n}^3 \cdot \log i\right)\right) \cdot 100 + \frac{\left(\log n \cdot {n}^{4}\right) \cdot \left(\log i \cdot \log i\right)}{\frac{i}{50}}\right) + \left(\left(\frac{100 \cdot \log n}{\frac{i \cdot i}{{n}^{4}}} + 50 \cdot \frac{{n}^{4}}{{i}^3}\right) + \frac{50}{3} \cdot \left(\frac{{n}^{4}}{i} \cdot {\left(\log n\right)}^3\right)\right)\right)\]
      3.8
    7. Applied final simplification
    8. Applied simplify to get
      \[\color{red}{\left(\left(\left(\frac{50}{3} \cdot \left({\left(\log i\right)}^3 \cdot \frac{{n}^{4}}{i}\right) + \frac{100 \cdot {n}^{4}}{\frac{i \cdot i}{\log i}}\right) + \frac{\log n \cdot \log n}{\frac{i}{{n}^3}} \cdot 50\right) + \left(\left(\frac{{n}^3 \cdot 100}{i \cdot i} + \frac{100 \cdot \left(n \cdot n\right)}{\frac{i}{\log i}}\right) + \left(\frac{50 \cdot {n}^3}{\frac{i}{\log i \cdot \log i}} + \frac{\left({n}^{4} \cdot 50\right) \cdot \left(\log n \cdot \log n\right)}{\frac{i}{\log i}}\right)\right)\right) - \left(\left(\left(\frac{\log n \cdot \left(n \cdot n\right)}{i} + \frac{\log n}{i} \cdot \left({n}^3 \cdot \log i\right)\right) \cdot 100 + \frac{\left(\log n \cdot {n}^{4}\right) \cdot \left(\log i \cdot \log i\right)}{\frac{i}{50}}\right) + \left(\left(\frac{100 \cdot \log n}{\frac{i \cdot i}{{n}^{4}}} + 50 \cdot \frac{{n}^{4}}{{i}^3}\right) + \frac{50}{3} \cdot \left(\frac{{n}^{4}}{i} \cdot {\left(\log n\right)}^3\right)\right)\right)} \leadsto \color{blue}{\left(\left(\frac{100 \cdot \left(n \cdot n\right)}{\frac{i}{\log i}} + \frac{{n}^3}{i} \cdot \frac{100}{i}\right) + \left(\left(\frac{50}{3} \cdot \left({\left(\log i\right)}^3 \cdot \frac{{n}^{4}}{i}\right) + \frac{{\left(\log n\right)}^2}{\frac{i}{{n}^3}} \cdot 50\right) + \frac{{n}^{4}}{\frac{i}{100}} \cdot \frac{\log i}{i}\right)\right) + \left(\left(\left(\left(\frac{50}{i} \cdot \left(\log i \cdot \log i\right)\right) \cdot {n}^3 + \frac{\left(50 \cdot {n}^{4}\right) \cdot {\left(\log n\right)}^2}{\frac{i}{\log i}}\right) - \left(\left({\left(\log n\right)}^3 \cdot \frac{50}{3}\right) \cdot \frac{{n}^{4}}{i} + \left(\frac{\frac{50}{i}}{i} \cdot \frac{{n}^{4}}{i} + \frac{{n}^{4}}{\frac{i}{100}} \cdot \frac{\log n}{i}\right)\right)\right) - \left(\frac{\left(50 \cdot {n}^{4}\right) \cdot \log n}{\frac{i}{\log i \cdot \log i}} + \left(\frac{\log n}{i} \cdot \left({n}^3 \cdot \log i\right) + \frac{\log n \cdot \left(n \cdot n\right)}{i}\right) \cdot 100\right)\right)}\]
      3.7
  1. Removed slow pow expressions

Original test:


(lambda ((i default) (n default))
  #:name "Compound Interest"
  (* 100 (/ (- (pow (+ 1 (/ i n)) n) 1) (/ i n)))
  #:target
  (* 100 (/ (- (exp (* n (if (= (+ 1 (/ i n)) 1) (/ i n) (/ (* (/ i n) (log (+ 1 (/ i n)))) (- (+ (/ i n) 1) 1))))) 1) (/ i n))))