\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
Test:
Compound Interest
Bits:
128 bits
Bits error versus i
Bits error versus n
Time: 23.9 s
Input Error: 51.9
Output Error: 8.0
Log: ⚲
Profile: 🕒
\(\begin{cases} \frac{100}{i} \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{1}{n}} & \text{when } i \le -1.1913389516717047 \cdot 10^{-21} \\ \left(i \cdot 50 + 100\right) \cdot n & \text{when } i \le 7.042074692004344 \cdot 10^{-197} \\ \left(i \cdot 50 + 100\right) \cdot n & \text{when } i \le 17205888601990438.0 \\ \frac{100}{i} \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{1}{n}} & \text{when } i \le 2.7108568158997583 \cdot 10^{+192} \\ \frac{\left(\frac{1}{2} \cdot e^{\frac{2}{i}}\right) \cdot \left(100 \cdot n\right)}{i \cdot e^{\frac{\frac{2}{i}}{i}}} & \text{when } i \le 2.2637111157736717 \cdot 10^{+287} \\ \frac{100}{i} \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{1}{n}} & \text{otherwise} \end{cases}\)

    if i < -1.1913389516717047e-21 or 2.2637111157736717e+287 < i

    1. Started with
      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
      28.6
    2. Using strategy rm
      28.6
    3. Applied div-inv to get
      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\color{red}{\frac{i}{n}}} \leadsto 100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\color{blue}{i \cdot \frac{1}{n}}}\]
      28.6
    4. Applied *-un-lft-identity to get
      \[100 \cdot \frac{\color{red}{{\left(1 + \frac{i}{n}\right)}^{n} - 1}}{i \cdot \frac{1}{n}} \leadsto 100 \cdot \frac{\color{blue}{1 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}}{i \cdot \frac{1}{n}}\]
      28.6
    5. Applied times-frac to get
      \[100 \cdot \color{red}{\frac{1 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{i \cdot \frac{1}{n}}} \leadsto 100 \cdot \color{blue}{\left(\frac{1}{i} \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{1}{n}}\right)}\]
      29.0
    6. Applied associate-*r* to get
      \[\color{red}{100 \cdot \left(\frac{1}{i} \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{1}{n}}\right)} \leadsto \color{blue}{\left(100 \cdot \frac{1}{i}\right) \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{1}{n}}}\]
      29.0
    7. Applied simplify to get
      \[\color{red}{\left(100 \cdot \frac{1}{i}\right)} \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{1}{n}} \leadsto \color{blue}{\frac{100}{i}} \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{1}{n}}\]
      29.0

    if -1.1913389516717047e-21 < i < 7.042074692004344e-197

    1. Started with
      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
      61.6
    2. Applied taylor to get
      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \leadsto 100 \cdot \frac{\left(\frac{1}{2} \cdot {i}^2 + \left(1 + i\right)\right) - 1}{\frac{i}{n}}\]
      61.3
    3. Taylor expanded around 0 to get
      \[100 \cdot \frac{\color{red}{\left(\frac{1}{2} \cdot {i}^2 + \left(1 + i\right)\right)} - 1}{\frac{i}{n}} \leadsto 100 \cdot \frac{\color{blue}{\left(\frac{1}{2} \cdot {i}^2 + \left(1 + i\right)\right)} - 1}{\frac{i}{n}}\]
      61.3
    4. Applied simplify to get
      \[\color{red}{100 \cdot \frac{\left(\frac{1}{2} \cdot {i}^2 + \left(1 + i\right)\right) - 1}{\frac{i}{n}}} \leadsto \color{blue}{\left(\frac{1}{2} \cdot i + 1\right) \cdot \frac{i \cdot 100}{\frac{i}{n}}}\]
      17.0
    5. Applied taylor to get
      \[\left(\frac{1}{2} \cdot i + 1\right) \cdot \frac{i \cdot 100}{\frac{i}{n}} \leadsto 100 \cdot n + 50 \cdot \left(n \cdot i\right)\]
      0.0
    6. Taylor expanded around 0 to get
      \[\color{red}{100 \cdot n + 50 \cdot \left(n \cdot i\right)} \leadsto \color{blue}{100 \cdot n + 50 \cdot \left(n \cdot i\right)}\]
      0.0
    7. Applied simplify to get
      \[\color{red}{100 \cdot n + 50 \cdot \left(n \cdot i\right)} \leadsto \color{blue}{\left(i \cdot 50 + 100\right) \cdot n}\]
      0.0

