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Target

Derivation

1. Split input into 3 regimes

2. if i < -0.6575761067144743

   1. Initial program 27.3

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

   2. Taylor expanded around inf 62.9

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

   3. Simplified18.0

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

   if -0.6575761067144743 < i < 6.882221154977176

   1. Initial program 57.7

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

   2. Taylor expanded around 0 25.2

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

   3. Simplified25.2

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

   4. Taylor expanded around -inf 9.4

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

   5. Simplified9.4

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

   if 6.882221154977176 < i

   1. Initial program 30.2

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

   3. Applied div-inv30.2

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

   4. Applied *-un-lft-identity30.2

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

   5. Applied times-frac30.2

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

   6. Simplified30.2

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

4. Final simplification14.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -0.6575761067144743:\\ \;\;\;\;\left({\left(\frac{i}{n}\right)}^{n} + -1\right) \cdot \frac{100}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 6.882221154977176:\\ \;\;\;\;100 \cdot \left(n + \left(i \cdot \frac{1}{6} + \frac{1}{2}\right) \cdot \left(n \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(\left(n \cdot {\left(\frac{i}{n} + 1\right)}^{n} - n\right) \cdot \frac{1}{i}\right)\\ \end{array}\]

Runtime

Time bar (total: 48.3s)Debug logProfile

herbie shell --seed 2018295 
(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))))