Error

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Target

Derivation

1. Split input into 4 regimes

2. if n < -8.343525586116674e+102 or -6.719525389220468e+95 < n < -2.8509751175228048e+79 or 1.0117590341155643e-139 < n

   1. Initial program 55.0

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

   2. Taylor expanded around 0 38.9

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

   3. Simplified38.8

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

   if -8.343525586116674e+102 < n < -6.719525389220468e+95

   1. Initial program 47.4

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

   3. Applied flip--47.4

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

   4. Taylor expanded around inf 63.5

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

   5. Simplified47.3

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

   if -2.8509751175228048e+79 < n < -4.888757877360074e-197

   1. Initial program 25.2

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

   3. Applied flip--25.2

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

   if -4.888757877360074e-197 < n < 1.0117590341155643e-139

   1. Initial program 28.8

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

   3. Applied flip--28.8

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

   4. Taylor expanded around -inf 43.1

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

   5. Simplified17.2

      \[\leadsto 100 \cdot \color{blue}{\left(\frac{n}{\frac{i}{1}} \cdot \left({\left(e^{n}\right)}^{\left(\log \left(\frac{-1}{n}\right) - \log \left(\frac{-1}{i}\right)\right)} + -1\right)\right)}\]
3. Recombined 4 regimes into one program.

4. Final simplification31.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -8.343525586116674 \cdot 10^{+102}:\\ \;\;\;\;100 \cdot \frac{\left(\frac{1}{6} \cdot i + \frac{1}{2}\right) \cdot \left(i \cdot i\right) + i}{\frac{i}{n}}\\ \mathbf{elif}\;n \le -6.719525389220468 \cdot 10^{+95}:\\ \;\;\;\;100 \cdot \left(\frac{n}{i} \cdot \left({\left(\frac{i}{n}\right)}^{n} + -1\right)\right)\\ \mathbf{elif}\;n \le -2.8509751175228048 \cdot 10^{+79}:\\ \;\;\;\;100 \cdot \frac{\left(\frac{1}{6} \cdot i + \frac{1}{2}\right) \cdot \left(i \cdot i\right) + i}{\frac{i}{n}}\\ \mathbf{elif}\;n \le -4.888757877360074 \cdot 10^{-197}:\\ \;\;\;\;100 \cdot \frac{\frac{{\left(\frac{i}{n} + 1\right)}^{n} \cdot {\left(\frac{i}{n} + 1\right)}^{n} - 1}{{\left(\frac{i}{n} + 1\right)}^{n} + 1}}{\frac{i}{n}}\\ \mathbf{elif}\;n \le 1.0117590341155643 \cdot 10^{-139}:\\ \;\;\;\;100 \cdot \left(\left({\left(e^{n}\right)}^{\left(\log \left(\frac{-1}{n}\right) - \log \left(\frac{-1}{i}\right)\right)} + -1\right) \cdot \frac{n}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\left(\frac{1}{6} \cdot i + \frac{1}{2}\right) \cdot \left(i \cdot i\right) + i}{\frac{i}{n}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019002 
(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

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