\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
Test:
Compound Interest
Bits:
128 bits
Bits error versus i
Bits error versus n
Time: 57.3 s
Input Error: 24.7
Output Error: 2.5
Log: ⚲
Profile: 🕒
\(\begin{cases} 100 \cdot \frac{{\left({\left(1 + \frac{i}{n}\right)}^{n}\right)}^{3} - {1}^{3}}{\frac{i}{n} \cdot \left({\left({\left(1 + \frac{i}{n}\right)}^{n}\right)}^2 + \left({1}^2 + {\left(1 + \frac{i}{n}\right)}^{n} \cdot 1\right)\right)} & \text{when } i \le -7.6554832f0 \\ \left({\left(e^{\frac{1}{24}}\right)}^{\left(i \cdot i\right)} \cdot 100\right) \cdot \left(n \cdot e^{\frac{1}{2} \cdot i}\right) & \text{when } i \le 0.23250379f0 \\ 100 \cdot \left(\left(\left(\frac{{n}^{4}}{i} \cdot \left({\left(\log i\right)}^3 \cdot \frac{1}{6}\right) + \left(\frac{1}{2} \cdot \frac{n \cdot n}{\frac{i}{n}}\right) \cdot \left(\log n \cdot \log n + \log i \cdot \log i\right)\right) + \left(\left(\frac{n}{\frac{i}{n}} \cdot \left(\frac{n}{i} + \log i\right) + \frac{\log i \cdot \left(\frac{1}{2} \cdot \log n\right)}{\frac{i}{{n}^{4} \cdot \log n}}\right) + \frac{{n}^{4}}{i} \cdot \left(\frac{\log i}{i} - \frac{\frac{1}{2}}{i \cdot i}\right)\right)\right) - \left(\left(\left(\frac{\frac{1}{6}}{i} \cdot {n}^{4}\right) \cdot {\left(\log n\right)}^3 + \frac{{n}^{4}}{i} \cdot \frac{\log n}{i}\right) + \left(\left(\frac{\log n \cdot {n}^3}{\frac{i}{\log i}} + \log n \cdot \frac{n}{\frac{i}{n}}\right) + \frac{\frac{1}{2} \cdot \left(\left(\log n \cdot \log i\right) \cdot \left({n}^{4} \cdot \log i\right)\right)}{i}\right)\right)\right) & \text{otherwise} \end{cases}\)

    if i < -7.6554832f0

    1. Started with
      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
      9.9
    2. Using strategy rm
      9.9
    3. Applied flip3-- to get
      \[100 \cdot \frac{\color{red}{{\left(1 + \frac{i}{n}\right)}^{n} - 1}}{\frac{i}{n}} \leadsto 100 \cdot \frac{\color{blue}{\frac{{\left({\left(1 + \frac{i}{n}\right)}^{n}\right)}^{3} - {1}^{3}}{{\left({\left(1 + \frac{i}{n}\right)}^{n}\right)}^2 + \left({1}^2 + {\left(1 + \frac{i}{n}\right)}^{n} \cdot 1\right)}}}{\frac{i}{n}}\]
      9.9
    4. Applied associate-/l/ to get
      \[100 \cdot \color{red}{\frac{\frac{{\left({\left(1 + \frac{i}{n}\right)}^{n}\right)}^{3} - {1}^{3}}{{\left({\left(1 + \frac{i}{n}\right)}^{n}\right)}^2 + \left({1}^2 + {\left(1 + \frac{i}{n}\right)}^{n} \cdot 1\right)}}{\frac{i}{n}}} \leadsto 100 \cdot \color{blue}{\frac{{\left({\left(1 + \frac{i}{n}\right)}^{n}\right)}^{3} - {1}^{3}}{\frac{i}{n} \cdot \left({\left({\left(1 + \frac{i}{n}\right)}^{n}\right)}^2 + \left({1}^2 + {\left(1 + \frac{i}{n}\right)}^{n} \cdot 1\right)\right)}}\]
      9.9

    if -7.6554832f0 < i < 0.23250379f0

    1. Started with
      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
      29.6
    2. Using strategy rm
      29.6
    3. Applied add-exp-log to get
