Compound Interest

Average Error: 42.4 → 20.1

Time: 31.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;i \le -1.310331411391640804347629725946168576314 \cdot 10^{-11}:\\ \;\;\;\;\frac{100 \cdot \left({\left(\frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 1.280488904274837516756893343241770257007 \cdot 10^{-213}:\\ \;\;\;\;100 \cdot \left(\left(\sqrt[3]{\frac{1}{i} \cdot \left(\left(\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right) \cdot n\right)} \cdot \sqrt[3]{\frac{1}{i} \cdot \left(\left(\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right) \cdot n\right)}\right) \cdot \sqrt[3]{\frac{1}{i} \cdot \left(\left(\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right) \cdot n\right)}\right)\\ \mathbf{elif}\;i \le 1.07855901895175032609094606924337049933 \cdot 10^{-163}:\\ \;\;\;\;100 \cdot \left(\frac{1}{i} \cdot \sqrt[3]{{\left(\left(\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right) \cdot n\right)}^{3}}\right)\\ \mathbf{elif}\;i \le 0.454201659190358431050782428428647108376:\\ \;\;\;\;100 \cdot \left(\frac{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{i} \cdot n\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{100 \cdot \left({\left(\frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}\\ \end{array}\]

C

double f(double i, double n) {
        double r141855 = 100.0;
        double r141856 = 1.0;
        double r141857 = i;
        double r141858 = n;
        double r141859 = r141857 / r141858;
        double r141860 = r141856 + r141859;
        double r141861 = pow(r141860, r141858);
        double r141862 = r141861 - r141856;
        double r141863 = r141862 / r141859;
        double r141864 = r141855 * r141863;
        return r141864;
}

↓

double f(double i, double n) {
        double r141865 = i;
        double r141866 = -1.3103314113916408e-11;
        bool r141867 = r141865 <= r141866;
        double r141868 = 100.0;
        double r141869 = n;
        double r141870 = r141865 / r141869;
        double r141871 = pow(r141870, r141869);
        double r141872 = 1.0;
        double r141873 = r141871 - r141872;
        double r141874 = r141868 * r141873;
        double r141875 = r141874 / r141870;
        double r141876 = 1.2804889042748375e-213;
        bool r141877 = r141865 <= r141876;
        double r141878 = 1.0;
        double r141879 = r141878 / r141865;
        double r141880 = r141872 * r141865;
        double r141881 = 0.5;
        double r141882 = 2.0;
        double r141883 = pow(r141865, r141882);
        double r141884 = r141881 * r141883;
        double r141885 = log(r141872);
        double r141886 = r141885 * r141869;
        double r141887 = r141884 + r141886;
        double r141888 = r141880 + r141887;
        double r141889 = r141883 * r141885;
        double r141890 = r141881 * r141889;
        double r141891 = r141888 - r141890;
        double r141892 = r141891 * r141869;
        double r141893 = r141879 * r141892;
        double r141894 = cbrt(r141893);
        double r141895 = r141894 * r141894;
        double r141896 = r141895 * r141894;
        double r141897 = r141868 * r141896;
        double r141898 = 1.0785590189517503e-163;
        bool r141899 = r141865 <= r141898;
        double r141900 = 3.0;
        double r141901 = pow(r141892, r141900);
        double r141902 = cbrt(r141901);
        double r141903 = r141879 * r141902;
        double r141904 = r141868 * r141903;
        double r141905 = 0.45420165919035843;
        bool r141906 = r141865 <= r141905;
        double r141907 = r141891 / r141865;
        double r141908 = r141907 * r141869;
        double r141909 = r141868 * r141908;
        double r141910 = r141906 ? r141909 : r141875;
        double r141911 = r141899 ? r141904 : r141910;
        double r141912 = r141877 ? r141897 : r141911;
        double r141913 = r141867 ? r141875 : r141912;
        return r141913;
}

Error

Bits error versus i

Bits error versus n

Target

Original	42.4
Target	42.3
Herbie	20.1

\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

1. Split input into 4 regimes

2. if i < -1.3103314113916408e-11 or 0.45420165919035843 < i

   1. Initial program 29.9

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

   3. Applied associate-*r/ 29.8

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

   4. Taylor expanded around inf 51.5

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

   5. Simplified 23.9

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

   if -1.3103314113916408e-11 < i < 1.2804889042748375e-213

   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 div-inv 49.8

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

   4. Applied *-un-lft-identity 49.8

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

   5. Applied times-frac 49.5

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

   6. Simplified 49.5

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

   7. Taylor expanded around 0 15.7

      \[\leadsto 100 \cdot \left(\frac{1}{i} \cdot \left(\color{blue}{\left(\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right)} \cdot n\right)\right)\]
   8. Using strategy rm

