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Target

Derivation

1. Split input into 3 regimes

2. if i < -0.9837857423532252

   1. Initial program 27.0

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

   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. Simplified18.3

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

   if -0.9837857423532252 < i < 70.53354974606698

   1. Initial program 57.5

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

   2. Taylor expanded around 0 24.9

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

   3. Simplified24.9

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

   4. Taylor expanded around -inf 9.2

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

   5. Simplified9.2

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

   if 70.53354974606698 < i

   1. Initial program 32.8

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

   2. Taylor expanded around 0 60.6

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

   3. Simplified60.6

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

   4. Taylor expanded around -inf 62.6

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

   5. Simplified62.6

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

   7. Applied add-log-exp62.6

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

   8. Applied add-log-exp55.6

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

   9. Applied sum-log55.6

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

   10. Simplified30.2

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

4. Final simplification14.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -0.9837857423532252:\\ \;\;\;\;\left({\left(\frac{i}{n}\right)}^{n} - 1\right) \cdot \left(\frac{100}{i} \cdot n\right)\\ \mathbf{elif}\;i \le 70.53354974606698:\\ \;\;\;\;100 \cdot n + \left(n \cdot i\right) \cdot \left(i \cdot \frac{50}{3} + 50\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left({\left(e^{n}\right)}^{\left(\frac{50}{3} \cdot \left(i \cdot i\right) + \left(100 + 50 \cdot i\right)\right)}\right)\\ \end{array}\]

Runtime

Time bar (total: 58.5s)Debug logProfile

herbie shell --seed 2018230 
(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))))