\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
Test:
Compound Interest
Bits:
128 bits
Bits error versus i
Bits error versus n
Time: 54.9 s
Input Error: 24.7
Output Error: 2.3
Log: ⚲
Profile: 🕒
\(\begin{cases} 100 \cdot \frac{{\left({\left(1 + \frac{i}{n}\right)}^{n}\right)}^{3} - {1}^{3}}{\frac{i}{n} \cdot \left({\left({\left(1 + \frac{i}{n}\right)}^{n}\right)}^2 + \left({1}^2 + {\left(1 + \frac{i}{n}\right)}^{n} \cdot 1\right)\right)} & \text{when } i \le -25.745825f0 \\ \left({\left(e^{\frac{1}{24}}\right)}^{\left(i \cdot i\right)} \cdot 100\right) \cdot \left(n \cdot e^{\frac{1}{2} \cdot i}\right) & \text{when } i \le 0.00046563518f0 \\ \frac{\left(\left(\left({n}^3 \cdot \frac{1}{6}\right) \cdot {\left(\log i\right)}^3 + \left(\frac{\log i}{\frac{\frac{i}{n}}{n \cdot n}} + \frac{n \cdot n}{i}\right)\right) + \left(\left(\left(\log n \cdot \log n\right) \cdot \left({n}^3 \cdot \frac{1}{2}\right) + n\right) \cdot \log i - \left(\log n \cdot \left(\log i \cdot \left(n \cdot n\right) + n\right) + \left(\left({n}^3 \cdot \frac{1}{6}\right) \cdot {\left(\log n\right)}^3 + \frac{{n}^3}{i} \cdot \log n\right)\right)\right)\right) - \left(\left(\frac{{n}^3}{{i}^2} + \left(\log i \cdot \log i\right) \cdot \left(\log n \cdot {n}^3\right)\right) \cdot \frac{1}{2} - \left(\log i \cdot \log i + \log n \cdot \log n\right) \cdot \left(\frac{1}{2} \cdot \left(n \cdot n\right)\right)\right)}{\frac{\frac{i}{100}}{n}} & \text{otherwise} \end{cases}\)

    if i < -25.745825f0

    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 -25.745825f0 < i < 0.00046563518f0

    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
      \[100 \cdot e^{\log \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\right)} \leadsto 100 \cdot e^{\frac{1}{24} \cdot {i}^2 + \left(\log n + \frac{1}{2} \cdot i\right)}\]
      17.3
    5. Taylor expanded around 0 to get
      \[100 \cdot e^{\color{red}{\frac{1}{24} \cdot {i}^2 + \left(\log n + \frac{1}{2} \cdot i\right)}} \leadsto 100 \cdot e^{\color{blue}{\frac{1}{24} \cdot {i}^2 + \left(\log n + \frac{1}{2} \cdot i\right)}}\]
      17.3
    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 0.00046563518f0 < i

    1. Started with
      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
      24.0
    2. Using strategy rm
      24.0
    3. Applied add-cube-cbrt to get
      \[100 \cdot \frac{\color{red}{{\left(1 + \frac{i}{n}\right)}^{n}} - 1}{\frac{i}{n}} \leadsto 100 \cdot \frac{\color{blue}{{\left(\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n}}\right)}^3} - 1}{\frac{i}{n}}\]
      24.2
    4. Applied taylor to get