   9. Applied add-cube-cbrt 16.3

      \[\leadsto 100 \cdot \color{blue}{\left(\left(\sqrt[3]{\frac{1}{i} \cdot \left(\left(\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right) \cdot n\right)} \cdot \sqrt[3]{\frac{1}{i} \cdot \left(\left(\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right) \cdot n\right)}\right) \cdot \sqrt[3]{\frac{1}{i} \cdot \left(\left(\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right) \cdot n\right)}\right)}\]

   if 1.2804889042748375e-213 < i < 1.0785590189517503e-163

   1. Initial program 49.0

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

   3. Applied div-inv 49.0

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

   4. Applied *-un-lft-identity 49.0

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

   5. Applied times-frac 48.6

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

   6. Simplified 48.6

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

   7. Taylor expanded around 0 15.7

      \[\leadsto 100 \cdot \left(\frac{1}{i} \cdot \left(\color{blue}{\left(\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right)} \cdot n\right)\right)\]
   8. Using strategy rm

   9. Applied add-cbrt-cube 39.2

      \[\leadsto 100 \cdot \left(\frac{1}{i} \cdot \left(\left(\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right) \cdot \color{blue}{\sqrt[3]{\left(n \cdot n\right) \cdot n}}\right)\right)\]

   10. Applied add-cbrt-cube 49.0

       \[\leadsto 100 \cdot \left(\frac{1}{i} \cdot \left(\color{blue}{\sqrt[3]{\left(\left(\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right) \cdot \left(\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right)\right) \cdot \left(\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right)}} \cdot \sqrt[3]{\left(n \cdot n\right) \cdot n}\right)\right)\]

   11. Applied cbrt-unprod 49.0

       \[\leadsto 100 \cdot \left(\frac{1}{i} \cdot \color{blue}{\sqrt[3]{\left(\left(\left(\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right) \cdot \left(\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right)\right) \cdot \left(\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right)\right) \cdot \left(\left(n \cdot n\right) \cdot n\right)}}\right)\]

   12. Simplified 27.1

       \[\leadsto 100 \cdot \left(\frac{1}{i} \cdot \sqrt[3]{\color{blue}{{\left(\left(\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right) \cdot n\right)}^{3}}}\right)\]

   if 1.0785590189517503e-163 < i < 0.45420165919035843

   1. Initial program 51.7

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

   3. Applied div-inv 51.7

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

   4. Applied *-un-lft-identity 51.7

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

   5. Applied times-frac 51.6

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

   6. Simplified 51.6

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

   7. Taylor expanded around 0 15.1

      \[\leadsto 100 \cdot \left(\frac{1}{i} \cdot \left(\color{blue}{\left(\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right)} \cdot n\right)\right)\]
   8. Using strategy rm

   9. Applied associate-*r* 18.2

      \[\leadsto 100 \cdot \color{blue}{\left(\left(\frac{1}{i} \cdot \left(\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right)\right) \cdot n\right)}\]

   10. Simplified 18.1

       \[\leadsto 100 \cdot \left(\color{blue}{\frac{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{i}} \cdot n\right)\]
3. Recombined 4 regimes into one program.

4. Final simplification 20.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -1.310331411391640804347629725946168576314 \cdot 10^{-11}:\\ \;\;\;\;\frac{100 \cdot \left({\left(\frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 1.280488904274837516756893343241770257007 \cdot 10^{-213}:\\ \;\;\;\;100 \cdot \left(\left(\sqrt[3]{\frac{1}{i} \cdot \left(\left(\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right) \cdot n\right)} \cdot \sqrt[3]{\frac{1}{i} \cdot \left(\left(\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right) \cdot n\right)}\right) \cdot \sqrt[3]{\frac{1}{i} \cdot \left(\left(\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right) \cdot n\right)}\right)\\ \mathbf{elif}\;i \le 1.07855901895175032609094606924337049933 \cdot 10^{-163}:\\ \;\;\;\;100 \cdot \left(\frac{1}{i} \cdot \sqrt[3]{{\left(\left(\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right) \cdot n\right)}^{3}}\right)\\ \mathbf{elif}\;i \le 0.454201659190358431050782428428647108376:\\ \;\;\;\;100 \cdot \left(\frac{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{i} \cdot n\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{100 \cdot \left({\left(\frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019294 
(FPCore (i n)
  :name "Compound Interest"
  :precision binary64

  :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))))