\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
Test:
Compound Interest
Bits:
128 bits
Bits error versus i
Bits error versus n
Time: 26.8 s
Input Error: 52.4
Output Error: 9.8
Log: ⚲
Profile: 🕒
\(\begin{cases} 100 \cdot \frac{\left(\sqrt{{\left(1 + \frac{i}{n}\right)}^{n}} + 1\right) \cdot \left(\sqrt{{\left(1 + \frac{i}{n}\right)}^{n}} - 1\right)}{\frac{i}{n}} & \text{when } i \le -0.09197678863246804 \\ \left(\left(i \cdot n\right) \cdot 100\right) \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot i\right) + n \cdot 100 & \text{when } i \le 18073.65682704277 \\ 100 \cdot \frac{\left(\sqrt{{\left(1 + \frac{i}{n}\right)}^{n}} + 1\right) \cdot \left(\sqrt{{\left(1 + \frac{i}{n}\right)}^{n}} - 1\right)}{\frac{i}{n}} & \text{when } i \le 7.979289197910273 \cdot 10^{+111} \\ \frac{\frac{100}{i}}{\frac{i}{n}} \cdot \left(i \cdot \frac{1}{2} + 1\right) & \text{otherwise} \end{cases}\)

    if i < -0.09197678863246804

    1. Started with
      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
      27.4
    2. Using strategy rm
      27.4
    3. Applied add-sqr-sqrt to get
      \[100 \cdot \frac{\color{red}{{\left(1 + \frac{i}{n}\right)}^{n}} - 1}{\frac{i}{n}} \leadsto 100 \cdot \frac{\color{blue}{{\left(\sqrt{{\left(1 + \frac{i}{n}\right)}^{n}}\right)}^2} - 1}{\frac{i}{n}}\]
      27.4
    4. Applied difference-of-sqr-1 to get
      \[100 \cdot \frac{\color{red}{{\left(\sqrt{{\left(1 + \frac{i}{n}\right)}^{n}}\right)}^2 - 1}}{\frac{i}{n}} \leadsto 100 \cdot \frac{\color{blue}{\left(\sqrt{{\left(1 + \frac{i}{n}\right)}^{n}} + 1\right) \cdot \left(\sqrt{{\left(1 + \frac{i}{n}\right)}^{n}} - 1\right)}}{\frac{i}{n}}\]
      27.4

    if -0.09197678863246804 < i < 18073.65682704277

    1. Started with
      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
      61.7
    2. Applied taylor to get
      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \leadsto 100 \cdot \frac{\frac{1}{2} \cdot {i}^2 + \left(i + \frac{1}{6} \cdot {i}^{3}\right)}{\frac{i}{n}}\]
      13.5
    3. Taylor expanded around 0 to get
      \[100 \cdot \frac{\color{red}{\frac{1}{2} \cdot {i}^2 + \left(i + \frac{1}{6} \cdot {i}^{3}\right)}}{\frac{i}{n}} \leadsto 100 \cdot \frac{\color{blue}{\frac{1}{2} \cdot {i}^2 + \left(i + \frac{1}{6} \cdot {i}^{3}\right)}}{\frac{i}{n}}\]
      13.5
    4. Applied simplify to get
      \[\color{red}{100 \cdot \frac{\frac{1}{2} \cdot {i}^2 + \left(i + \frac{1}{6} \cdot {i}^{3}\right)}{\frac{i}{n}}} \leadsto \color{blue}{\frac{100}{\frac{i}{n}} \cdot \left(i + \left(i \cdot i\right) \cdot \left(\frac{1}{6} \cdot i + \frac{1}{2}\right)\right)}\]
      14.6
    5. Applied taylor to get
      \[\frac{100}{\frac{i}{n}} \cdot \left(i + \left(i \cdot i\right) \cdot \left(\frac{1}{6} \cdot i + \frac{1}{2}\right)\right) \leadsto \frac{100}{\frac{i}{n}} \cdot \left(i + \left(i \cdot i\right) \cdot \left(\frac{1}{6} \cdot i + \frac{1}{2}\right)\right)\]
      14.6
    6. Taylor expanded around 0 to get
      \[\frac{100}{\frac{i}{n}} \cdot \left(i + \left(i \cdot i\right) \cdot \left(\color{red}{\frac{1}{6} \cdot i} + \frac{1}{2}\right)\right) \leadsto \frac{100}{\frac{i}{n}} \cdot \left(i + \left(i \cdot i\right) \cdot \left(\color{blue}{\frac{1}{6} \cdot i} + \frac{1}{2}\right)\right)\]
      14.6
    7. Applied simplify to get
      \[\frac{100}{\frac{i}{n}} \cdot \left(i + \left(i \cdot i\right) \cdot \left(\frac{1}{6} \cdot i + \frac{1}{2}\right)\right) \leadsto \frac{i \cdot 100}{\frac{i}{n}} \cdot \left(i \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot i\right)\right) + \frac{i \cdot 100}{\frac{i}{n}}\]
      13.9
    8. Applied final simplification
    9. Applied simplify to get
      \[\color{red}{\frac{i \cdot 100}{\frac{i}{n}} \cdot \left(i \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot i\right)\right) + \frac{i \cdot 100}{\frac{i}{n}}} \leadsto \color{blue}{\left(\left(i \cdot n\right) \cdot 100\right) \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot i\right) + n \cdot 100}\]
      0.0

