Error

Target

Derivation

1. Split input into 4 regimes

2. if i < -3.601052710078104e-08 or 2.1617355275962025e-278 < i < 1.7125708079176692e-169

   1. Initial program 35.6

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

   2. Simplified35.5

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

   4. Applied add-exp-log35.5

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

   5. Applied pow-exp35.5

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

   6. Simplified21.0

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

   8. Applied add-cube-cbrt21.4

      \[\leadsto \frac{(100 \cdot \left(e^{n \cdot \log_* (1 + \frac{i}{n})}\right) + -100)_*}{\color{blue}{\left(\sqrt[3]{\frac{i}{n}} \cdot \sqrt[3]{\frac{i}{n}}\right) \cdot \sqrt[3]{\frac{i}{n}}}}\]

   9. Applied *-un-lft-identity21.4

      \[\leadsto \frac{\color{blue}{1 \cdot (100 \cdot \left(e^{n \cdot \log_* (1 + \frac{i}{n})}\right) + -100)_*}}{\left(\sqrt[3]{\frac{i}{n}} \cdot \sqrt[3]{\frac{i}{n}}\right) \cdot \sqrt[3]{\frac{i}{n}}}\]

   10. Applied times-frac21.4

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

   12. Applied add-cbrt-cube21.5

       \[\leadsto \frac{1}{\color{blue}{\sqrt[3]{\left(\sqrt[3]{\frac{i}{n}} \cdot \sqrt[3]{\frac{i}{n}}\right) \cdot \sqrt[3]{\frac{i}{n}}}} \cdot \sqrt[3]{\frac{i}{n}}} \cdot \frac{(100 \cdot \left(e^{n \cdot \log_* (1 + \frac{i}{n})}\right) + -100)_*}{\sqrt[3]{\frac{i}{n}}}\]

   if -3.601052710078104e-08 < i < 2.1617355275962025e-278 or 1.7125708079176692e-169 < i < 1.1718846182195001e-20

   1. Initial program 50.5

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

   2. Simplified50.5

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

   3. Taylor expanded around 0 32.6

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

   4. Simplified32.6

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

   if 1.1718846182195001e-20 < i < 1.614908515676577e+230

   1. Initial program 35.6

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

   2. Simplified35.6

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

   4. Applied add-exp-log47.4

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

   5. Applied pow-exp47.4

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

   6. Simplified43.8

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

   8. Applied fma-udef43.9

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

   9. Taylor expanded around 0 19.8

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

   10. Simplified19.8

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

   if 1.614908515676577e+230 < i

   1. Initial program 31.0

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

   2. Simplified30.9

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

   4. Applied div-inv31.0

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

   5. Applied *-un-lft-identity31.0

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

   6. Applied times-frac31.0

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

   7. Simplified31.0

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

4. Final simplification27.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -3.601052710078104 \cdot 10^{-08}:\\ \;\;\;\;\frac{1}{\sqrt[3]{\left(\sqrt[3]{\frac{i}{n}} \cdot \sqrt[3]{\frac{i}{n}}\right) \cdot \sqrt[3]{\frac{i}{n}}} \cdot \sqrt[3]{\frac{i}{n}}} \cdot \frac{(100 \cdot \left(e^{n \cdot \log_* (1 + \frac{i}{n})}\right) + -100)_*}{\sqrt[3]{\frac{i}{n}}}\\ \mathbf{elif}\;i \le 2.1617355275962025 \cdot 10^{-278}:\\ \;\;\;\;\frac{i \cdot (i \cdot \left((i \cdot \frac{50}{3} + 50)_*\right) + 100)_*}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 1.7125708079176692 \cdot 10^{-169}:\\ \;\;\;\;\frac{1}{\sqrt[3]{\left(\sqrt[3]{\frac{i}{n}} \cdot \sqrt[3]{\frac{i}{n}}\right) \cdot \sqrt[3]{\frac{i}{n}}} \cdot \sqrt[3]{\frac{i}{n}}} \cdot \frac{(100 \cdot \left(e^{n \cdot \log_* (1 + \frac{i}{n})}\right) + -100)_*}{\sqrt[3]{\frac{i}{n}}}\\ \mathbf{elif}\;i \le 1.1718846182195001 \cdot 10^{-20}:\\ \;\;\;\;\frac{i \cdot (i \cdot \left((i \cdot \frac{50}{3} + 50)_*\right) + 100)_*}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 1.614908515676577 \cdot 10^{+230}:\\ \;\;\;\;\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(n \cdot \left(\left(n \cdot \log i\right) \cdot \left(n \cdot \log i\right)\right)\right) \cdot \log i\right) + \left((100 \cdot \left(n \cdot \log i\right) + \left(50 \cdot \left(\log i \cdot \left(n \cdot \left(\left(n \cdot \log n\right) \cdot \left(n \cdot \log n\right)\right)\right)\right) + 50 \cdot \left(\left(n \cdot \log i\right) \cdot \left(n \cdot \log i\right)\right)\right))_*\right))_*\right))_* - \left((\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(\left(n \cdot \log n\right) \cdot 100\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{100}{3} + \left(\left(\log n \cdot \left(n \cdot n\right)\right) \cdot \left(50 \cdot \log i\right)\right))_* + \left(\log n \cdot \left(n \cdot n\right)\right) \cdot \left(50 \cdot \log i\right)\right)\right)}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\left((\left({\left(1 + \frac{i}{n}\right)}^{n}\right) \cdot 100 + -100)_* \cdot n\right) \cdot \frac{1}{i}\\ \end{array}\]

Reproduce

herbie shell --seed 2019091 +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))))