Compound Interest

Average Error: 43.1 → 34.1

Time: 15.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;n \le -8.046837769241036 \cdot 10^{81}:\\ \;\;\;\;100 \cdot \frac{\mathsf{fma}\left(i, 1, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right)}{\frac{i}{n}}\\ \mathbf{elif}\;n \le -4.5306115068498307 \cdot 10^{56}:\\ \;\;\;\;\frac{100}{i} \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{1}{n}}\\ \mathbf{elif}\;n \le -15831.5864445897478:\\ \;\;\;\;100 \cdot \frac{\mathsf{fma}\left(i, 1, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right)}{\frac{i}{n}}\\ \mathbf{elif}\;n \le 8.1979849820500912 \cdot 10^{-194}:\\ \;\;\;\;\left(100 \cdot \frac{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} - 1} \cdot \sqrt[3]{\log \left(e^{{\left(1 + \frac{i}{n}\right)}^{n} - 1}\right)}}{i}\right) \cdot \frac{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} - 1}}{\frac{1}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\mathsf{fma}\left(i, 1, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right)}{\frac{i}{n}}\\ \end{array}\]

C

double f(double i, double n) {
        double r128823 = 100.0;
        double r128824 = 1.0;
        double r128825 = i;
        double r128826 = n;
        double r128827 = r128825 / r128826;
        double r128828 = r128824 + r128827;
        double r128829 = pow(r128828, r128826);
        double r128830 = r128829 - r128824;
        double r128831 = r128830 / r128827;
        double r128832 = r128823 * r128831;
        return r128832;
}

↓

double f(double i, double n) {
        double r128833 = n;
        double r128834 = -8.046837769241036e+81;
        bool r128835 = r128833 <= r128834;
        double r128836 = 100.0;
        double r128837 = i;
        double r128838 = 1.0;
        double r128839 = 0.5;
        double r128840 = 2.0;
        double r128841 = pow(r128837, r128840);
        double r128842 = log(r128838);
        double r128843 = r128842 * r128833;
        double r128844 = fma(r128839, r128841, r128843);
        double r128845 = r128841 * r128842;
        double r128846 = r128839 * r128845;
        double r128847 = r128844 - r128846;
        double r128848 = fma(r128837, r128838, r128847);
        double r128849 = r128837 / r128833;
        double r128850 = r128848 / r128849;
        double r128851 = r128836 * r128850;
        double r128852 = -4.530611506849831e+56;
        bool r128853 = r128833 <= r128852;
        double r128854 = r128836 / r128837;
        double r128855 = r128838 + r128849;
        double r128856 = pow(r128855, r128833);
        double r128857 = r128856 - r128838;
        double r128858 = 1.0;
        double r128859 = r128858 / r128833;
        double r128860 = r128857 / r128859;
        double r128861 = r128854 * r128860;
        double r128862 = -15831.586444589748;
        bool r128863 = r128833 <= r128862;
        double r128864 = 8.197984982050091e-194;
        bool r128865 = r128833 <= r128864;
        double r128866 = cbrt(r128857);
        double r128867 = exp(r128857);
        double r128868 = log(r128867);
        double r128869 = cbrt(r128868);
        double r128870 = r128866 * r128869;
        double r128871 = r128870 / r128837;
        double r128872 = r128836 * r128871;
        double r128873 = r128866 / r128859;
        double r128874 = r128872 * r128873;
        double r128875 = r128865 ? r128874 : r128851;
        double r128876 = r128863 ? r128851 : r128875;
        double r128877 = r128853 ? r128861 : r128876;
        double r128878 = r128835 ? r128851 : r128877;
        return r128878;
}

Error

Bits error versus i

Bits error versus n

Target

Original	43.1
Target	43.0
Herbie	34.1

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

Derivation

1. Split input into 3 regimes

2. if n < -8.046837769241036e+81 or -4.530611506849831e+56 < n < -15831.586444589748 or 8.197984982050091e-194 < n

   1. Initial program 53.1

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

   2. Taylor expanded around 0 39.4

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

   3. Simplified 39.4

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

   if -8.046837769241036e+81 < n < -4.530611506849831e+56

   1. Initial program 33.4

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

   3. Applied div-inv 33.5

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

   4. Applied *-un-lft-identity 33.5

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

   5. Applied times-frac 33.3

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

   6. Applied associate-*r* 33.3

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

   7. Simplified 33.2

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

   if -15831.586444589748 < n < 8.197984982050091e-194

   1. Initial program 21.8

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

   3. Applied div-inv 21.8

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

   4. Applied add-cube-cbrt 21.8

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

   5. Applied times-frac 22.3

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

   6. Applied associate-*r* 22.4

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

   8. Applied add-cbrt-cube 22.4

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

   9. Simplified 22.4

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

   11. Applied add-log-exp 22.4

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

   12. Applied add-log-exp 22.4

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

   13. Applied diff-log 22.4

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

   14. Simplified 22.4

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

4. Final simplification 34.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -8.046837769241036 \cdot 10^{81}:\\ \;\;\;\;100 \cdot \frac{\mathsf{fma}\left(i, 1, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right)}{\frac{i}{n}}\\ \mathbf{elif}\;n \le -4.5306115068498307 \cdot 10^{56}:\\ \;\;\;\;\frac{100}{i} \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{1}{n}}\\ \mathbf{elif}\;n \le -15831.5864445897478:\\ \;\;\;\;100 \cdot \frac{\mathsf{fma}\left(i, 1, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right)}{\frac{i}{n}}\\ \mathbf{elif}\;n \le 8.1979849820500912 \cdot 10^{-194}:\\ \;\;\;\;\left(100 \cdot \frac{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} - 1} \cdot \sqrt[3]{\log \left(e^{{\left(1 + \frac{i}{n}\right)}^{n} - 1}\right)}}{i}\right) \cdot \frac{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} - 1}}{\frac{1}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\mathsf{fma}\left(i, 1, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right)}{\frac{i}{n}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020083 +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))))