    if 18073.65682704277 < i < 7.979289197910273e+111

    1. Started with
      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
      32.5
    2. Using strategy rm
      32.5
    3. Applied add-sqr-sqrt to get
      \[100 \cdot \frac{\color{red}{{\left(1 + \frac{i}{n}\right)}^{n}} - 1}{\frac{i}{n}} \leadsto 100 \cdot \frac{\color{blue}{{\left(\sqrt{{\left(1 + \frac{i}{n}\right)}^{n}}\right)}^2} - 1}{\frac{i}{n}}\]
      32.6
    4. Applied difference-of-sqr-1 to get
      \[100 \cdot \frac{\color{red}{{\left(\sqrt{{\left(1 + \frac{i}{n}\right)}^{n}}\right)}^2 - 1}}{\frac{i}{n}} \leadsto 100 \cdot \frac{\color{blue}{\left(\sqrt{{\left(1 + \frac{i}{n}\right)}^{n}} + 1\right) \cdot \left(\sqrt{{\left(1 + \frac{i}{n}\right)}^{n}} - 1\right)}}{\frac{i}{n}}\]
      32.6

    if 7.979289197910273e+111 < i

    1. Started with
      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
      51.4
    2. Applied taylor to get
      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \leadsto 100 \cdot \frac{\left(\frac{1}{2} \cdot {i}^2 + \left(1 + i\right)\right) - 1}{\frac{i}{n}}\]
      59.4
    3. Taylor expanded around 0 to get
      \[100 \cdot \frac{\color{red}{\left(\frac{1}{2} \cdot {i}^2 + \left(1 + i\right)\right)} - 1}{\frac{i}{n}} \leadsto 100 \cdot \frac{\color{blue}{\left(\frac{1}{2} \cdot {i}^2 + \left(1 + i\right)\right)} - 1}{\frac{i}{n}}\]
      59.4
    4. Applied simplify to get
      \[\color{red}{100 \cdot \frac{\left(\frac{1}{2} \cdot {i}^2 + \left(1 + i\right)\right) - 1}{\frac{i}{n}}} \leadsto \color{blue}{\left(\frac{1}{2} \cdot i + 1\right) \cdot \frac{i \cdot 100}{\frac{i}{n}}}\]
      41.8
    5. Using strategy rm
      41.8
    6. Applied add-cube-cbrt to get
      \[\left(\frac{1}{2} \cdot i + 1\right) \cdot \frac{i \cdot 100}{\color{red}{\frac{i}{n}}} \leadsto \left(\frac{1}{2} \cdot i + 1\right) \cdot \frac{i \cdot 100}{\color{blue}{{\left(\sqrt[3]{\frac{i}{n}}\right)}^3}}\]
      41.8
    7. Applied add-cube-cbrt to get
      \[\left(\frac{1}{2} \cdot i + 1\right) \cdot \frac{\color{red}{i \cdot 100}}{{\left(\sqrt[3]{\frac{i}{n}}\right)}^3} \leadsto \left(\frac{1}{2} \cdot i + 1\right) \cdot \frac{\color{blue}{{\left(\sqrt[3]{i \cdot 100}\right)}^3}}{{\left(\sqrt[3]{\frac{i}{n}}\right)}^3}\]
      41.8
    8. Applied cube-undiv to get
      \[\left(\frac{1}{2} \cdot i + 1\right) \cdot \color{red}{\frac{{\left(\sqrt[3]{i \cdot 100}\right)}^3}{{\left(\sqrt[3]{\frac{i}{n}}\right)}^3}} \leadsto \left(\frac{1}{2} \cdot i + 1\right) \cdot \color{blue}{{\left(\frac{\sqrt[3]{i \cdot 100}}{\sqrt[3]{\frac{i}{n}}}\right)}^3}\]
      41.8
    9. Applied taylor to get
      \[\left(\frac{1}{2} \cdot i + 1\right) \cdot {\left(\frac{\sqrt[3]{i \cdot 100}}{\sqrt[3]{\frac{i}{n}}}\right)}^3 \leadsto \left(\frac{1}{2} \cdot i + 1\right) \cdot {\left(\frac{\sqrt[3]{\frac{100}{i}}}{\sqrt[3]{\frac{i}{n}}}\right)}^3\]
      19.2
    10. Taylor expanded around inf to get
      \[\left(\frac{1}{2} \cdot i + 1\right) \cdot {\left(\frac{\color{red}{\sqrt[3]{\frac{100}{i}}}}{\sqrt[3]{\frac{i}{n}}}\right)}^3 \leadsto \left(\frac{1}{2} \cdot i + 1\right) \cdot {\left(\frac{\color{blue}{\sqrt[3]{\frac{100}{i}}}}{\sqrt[3]{\frac{i}{n}}}\right)}^3\]
      19.2
    11. Applied simplify to get
      \[\left(\frac{1}{2} \cdot i + 1\right) \cdot {\left(\frac{\sqrt[3]{\frac{100}{i}}}{\sqrt[3]{\frac{i}{n}}}\right)}^3 \leadsto \frac{\frac{100}{i}}{\frac{i}{n}} \cdot \left(i \cdot \frac{1}{2} + 1\right)\]
      18.7
    12. 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))))