Error

Try it out

Target

Derivation

1. Split input into 3 regimes

2. if n < -2.2020953495892706e+82 or -1.1970289205805727e+28 < n < -660.4789951886963 or 3.8746181704323983e-106 < n

   1. Initial program 53.7

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
   2. Using strategy rm

   3. Applied div-sub53.6

      \[\leadsto 100 \cdot \color{blue}{\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \frac{1}{\frac{i}{n}}\right)}\]

   4. Taylor expanded around 0 21.0

      \[\leadsto 100 \cdot \color{blue}{\left(\frac{1}{6} \cdot \left({i}^{2} \cdot n\right) + \left(n + \frac{1}{2} \cdot \left(i \cdot n\right)\right)\right)}\]

   5. Simplified20.9

      \[\leadsto 100 \cdot \color{blue}{\left(n + \left(i \cdot n\right) \cdot \left(\frac{1}{6} \cdot i + \frac{1}{2}\right)\right)}\]
   6. Using strategy rm

   7. Applied add-cube-cbrt20.9

      \[\leadsto 100 \cdot \left(n + \color{blue}{\left(\sqrt[3]{\left(i \cdot n\right) \cdot \left(\frac{1}{6} \cdot i + \frac{1}{2}\right)} \cdot \sqrt[3]{\left(i \cdot n\right) \cdot \left(\frac{1}{6} \cdot i + \frac{1}{2}\right)}\right) \cdot \sqrt[3]{\left(i \cdot n\right) \cdot \left(\frac{1}{6} \cdot i + \frac{1}{2}\right)}}\right)\]

   if -2.2020953495892706e+82 < n < -1.1970289205805727e+28

   1. Initial program 32.5

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
   2. Using strategy rm

   3. Applied div-sub32.5

      \[\leadsto 100 \cdot \color{blue}{\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \frac{1}{\frac{i}{n}}\right)}\]
   4. Using strategy rm

   5. Applied associate-/r/32.4

      \[\leadsto 100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \color{blue}{\frac{1}{i} \cdot n}\right)\]

   6. Applied associate-/r/32.5

      \[\leadsto 100 \cdot \left(\color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i} \cdot n} - \frac{1}{i} \cdot n\right)\]

   7. Applied distribute-rgt-out--32.4

      \[\leadsto 100 \cdot \color{blue}{\left(n \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i} - \frac{1}{i}\right)\right)}\]

   if -660.4789951886963 < n < 3.8746181704323983e-106

   1. Initial program 26.1

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]

   2. Taylor expanded around 0 20.0

      \[\leadsto \color{blue}{0}\]
3. Recombined 3 regimes into one program.

4. Final simplification21.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -2.2020953495892706 \cdot 10^{+82}:\\ \;\;\;\;\left(\sqrt[3]{\left(\frac{1}{6} \cdot i + \frac{1}{2}\right) \cdot \left(i \cdot n\right)} \cdot \left(\sqrt[3]{\left(\frac{1}{6} \cdot i + \frac{1}{2}\right) \cdot \left(i \cdot n\right)} \cdot \sqrt[3]{\left(\frac{1}{6} \cdot i + \frac{1}{2}\right) \cdot \left(i \cdot n\right)}\right) + n\right) \cdot 100\\ \mathbf{elif}\;n \le -1.1970289205805727 \cdot 10^{+28}:\\ \;\;\;\;100 \cdot \left(n \cdot \left(\frac{{\left(\frac{i}{n} + 1\right)}^{n}}{i} - \frac{1}{i}\right)\right)\\ \mathbf{elif}\;n \le -660.4789951886963 \lor \neg \left(n \le 3.8746181704323983 \cdot 10^{-106}\right):\\ \;\;\;\;\left(\sqrt[3]{\left(\frac{1}{6} \cdot i + \frac{1}{2}\right) \cdot \left(i \cdot n\right)} \cdot \left(\sqrt[3]{\left(\frac{1}{6} \cdot i + \frac{1}{2}\right) \cdot \left(i \cdot n\right)} \cdot \sqrt[3]{\left(\frac{1}{6} \cdot i + \frac{1}{2}\right) \cdot \left(i \cdot n\right)}\right) + n\right) \cdot 100\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Reproduce

herbie shell --seed 2019007 
(FPCore (i n)
  :name "Compound Interest"

  :herbie-target
  (* 100 (/ (- (exp (* n (if (== (+ 1 (/ i n)) 1) (/ i n) (/ (* (/ i n) (log (+ 1 (/ i n)))) (- (+ (/ i n) 1) 1))))) 1) (/ i n)))

  (* 100 (/ (- (pow (+ 1 (/ i n)) n) 1) (/ i n))))

Details

Time bar (total: 26.6s)Debug log