    if 7.042074692004344e-197 < i < 17205888601990438.0

    1. Started with
      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
      61.8
    2. Applied taylor to get
      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \leadsto 100 \cdot \frac{\left(\frac{1}{2} \cdot {i}^2 + \left(1 + i\right)\right) - 1}{\frac{i}{n}}\]
      53.4
    3. Taylor expanded around 0 to get
      \[100 \cdot \frac{\color{red}{\left(\frac{1}{2} \cdot {i}^2 + \left(1 + i\right)\right)} - 1}{\frac{i}{n}} \leadsto 100 \cdot \frac{\color{blue}{\left(\frac{1}{2} \cdot {i}^2 + \left(1 + i\right)\right)} - 1}{\frac{i}{n}}\]
      53.4
    4. Applied simplify to get
      \[\color{red}{100 \cdot \frac{\left(\frac{1}{2} \cdot {i}^2 + \left(1 + i\right)\right) - 1}{\frac{i}{n}}} \leadsto \color{blue}{\left(\frac{1}{2} \cdot i + 1\right) \cdot \frac{i \cdot 100}{\frac{i}{n}}}\]
      7.9
    5. Applied taylor to get
      \[\left(\frac{1}{2} \cdot i + 1\right) \cdot \frac{i \cdot 100}{\frac{i}{n}} \leadsto 100 \cdot n + 50 \cdot \left(n \cdot i\right)\]
      0.1
    6. Taylor expanded around 0 to get
      \[\color{red}{100 \cdot n + 50 \cdot \left(n \cdot i\right)} \leadsto \color{blue}{100 \cdot n + 50 \cdot \left(n \cdot i\right)}\]
      0.1
    7. Applied simplify to get
      \[\color{red}{100 \cdot n + 50 \cdot \left(n \cdot i\right)} \leadsto \color{blue}{\left(i \cdot 50 + 100\right) \cdot n}\]
      0.1

    if 17205888601990438.0 < i < 2.7108568158997583e+192

    1. Started with
      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
      31.3
    2. Using strategy rm
      31.3
    3. Applied div-inv to get
      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\color{red}{\frac{i}{n}}} \leadsto 100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\color{blue}{i \cdot \frac{1}{n}}}\]
      31.3
    4. Applied *-un-lft-identity to get
      \[100 \cdot \frac{\color{red}{{\left(1 + \frac{i}{n}\right)}^{n} - 1}}{i \cdot \frac{1}{n}} \leadsto 100 \cdot \frac{\color{blue}{1 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}}{i \cdot \frac{1}{n}}\]
      31.3
    5. Applied times-frac to get
      \[100 \cdot \color{red}{\frac{1 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{i \cdot \frac{1}{n}}} \leadsto 100 \cdot \color{blue}{\left(\frac{1}{i} \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{1}{n}}\right)}\]
      31.3
    6. Applied associate-*r* to get
      \[\color{red}{100 \cdot \left(\frac{1}{i} \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{1}{n}}\right)} \leadsto \color{blue}{\left(100 \cdot \frac{1}{i}\right) \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{1}{n}}}\]
      31.3
    7. Applied simplify to get
      \[\color{red}{\left(100 \cdot \frac{1}{i}\right)} \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{1}{n}} \leadsto \color{blue}{\frac{100}{i}} \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{1}{n}}\]
      31.3

