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Target

Derivation

1. Split input into 3 regimes

2. if n < -1.5274599045984505e-107

   1. Initial program 39.4

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

   3. Applied pow-to-exp45.4

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

   4. Applied expm1-def41.1

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

   5. Simplified26.1

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

   7. Applied div-inv26.1

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

   8. Applied *-un-lft-identity26.1

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

   9. Applied times-frac25.4

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

   10. Applied associate-*r*25.4

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

   11. Simplified25.4

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

   if -1.5274599045984505e-107 < n < 2.195458028809774e-180

   1. Initial program 22.8

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

   3. Applied pow-to-exp22.8

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

   4. Applied expm1-def13.4

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

   5. Simplified17.5

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

   6. Taylor expanded around 0 14.4

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

   if 2.195458028809774e-180 < n

   1. Initial program 57.5

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

   3. Applied pow-to-exp57.5

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

   4. Applied expm1-def51.1

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

   5. Simplified15.6

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

   7. Applied *-commutative15.6

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

4. Final simplification19.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -1.5274599045984505 \cdot 10^{-107}:\\ \;\;\;\;\left((e^{\log_* (1 + \frac{i}{n}) \cdot n} - 1)^* \cdot n\right) \cdot \left(100 \cdot \frac{1}{i}\right)\\ \mathbf{elif}\;n \le 2.195458028809774 \cdot 10^{-180}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{(e^{\log_* (1 + \frac{i}{n}) \cdot n} - 1)^*}{\frac{i}{n}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019004 +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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