Error

Target

Derivation

1. Split input into 3 regimes

2. if (/ (- (pow (+ 1 (/ i n)) n) 1) (/ i n)) < 7.252615379260614e-269

   1. Initial program 45.2

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

   3. Applied add-exp-log45.2

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

   4. Applied pow-exp45.2

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

   5. Applied expm1-def37.5

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

   6. Simplified0.3

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

   8. Applied associate-*r/0.3

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

   if 7.252615379260614e-269 < (/ (- (pow (+ 1 (/ i n)) n) 1) (/ i n)) < 95524185912919.44

   1. Initial program 3.5

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

   3. Applied associate-*r/3.4

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

   if 95524185912919.44 < (/ (- (pow (+ 1 (/ i n)) n) 1) (/ i n))

   1. Initial program 61.8

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

   3. Applied add-exp-log61.8

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

   4. Applied pow-exp61.8

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

   5. Applied expm1-def61.8

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

   6. Simplified61.8

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

   8. Applied *-un-lft-identity61.8

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

   9. Applied associate-/l*61.8

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

   10. Taylor expanded around 0 13.8

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

   11. Simplified0.4

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

4. Final simplification0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{\frac{i}{n}} \le 7.252615379260614 \cdot 10^{-269}:\\ \;\;\;\;\frac{100 \cdot (e^{\log_* (1 + \frac{i}{n}) \cdot n} - 1)^*}{\frac{i}{n}}\\ \mathbf{elif}\;\frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{\frac{i}{n}} \le 95524185912919.44:\\ \;\;\;\;\frac{100 \cdot \left({\left(\frac{i}{n} + 1\right)}^{n} - 1\right)}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{(\left(\frac{i}{n}\right) \cdot \left(\frac{\frac{1}{2}}{n} - \frac{1}{2}\right) + \left(\frac{1}{n}\right))_*} \cdot 100\\ \end{array}\]

Runtime

Time bar (total: 39.1s)Debug logProfile

herbie shell --seed 2018263 +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))))