\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
Test:
Compound Interest
Bits:
128 bits
Bits error versus i
Bits error versus n
Time: 51.5 s
Input Error: 50.9
Output Error: 2.9
Log: ⚲
Profile: 🕒
\(\begin{cases} \frac{(e^{n \cdot \log_* (1 + \frac{i}{n})} - 1)^*}{\frac{\frac{i}{n}}{100}} & \text{when } i \le -1.276867142421044 \cdot 10^{-101} \\ \frac{100 \cdot n}{\frac{1}{i}} \cdot (i * \frac{1}{6} + \frac{1}{2})_* + \frac{100 \cdot n}{1} & \text{when } i \le 8.327500089582464 \cdot 10^{-165} \\ \frac{(e^{n \cdot \log_* (1 + \frac{i}{n})} - 1)^*}{\frac{\frac{i}{n}}{100}} & \text{when } i \le 3.858341026223498 \cdot 10^{+89} \\ \left(\left((\frac{50}{3} * \left({\left(\log i\right)}^3 \cdot \frac{{n}^{4}}{i}\right) + \left(\frac{100 \cdot {n}^{4}}{\frac{i \cdot i}{\log i}}\right))_* + (50 * \left(\frac{\log n \cdot \log n}{\frac{i}{{n}^3}}\right) + \left(\frac{n \cdot \left(n \cdot 100\right)}{\frac{i}{\log i}}\right))_*\right) + (\left(\frac{{n}^{4} \cdot \left(\log n \cdot \log n\right)}{\frac{i}{\log i}}\right) * 50 + \left((\left(\frac{{n}^3}{i} \cdot \left(\log i \cdot \log i\right)\right) * 50 + \left(\frac{100}{\frac{i \cdot i}{{n}^3}}\right))_*\right))_*\right) - \left((100 * \left((\left(\frac{n \cdot n}{i}\right) * \left(\log n\right) + \left(\frac{{n}^3 \cdot \log n}{\frac{i}{\log i}}\right))_*\right) + \left(\frac{\left(\log n \cdot {n}^{4}\right) \cdot \left(\log i \cdot \log i\right)}{\frac{i}{50}}\right))_* + (50 * \left(\frac{{n}^{4}}{{i}^3}\right) + \left((\left(\frac{{n}^{4}}{i} \cdot {\left(\log n\right)}^3\right) * \frac{50}{3} + \left(\frac{100 \cdot {n}^{4}}{\frac{i \cdot i}{\log n}}\right))_*\right))_*\right) & \text{otherwise} \end{cases}\)

    if i < -1.276867142421044e-101

    1. Started with
      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
      35.9
    2. Using strategy rm
      35.9
    3. Applied add-exp-log to get
      \[100 \cdot \frac{{\color{red}{\left(1 + \frac{i}{n}\right)}}^{n} - 1}{\frac{i}{n}} \leadsto 100 \cdot \frac{{\color{blue}{\left(e^{\log \left(1 + \frac{i}{n}\right)}\right)}}^{n} - 1}{\frac{i}{n}}\]
      35.9
    4. Applied pow-exp to get
      \[100 \cdot \frac{\color{red}{{\left(e^{\log \left(1 + \frac{i}{n}\right)}\right)}^{n}} - 1}{\frac{i}{n}} \leadsto 100 \cdot \frac{\color{blue}{e^{\log \left(1 + \frac{i}{n}\right) \cdot n}} - 1}{\frac{i}{n}}\]
      35.9
    5. Applied simplify to get
      \[100 \cdot \frac{e^{\color{red}{\log \left(1 + \frac{i}{n}\right) \cdot n}} - 1}{\frac{i}{n}} \leadsto 100 \cdot \frac{e^{\color{blue}{\log_* (1 + \frac{i}{n}) \cdot n}} - 1}{\frac{i}{n}}\]
      19.3
    6. Applied taylor to get
      \[100 \cdot \frac{e^{\log_* (1 + \frac{i}{n}) \cdot n} - 1}{\frac{i}{n}} \leadsto 100 \cdot \frac{e^{n \cdot \log_* (1 + \frac{i}{n})} - 1}{\frac{i}{n}}\]
      19.3
    7. Taylor expanded around 0 to get
      \[100 \cdot \frac{e^{\color{red}{n \cdot \log_* (1 + \frac{i}{n})}} - 1}{\frac{i}{n}} \leadsto 100 \cdot \frac{e^{\color{blue}{n \cdot \log_* (1 + \frac{i}{n})}} - 1}{\frac{i}{n}}\]
      19.3
    8. Applied simplify to get
      \[100 \cdot \frac{e^{n \cdot \log_* (1 + \frac{i}{n})} - 1}{\frac{i}{n}} \leadsto \frac{(e^{n \cdot \log_* (1 + \frac{i}{n})} - 1)^*}{\frac{\frac{i}{n}}{100}}\]
      1.9
    9. Applied final simplification

