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Target

Derivation

1. Split input into 2 regimes

2. if n < -0.49724335821818005 or 8.31291788103925e-125 < n

   1. Initial program 52.1

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

   2. Taylor expanded around 0 38.0

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

   3. Simplified38.0

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

   if -0.49724335821818005 < n < 8.31291788103925e-125

   1. Initial program 25.9

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

   2. Taylor expanded around 0 19.2

      \[\leadsto \color{blue}{0}\]
3. Recombined 2 regimes into one program.

4. Final simplification31.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -0.49724335821818005 \lor \neg \left(n \le 8.31291788103925 \cdot 10^{-125}\right):\\ \;\;\;\;100 \cdot \frac{\left(\frac{1}{6} \cdot i + \frac{1}{2}\right) \cdot \left(i \cdot i\right) + i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Reproduce

herbie shell --seed 2019008 
(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

Time bar (total: 25.6s)Debug log