    if 2.7108568158997583e+192 < i < 2.2637111157736717e+287

    1. Started with
      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
      50.4
    2. Applied taylor to get
      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \leadsto 100 \cdot \frac{\left(\frac{1}{2} \cdot {i}^2 + \left(1 + i\right)\right) - 1}{\frac{i}{n}}\]
      62.7
    3. Taylor expanded around 0 to get
      \[100 \cdot \frac{\color{red}{\left(\frac{1}{2} \cdot {i}^2 + \left(1 + i\right)\right)} - 1}{\frac{i}{n}} \leadsto 100 \cdot \frac{\color{blue}{\left(\frac{1}{2} \cdot {i}^2 + \left(1 + i\right)\right)} - 1}{\frac{i}{n}}\]
      62.7
    4. Applied simplify to get
      \[\color{red}{100 \cdot \frac{\left(\frac{1}{2} \cdot {i}^2 + \left(1 + i\right)\right) - 1}{\frac{i}{n}}} \leadsto \color{blue}{\left(\frac{1}{2} \cdot i + 1\right) \cdot \frac{i \cdot 100}{\frac{i}{n}}}\]
      37.8
    5. Using strategy rm
      37.8
    6. Applied add-exp-log to get
      \[\left(\frac{1}{2} \cdot i + 1\right) \cdot \frac{i \cdot 100}{\color{red}{\frac{i}{n}}} \leadsto \left(\frac{1}{2} \cdot i + 1\right) \cdot \frac{i \cdot 100}{\color{blue}{e^{\log \left(\frac{i}{n}\right)}}}\]
      50.1
    7. Applied add-exp-log to get
      \[\left(\frac{1}{2} \cdot i + 1\right) \cdot \frac{\color{red}{i \cdot 100}}{e^{\log \left(\frac{i}{n}\right)}} \leadsto \left(\frac{1}{2} \cdot i + 1\right) \cdot \frac{\color{blue}{e^{\log \left(i \cdot 100\right)}}}{e^{\log \left(\frac{i}{n}\right)}}\]
      50.1
    8. Applied div-exp to get
      \[\left(\frac{1}{2} \cdot i + 1\right) \cdot \color{red}{\frac{e^{\log \left(i \cdot 100\right)}}{e^{\log \left(\frac{i}{n}\right)}}} \leadsto \left(\frac{1}{2} \cdot i + 1\right) \cdot \color{blue}{e^{\log \left(i \cdot 100\right) - \log \left(\frac{i}{n}\right)}}\]
      50.1
    9. Applied add-exp-log to get
      \[\color{red}{\left(\frac{1}{2} \cdot i + 1\right)} \cdot e^{\log \left(i \cdot 100\right) - \log \left(\frac{i}{n}\right)} \leadsto \color{blue}{e^{\log \left(\frac{1}{2} \cdot i + 1\right)}} \cdot e^{\log \left(i \cdot 100\right) - \log \left(\frac{i}{n}\right)}\]
      50.1
    10. Applied prod-exp to get
      \[\color{red}{e^{\log \left(\frac{1}{2} \cdot i + 1\right)} \cdot e^{\log \left(i \cdot 100\right) - \log \left(\frac{i}{n}\right)}} \leadsto \color{blue}{e^{\log \left(\frac{1}{2} \cdot i + 1\right) + \left(\log \left(i \cdot 100\right) - \log \left(\frac{i}{n}\right)\right)}}\]
      50.1
    11. Applied simplify to get
      \[e^{\color{red}{\log \left(\frac{1}{2} \cdot i + 1\right) + \left(\log \left(i \cdot 100\right) - \log \left(\frac{i}{n}\right)\right)}} \leadsto e^{\color{blue}{\left(\log n + \log 100\right) + \log \left(i \cdot \frac{1}{2} + 1\right)}}\]
      62.8
    12. Applied taylor to get
      \[e^{\left(\log n + \log 100\right) + \log \left(i \cdot \frac{1}{2} + 1\right)} \leadsto e^{\left(\log n + \log 100\right) + \left(\left(\log \frac{1}{2} + 2 \cdot \frac{1}{i}\right) - \left(2 \cdot \frac{1}{{i}^2} + \log i\right)\right)}\]
      34.5
    13. Taylor expanded around inf to get
      \[e^{\left(\log n + \log 100\right) + \color{red}{\left(\left(\log \frac{1}{2} + 2 \cdot \frac{1}{i}\right) - \left(2 \cdot \frac{1}{{i}^2} + \log i\right)\right)}} \leadsto e^{\left(\log n + \log 100\right) + \color{blue}{\left(\left(\log \frac{1}{2} + 2 \cdot \frac{1}{i}\right) - \left(2 \cdot \frac{1}{{i}^2} + \log i\right)\right)}}\]
      34.5
    14. Applied simplify to get
      \[e^{\left(\log n + \log 100\right) + \left(\left(\log \frac{1}{2} + 2 \cdot \frac{1}{i}\right) - \left(2 \cdot \frac{1}{{i}^2} + \log i\right)\right)} \leadsto \frac{\left(\frac{1}{2} \cdot e^{\frac{2}{i}}\right) \cdot \left(100 \cdot n\right)}{i \cdot e^{\frac{\frac{2}{i}}{i}}}\]
      0.3
    15. Applied final simplification
  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))))