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Target

Derivation

1. Split input into 3 regimes

2. if n < -1.8862076621361236e+175 or 1043400119.5721545 < n

   1. Initial program 58.0

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

   2. Initial simplification58.0

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

   3. Taylor expanded around -inf 46.0

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

   if -1.8862076621361236e+175 < n < 3.9851971892074e-310

   1. Initial program 37.0

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

   2. Initial simplification37.3

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

   4. Applied div-inv39.1

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

   if 3.9851971892074e-310 < n < 1043400119.5721545

   1. Initial program 49.6

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

   2. Initial simplification49.6

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

   3. Taylor expanded around inf 37.5

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

   4. Simplified49.9

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

   6. Applied associate-*l/50.0

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

   7. Applied sub-div49.9

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

   8. Taylor expanded around 0 30.2

      \[\leadsto \frac{\color{blue}{\left(\frac{50}{3} \cdot \left({n}^{4} \cdot \left(\log i \cdot {\left(\log n\right)}^{2}\right)\right) + \left(50 \cdot \left({n}^{3} \cdot {\left(\log n\right)}^{2}\right) + \left(50 \cdot \left({n}^{3} \cdot {\left(\log i\right)}^{2}\right) + \left(\frac{100}{3} \cdot \left({n}^{4} \cdot \left({\left(\log n\right)}^{2} \cdot \log i\right)\right) + \left(\frac{50}{3} \cdot \left({n}^{4} \cdot {\left(\log i\right)}^{3}\right) + 100 \cdot \left({n}^{2} \cdot \log i\right)\right)\right)\right)\right)\right) - \left(50 \cdot \left({n}^{3} \cdot \left(\log n \cdot \log i\right)\right) + \left(50 \cdot \left({n}^{3} \cdot \left(\log i \cdot \log n\right)\right) + \left(100 \cdot \left({n}^{2} \cdot \log n\right) + \left(\frac{50}{3} \cdot \left({n}^{4} \cdot {\left(\log n\right)}^{3}\right) + \left(\frac{50}{3} \cdot \left({n}^{4} \cdot \left(\log n \cdot {\left(\log i\right)}^{2}\right)\right) + \frac{100}{3} \cdot \left({n}^{4} \cdot \left({\left(\log i\right)}^{2} \cdot \log n\right)\right)\right)\right)\right)\right)\right)}}{i}\]

   9. Simplified30.2

      \[\leadsto \frac{\color{blue}{\left(\left(\frac{50}{3} \cdot \left(\left({n}^{4} \cdot \log i\right) \cdot \left(\log n \cdot \log n\right)\right) + \left(\left(50 \cdot n\right) \cdot \left(n \cdot n\right)\right) \cdot \left(\log n \cdot \log n + \log i \cdot \log i\right)\right) + \left(\left({n}^{4} \cdot \log i\right) \cdot \left(\left(\log n \cdot \log n\right) \cdot \frac{100}{3}\right) + \left(\left({n}^{4} \cdot \frac{50}{3}\right) \cdot {\left(\log i\right)}^{3} + \log i \cdot \left(100 \cdot \left(n \cdot n\right)\right)\right)\right)\right) - \left(\left(\left(\log i \cdot \log n\right) \cdot \left(\left(50 \cdot n\right) \cdot \left(n \cdot n\right) + \left(50 \cdot n\right) \cdot \left(n \cdot n\right)\right) + \left({\left(\log n\right)}^{3} \cdot \left({n}^{4} \cdot \frac{50}{3}\right) + \left(\left(n \cdot n\right) \cdot \log n\right) \cdot 100\right)\right) + \left(\left(\log i \cdot \log i\right) \cdot \left({n}^{4} \cdot \log n\right)\right) \cdot 50\right)}}{i}\]
3. Recombined 3 regimes into one program.

4. Final simplification39.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -1.8862076621361236 \cdot 10^{+175}:\\ \;\;\;\;\frac{\left(100 \cdot e^{i} - 100\right) \cdot n}{i}\\ \mathbf{elif}\;n \le 3.9851971892074 \cdot 10^{-310}:\\ \;\;\;\;{\left(1 + \frac{i}{n}\right)}^{n} \cdot \frac{100 \cdot n}{i} - \left(100 \cdot n\right) \cdot \frac{1}{i}\\ \mathbf{elif}\;n \le 1043400119.5721545:\\ \;\;\;\;\frac{\left(\left(\left(\log n \cdot \log n + \log i \cdot \log i\right) \cdot \left(\left(n \cdot n\right) \cdot \left(n \cdot 50\right)\right) + \left(\left(\log i \cdot {n}^{4}\right) \cdot \left(\log n \cdot \log n\right)\right) \cdot \frac{50}{3}\right) + \left(\left(\log i \cdot {n}^{4}\right) \cdot \left(\frac{100}{3} \cdot \left(\log n \cdot \log n\right)\right) + \left(\log i \cdot \left(100 \cdot \left(n \cdot n\right)\right) + \left(\frac{50}{3} \cdot {n}^{4}\right) \cdot {\left(\log i\right)}^{3}\right)\right)\right) - \left(\left(\left(\left(n \cdot n\right) \cdot \left(n \cdot 50\right) + \left(n \cdot n\right) \cdot \left(n \cdot 50\right)\right) \cdot \left(\log n \cdot \log i\right) + \left({\left(\log n\right)}^{3} \cdot \left(\frac{50}{3} \cdot {n}^{4}\right) + \left(\log n \cdot \left(n \cdot n\right)\right) \cdot 100\right)\right) + \left(\left(\log n \cdot {n}^{4}\right) \cdot \left(\log i \cdot \log i\right)\right) \cdot 50\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(100 \cdot e^{i} - 100\right) \cdot n}{i}\\ \end{array}\]

Runtime

Time bar (total: 37.1s)Debug logProfile

herbie shell --seed 2018285 
(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))))