Average Error: 42.6 → 21.6
Time: 39.8s
Precision: 64
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le -1.77715864264781547 \cdot 10^{-6}:\\ \;\;\;\;100 \cdot \frac{\frac{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} - 1 \cdot 1}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 2.88216887028242915 \cdot 10^{-18}:\\ \;\;\;\;100 \cdot \frac{\frac{1}{\sqrt[3]{i} \cdot \sqrt[3]{i}}}{\frac{\frac{1}{n}}{\frac{\sqrt[3]{\mathsf{fma}\left(1, i, \mathsf{fma}\left(0.5, i \cdot i, \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)} \cdot \sqrt[3]{\mathsf{fma}\left(1, i, \mathsf{fma}\left(0.5, i \cdot i, \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}}{\sqrt[3]{\sqrt[3]{i} \cdot \sqrt[3]{i}}} \cdot \frac{\sqrt[3]{\mathsf{fma}\left(1, i, \mathsf{fma}\left(0.5, i \cdot i, \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}}{\sqrt[3]{\sqrt[3]{i}}}}}\\ \mathbf{elif}\;i \le 2.1606840039657686 \cdot 10^{162}:\\ \;\;\;\;100 \cdot \frac{\frac{\frac{{\left({\left(1 + \frac{i}{n}\right)}^{n}\right)}^{3} - {1}^{3}}{1 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} + 1\right) + {\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)}}}{i}}{\frac{1}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\mathsf{fma}\left(1, i, \mathsf{fma}\left(\log 1, n, 1\right)\right) - 1}{\frac{i}{n}}\\ \end{array}\]
100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}
↓
\begin{array}{l}
\mathbf{if}\;i \le -1.77715864264781547 \cdot 10^{-6}:\\
\;\;\;\;100 \cdot \frac{\frac{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} - 1 \cdot 1}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{\frac{i}{n}}\\

\mathbf{elif}\;i \le 2.88216887028242915 \cdot 10^{-18}:\\
\;\;\;\;100 \cdot \frac{\frac{1}{\sqrt[3]{i} \cdot \sqrt[3]{i}}}{\frac{\frac{1}{n}}{\frac{\sqrt[3]{\mathsf{fma}\left(1, i, \mathsf{fma}\left(0.5, i \cdot i, \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)} \cdot \sqrt[3]{\mathsf{fma}\left(1, i, \mathsf{fma}\left(0.5, i \cdot i, \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}}{\sqrt[3]{\sqrt[3]{i} \cdot \sqrt[3]{i}}} \cdot \frac{\sqrt[3]{\mathsf{fma}\left(1, i, \mathsf{fma}\left(0.5, i \cdot i, \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}}{\sqrt[3]{\sqrt[3]{i}}}}}\\

\mathbf{elif}\;i \le 2.1606840039657686 \cdot 10^{162}:\\
\;\;\;\;100 \cdot \frac{\frac{\frac{{\left({\left(1 + \frac{i}{n}\right)}^{n}\right)}^{3} - {1}^{3}}{1 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} + 1\right) + {\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)}}}{i}}{\frac{1}{n}}\\

\mathbf{else}:\\
\;\;\;\;100 \cdot \frac{\mathsf{fma}\left(1, i, \mathsf{fma}\left(\log 1, n, 1\right)\right) - 1}{\frac{i}{n}}\\

