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Target

Derivation

1. Split input into 3 regimes

2. if n < -2.3679928838256e+52 or 0.0006875070209031384 < n

   1. Initial program 52.7

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

   3. Applied add-exp-log52.7

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

   4. Applied expm1-def52.7

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

   5. Simplified24.3

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

   7. Applied associate-/r/25.5

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

   8. Applied associate-*r*25.5

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

   9. Taylor expanded around 0 4.3

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

   if -2.3679928838256e+52 < n < -5.719087173721921e-256 or 1.3901384564376271e-241 < n < 0.0006875070209031384

   1. Initial program 45.0

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

   3. Applied add-exp-log45.0

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

   4. Applied expm1-def45.0

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

   5. Simplified5.8

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

   7. Applied associate-/r/6.2

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

   8. Applied associate-*r*6.1

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

   if -5.719087173721921e-256 < n < 1.3901384564376271e-241

   1. Initial program 23.5

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

   3. Applied add-exp-log23.5

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

   4. Applied expm1-def23.5

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

   5. Simplified14.4

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

   6. Taylor expanded around 0 9.2

      \[\leadsto \color{blue}{0}\]
3. Recombined 3 regimes into one program.

4. Final simplification5.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -2.3679928838256 \cdot 10^{+52}:\\ \;\;\;\;\left(100 \cdot \frac{(e^{i} - 1)^*}{i}\right) \cdot n\\ \mathbf{elif}\;n \le -5.719087173721921 \cdot 10^{-256}:\\ \;\;\;\;n \cdot \left(\frac{(e^{\log_* (1 + \frac{i}{n}) \cdot n} - 1)^*}{i} \cdot 100\right)\\ \mathbf{elif}\;n \le 1.3901384564376271 \cdot 10^{-241}:\\ \;\;\;\;0\\ \mathbf{elif}\;n \le 0.0006875070209031384:\\ \;\;\;\;n \cdot \left(\frac{(e^{\log_* (1 + \frac{i}{n}) \cdot n} - 1)^*}{i} \cdot 100\right)\\ \mathbf{else}:\\ \;\;\;\;\left(100 \cdot \frac{(e^{i} - 1)^*}{i}\right) \cdot n\\ \end{array}\]

Runtime

Time bar (total: 33.6s)Debug logProfile

herbie shell --seed 2018295 +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))))