    if -1.276867142421044e-101 < i < 8.327500089582464e-165

    1. Started with
      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
      61.5
    2. Applied taylor to get
      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \leadsto 100 \cdot \frac{\frac{1}{2} \cdot {i}^2 + \left(i + \frac{1}{6} \cdot {i}^{3}\right)}{\frac{i}{n}}\]
      18.9
    3. Taylor expanded around 0 to get
      \[100 \cdot \frac{\color{red}{\frac{1}{2} \cdot {i}^2 + \left(i + \frac{1}{6} \cdot {i}^{3}\right)}}{\frac{i}{n}} \leadsto 100 \cdot \frac{\color{blue}{\frac{1}{2} \cdot {i}^2 + \left(i + \frac{1}{6} \cdot {i}^{3}\right)}}{\frac{i}{n}}\]
      18.9
    4. Applied simplify to get
      \[\color{red}{100 \cdot \frac{\frac{1}{2} \cdot {i}^2 + \left(i + \frac{1}{6} \cdot {i}^{3}\right)}{\frac{i}{n}}} \leadsto \color{blue}{\frac{n \cdot 100}{i} \cdot (\left(i \cdot i\right) * \left((i * \frac{1}{6} + \frac{1}{2})_*\right) + i)_*}\]
      20.1
    5. Using strategy rm
      20.1
    6. Applied fma-udef to get
      \[\frac{n \cdot 100}{i} \cdot \color{red}{(\left(i \cdot i\right) * \left((i * \frac{1}{6} + \frac{1}{2})_*\right) + i)_*} \leadsto \frac{n \cdot 100}{i} \cdot \color{blue}{\left(\left(i \cdot i\right) \cdot (i * \frac{1}{6} + \frac{1}{2})_* + i\right)}\]
      20.1
    7. Applied distribute-lft-in to get
      \[\color{red}{\frac{n \cdot 100}{i} \cdot \left(\left(i \cdot i\right) \cdot (i * \frac{1}{6} + \frac{1}{2})_* + i\right)} \leadsto \color{blue}{\frac{n \cdot 100}{i} \cdot \left(\left(i \cdot i\right) \cdot (i * \frac{1}{6} + \frac{1}{2})_*\right) + \frac{n \cdot 100}{i} \cdot i}\]
      20.7
    8. Applied simplify to get
      \[\color{red}{\frac{n \cdot 100}{i} \cdot \left(\left(i \cdot i\right) \cdot (i * \frac{1}{6} + \frac{1}{2})_*\right)} + \frac{n \cdot 100}{i} \cdot i \leadsto \color{blue}{\frac{100 \cdot n}{\frac{1}{i}} \cdot (i * \frac{1}{6} + \frac{1}{2})_*} + \frac{n \cdot 100}{i} \cdot i\]
      20.1
    9. Applied simplify to get
      \[\frac{100 \cdot n}{\frac{1}{i}} \cdot (i * \frac{1}{6} + \frac{1}{2})_* + \color{red}{\frac{n \cdot 100}{i} \cdot i} \leadsto \frac{100 \cdot n}{\frac{1}{i}} \cdot (i * \frac{1}{6} + \frac{1}{2})_* + \color{blue}{\frac{100 \cdot n}{1}}\]
      0.0

