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Target

Derivation

1. Split input into 5 regimes

2. if n < -8.764753048170504e+49

   1. Initial program 45.8

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

   3. Applied add-exp-log53.6

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

   4. Applied pow-exp53.6

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

   5. Applied expm1-def53.6

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

   6. Simplified27.4

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

   7. Taylor expanded around inf 42.5

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

   8. Simplified27.0

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

   if -8.764753048170504e+49 < n < -3.869064973251871e-99

   1. Initial program 25.7

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

   3. Applied associate-*r/25.6

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

   if -3.869064973251871e-99 < n < -2.549728235978189e-174

   1. Initial program 18.1

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

   3. Applied add-exp-log18.1

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

   4. Applied pow-exp18.1

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

   5. Applied expm1-def2.0

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

   6. Simplified23.3

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

   8. Applied add-sqr-sqrt25.4

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

   9. Applied *-un-lft-identity25.4

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

   10. Applied times-frac25.4

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

   if -2.549728235978189e-174 < n < 5.831751074444269e-195

   1. Initial program 22.6

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

   3. Applied add-exp-log22.6

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

   4. Applied pow-exp22.6

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

   5. Applied expm1-def15.2

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

   6. Simplified15.0

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

   7. Taylor expanded around 0 12.1

      \[\leadsto \color{blue}{0}\]

   if 5.831751074444269e-195 < n

   1. Initial program 57.2

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

   3. Applied add-exp-log57.2

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

   4. Applied pow-exp57.2

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

   5. Applied expm1-def50.7

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

   6. Simplified15.5

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

   8. Applied add-sqr-sqrt15.8

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

   9. Applied associate-*r*15.8

      \[\leadsto \color{blue}{\left(100 \cdot \sqrt{\frac{(e^{n \cdot \log_* (1 + \frac{i}{n})} - 1)^*}{\frac{i}{n}}}\right) \cdot \sqrt{\frac{(e^{n \cdot \log_* (1 + \frac{i}{n})} - 1)^*}{\frac{i}{n}}}}\]
3. Recombined 5 regimes into one program.

4. Final simplification20.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -8.764753048170504 \cdot 10^{+49}:\\ \;\;\;\;100 \cdot \frac{(e^{i} - 1)^*}{\frac{i}{n}}\\ \mathbf{elif}\;n \le -3.869064973251871 \cdot 10^{-99}:\\ \;\;\;\;\frac{\left({\left(\frac{i}{n} + 1\right)}^{n} - 1\right) \cdot 100}{\frac{i}{n}}\\ \mathbf{elif}\;n \le -2.549728235978189 \cdot 10^{-174}:\\ \;\;\;\;100 \cdot \left(\frac{(e^{\log_* (1 + \frac{i}{n}) \cdot n} - 1)^*}{\sqrt{\frac{i}{n}}} \cdot \frac{1}{\sqrt{\frac{i}{n}}}\right)\\ \mathbf{elif}\;n \le 5.831751074444269 \cdot 10^{-195}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{(e^{\log_* (1 + \frac{i}{n}) \cdot n} - 1)^*}{\frac{i}{n}}} \cdot \left(100 \cdot \sqrt{\frac{(e^{\log_* (1 + \frac{i}{n}) \cdot n} - 1)^*}{\frac{i}{n}}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2018362 +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))))

Details

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