Error

Target

Derivation

1. Split input into 3 regimes

2. if i < -2.643994791550655e-18

   1. Initial program 30.6

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

   2. Simplified30.6

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

   4. Applied add-exp-log30.6

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

   5. Applied pow-exp30.6

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

   6. Simplified7.3

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

   8. Applied associate-/r/7.9

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

   10. Applied fma-udef8.0

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

   if -2.643994791550655e-18 < i < 2.5132717108214893e-17

   1. Initial program 49.6

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

   2. Simplified49.6

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

   3. Taylor expanded around 0 33.5

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

   4. Simplified33.5

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

   if 2.5132717108214893e-17 < i

   1. Initial program 35.2

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

   2. Simplified35.2

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

   4. Applied add-exp-log52.0

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

   5. Applied pow-exp52.0

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

   6. Simplified48.7

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

   8. Applied associate-/r/48.7

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

   9. Taylor expanded around 0 22.8

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

   10. Simplified22.8

       \[\leadsto \frac{\color{blue}{(50 \cdot \left(\left(n \cdot \log n\right) \cdot \left(n \cdot \log n\right)\right) + \left((\frac{50}{3} \cdot \left(\log i \cdot \left(n \cdot \left(\left(n \cdot \log i\right) \cdot \left(n \cdot \log i\right)\right)\right)\right) + \left((100 \cdot \left(n \cdot \log i\right) + \left(\left(\left(n \cdot \log i\right) \cdot \left(n \cdot \log i\right)\right) \cdot 50 + \left(\log i \cdot \left(n \cdot \left(\left(n \cdot \log n\right) \cdot \left(n \cdot \log n\right)\right)\right)\right) \cdot 50\right))_*\right))_*\right))_* - \left(\left((\left(\left(\log i \cdot \log i\right) \cdot \left(\left(n \cdot n\right) \cdot \left(n \cdot \log n\right)\right)\right) \cdot \frac{100}{3} + \left(\left(\left(n \cdot n\right) \cdot \log n\right) \cdot \left(\log i \cdot 50\right)\right))_* + \left(\left(n \cdot n\right) \cdot \log n\right) \cdot \left(\log i \cdot 50\right)\right) + (\frac{50}{3} \cdot \left(\log n \cdot \left(n \cdot \left(\left(n \cdot \log n\right) \cdot \left(n \cdot \log n\right)\right)\right)\right) + \left((\left(\left(\log i \cdot \log i\right) \cdot \left(\left(n \cdot n\right) \cdot \left(n \cdot \log n\right)\right)\right) \cdot \frac{50}{3} + \left(100 \cdot \left(n \cdot \log n\right)\right))_*\right))_*\right)}}{i} \cdot n\]
3. Recombined 3 regimes into one program.

4. Final simplification25.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -2.643994791550655 \cdot 10^{-18}:\\ \;\;\;\;n \cdot \frac{-100 + e^{\log_* (1 + \frac{i}{n}) \cdot n} \cdot 100}{i}\\ \mathbf{elif}\;i \le 2.5132717108214893 \cdot 10^{-17}:\\ \;\;\;\;\frac{i \cdot (i \cdot \left((i \cdot \frac{50}{3} + 50)_*\right) + 100)_*}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{(50 \cdot \left(\left(n \cdot \log n\right) \cdot \left(n \cdot \log n\right)\right) + \left((\frac{50}{3} \cdot \left(\left(\left(\left(n \cdot \log i\right) \cdot \left(n \cdot \log i\right)\right) \cdot n\right) \cdot \log i\right) + \left((100 \cdot \left(n \cdot \log i\right) + \left(\left(\left(n \cdot \log i\right) \cdot \left(n \cdot \log i\right)\right) \cdot 50 + \left(\log i \cdot \left(n \cdot \left(\left(n \cdot \log n\right) \cdot \left(n \cdot \log n\right)\right)\right)\right) \cdot 50\right))_*\right))_*\right))_* - \left((\frac{50}{3} \cdot \left(\left(n \cdot \left(\left(n \cdot \log n\right) \cdot \left(n \cdot \log n\right)\right)\right) \cdot \log n\right) + \left((\left(\left(\log i \cdot \log i\right) \cdot \left(\left(n \cdot \log n\right) \cdot \left(n \cdot n\right)\right)\right) \cdot \frac{50}{3} + \left(100 \cdot \left(n \cdot \log n\right)\right))_*\right))_* + \left((\left(\left(\log i \cdot \log i\right) \cdot \left(\left(n \cdot \log n\right) \cdot \left(n \cdot n\right)\right)\right) \cdot \frac{100}{3} + \left(\left(\log n \cdot \left(n \cdot n\right)\right) \cdot \left(\log i \cdot 50\right)\right))_* + \left(\log n \cdot \left(n \cdot n\right)\right) \cdot \left(\log i \cdot 50\right)\right)\right)}{i} \cdot n\\ \end{array}\]

Reproduce

herbie shell --seed 2019090 +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))))