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Target

Derivation

1. Split input into 4 regimes

2. if i < -1.013284235712507

   1. Initial program 28.8

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

   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. Simplified19.1

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

   if -1.013284235712507 < i < 14617361289733.344

   1. Initial program 48.8

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

   2. Taylor expanded around 0 33.6

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

   3. Simplified33.6

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

   5. Applied div-inv33.7

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

   6. Applied add-cube-cbrt34.1

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

   7. Applied times-frac18.1

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

   8. Applied associate-*r*18.1

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

   9. Simplified18.1

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

   10. Taylor expanded around 0 41.0

       \[\leadsto \left(100 \cdot \color{blue}{\left(\frac{1}{3} \cdot {\left({i}^{2}\right)}^{\frac{1}{3}} + \left({\left(\frac{1}{i}\right)}^{\frac{1}{3}} + \frac{1}{12} \cdot {\left({i}^{5}\right)}^{\frac{1}{3}}\right)\right)}\right) \cdot \left(n \cdot \sqrt[3]{i + \left(i \cdot \frac{1}{6} + \frac{1}{2}\right) \cdot \left(i \cdot i\right)}\right)\]

   11. Simplified17.7

       \[\leadsto \left(100 \cdot \color{blue}{\left(\sqrt[3]{i \cdot i} \cdot \frac{1}{3} + \left(\frac{1}{12} \cdot \sqrt[3]{{i}^{5}} + \sqrt[3]{\frac{1}{i}}\right)\right)}\right) \cdot \left(n \cdot \sqrt[3]{i + \left(i \cdot \frac{1}{6} + \frac{1}{2}\right) \cdot \left(i \cdot i\right)}\right)\]

   12. Taylor expanded around 0 41.0

       \[\leadsto \left(100 \cdot \left(\sqrt[3]{i \cdot i} \cdot \frac{1}{3} + \left(\frac{1}{12} \cdot \sqrt[3]{{i}^{5}} + \sqrt[3]{\frac{1}{i}}\right)\right)\right) \cdot \left(n \cdot \color{blue}{\left({i}^{\frac{1}{3}} + \left(\frac{1}{6} \cdot {\left({i}^{4}\right)}^{\frac{1}{3}} + \frac{1}{36} \cdot {\left({i}^{7}\right)}^{\frac{1}{3}}\right)\right)}\right)\]

   13. Simplified17.7

       \[\leadsto \left(100 \cdot \left(\sqrt[3]{i \cdot i} \cdot \frac{1}{3} + \left(\frac{1}{12} \cdot \sqrt[3]{{i}^{5}} + \sqrt[3]{\frac{1}{i}}\right)\right)\right) \cdot \left(n \cdot \color{blue}{\left(\left(\frac{1}{6} \cdot \sqrt[3]{{i}^{4}} + \sqrt[3]{{i}^{7}} \cdot \frac{1}{36}\right) + \sqrt[3]{i}\right)}\right)\]

   if 14617361289733.344 < i < 6.232732970873983e+189

   1. Initial program 29.3

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

   2. Initial simplification29.3

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

   if 6.232732970873983e+189 < 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 31.0

      \[\leadsto \color{blue}{0}\]
3. Recombined 4 regimes into one program.

4. Final simplification19.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -1.013284235712507:\\ \;\;\;\;\left({\left(\frac{i}{n}\right)}^{n} + -1\right) \cdot \frac{100}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 14617361289733.344:\\ \;\;\;\;\left(n \cdot \left(\sqrt[3]{i} + \left(\sqrt[3]{{i}^{7}} \cdot \frac{1}{36} + \sqrt[3]{{i}^{4}} \cdot \frac{1}{6}\right)\right)\right) \cdot \left(100 \cdot \left(\frac{1}{3} \cdot \sqrt[3]{i \cdot i} + \left(\sqrt[3]{{i}^{5}} \cdot \frac{1}{12} + \sqrt[3]{\frac{1}{i}}\right)\right)\right)\\ \mathbf{elif}\;i \le 6.232732970873983 \cdot 10^{+189}:\\ \;\;\;\;{\left(\frac{i}{n} + 1\right)}^{n} \cdot \frac{100 \cdot n}{i} - \frac{100 \cdot n}{i}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Runtime

Time bar (total: 53.1s)Debug logProfile

herbie shell --seed 2018352 
(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))))