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Target

Derivation

1. Split input into 3 regimes

2. if (+ 1 (* 1/2 i)) < -3.5964007201178633e+21 or 1.00000000000054 < (+ 1 (* 1/2 i)) < 2.3039825294862687e+153

   1. Initial program 28.3

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

   3. Applied flip--28.3

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

   4. Applied simplify28.3

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

   if -3.5964007201178633e+21 < (+ 1 (* 1/2 i)) < 1.00000000000054

   1. Initial program 57.3

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

   2. Taylor expanded around 0 57.5

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

   3. Applied simplify26.4

      \[\leadsto \color{blue}{\frac{i \cdot \frac{1}{2} + 1}{\frac{\frac{i}{n}}{100 \cdot i}}}\]
   4. Using strategy rm

   5. Applied *-un-lft-identity26.4

      \[\leadsto \frac{i \cdot \frac{1}{2} + 1}{\frac{\color{blue}{1 \cdot \frac{i}{n}}}{100 \cdot i}}\]

   6. Applied times-frac26.4

      \[\leadsto \frac{i \cdot \frac{1}{2} + 1}{\color{blue}{\frac{1}{100} \cdot \frac{\frac{i}{n}}{i}}}\]

   7. Applied add-cube-cbrt26.4

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

   8. Applied times-frac26.3

      \[\leadsto \color{blue}{\frac{\sqrt[3]{i \cdot \frac{1}{2} + 1} \cdot \sqrt[3]{i \cdot \frac{1}{2} + 1}}{\frac{1}{100}} \cdot \frac{\sqrt[3]{i \cdot \frac{1}{2} + 1}}{\frac{\frac{i}{n}}{i}}}\]

   9. Applied simplify26.3

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

   10. Applied simplify10.8

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

   11. Taylor expanded around 0 10.7

       \[\leadsto \color{blue}{\left(\left(100 + \frac{100}{3} \cdot i\right) - \frac{25}{9} \cdot {i}^{2}\right)} \cdot \left(\sqrt[3]{1 + \frac{1}{2} \cdot i} \cdot n\right)\]

   if 2.3039825294862687e+153 < (+ 1 (* 1/2 i))

   1. Initial program 32.3

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

   2. Taylor expanded around 0 63.2

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

   3. Applied simplify35.1

      \[\leadsto \color{blue}{\frac{i \cdot \frac{1}{2} + 1}{\frac{\frac{i}{n}}{100 \cdot i}}}\]
   4. Using strategy rm

   5. Applied add-exp-log35.1

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

   6. Applied add-exp-log34.7

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

   7. Applied div-exp34.7

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

Runtime

Time bar (total: 6.9m)Debug logProfile

herbie shell --seed '#(1072967564 1937075727 894099792 790700740 1036514779 1027793188)' 
(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))))