Error

Target

Derivation

1. Split input into 3 regimes

2. if (/ (- (pow (+ 1 (/ i n)) n) 1) (/ i n)) < 2.9187789820550343e-272

   1. Initial program 39.1

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

   3. Applied pow-to-exp39.1

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

   4. Applied expm1-def31.6

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

   5. Simplified7.4

      \[\leadsto 100 \cdot \frac{(e^{\color{blue}{n \cdot \log_* (1 + \frac{i}{n})}} - 1)^*}{\frac{i}{n}}\]
   6. Using strategy rm

   7. Applied associate-*r/7.4

      \[\leadsto \color{blue}{\frac{100 \cdot (e^{n \cdot \log_* (1 + \frac{i}{n})} - 1)^*}{\frac{i}{n}}}\]

   if 2.9187789820550343e-272 < (/ (- (pow (+ 1 (/ i n)) n) 1) (/ i n)) < 34072.35741418967

   1. Initial program 3.2

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

   2. Taylor expanded around inf 58.4

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

   3. Simplified3.8

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

   if 34072.35741418967 < (/ (- (pow (+ 1 (/ i n)) n) 1) (/ i n))

   1. Initial program 61.0

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

   3. Applied pow-to-exp61.0

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

   4. Applied expm1-def61.0

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

   5. Simplified61.0

      \[\leadsto 100 \cdot \frac{(e^{\color{blue}{n \cdot \log_* (1 + \frac{i}{n})}} - 1)^*}{\frac{i}{n}}\]
   6. Using strategy rm

   7. Applied associate-*r/61.0

      \[\leadsto \color{blue}{\frac{100 \cdot (e^{n \cdot \log_* (1 + \frac{i}{n})} - 1)^*}{\frac{i}{n}}}\]
   8. Using strategy rm

   9. Applied associate-/l*61.0

      \[\leadsto \color{blue}{\frac{100}{\frac{\frac{i}{n}}{(e^{n \cdot \log_* (1 + \frac{i}{n})} - 1)^*}}}\]

   10. Taylor expanded around 0 15.1

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

   11. Simplified1.1

       \[\leadsto \frac{100}{\color{blue}{(\left(\frac{i}{n}\right) \cdot \left(\frac{\frac{1}{2}}{n} - \frac{1}{2}\right) + \left(\frac{1}{n}\right))_*}}\]
3. Recombined 3 regimes into one program.

4. Final simplification5.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{\frac{i}{n}} \le 2.9187789820550343 \cdot 10^{-272}:\\ \;\;\;\;\frac{100 \cdot (e^{\log_* (1 + \frac{i}{n}) \cdot n} - 1)^*}{\frac{i}{n}}\\ \mathbf{elif}\;\frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{\frac{i}{n}} \le 34072.35741418967:\\ \;\;\;\;\frac{(\left({\left(\frac{i}{n}\right)}^{n}\right) \cdot 100 + -100)_*}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{100}{(\left(\frac{i}{n}\right) \cdot \left(\frac{\frac{1}{2}}{n} - \frac{1}{2}\right) + \left(\frac{1}{n}\right))_*}\\ \end{array}\]

Runtime

Time bar (total: 35.0s)Debug logProfile

herbie shell --seed 2018352 +o rules:numerics
(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))))