Error

Target

Derivation

1. Split input into 3 regimes

2. if (/ (- (pow (+ 1 (/ i n)) n) 1) (/ i n)) < 4.599814456066394e-286

   1. Initial program 40.2

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

   3. Applied add-exp-log40.2

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

   4. Applied expm1-def40.2

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

   5. Simplified6.8

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

   7. Applied *-commutative6.8

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

   if 4.599814456066394e-286 < (/ (- (pow (+ 1 (/ i n)) n) 1) (/ i n)) < 515595789703.47595

   1. Initial program 3.1

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

   3. Applied add-exp-log3.1

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

   4. Applied expm1-def3.1

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

   5. Simplified51.2

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

   7. Applied *-commutative51.2

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

   8. Taylor expanded around inf 58.4

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

   9. Simplified3.9

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

   if 515595789703.47595 < (/ (- (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

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

   4. Applied expm1-def61.8

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

   5. Simplified61.8

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

   6. Taylor expanded around 0 48.1

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

4. Final simplification15.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{\frac{i}{n}} \le 4.599814456066394 \cdot 10^{-286}:\\ \;\;\;\;100 \cdot \frac{(e^{n \cdot \log_* (1 + \frac{i}{n})} - 1)^*}{\frac{i}{n}}\\ \mathbf{elif}\;\frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{\frac{i}{n}} \le 515595789703.47595:\\ \;\;\;\;\frac{(\left({\left(\frac{i}{n}\right)}^{n}\right) \cdot 100 + -100)_*}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Reproduce

herbie shell --seed 2019008 +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: 24.2s)Debug log