    if 8.327500089582464e-165 < i < 3.858341026223498e+89

    1. Started with
      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
      51.2
    2. Using strategy rm
      51.2
    3. Applied add-exp-log to get
      \[100 \cdot \frac{{\color{red}{\left(1 + \frac{i}{n}\right)}}^{n} - 1}{\frac{i}{n}} \leadsto 100 \cdot \frac{{\color{blue}{\left(e^{\log \left(1 + \frac{i}{n}\right)}\right)}}^{n} - 1}{\frac{i}{n}}\]
      51.2
    4. Applied pow-exp to get
      \[100 \cdot \frac{\color{red}{{\left(e^{\log \left(1 + \frac{i}{n}\right)}\right)}^{n}} - 1}{\frac{i}{n}} \leadsto 100 \cdot \frac{\color{blue}{e^{\log \left(1 + \frac{i}{n}\right) \cdot n}} - 1}{\frac{i}{n}}\]
      51.2
    5. Applied simplify to get
      \[100 \cdot \frac{e^{\color{red}{\log \left(1 + \frac{i}{n}\right) \cdot n}} - 1}{\frac{i}{n}} \leadsto 100 \cdot \frac{e^{\color{blue}{\log_* (1 + \frac{i}{n}) \cdot n}} - 1}{\frac{i}{n}}\]
      48.7
    6. Applied taylor to get
      \[100 \cdot \frac{e^{\log_* (1 + \frac{i}{n}) \cdot n} - 1}{\frac{i}{n}} \leadsto 100 \cdot \frac{e^{n \cdot \log_* (1 + \frac{i}{n})} - 1}{\frac{i}{n}}\]
      48.7
    7. Taylor expanded around 0 to get
      \[100 \cdot \frac{e^{\color{red}{n \cdot \log_* (1 + \frac{i}{n})}} - 1}{\frac{i}{n}} \leadsto 100 \cdot \frac{e^{\color{blue}{n \cdot \log_* (1 + \frac{i}{n})}} - 1}{\frac{i}{n}}\]
      48.7
    8. Applied simplify to get
      \[100 \cdot \frac{e^{n \cdot \log_* (1 + \frac{i}{n})} - 1}{\frac{i}{n}} \leadsto \frac{(e^{n \cdot \log_* (1 + \frac{i}{n})} - 1)^*}{\frac{\frac{i}{n}}{100}}\]
      8.8
    9. Applied final simplification