      \[100 \cdot \color{red}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}} \leadsto 100 \cdot \color{blue}{e^{\log \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\right)}}\]
      29.8
    4. Applied taylor to get
      \[100 \cdot e^{\log \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\right)} \leadsto 100 \cdot e^{\frac{1}{24} \cdot {i}^2 + \left(\log n + \frac{1}{2} \cdot i\right)}\]
      17.4
    5. Taylor expanded around 0 to get
      \[100 \cdot e^{\color{red}{\frac{1}{24} \cdot {i}^2 + \left(\log n + \frac{1}{2} \cdot i\right)}} \leadsto 100 \cdot e^{\color{blue}{\frac{1}{24} \cdot {i}^2 + \left(\log n + \frac{1}{2} \cdot i\right)}}\]
      17.4
    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 0.23250379f0 < i

    1. Started with
      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
      23.8
    2. Using strategy rm
      23.8
    3. Applied add-cbrt-cube to get
      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\color{red}{\frac{i}{n}}} \leadsto 100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\color{blue}{\sqrt[3]{{\left(\frac{i}{n}\right)}^3}}}\]
      25.2
    4. Applied add-cbrt-cube to get
      \[100 \cdot \frac{\color{red}{{\left(1 + \frac{i}{n}\right)}^{n} - 1}}{\sqrt[3]{{\left(\frac{i}{n}\right)}^3}} \leadsto 100 \cdot \frac{\color{blue}{\sqrt[3]{{\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}^3}}}{\sqrt[3]{{\left(\frac{i}{n}\right)}^3}}\]
      25.2
    5. Applied cbrt-undiv to get
      \[100 \cdot \color{red}{\frac{\sqrt[3]{{\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}^3}}{\sqrt[3]{{\left(\frac{i}{n}\right)}^3}}} \leadsto 100 \cdot \color{blue}{\sqrt[3]{\frac{{\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}^3}{{\left(\frac{i}{n}\right)}^3}}}\]
      25.3
    6. Applied simplify to get
      \[100 \cdot \sqrt[3]{\color{red}{\frac{{\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}^3}{{\left(\frac{i}{n}\right)}^3}}} \leadsto 100 \cdot \sqrt[3]{\color{blue}{{\left(\frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{\frac{i}{n}}\right)}^3}}\]
      25.2
    7. Applied taylor to get
      \[100 \cdot \sqrt[3]{{\left(\frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{\frac{i}{n}}\right)}^3} \leadsto 100 \cdot \left(\left(\frac{1}{2} \cdot \frac{{n}^{3} \cdot {\left(\log i\right)}^2}{i} + \left(\frac{1}{2} \cdot \frac{{n}^{3} \cdot {\left(\log n\right)}^2}{i} + \left(\frac{1}{6} \cdot \frac{{n}^{4} \cdot {\left(\log i\right)}^{3}}{i} + \left(\frac{{n}^{4} \cdot \log i}{{i}^2} + \left(\frac{{n}^2 \cdot \log i}{i} + \left(\frac{{n}^{3}}{{i}^2} + \frac{1}{2} \cdot \frac{{\left(\log n\right)}^2 \cdot \left({n}^{4} \cdot \log i\right)}{i}\right)\right)\right)\right)\right)\right) - \left(\frac{1}{2} \cdot \frac{{n}^{4}}{{i}^{3}} + \left(\frac{\log n \cdot {n}^{4}}{{i}^2} + \left(\frac{1}{6} \cdot \frac{{\left(\log n\right)}^{3} \cdot {n}^{4}}{i} + \left(\frac{{n}^2 \cdot \log n}{i} + \left(\frac{1}{2} \cdot \frac{\log n \cdot \left({n}^{4} \cdot {\left(\log i\right)}^2\right)}{i} + \frac{{n}^{3} \cdot \left(\log n \cdot \log i\right)}{i}\right)\right)\right)\right)\right)\right)\]
