Error

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Target

Derivation

1. Split input into 3 regimes

2. if n < -1.8564747933458696 or 7.104375322685901e-09 < n

   1. Initial program 51.0

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

   3. Applied add-exp-log51.0

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

   4. Applied expm1-def51.0

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

   5. Simplified22.9

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

   7. Applied associate-/r/24.0

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

   8. Taylor expanded around 0 4.9

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

   if -1.8564747933458696 < n < 2.7268656769834012e-222

   1. Initial program 18.6

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

   3. Applied add-exp-log18.6

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

   4. Applied expm1-def18.6

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

   5. Simplified22.9

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

   6. Taylor expanded around 0 14.8

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

   if 2.7268656769834012e-222 < n < 7.104375322685901e-09

   1. Initial program 53.0

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

   3. Applied add-exp-log53.0

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

   4. Applied expm1-def53.0

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

   5. Simplified9.4

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

   7. Applied div-inv9.5

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

   8. Applied associate-*r*9.4

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

   9. Simplified9.4

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

4. Final simplification8.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -1.8564747933458696:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{(e^{i} - 1)^*}{i}\right)\\ \mathbf{elif}\;n \le 2.7268656769834012 \cdot 10^{-222}:\\ \;\;\;\;0\\ \mathbf{elif}\;n \le 7.104375322685901 \cdot 10^{-09}:\\ \;\;\;\;\left(100 \cdot (e^{\log_* (1 + \frac{i}{n}) \cdot n} - 1)^*\right) \cdot \frac{n}{i}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{(e^{i} - 1)^*}{i}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019007 +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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