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Target

Derivation

1. Split input into 3 regimes

2. if i < -0.00011372447818561648

   1. Initial program 28.4

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

   2. Initial simplification29.1

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

   3. Taylor expanded around -inf 12.8

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

   if -0.00011372447818561648 < i < 1.0296685150950594e-11

   1. Initial program 50.2

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

   2. Taylor expanded around 0 33.2

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

   3. Simplified33.2

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

   if 1.0296685150950594e-11 < i

   1. Initial program 33.0

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

   2. Taylor expanded around inf 32.2

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

   3. Simplified33.0

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

   4. Taylor expanded around 0 23.3

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

   5. Simplified23.3

      \[\leadsto \frac{100}{\frac{i}{n}} \cdot \color{blue}{\left(\left(\left(\left(\left(\frac{1}{6} \cdot n\right) \cdot \left(n \cdot n\right)\right) \cdot {\left(\log i\right)}^{3} + \left(\left(n \cdot n\right) \cdot \frac{1}{2}\right) \cdot \left(\log n \cdot \log n\right)\right) + \left(\left(\left(\log i \cdot {n}^{3}\right) \cdot \left(\log n \cdot \log n\right)\right) \cdot \frac{1}{2} + \left(\left(\log i \cdot \log i\right) \cdot \left(\left(n \cdot n\right) \cdot \frac{1}{2}\right) + n \cdot \log i\right)\right)\right) - \left(\left(\left(\left(\frac{1}{3} \cdot n\right) \cdot \left(n \cdot n\right)\right) \cdot \left(\log n \cdot \left(\log i \cdot \log i\right)\right) + \left(\left(\log i \cdot \left(n \cdot n\right)\right) \cdot \log n\right) \cdot 1\right) + \left(\left({\left(\log n\right)}^{3} \cdot \left(\left(\frac{1}{6} \cdot n\right) \cdot \left(n \cdot n\right)\right) + n \cdot \log n\right) + \left(\left(\frac{1}{6} \cdot n\right) \cdot \left(n \cdot n\right)\right) \cdot \left(\log n \cdot \left(\log i \cdot \log i\right)\right)\right)\right)\right)}\]
3. Recombined 3 regimes into one program.

4. Final simplification27.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -0.00011372447818561648:\\ \;\;\;\;\frac{n \cdot \left(100 \cdot e^{i} - 100\right)}{i}\\ \mathbf{elif}\;i \le 1.0296685150950594 \cdot 10^{-11}:\\ \;\;\;\;\frac{\left(i \cdot i\right) \cdot \left(\frac{1}{6} \cdot i + \frac{1}{2}\right) + i}{\frac{i}{n}} \cdot 100\\ \mathbf{else}:\\ \;\;\;\;\frac{100}{\frac{i}{n}} \cdot \left(\left(\left(\left(\log n \cdot \log n\right) \cdot \left(\frac{1}{2} \cdot \left(n \cdot n\right)\right) + {\left(\log i\right)}^{3} \cdot \left(\left(n \cdot n\right) \cdot \left(n \cdot \frac{1}{6}\right)\right)\right) + \left(\left(\left(\log n \cdot \log n\right) \cdot \left({n}^{3} \cdot \log i\right)\right) \cdot \frac{1}{2} + \left(\log i \cdot n + \left(\log i \cdot \log i\right) \cdot \left(\frac{1}{2} \cdot \left(n \cdot n\right)\right)\right)\right)\right) - \left(\left(\left(\left(\log i \cdot \log i\right) \cdot \log n\right) \cdot \left(\left(n \cdot n\right) \cdot \left(n \cdot \frac{1}{6}\right)\right) + \left({\left(\log n\right)}^{3} \cdot \left(\left(n \cdot n\right) \cdot \left(n \cdot \frac{1}{6}\right)\right) + n \cdot \log n\right)\right) + \left(\left(\left(\log i \cdot \log i\right) \cdot \log n\right) \cdot \left(\left(n \cdot n\right) \cdot \left(n \cdot \frac{1}{3}\right)\right) + \left(\left(n \cdot n\right) \cdot \log i\right) \cdot \log n\right)\right)\right)\\ \end{array}\]

Runtime

Time bar (total: 46.9s)Debug logProfile

herbie shell --seed 2018304 
(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))))