Compound Interest

Average Error: 43.1 → 21.4

Time: 24.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;i \le -0.1040807419505014597138625731531647033989:\\ \;\;\;\;100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\\ \mathbf{elif}\;i \le -1.431352546837178178390578970408631011162 \cdot 10^{-255}:\\ \;\;\;\;\left(\sqrt[3]{\frac{100}{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{100}{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{100}{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)}\\ \mathbf{elif}\;i \le 3.912989910623956103641462485659015187104 \cdot 10^{-198}:\\ \;\;\;\;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{elif}\;i \le 18.6629372047199559858654538402333855629:\\ \;\;\;\;\left(\sqrt[3]{\frac{100}{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{100}{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{100}{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)}\\ \mathbf{elif}\;i \le 5.064717226083425690234124091873168768786 \cdot 10^{144}:\\ \;\;\;\;\left(100 \cdot \frac{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} - 1} \cdot \sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} - 1}}{i}\right) \cdot \frac{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} - 1}}{\frac{1}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\left(1 \cdot i + \left(\log 1 \cdot n + 1\right)\right) - 1}{\frac{i}{n}}\\ \end{array}\]

C

double f(double i, double n) {
        double r104814 = 100.0;
        double r104815 = 1.0;
        double r104816 = i;
        double r104817 = n;
        double r104818 = r104816 / r104817;
        double r104819 = r104815 + r104818;
        double r104820 = pow(r104819, r104817);
        double r104821 = r104820 - r104815;
        double r104822 = r104821 / r104818;
        double r104823 = r104814 * r104822;
        return r104823;
}

↓

double f(double i, double n) {
        double r104824 = i;
        double r104825 = -0.10408074195050146;
        bool r104826 = r104824 <= r104825;
        double r104827 = 100.0;
        double r104828 = 1.0;
        double r104829 = n;
        double r104830 = r104824 / r104829;
        double r104831 = r104828 + r104830;
        double r104832 = pow(r104831, r104829);
        double r104833 = r104832 - r104828;
        double r104834 = r104833 / r104830;
        double r104835 = r104827 * r104834;
        double r104836 = -1.4313525468371782e-255;
        bool r104837 = r104824 <= r104836;
        double r104838 = r104827 / r104824;
        double r104839 = r104828 * r104824;
        double r104840 = 0.5;
        double r104841 = 2.0;
        double r104842 = pow(r104824, r104841);
        double r104843 = r104840 * r104842;
        double r104844 = log(r104828);
        double r104845 = r104844 * r104829;
        double r104846 = r104843 + r104845;
        double r104847 = r104839 + r104846;
        double r104848 = r104842 * r104844;
        double r104849 = r104840 * r104848;
        double r104850 = r104847 - r104849;
        double r104851 = r104850 * r104829;
        double r104852 = r104838 * r104851;
        double r104853 = cbrt(r104852);
        double r104854 = r104853 * r104853;
        double r104855 = r104854 * r104853;
        double r104856 = 3.912989910623956e-198;
        bool r104857 = r104824 <= r104856;
        double r104858 = r104850 / r104824;
        double r104859 = r104858 * r104829;
        double r104860 = r104827 * r104859;
        double r104861 = 18.662937204719956;
        bool r104862 = r104824 <= r104861;
        double r104863 = 5.064717226083426e+144;
        bool r104864 = r104824 <= r104863;
        double r104865 = cbrt(r104833);
        double r104866 = r104865 * r104865;
        double r104867 = r104866 / r104824;
        double r104868 = r104827 * r104867;
        double r104869 = 1.0;
        double r104870 = r104869 / r104829;
        double r104871 = r104865 / r104870;
        double r104872 = r104868 * r104871;
        double r104873 = r104845 + r104869;
        double r104874 = r104839 + r104873;
        double r104875 = r104874 - r104828;
        double r104876 = r104875 / r104830;
        double r104877 = r104827 * r104876;
        double r104878 = r104864 ? r104872 : r104877;
        double r104879 = r104862 ? r104855 : r104878;
        double r104880 = r104857 ? r104860 : r104879;
        double r104881 = r104837 ? r104855 : r104880;
        double r104882 = r104826 ? r104835 : r104881;
        return r104882;
}

Error

Bits error versus i

Bits error versus n

Target

Original	43.1
Target	42.7
Herbie	21.4

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

Derivation

1. Split input into 5 regimes

2. if i < -0.10408074195050146

   1. Initial program 29.8

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

   if -0.10408074195050146 < i < -1.4313525468371782e-255 or 3.912989910623956e-198 < i < 18.662937204719956

   1. Initial program 51.0

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

   2. Taylor expanded around 0 30.6

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

   4. Applied div-inv 30.7

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

   5. Applied *-un-lft-identity 30.7

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

   6. Applied times-frac 15.6

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

   7. Applied associate-*r* 15.7

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

   8. Simplified 15.7

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

   10. Applied add-cube-cbrt 16.2

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

   11. Simplified 16.2

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

   12. Simplified 16.2

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

   if -1.4313525468371782e-255 < i < 3.912989910623956e-198

   1. Initial program 49.3

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

   2. Taylor expanded around 0 40.8

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

   4. Applied associate-/r/ 14.4

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

   if 18.662937204719956 < i < 5.064717226083426e+144

   1. Initial program 30.0

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

   3. Applied div-inv 30.0

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

   4. Applied add-cube-cbrt 30.0

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

   5. Applied times-frac 30.0

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

   6. Applied associate-*r* 30.0

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

   if 5.064717226083426e+144 < i

   1. Initial program 30.1

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

   2. Taylor expanded around 0 36.9

      \[\leadsto 100 \cdot \frac{\color{blue}{\left(1 \cdot i + \left(\log 1 \cdot n + 1\right)\right)} - 1}{\frac{i}{n}}\]
3. Recombined 5 regimes into one program.

4. Final simplification 21.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -0.1040807419505014597138625731531647033989:\\ \;\;\;\;100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\\ \mathbf{elif}\;i \le -1.431352546837178178390578970408631011162 \cdot 10^{-255}:\\ \;\;\;\;\left(\sqrt[3]{\frac{100}{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{100}{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{100}{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)}\\ \mathbf{elif}\;i \le 3.912989910623956103641462485659015187104 \cdot 10^{-198}:\\ \;\;\;\;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{elif}\;i \le 18.6629372047199559858654538402333855629:\\ \;\;\;\;\left(\sqrt[3]{\frac{100}{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{100}{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{100}{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)}\\ \mathbf{elif}\;i \le 5.064717226083425690234124091873168768786 \cdot 10^{144}:\\ \;\;\;\;\left(100 \cdot \frac{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} - 1} \cdot \sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} - 1}}{i}\right) \cdot \frac{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} - 1}}{\frac{1}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\left(1 \cdot i + \left(\log 1 \cdot n + 1\right)\right) - 1}{\frac{i}{n}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019235 
(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))))