Error

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Target

Derivation

1. Split input into 3 regimes

2. if i < -0.0006916119981075335

   1. Initial program 28.5

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

   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. Simplified18.7

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

   if -0.0006916119981075335 < i < 3.5996068140721376e-62

   1. Initial program 49.3

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

   3. Applied div-sub49.3

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

   4. Taylor expanded around 0 17.1

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

   5. Simplified17.1

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

   7. Applied distribute-rgt-in17.1

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

   8. Simplified17.1

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

   if 3.5996068140721376e-62 < i

   1. Initial program 37.0

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

   3. Applied associate-/r/37.1

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

   4. Applied associate-*r*37.1

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

4. Final simplification21.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -0.0006916119981075335:\\ \;\;\;\;\left({\left(\frac{i}{n}\right)}^{n} + -1\right) \cdot \frac{100}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 3.5996068140721376 \cdot 10^{-62}:\\ \;\;\;\;\left(n \cdot i\right) \cdot \left(\frac{50}{3} \cdot i + 50\right) + 100 \cdot n\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 \cdot \frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{i}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019004 
(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: 27.5s)Debug log