      \[100 \cdot \frac{{\left(\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n}}\right)}^3 - 1}{\frac{i}{n}} \leadsto 100 \cdot \frac{\left(\frac{1}{2} \cdot \left({n}^2 \cdot {\left(\log n\right)}^2\right) + \left(\frac{1}{2} \cdot \left({n}^2 \cdot {\left(\log i\right)}^2\right) + \left(n \cdot \log i + \left(\frac{1}{2} \cdot \left({n}^{3} \cdot \left({\left(\log n\right)}^2 \cdot \log i\right)\right) + \left(\frac{1}{6} \cdot \left({n}^{3} \cdot {\left(\log i\right)}^{3}\right) + \left(\frac{{n}^2}{i} + \frac{{n}^{3} \cdot \log i}{i}\right)\right)\right)\right)\right)\right) - \left({n}^2 \cdot \left(\log n \cdot \log i\right) + \left(n \cdot \log n + \left(\frac{1}{2} \cdot \left({n}^{3} \cdot \left(\log n \cdot {\left(\log i\right)}^2\right)\right) + \left(\frac{1}{2} \cdot \frac{{n}^{3}}{{i}^2} + \left(\frac{{n}^{3} \cdot \log n}{i} + \frac{1}{6} \cdot \left({n}^{3} \cdot {\left(\log n\right)}^{3}\right)\right)\right)\right)\right)\right)}{\frac{i}{n}}\]
      1.2
    5. Taylor expanded around 0 to get
      \[100 \cdot \frac{\color{red}{\left(\frac{1}{2} \cdot \left({n}^2 \cdot {\left(\log n\right)}^2\right) + \left(\frac{1}{2} \cdot \left({n}^2 \cdot {\left(\log i\right)}^2\right) + \left(n \cdot \log i + \left(\frac{1}{2} \cdot \left({n}^{3} \cdot \left({\left(\log n\right)}^2 \cdot \log i\right)\right) + \left(\frac{1}{6} \cdot \left({n}^{3} \cdot {\left(\log i\right)}^{3}\right) + \left(\frac{{n}^2}{i} + \frac{{n}^{3} \cdot \log i}{i}\right)\right)\right)\right)\right)\right) - \left({n}^2 \cdot \left(\log n \cdot \log i\right) + \left(n \cdot \log n + \left(\frac{1}{2} \cdot \left({n}^{3} \cdot \left(\log n \cdot {\left(\log i\right)}^2\right)\right) + \left(\frac{1}{2} \cdot \frac{{n}^{3}}{{i}^2} + \left(\frac{{n}^{3} \cdot \log n}{i} + \frac{1}{6} \cdot \left({n}^{3} \cdot {\left(\log n\right)}^{3}\right)\right)\right)\right)\right)\right)}}{\frac{i}{n}} \leadsto 100 \cdot \frac{\color{blue}{\left(\frac{1}{2} \cdot \left({n}^2 \cdot {\left(\log n\right)}^2\right) + \left(\frac{1}{2} \cdot \left({n}^2 \cdot {\left(\log i\right)}^2\right) + \left(n \cdot \log i + \left(\frac{1}{2} \cdot \left({n}^{3} \cdot \left({\left(\log n\right)}^2 \cdot \log i\right)\right) + \left(\frac{1}{6} \cdot \left({n}^{3} \cdot {\left(\log i\right)}^{3}\right) + \left(\frac{{n}^2}{i} + \frac{{n}^{3} \cdot \log i}{i}\right)\right)\right)\right)\right)\right) - \left({n}^2 \cdot \left(\log n \cdot \log i\right) + \left(n \cdot \log n + \left(\frac{1}{2} \cdot \left({n}^{3} \cdot \left(\log n \cdot {\left(\log i\right)}^2\right)\right) + \left(\frac{1}{2} \cdot \frac{{n}^{3}}{{i}^2} + \left(\frac{{n}^{3} \cdot \log n}{i} + \frac{1}{6} \cdot \left({n}^{3} \cdot {\left(\log n\right)}^{3}\right)\right)\right)\right)\right)\right)}}{\frac{i}{n}}\]
      1.2
    6. Applied simplify to get
      \[100 \cdot \frac{\left(\frac{1}{2} \cdot \left({n}^2 \cdot {\left(\log n\right)}^2\right) + \left(\frac{1}{2} \cdot \left({n}^2 \cdot {\left(\log i\right)}^2\right) + \left(n \cdot \log i + \left(\frac{1}{2} \cdot \left({n}^{3} \cdot \left({\left(\log n\right)}^2 \cdot \log i\right)\right) + \left(\frac{1}{6} \cdot \left({n}^{3} \cdot {\left(\log i\right)}^{3}\right) + \left(\frac{{n}^2}{i} + \frac{{n}^{3} \cdot \log i}{i}\right)\right)\right)\right)\right)\right) - \left({n}^2 \cdot \left(\log n \cdot \log i\right) + \left(n \cdot \log n + \left(\frac{1}{2} \cdot \left({n}^{3} \cdot \left(\log n \cdot {\left(\log i\right)}^2\right)\right) + \left(\frac{1}{2} \cdot \frac{{n}^{3}}{{i}^2} + \left(\frac{{n}^{3} \cdot \log n}{i} + \frac{1}{6} \cdot \left({n}^{3} \cdot {\left(\log n\right)}^{3}\right)\right)\right)\right)\right)\right)}{\frac{i}{n}} \leadsto \frac{\left(\left(\frac{1}{2} \cdot n\right) \cdot n\right) \cdot \left(\log n \cdot \log n + \log i \cdot \log i\right) + \left(\left(\left(\left(\frac{n}{i} \cdot n + \frac{\log i \cdot {n}^3}{i}\right) + {\left(\log i\right)}^3 \cdot \left(\frac{1}{6} \cdot {n}^3\right)\right) + \log i \cdot \left(n + \left(\frac{1}{2} \cdot {n}^3\right) \cdot \left(\log n \cdot \log n\right)\right)\right) - \left(\left(\left(\log i \cdot \left(\left(n \cdot n\right) \cdot \log n\right) + \log n \cdot n\right) + \left(\left(\frac{1}{6} \cdot {n}^3\right) \cdot {\left(\log n\right)}^3 + \frac{{n}^3}{i} \cdot \log n\right)\right) + \left(\left(\log i \cdot \log i\right) \cdot \left({n}^3 \cdot \log n\right) + \frac{{n}^3}{i \cdot i}\right) \cdot \frac{1}{2}\right)\right)}{\frac{\frac{i}{n}}{100}}\]
      1.2
    7. Applied final simplification
    8. Applied simplify to get
      \[\color{red}{\frac{\left(\left(\frac{1}{2} \cdot n\right) \cdot n\right) \cdot \left(\log n \cdot \log n + \log i \cdot \log i\right) + \left(\left(\left(\left(\frac{n}{i} \cdot n + \frac{\log i \cdot {n}^3}{i}\right) + {\left(\log i\right)}^3 \cdot \left(\frac{1}{6} \cdot {n}^3\right)\right) + \log i \cdot \left(n + \left(\frac{1}{2} \cdot {n}^3\right) \cdot \left(\log n \cdot \log n\right)\right)\right) - \left(\left(\left(\log i \cdot \left(\left(n \cdot n\right) \cdot \log n\right) + \log n \cdot n\right) + \left(\left(\frac{1}{6} \cdot {n}^3\right) \cdot {\left(\log n\right)}^3 + \frac{{n}^3}{i} \cdot \log n\right)\right) + \left(\left(\log i \cdot \log i\right) \cdot \left({n}^3 \cdot \log n\right) + \frac{{n}^3}{i \cdot i}\right) \cdot \frac{1}{2}\right)\right)}{\frac{\frac{i}{n}}{100}}} \leadsto \color{blue}{\frac{\left(\left(\left({n}^3 \cdot \frac{1}{6}\right) \cdot {\left(\log i\right)}^3 + \left(\frac{\log i}{\frac{\frac{i}{n}}{n \cdot n}} + \frac{n \cdot n}{i}\right)\right) + \left(\left(\left(\log n \cdot \log n\right) \cdot \left({n}^3 \cdot \frac{1}{2}\right) + n\right) \cdot \log i - \left(\log n \cdot \left(\log i \cdot \left(n \cdot n\right) + n\right) + \left(\left({n}^3 \cdot \frac{1}{6}\right) \cdot {\left(\log n\right)}^3 + \frac{{n}^3}{i} \cdot \log n\right)\right)\right)\right) - \left(\left(\frac{{n}^3}{{i}^2} + \left(\log i \cdot \log i\right) \cdot \left(\log n \cdot {n}^3\right)\right) \cdot \frac{1}{2} - \left(\log i \cdot \log i + \log n \cdot \log n\right) \cdot \left(\frac{1}{2} \cdot \left(n \cdot n\right)\right)\right)}{\frac{\frac{i}{100}}{n}}}\]
      1.0
  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))))