    if 3.858341026223498e+89 < i

    1. Started with
      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
      53.2
    2. Using strategy rm
      53.2
    3. Applied add-cbrt-cube to get
      \[\color{red}{100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}} \leadsto \color{blue}{\sqrt[3]{{\left(100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\right)}^3}}\]
      54.4
    4. Applied taylor to get
      \[\sqrt[3]{{\left(100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\right)}^3} \leadsto \left(100 \cdot \frac{{n}^{4} \cdot \log i}{{i}^2} + \left(\frac{50}{3} \cdot \frac{{n}^{4} \cdot {\left(\log i\right)}^{3}}{i} + \left(50 \cdot \frac{{\left(\log n\right)}^2 \cdot {n}^{3}}{i} + \left(100 \cdot \frac{{n}^2 \cdot \log i}{i} + \left(100 \cdot \frac{{n}^{3}}{{i}^2} + \left(50 \cdot \frac{{n}^{3} \cdot {\left(\log i\right)}^2}{i} + 50 \cdot \frac{{n}^{4} \cdot \left({\left(\log n\right)}^2 \cdot \log i\right)}{i}\right)\right)\right)\right)\right)\right) - \left(50 \cdot \frac{{n}^{4}}{{i}^{3}} + \left(100 \cdot \frac{{n}^{4} \cdot \log n}{{i}^2} + \left(\frac{50}{3} \cdot \frac{{n}^{4} \cdot {\left(\log n\right)}^{3}}{i} + \left(100 \cdot \frac{{n}^2 \cdot \log n}{i} + \left(100 \cdot \frac{\log n \cdot \left({n}^{3} \cdot \log i\right)}{i} + 50 \cdot \frac{{n}^{4} \cdot \left(\log n \cdot {\left(\log i\right)}^2\right)}{i}\right)\right)\right)\right)\right)\]
      7.4
    5. Taylor expanded around 0 to get
      \[\color{red}{\left(100 \cdot \frac{{n}^{4} \cdot \log i}{{i}^2} + \left(\frac{50}{3} \cdot \frac{{n}^{4} \cdot {\left(\log i\right)}^{3}}{i} + \left(50 \cdot \frac{{\left(\log n\right)}^2 \cdot {n}^{3}}{i} + \left(100 \cdot \frac{{n}^2 \cdot \log i}{i} + \left(100 \cdot \frac{{n}^{3}}{{i}^2} + \left(50 \cdot \frac{{n}^{3} \cdot {\left(\log i\right)}^2}{i} + 50 \cdot \frac{{n}^{4} \cdot \left({\left(\log n\right)}^2 \cdot \log i\right)}{i}\right)\right)\right)\right)\right)\right) - \left(50 \cdot \frac{{n}^{4}}{{i}^{3}} + \left(100 \cdot \frac{{n}^{4} \cdot \log n}{{i}^2} + \left(\frac{50}{3} \cdot \frac{{n}^{4} \cdot {\left(\log n\right)}^{3}}{i} + \left(100 \cdot \frac{{n}^2 \cdot \log n}{i} + \left(100 \cdot \frac{\log n \cdot \left({n}^{3} \cdot \log i\right)}{i} + 50 \cdot \frac{{n}^{4} \cdot \left(\log n \cdot {\left(\log i\right)}^2\right)}{i}\right)\right)\right)\right)\right)} \leadsto \color{blue}{\left(100 \cdot \frac{{n}^{4} \cdot \log i}{{i}^2} + \left(\frac{50}{3} \cdot \frac{{n}^{4} \cdot {\left(\log i\right)}^{3}}{i} + \left(50 \cdot \frac{{\left(\log n\right)}^2 \cdot {n}^{3}}{i} + \left(100 \cdot \frac{{n}^2 \cdot \log i}{i} + \left(100 \cdot \frac{{n}^{3}}{{i}^2} + \left(50 \cdot \frac{{n}^{3} \cdot {\left(\log i\right)}^2}{i} + 50 \cdot \frac{{n}^{4} \cdot \left({\left(\log n\right)}^2 \cdot \log i\right)}{i}\right)\right)\right)\right)\right)\right) - \left(50 \cdot \frac{{n}^{4}}{{i}^{3}} + \left(100 \cdot \frac{{n}^{4} \cdot \log n}{{i}^2} + \left(\frac{50}{3} \cdot \frac{{n}^{4} \cdot {\left(\log n\right)}^{3}}{i} + \left(100 \cdot \frac{{n}^2 \cdot \log n}{i} + \left(100 \cdot \frac{\log n \cdot \left({n}^{3} \cdot \log i\right)}{i} + 50 \cdot \frac{{n}^{4} \cdot \left(\log n \cdot {\left(\log i\right)}^2\right)}{i}\right)\right)\right)\right)\right)}\]
      7.4
    6. Applied simplify to get
      \[\left(100 \cdot \frac{{n}^{4} \cdot \log i}{{i}^2} + \left(\frac{50}{3} \cdot \frac{{n}^{4} \cdot {\left(\log i\right)}^{3}}{i} + \left(50 \cdot \frac{{\left(\log n\right)}^2 \cdot {n}^{3}}{i} + \left(100 \cdot \frac{{n}^2 \cdot \log i}{i} + \left(100 \cdot \frac{{n}^{3}}{{i}^2} + \left(50 \cdot \frac{{n}^{3} \cdot {\left(\log i\right)}^2}{i} + 50 \cdot \frac{{n}^{4} \cdot \left({\left(\log n\right)}^2 \cdot \log i\right)}{i}\right)\right)\right)\right)\right)\right) - \left(50 \cdot \frac{{n}^{4}}{{i}^{3}} + \left(100 \cdot \frac{{n}^{4} \cdot \log n}{{i}^2} + \left(\frac{50}{3} \cdot \frac{{n}^{4} \cdot {\left(\log n\right)}^{3}}{i} + \left(100 \cdot \frac{{n}^2 \cdot \log n}{i} + \left(100 \cdot \frac{\log n \cdot \left({n}^{3} \cdot \log i\right)}{i} + 50 \cdot \frac{{n}^{4} \cdot \left(\log n \cdot {\left(\log i\right)}^2\right)}{i}\right)\right)\right)\right)\right) \leadsto \left(\left((\frac{50}{3} * \left({\left(\log i\right)}^3 \cdot \frac{{n}^{4}}{i}\right) + \left(\frac{100 \cdot {n}^{4}}{\frac{i \cdot i}{\log i}}\right))_* + (50 * \left(\frac{\log n \cdot \log n}{\frac{i}{{n}^3}}\right) + \left(\frac{n \cdot \left(n \cdot 100\right)}{\frac{i}{\log i}}\right))_*\right) + (\left(\frac{{n}^{4} \cdot \left(\log n \cdot \log n\right)}{\frac{i}{\log i}}\right) * 50 + \left((\left(\frac{{n}^3}{i} \cdot \left(\log i \cdot \log i\right)\right) * 50 + \left(\frac{100}{\frac{i \cdot i}{{n}^3}}\right))_*\right))_*\right) - \left((100 * \left((\left(\frac{n \cdot n}{i}\right) * \left(\log n\right) + \left(\frac{{n}^3 \cdot \log n}{\frac{i}{\log i}}\right))_*\right) + \left(\frac{\left(\log n \cdot {n}^{4}\right) \cdot \left(\log i \cdot \log i\right)}{\frac{i}{50}}\right))_* + (50 * \left(\frac{{n}^{4}}{{i}^3}\right) + \left((\left(\frac{{n}^{4}}{i} \cdot {\left(\log n\right)}^3\right) * \frac{50}{3} + \left(\frac{100 \cdot {n}^{4}}{\frac{i \cdot i}{\log n}}\right))_*\right))_*\right)\]
      7.7
    7. 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))))