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Target

Derivation

1. Split input into 3 regimes

2. if n < -0.5048727192312397 or 0.0021140823519030126 < n

   1. Initial program 50.9

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

   3. Applied add-exp-log50.9

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

   4. Applied expm1-def50.9

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

   5. Simplified23.3

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

   7. Applied associate-*r/23.3

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

   9. Applied associate-/r/24.5

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

   10. Taylor expanded around 0 5.0

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

   if -0.5048727192312397 < n < -1.4243394807001e-310

   1. Initial program 16.5

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

   3. Applied add-exp-log16.5

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

   4. Applied expm1-def16.5

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

   5. Simplified19.7

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

   7. Applied associate-*r/19.6

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

   9. Applied add-cbrt-cube10.0

      \[\leadsto \frac{100 \cdot (e^{\color{blue}{\sqrt[3]{\left(\log_* (1 + \frac{i}{n}) \cdot \log_* (1 + \frac{i}{n})\right) \cdot \log_* (1 + \frac{i}{n})}} \cdot n} - 1)^*}{\frac{i}{n}}\]

   if -1.4243394807001e-310 < n < 0.0021140823519030126

   1. Initial program 49.1

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

   3. Applied add-exp-log49.1

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

   4. Applied expm1-def49.1

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

   5. Simplified12.9

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

   7. Applied associate-*r/12.9

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

   9. Applied associate-/r/12.8

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

   11. Applied associate-/l*12.8

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

4. Final simplification7.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -0.5048727192312397:\\ \;\;\;\;\frac{100 \cdot (e^{i} - 1)^*}{i} \cdot n\\ \mathbf{elif}\;n \le -1.4243394807001 \cdot 10^{-310}:\\ \;\;\;\;\frac{100 \cdot (e^{\sqrt[3]{\left(\log_* (1 + \frac{i}{n}) \cdot \log_* (1 + \frac{i}{n})\right) \cdot \log_* (1 + \frac{i}{n})} \cdot n} - 1)^*}{\frac{i}{n}}\\ \mathbf{elif}\;n \le 0.0021140823519030126:\\ \;\;\;\;n \cdot \frac{100}{\frac{i}{(e^{\log_* (1 + \frac{i}{n}) \cdot n} - 1)^*}}\\ \mathbf{else}:\\ \;\;\;\;\frac{100 \cdot (e^{i} - 1)^*}{i} \cdot n\\ \end{array}\]

Reproduce

herbie shell --seed 2019005 +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

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