Error

Target

Derivation

1. Split input into 4 regimes

2. if i < -1.8322447925192374e-58

   1. Initial program 33.9

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

   3. Applied pow-to-exp33.9

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

   4. Applied expm1-def27.2

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

   5. Simplified0.9

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

   7. Applied clear-num1.2

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

   if -1.8322447925192374e-58 < i < 4.7404880186548754e-268

   1. Initial program 49.3

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

   3. Applied pow-to-exp49.3

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

   4. Applied expm1-def44.9

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

   5. Simplified30.0

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

   7. Applied clear-num30.2

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

   8. Taylor expanded around 0 14.1

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

   9. Simplified13.3

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

   11. Applied un-div-inv13.4

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

   if 4.7404880186548754e-268 < i < 3.35176024751346e-09

   1. Initial program 49.8

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

   3. Applied pow-to-exp49.8

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

   4. Applied expm1-def43.9

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

   5. Simplified21.8

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

   7. Applied clear-num22.0

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

   8. Taylor expanded around 0 13.1

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

   9. Simplified12.1

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

   11. Applied log1p-expm1-u14.7

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

   if 3.35176024751346e-09 < i

   1. Initial program 32.8

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

   2. Taylor expanded around inf 30.4

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

   3. Simplified32.9

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

4. Final simplification12.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -1.8322447925192374 \cdot 10^{-58}:\\ \;\;\;\;100 \cdot \frac{1}{\frac{\frac{i}{n}}{(e^{\log_* (1 + \frac{i}{n}) \cdot n} - 1)^*}}\\ \mathbf{elif}\;i \le 4.7404880186548754 \cdot 10^{-268}:\\ \;\;\;\;\frac{100}{(\left(\frac{i}{n}\right) \cdot \left(\frac{\frac{1}{2}}{n} - \frac{1}{2}\right) + \left(\frac{1}{n}\right))_*}\\ \mathbf{elif}\;i \le 3.35176024751346 \cdot 10^{-09}:\\ \;\;\;\;100 \cdot \frac{1}{\log_* (1 + (e^{(\left(\frac{i}{n}\right) \cdot \left(\frac{\frac{1}{2}}{n} - \frac{1}{2}\right) + \left(\frac{1}{n}\right))_*} - 1)^*)}\\ \mathbf{else}:\\ \;\;\;\;\frac{(\left({\left(\frac{i}{n}\right)}^{n}\right) \cdot 100 + -100)_*}{\frac{i}{n}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019002 +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: 26.4s)Debug log