Error

Target

Derivation

1. Split input into 3 regimes

2. if i < -9.786895623065357e-113

   1. Initial program 35.6

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

   3. Applied pow-to-exp35.6

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

   4. Applied expm1-def29.6

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

   5. Simplified2.6

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

   7. Applied associate-*r/2.5

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

   9. Applied clear-num2.8

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

   if -9.786895623065357e-113 < i < 9.869555751605711e-90 or 7.239073862837944e+78 < i

   1. Initial program 44.9

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

   3. Applied pow-to-exp49.5

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

   4. Applied expm1-def45.3

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

   5. Simplified34.0

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

   7. Applied associate-*r/34.0

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

   9. Applied clear-num34.2

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

   10. Taylor expanded around 0 21.6

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

   11. Simplified15.9

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

   if 9.869555751605711e-90 < i < 7.239073862837944e+78

   1. Initial program 46.4

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

   3. Applied pow-to-exp47.4

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

   4. Applied expm1-def35.8

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

   5. Simplified6.4

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

   7. Applied associate-/r/7.0

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

   8. Applied associate-*r*6.9

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

4. Final simplification10.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -9.786895623065357 \cdot 10^{-113}:\\ \;\;\;\;\frac{1}{\frac{\frac{i}{n}}{(e^{\log_* (1 + \frac{i}{n}) \cdot n} - 1)^* \cdot 100}}\\ \mathbf{elif}\;i \le 9.869555751605711 \cdot 10^{-90} \lor \neg \left(i \le 7.239073862837944 \cdot 10^{+78}\right):\\ \;\;\;\;\frac{1}{(\left(\frac{\frac{1}{200}}{n}\right) \cdot \left(\frac{i}{n} - i\right) + \left(\frac{\frac{1}{100}}{n}\right))_*}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 \cdot \frac{(e^{\log_* (1 + \frac{i}{n}) \cdot n} - 1)^*}{i}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2018365 +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

Time bar (total: 17.7s)Debug log