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Target

Derivation

1. Split input into 4 regimes

2. if i < -12.035440265493026

   1. Initial program 27.9

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

   2. Taylor expanded around inf 62.9

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

   3. Simplified18.4

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

   if -12.035440265493026 < i < -6.979401657817147e-276

   1. Initial program 51.1

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

   3. Applied associate-/r/51.4

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

   4. Applied associate-*r*51.4

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

   5. Taylor expanded around 0 16.5

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

   6. Simplified16.5

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

   8. Applied associate-*r/16.6

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

   9. Applied associate-*l/15.2

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

   if -6.979401657817147e-276 < i < 9.950093919193757e-43

   1. Initial program 49.9

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

   3. Applied associate-/r/50.1

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

   4. Applied associate-*r*50.1

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

   5. Taylor expanded around 0 15.6

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

   6. Simplified15.6

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

   7. Taylor expanded around -inf 15.6

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

   8. Simplified15.6

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

   if 9.950093919193757e-43 < i

   1. Initial program 35.4

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

   3. Applied associate-/r/35.4

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

   4. Applied associate-*r*35.4

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

4. Final simplification19.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -12.035440265493026:\\ \;\;\;\;100 \cdot \frac{{\left(\frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\\ \mathbf{elif}\;i \le -6.979401657817147 \cdot 10^{-276}:\\ \;\;\;\;\frac{n \cdot \left(100 \cdot \left(\left(i \cdot \frac{1}{6} + \frac{1}{2}\right) \cdot \left(i \cdot i\right) + i\right)\right)}{i}\\ \mathbf{elif}\;i \le 9.950093919193757 \cdot 10^{-43}:\\ \;\;\;\;\left(n \cdot i\right) \cdot \left(\frac{50}{3} \cdot i + 50\right) + 100 \cdot n\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 \cdot \frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{i}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019005 
(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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