      2.0
    8. Taylor expanded around 0 to get
      \[100 \cdot \color{red}{\left(\left(\frac{1}{2} \cdot \frac{{n}^{3} \cdot {\left(\log i\right)}^2}{i} + \left(\frac{1}{2} \cdot \frac{{n}^{3} \cdot {\left(\log n\right)}^2}{i} + \left(\frac{1}{6} \cdot \frac{{n}^{4} \cdot {\left(\log i\right)}^{3}}{i} + \left(\frac{{n}^{4} \cdot \log i}{{i}^2} + \left(\frac{{n}^2 \cdot \log i}{i} + \left(\frac{{n}^{3}}{{i}^2} + \frac{1}{2} \cdot \frac{{\left(\log n\right)}^2 \cdot \left({n}^{4} \cdot \log i\right)}{i}\right)\right)\right)\right)\right)\right) - \left(\frac{1}{2} \cdot \frac{{n}^{4}}{{i}^{3}} + \left(\frac{\log n \cdot {n}^{4}}{{i}^2} + \left(\frac{1}{6} \cdot \frac{{\left(\log n\right)}^{3} \cdot {n}^{4}}{i} + \left(\frac{{n}^2 \cdot \log n}{i} + \left(\frac{1}{2} \cdot \frac{\log n \cdot \left({n}^{4} \cdot {\left(\log i\right)}^2\right)}{i} + \frac{{n}^{3} \cdot \left(\log n \cdot \log i\right)}{i}\right)\right)\right)\right)\right)\right)} \leadsto 100 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \frac{{n}^{3} \cdot {\left(\log i\right)}^2}{i} + \left(\frac{1}{2} \cdot \frac{{n}^{3} \cdot {\left(\log n\right)}^2}{i} + \left(\frac{1}{6} \cdot \frac{{n}^{4} \cdot {\left(\log i\right)}^{3}}{i} + \left(\frac{{n}^{4} \cdot \log i}{{i}^2} + \left(\frac{{n}^2 \cdot \log i}{i} + \left(\frac{{n}^{3}}{{i}^2} + \frac{1}{2} \cdot \frac{{\left(\log n\right)}^2 \cdot \left({n}^{4} \cdot \log i\right)}{i}\right)\right)\right)\right)\right)\right) - \left(\frac{1}{2} \cdot \frac{{n}^{4}}{{i}^{3}} + \left(\frac{\log n \cdot {n}^{4}}{{i}^2} + \left(\frac{1}{6} \cdot \frac{{\left(\log n\right)}^{3} \cdot {n}^{4}}{i} + \left(\frac{{n}^2 \cdot \log n}{i} + \left(\frac{1}{2} \cdot \frac{\log n \cdot \left({n}^{4} \cdot {\left(\log i\right)}^2\right)}{i} + \frac{{n}^{3} \cdot \left(\log n \cdot \log i\right)}{i}\right)\right)\right)\right)\right)\right)}\]
      2.0
    9. Applied simplify to get
      \[100 \cdot \left(\left(\frac{1}{2} \cdot \frac{{n}^{3} \cdot {\left(\log i\right)}^2}{i} + \left(\frac{1}{2} \cdot \frac{{n}^{3} \cdot {\left(\log n\right)}^2}{i} + \left(\frac{1}{6} \cdot \frac{{n}^{4} \cdot {\left(\log i\right)}^{3}}{i} + \left(\frac{{n}^{4} \cdot \log i}{{i}^2} + \left(\frac{{n}^2 \cdot \log i}{i} + \left(\frac{{n}^{3}}{{i}^2} + \frac{1}{2} \cdot \frac{{\left(\log n\right)}^2 \cdot \left({n}^{4} \cdot \log i\right)}{i}\right)\right)\right)\right)\right)\right) - \left(\frac{1}{2} \cdot \frac{{n}^{4}}{{i}^{3}} + \left(\frac{\log n \cdot {n}^{4}}{{i}^2} + \left(\frac{1}{6} \cdot \frac{{\left(\log n\right)}^{3} \cdot {n}^{4}}{i} + \left(\frac{{n}^2 \cdot \log n}{i} + \left(\frac{1}{2} \cdot \frac{\log n \cdot \left({n}^{4} \cdot {\left(\log i\right)}^2\right)}{i} + \frac{{n}^{3} \cdot \left(\log n \cdot \log i\right)}{i}\right)\right)\right)\right)\right)\right) \leadsto \left(\left(\frac{1}{2} \cdot \left(\frac{{n}^3}{i} \cdot {\left(\log i\right)}^2 + \frac{{n}^3}{i} \cdot {\left(\log n\right)}^2\right) + \left(\frac{{n}^{4}}{i} \cdot {\left(\log i\right)}^3\right) \cdot \frac{1}{6}\right) + \left(\left(\left(\left(\frac{\left(\frac{1}{2} \cdot \log n\right) \cdot \left(\log n \cdot {n}^{4}\right)}{\frac{i}{\log i}} + \frac{{n}^3}{i \cdot i}\right) + \left(\frac{n \cdot n}{i} \cdot \log i + \frac{{n}^{4}}{\frac{i \cdot i}{\log i}}\right)\right) - \frac{{n}^{4}}{i} \cdot \frac{\frac{1}{2}}{i \cdot i}\right) - \left(\left(\left(\log n \cdot \frac{n \cdot n}{i} + \frac{\log n \cdot {n}^3}{\frac{i}{\log i}}\right) + \frac{\left(\frac{1}{2} \cdot \log n\right) \cdot \left(\log i \cdot \left({n}^{4} \cdot \log i\right)\right)}{i}\right) + \left(\frac{\frac{1}{6} \cdot {\left(\log n\right)}^3}{\frac{i}{{n}^{4}}} + \frac{{n}^{4}}{i} \cdot \frac{\log n}{i}\right)\right)\right)\right) \cdot 100\]
      2.7
    10. Applied final simplification
    11. Applied simplify to get
      \[\color{red}{\left(\left(\frac{1}{2} \cdot \left(\frac{{n}^3}{i} \cdot {\left(\log i\right)}^2 + \frac{{n}^3}{i} \cdot {\left(\log n\right)}^2\right) + \left(\frac{{n}^{4}}{i} \cdot {\left(\log i\right)}^3\right) \cdot \frac{1}{6}\right) + \left(\left(\left(\left(\frac{\left(\frac{1}{2} \cdot \log n\right) \cdot \left(\log n \cdot {n}^{4}\right)}{\frac{i}{\log i}} + \frac{{n}^3}{i \cdot i}\right) + \left(\frac{n \cdot n}{i} \cdot \log i + \frac{{n}^{4}}{\frac{i \cdot i}{\log i}}\right)\right) - \frac{{n}^{4}}{i} \cdot \frac{\frac{1}{2}}{i \cdot i}\right) - \left(\left(\left(\log n \cdot \frac{n \cdot n}{i} + \frac{\log n \cdot {n}^3}{\frac{i}{\log i}}\right) + \frac{\left(\frac{1}{2} \cdot \log n\right) \cdot \left(\log i \cdot \left({n}^{4} \cdot \log i\right)\right)}{i}\right) + \left(\frac{\frac{1}{6} \cdot {\left(\log n\right)}^3}{\frac{i}{{n}^{4}}} + \frac{{n}^{4}}{i} \cdot \frac{\log n}{i}\right)\right)\right)\right) \cdot 100} \leadsto \color{blue}{100 \cdot \left(\left(\left(\frac{{n}^{4}}{i} \cdot \left({\left(\log i\right)}^3 \cdot \frac{1}{6}\right) + \left(\frac{1}{2} \cdot \frac{n \cdot n}{\frac{i}{n}}\right) \cdot \left(\log n \cdot \log n + \log i \cdot \log i\right)\right) + \left(\left(\frac{n}{\frac{i}{n}} \cdot \left(\frac{n}{i} + \log i\right) + \frac{\log i \cdot \left(\frac{1}{2} \cdot \log n\right)}{\frac{i}{{n}^{4} \cdot \log n}}\right) + \frac{{n}^{4}}{i} \cdot \left(\frac{\log i}{i} - \frac{\frac{1}{2}}{i \cdot i}\right)\right)\right) - \left(\left(\left(\frac{\frac{1}{6}}{i} \cdot {n}^{4}\right) \cdot {\left(\log n\right)}^3 + \frac{{n}^{4}}{i} \cdot \frac{\log n}{i}\right) + \left(\left(\frac{\log n \cdot {n}^3}{\frac{i}{\log i}} + \log n \cdot \frac{n}{\frac{i}{n}}\right) + \frac{\frac{1}{2} \cdot \left(\left(\log n \cdot \log i\right) \cdot \left({n}^{4} \cdot \log i\right)\right)}{i}\right)\right)\right)}\]
      2.8
  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))))