Error

Target

Derivation

1. Split input into 2 regimes

2. if i < -3.3094449476994233e-149 or 8.195592385399544e-84 < i < 8.161881343676091e+113

   1. Initial program 40.3

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

   3. Applied add-exp-log40.3

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

   4. Applied expm1-def40.3

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

   5. Simplified4.2

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

   7. Applied associate-*r/4.2

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

   if -3.3094449476994233e-149 < i < 8.195592385399544e-84 or 8.161881343676091e+113 < i

   1. Initial program 54.5

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

   3. Applied add-exp-log54.5

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

   4. Applied expm1-def54.5

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

   5. Simplified27.4

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

   7. Applied associate-*r/27.5

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

   9. Applied clear-num27.7

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

   10. Taylor expanded around 0 16.7

       \[\leadsto \frac{1}{\color{blue}{\left(\frac{1}{200} \cdot \frac{i}{{n}^{2}} + \frac{1}{100} \cdot \frac{1}{n}\right) - \frac{1}{200} \cdot \frac{i}{n}}}\]

   11. Simplified7.1

       \[\leadsto \frac{1}{\color{blue}{(\left(\frac{\frac{1}{200}}{n}\right) \cdot \left(\frac{i}{n} - i\right) + \left(\frac{\frac{1}{100}}{n}\right))_*}}\]
3. Recombined 2 regimes into one program.

4. Final simplification5.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -3.3094449476994233 \cdot 10^{-149} \lor \neg \left(i \le 8.195592385399544 \cdot 10^{-84}\right) \land i \le 8.161881343676091 \cdot 10^{+113}:\\ \;\;\;\;\frac{(e^{\log_* (1 + \frac{i}{n}) \cdot n} - 1)^* \cdot 100}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{(\left(\frac{\frac{1}{200}}{n}\right) \cdot \left(\frac{i}{n} - i\right) + \left(\frac{\frac{1}{100}}{n}\right))_*}\\ \end{array}\]

Runtime

Time bar (total: 29.9s)Debug logProfile

herbie shell --seed 2018290 +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))))