Error

Target

Derivation

1. Split input into 4 regimes

2. if i < -4.900146667805007e-05

   1. Initial program 28.2

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

   2. Simplified28.2

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

   4. Applied add-exp-log28.2

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

   5. Applied pow-exp28.2

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

   6. Simplified5.6

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

   8. Applied div-inv5.7

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

   10. Applied add-log-exp5.7

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

   if -4.900146667805007e-05 < i < 4.8794609297355254e-23

   1. Initial program 50.0

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

   2. Simplified50.0

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

   3. Taylor expanded around 0 33.7

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

   4. Simplified33.7

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

   if 4.8794609297355254e-23 < i < 7.399798187623056e+238 or 1.6692294268394334e+295 < i

   1. Initial program 33.8

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

   2. Simplified33.8

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

   4. Applied add-exp-log48.4

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

   5. Applied pow-exp48.4

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

   6. Simplified44.7

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

   7. Taylor expanded around 0 23.6

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

   8. Simplified23.6

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

   if 7.399798187623056e+238 < i < 1.6692294268394334e+295

   1. Initial program 33.5

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

   2. Simplified33.5

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

   4. Applied div-inv33.5

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

   5. Applied *-un-lft-identity33.5

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

   6. Applied times-frac33.5

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

   7. Simplified33.5

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

4. Final simplification25.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -4.900146667805007 \cdot 10^{-05}:\\ \;\;\;\;\log \left(e^{(\left(e^{n \cdot \log_* (1 + \frac{i}{n})}\right) \cdot 100 + -100)_*}\right) \cdot \frac{1}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 4.8794609297355254 \cdot 10^{-23}:\\ \;\;\;\;\frac{(\left((i \cdot \frac{50}{3} + 50)_*\right) \cdot i + 100)_* \cdot i}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 7.399798187623056 \cdot 10^{+238}:\\ \;\;\;\;\frac{(\left(50 \cdot \left(n \cdot n\right)\right) \cdot \left(\log n \cdot \log n\right) + \left(\left(\left(\left(\left(n \cdot n\right) \cdot \frac{50}{3}\right) \cdot n\right) \cdot \left(\left(\log i \cdot \log i\right) \cdot \log i\right) - (\left(\left(n \cdot n\right) \cdot \left(\frac{100}{3} \cdot n\right)\right) \cdot \left(\log n \cdot \left(\log i \cdot \log i\right)\right) + \left((\left(\left(\left(n \cdot n\right) \cdot \frac{50}{3}\right) \cdot n\right) \cdot \left(\left(\log n \cdot \log n\right) \cdot \log n\right) + \left((\left(\left(\left(n \cdot n\right) \cdot \frac{50}{3}\right) \cdot n\right) \cdot \left(\log n \cdot \left(\log i \cdot \log i\right)\right) + \left(\left(\log n \cdot n\right) \cdot 100\right))_*\right))_* + 100 \cdot \left(\left(\log n \cdot \left(n \cdot n\right)\right) \cdot \log i\right)\right))_*\right) + (100 \cdot \left(\log i \cdot n\right) + \left(\left(\log i \cdot \left(50 \cdot \left(n \cdot n\right)\right)\right) \cdot \log i + \left(\left(n \cdot \left(n \cdot n\right)\right) \cdot \left(\log i \cdot \left(\log n \cdot \log n\right)\right)\right) \cdot 50\right))_*\right))_*}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 1.6692294268394334 \cdot 10^{+295}:\\ \;\;\;\;\frac{1}{i} \cdot \left(n \cdot (\left({\left(\frac{i}{n} + 1\right)}^{n}\right) \cdot 100 + -100)_*\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{(\left(50 \cdot \left(n \cdot n\right)\right) \cdot \left(\log n \cdot \log n\right) + \left(\left(\left(\left(\left(n \cdot n\right) \cdot \frac{50}{3}\right) \cdot n\right) \cdot \left(\left(\log i \cdot \log i\right) \cdot \log i\right) - (\left(\left(n \cdot n\right) \cdot \left(\frac{100}{3} \cdot n\right)\right) \cdot \left(\log n \cdot \left(\log i \cdot \log i\right)\right) + \left((\left(\left(\left(n \cdot n\right) \cdot \frac{50}{3}\right) \cdot n\right) \cdot \left(\left(\log n \cdot \log n\right) \cdot \log n\right) + \left((\left(\left(\left(n \cdot n\right) \cdot \frac{50}{3}\right) \cdot n\right) \cdot \left(\log n \cdot \left(\log i \cdot \log i\right)\right) + \left(\left(\log n \cdot n\right) \cdot 100\right))_*\right))_* + 100 \cdot \left(\left(\log n \cdot \left(n \cdot n\right)\right) \cdot \log i\right)\right))_*\right) + (100 \cdot \left(\log i \cdot n\right) + \left(\left(\log i \cdot \left(50 \cdot \left(n \cdot n\right)\right)\right) \cdot \log i + \left(\left(n \cdot \left(n \cdot n\right)\right) \cdot \left(\log i \cdot \left(\log n \cdot \log n\right)\right)\right) \cdot 50\right))_*\right))_*}{\frac{i}{n}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019093 +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))))