Error

Target

Derivation

1. Split input into 4 regimes

2. if i < -4.791923504768154e-07

   1. Initial program 28.0

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

   3. Applied add-exp-log28.0

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

   4. Applied pow-exp28.0

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

   5. Applied simplify5.9

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

   if -4.791923504768154e-07 < i < 0.008058758605651986

   1. Initial program 58.3

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

   2. Taylor expanded around 0 57.8

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

   3. Applied simplify26.5

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

   5. Applied pow126.5

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

   6. Applied pow126.5

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

   7. Applied pow-prod-down26.5

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

   8. Applied simplify8.5

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

   if 0.008058758605651986 < i < 8.06711543933211e+59

   1. Initial program 35.2

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

   3. Applied add-exp-log38.0

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

   4. Applied pow-exp38.0

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

   5. Applied expm1-def13.8

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

   if 8.06711543933211e+59 < i

   1. Initial program 31.3

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

   2. Taylor expanded around inf 27.7

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

   3. Applied simplify27.6

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

4. Applied simplify10.2

   \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;i \le -4.791923504768154 \cdot 10^{-07}:\\ \;\;\;\;100 \cdot \frac{e^{n \cdot \log_* (1 + \frac{i}{n})} - 1}{\frac{i}{n}}\\ \mathbf{if}\;i \le 0.008058758605651986:\\ \;\;\;\;n \cdot \left(100 \cdot (\frac{1}{2} \cdot i + 1)_*\right)\\ \mathbf{if}\;i \le 8.06711543933211 \cdot 10^{+59}:\\ \;\;\;\;100 \cdot \frac{(e^{n \cdot \log \left(1 + \frac{i}{n}\right)} - 1)^*}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{{i}^{n}}{{n}^{n}} - 1\right) \cdot \left(\frac{100}{i} \cdot n\right)\\ \end{array}}\]

Runtime

Time bar (total: 9.3m)Debug logProfile

herbie shell --seed 2018167 +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))))