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Target

Derivation

1. Split input into 3 regimes

2. if i < -0.4135341170383525

   1. Initial program 28.6

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

   2. Taylor expanded around inf 62.9

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

   3. Simplified18.3

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

   if -0.4135341170383525 < i < 8.252560782670268e+24

   1. Initial program 57.5

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

   2. Taylor expanded around 0 25.7

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

   3. Simplified25.7

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

   4. Taylor expanded around inf 10.0

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

   5. Simplified10.0

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

   if 8.252560782670268e+24 < i

   1. Initial program 31.4

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

   2. Taylor expanded around 0 61.2

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

   3. Simplified61.2

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

   4. Taylor expanded around inf 62.8

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

   5. Simplified62.7

      \[\leadsto \color{blue}{\left(i \cdot n\right) \cdot \left(50 + \frac{50}{3} \cdot i\right) + 100 \cdot n}\]
   6. Using strategy rm

   7. Applied add-log-exp62.8

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

   8. Applied add-log-exp57.7

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

   9. Applied sum-log57.7

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

   10. Simplified31.5

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

4. Final simplification14.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -0.4135341170383525:\\ \;\;\;\;100 \cdot \frac{{\left(\frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 8.252560782670268 \cdot 10^{+24}:\\ \;\;\;\;100 \cdot n + \left(\frac{50}{3} \cdot i + 50\right) \cdot \left(n \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left({\left(e^{n}\right)}^{\left(\left(i \cdot i\right) \cdot \frac{50}{3} + \left(100 + i \cdot 50\right)\right)}\right)\\ \end{array}\]

Runtime

Time bar (total: 44.6s)Debug logProfile

herbie shell --seed 2018235 
(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))))