Error

Target

Derivation

1. Split input into 4 regimes

2. if i < -3.0747822661954016e-06

   1. Initial program 27.5

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

   3. Applied add-exp-log27.5

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

   4. Applied pow-exp27.5

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

   5. Applied simplify5.9

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

   if -3.0747822661954016e-06 < i < 0.05659584536834399

   1. Initial program 57.4

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

   2. Taylor expanded around 0 26.2

      \[\leadsto 100 \cdot \frac{\color{blue}{\frac{1}{6} \cdot {i}^{3} + \left(\frac{1}{2} \cdot {i}^{2} + i\right)}}{\frac{i}{n}}\]

   3. Applied simplify27.6

      \[\leadsto \color{blue}{(\left((\frac{1}{6} \cdot i + \frac{1}{2})_*\right) \cdot \left(i \cdot i\right) + i)_* \cdot \frac{100}{\frac{i}{n}}}\]
   4. Using strategy rm

   5. Applied pow127.6

      \[\leadsto (\left((\frac{1}{6} \cdot i + \frac{1}{2})_*\right) \cdot \left(i \cdot i\right) + i)_* \cdot \color{blue}{{\left(\frac{100}{\frac{i}{n}}\right)}^{1}}\]

   6. Applied pow127.6

      \[\leadsto \color{blue}{{\left((\left((\frac{1}{6} \cdot i + \frac{1}{2})_*\right) \cdot \left(i \cdot i\right) + i)_*\right)}^{1}} \cdot {\left(\frac{100}{\frac{i}{n}}\right)}^{1}\]

   7. Applied pow-prod-down27.6

      \[\leadsto \color{blue}{{\left((\left((\frac{1}{6} \cdot i + \frac{1}{2})_*\right) \cdot \left(i \cdot i\right) + i)_* \cdot \frac{100}{\frac{i}{n}}\right)}^{1}}\]

   8. Applied simplify9.7

      \[\leadsto {\color{blue}{\left(\left(\frac{100}{1} \cdot n\right) \cdot (i \cdot \left((\frac{1}{6} \cdot i + \frac{1}{2})_*\right) + 1)_*\right)}}^{1}\]

   if 0.05659584536834399 < i < 6.973980633362326e+141

   1. Initial program 28.7

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

   3. Applied add-exp-log37.6

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

   4. Applied pow-exp37.6

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

   5. Applied expm1-def20.0

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

   if 6.973980633362326e+141 < i

   1. Initial program 31.4

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

   3. Applied associate-*r/31.4

      \[\leadsto \color{blue}{\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}}\]
3. Recombined 4 regimes into one program.

4. Applied simplify11.1

   \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;i \le -3.0747822661954016 \cdot 10^{-06}:\\ \;\;\;\;100 \cdot \frac{e^{\log_* (1 + \frac{i}{n}) \cdot n} - 1}{\frac{i}{n}}\\ \mathbf{if}\;i \le 0.05659584536834399:\\ \;\;\;\;\left(100 \cdot n\right) \cdot (i \cdot \left((\frac{1}{6} \cdot i + \frac{1}{2})_*\right) + 1)_*\\ \mathbf{if}\;i \le 6.973980633362326 \cdot 10^{+141}:\\ \;\;\;\;100 \cdot \frac{(e^{n \cdot \log \left(1 + \frac{i}{n}\right)} - 1)^*}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right) \cdot 100}{\frac{i}{n}}\\ \end{array}}\]

Runtime

Time bar (total: 2.3m)Debug logProfile

herbie shell --seed 2020178 +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))))