\end{array}
double f(double i, double n) {
        double r214860 = 100.0;
        double r214861 = 1.0;
        double r214862 = i;
        double r214863 = n;
        double r214864 = r214862 / r214863;
        double r214865 = r214861 + r214864;
        double r214866 = pow(r214865, r214863);
        double r214867 = r214866 - r214861;
        double r214868 = r214867 / r214864;
        double r214869 = r214860 * r214868;
        return r214869;
}
↓
double f(double i, double n) {
        double r214870 = i;
        double r214871 = -1.7771586426478155e-06;
        bool r214872 = r214870 <= r214871;
        double r214873 = 100.0;
        double r214874 = 1.0;
        double r214875 = n;
        double r214876 = r214870 / r214875;
        double r214877 = r214874 + r214876;
        double r214878 = 2.0;
        double r214879 = r214878 * r214875;
        double r214880 = pow(r214877, r214879);
        double r214881 = r214874 * r214874;
        double r214882 = r214880 - r214881;
        double r214883 = pow(r214877, r214875);
        double r214884 = r214883 + r214874;
        double r214885 = r214882 / r214884;
        double r214886 = r214885 / r214876;
        double r214887 = r214873 * r214886;
        double r214888 = 2.882168870282429e-18;
        bool r214889 = r214870 <= r214888;
        double r214890 = 1.0;
        double r214891 = cbrt(r214870);
        double r214892 = r214891 * r214891;
        double r214893 = r214890 / r214892;
        double r214894 = r214890 / r214875;
        double r214895 = 0.5;
        double r214896 = r214870 * r214870;
        double r214897 = log(r214874);
        double r214898 = r214897 * r214875;
        double r214899 = fma(r214895, r214896, r214898);
        double r214900 = fma(r214874, r214870, r214899);
        double r214901 = pow(r214870, r214878);
        double r214902 = r214901 * r214897;
        double r214903 = r214895 * r214902;
        double r214904 = r214900 - r214903;
        double r214905 = cbrt(r214904);
        double r214906 = r214905 * r214905;
        double r214907 = cbrt(r214892);
        double r214908 = r214906 / r214907;
        double r214909 = cbrt(r214891);
        double r214910 = r214905 / r214909;
        double r214911 = r214908 * r214910;
        double r214912 = r214894 / r214911;
        double r214913 = r214893 / r214912;
        double r214914 = r214873 * r214913;
        double r214915 = 2.1606840039657686e+162;
        bool r214916 = r214870 <= r214915;
        double r214917 = 3.0;
        double r214918 = pow(r214883, r214917);
        double r214919 = pow(r214874, r214917);
        double r214920 = r214918 - r214919;
        double r214921 = r214874 * r214884;
        double r214922 = r214921 + r214880;
        double r214923 = r214920 / r214922;
        double r214924 = r214923 / r214870;
        double r214925 = r214924 / r214894;
        double r214926 = r214873 * r214925;
        double r214927 = fma(r214897, r214875, r214890);
        double r214928 = fma(r214874, r214870, r214927);
        double r214929 = r214928 - r214874;
        double r214930 = r214929 / r214876;
        double r214931 = r214873 * r214930;
        double r214932 = r214916 ? r214926 : r214931;
        double r214933 = r214889 ? r214914 : r214932;
        double r214934 = r214872 ? r214887 : r214933;
        return r214934;
}

Error

Bits error versus i

Bits error versus n

Target

Original42.6
Target42.5
Herbie21.6
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 4 regimes

  2. if i < -1.7771586426478155e-06

    1. Initial program 27.7

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

    3. Applied flip--27.7

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

    4. Simplified27.7

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

    if -1.7771586426478155e-06 < i < 2.882168870282429e-18

    1. Initial program 50.5

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

    3. Applied div-inv50.5

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

    4. Applied associate-/r*50.2

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

    5. Taylor expanded around 0 16.7

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

    6. Simplified16.7

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

    8. Applied add-cube-cbrt17.6

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

    9. Applied *-un-lft-identity17.6

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

    10. Applied times-frac17.6

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

    11. Applied associate-/l*16.1

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

    13. Applied add-cube-cbrt16.2

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

    14. Applied cbrt-prod16.3

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

    15. Applied add-cube-cbrt15.8

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

    16. Applied times-frac15.8

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

    if 2.882168870282429e-18 < i < 2.1606840039657686e+162

    1. Initial program 36.1

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

    3. Applied div-inv36.1

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

    4. Applied associate-/r*36.1

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

    6. Applied flip3--36.1

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

    7. Simplified36.1

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

    if 2.1606840039657686e+162 < i

    1. Initial program 31.9

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

    2. Taylor expanded around 0 35.5

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

    3. Simplified35.5

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

  4. Final simplification21.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -1.77715864264781547 \cdot 10^{-6}:\\ \;\;\;\;100 \cdot \frac{\frac{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} - 1 \cdot 1}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 2.88216887028242915 \cdot 10^{-18}:\\ \;\;\;\;100 \cdot \frac{\frac{1}{\sqrt[3]{i} \cdot \sqrt[3]{i}}}{\frac{\frac{1}{n}}{\frac{\sqrt[3]{\mathsf{fma}\left(1, i, \mathsf{fma}\left(0.5, i \cdot i, \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)} \cdot \sqrt[3]{\mathsf{fma}\left(1, i, \mathsf{fma}\left(0.5, i \cdot i, \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}}{\sqrt[3]{\sqrt[3]{i} \cdot \sqrt[3]{i}}} \cdot \frac{\sqrt[3]{\mathsf{fma}\left(1, i, \mathsf{fma}\left(0.5, i \cdot i, \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}}{\sqrt[3]{\sqrt[3]{i}}}}}\\ \mathbf{elif}\;i \le 2.1606840039657686 \cdot 10^{162}:\\ \;\;\;\;100 \cdot \frac{\frac{\frac{{\left({\left(1 + \frac{i}{n}\right)}^{n}\right)}^{3} - {1}^{3}}{1 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} + 1\right) + {\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)}}}{i}}{\frac{1}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\mathsf{fma}\left(1, i, \mathsf{fma}\left(\log 1, n, 1\right)\right) - 1}{\frac{i}{n}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020042 +o rules:numerics
(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))))