Compound Interest

Average Error: 43.0 → 24.4

Time: 15.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;n \le -3.55885888884481173 \cdot 10^{195}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(i, 1, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right) \cdot n\right) \cdot 100}{i}\\ \mathbf{elif}\;n \le 2.5276530280957779 \cdot 10^{-300}:\\ \;\;\;\;\frac{100}{i} \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} \cdot n - 1 \cdot n\right)\\ \mathbf{elif}\;n \le 1.0241992467778217 \cdot 10^{-176}:\\ \;\;\;\;100 \cdot \frac{\left(e^{\left(\log \left(\frac{1}{n}\right) - \log \left(\frac{1}{i}\right)\right) \cdot n} - 1\right) \cdot n}{i}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{100}{i} \cdot \frac{\sqrt[3]{\mathsf{fma}\left(i, 1, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right)} \cdot \sqrt[3]{\mathsf{fma}\left(i, 1, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right)}}{\sqrt{\frac{1}{n}}}\right) \cdot \frac{\sqrt[3]{\mathsf{fma}\left(i, 1, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right)}}{\sqrt{\frac{1}{n}}}\\ \end{array}\]

C

double f(double i, double n) {
        double r125824 = 100.0;
        double r125825 = 1.0;
        double r125826 = i;
        double r125827 = n;
        double r125828 = r125826 / r125827;
        double r125829 = r125825 + r125828;
        double r125830 = pow(r125829, r125827);
        double r125831 = r125830 - r125825;
        double r125832 = r125831 / r125828;
        double r125833 = r125824 * r125832;
        return r125833;
}

↓

double f(double i, double n) {
        double r125834 = n;
        double r125835 = -3.5588588888448117e+195;
        bool r125836 = r125834 <= r125835;
        double r125837 = i;
        double r125838 = 1.0;
        double r125839 = 0.5;
        double r125840 = 2.0;
        double r125841 = pow(r125837, r125840);
        double r125842 = log(r125838);
        double r125843 = r125842 * r125834;
        double r125844 = fma(r125839, r125841, r125843);
        double r125845 = r125841 * r125842;
        double r125846 = r125839 * r125845;
        double r125847 = r125844 - r125846;
        double r125848 = fma(r125837, r125838, r125847);
        double r125849 = r125848 * r125834;
        double r125850 = 100.0;
        double r125851 = r125849 * r125850;
        double r125852 = r125851 / r125837;
        double r125853 = 2.527653028095778e-300;
        bool r125854 = r125834 <= r125853;
        double r125855 = r125850 / r125837;
        double r125856 = r125837 / r125834;
        double r125857 = r125838 + r125856;
        double r125858 = pow(r125857, r125834);
        double r125859 = r125858 * r125834;
        double r125860 = r125838 * r125834;
        double r125861 = r125859 - r125860;
        double r125862 = r125855 * r125861;
        double r125863 = 1.0241992467778217e-176;
        bool r125864 = r125834 <= r125863;
        double r125865 = 1.0;
        double r125866 = r125865 / r125834;
        double r125867 = log(r125866);
        double r125868 = r125865 / r125837;
        double r125869 = log(r125868);
        double r125870 = r125867 - r125869;
        double r125871 = r125870 * r125834;
        double r125872 = exp(r125871);
        double r125873 = r125872 - r125838;
        double r125874 = r125873 * r125834;
        double r125875 = r125874 / r125837;
        double r125876 = r125850 * r125875;
        double r125877 = cbrt(r125848);
        double r125878 = r125877 * r125877;
        double r125879 = sqrt(r125866);
        double r125880 = r125878 / r125879;
        double r125881 = r125855 * r125880;
        double r125882 = r125877 / r125879;
        double r125883 = r125881 * r125882;
        double r125884 = r125864 ? r125876 : r125883;
        double r125885 = r125854 ? r125862 : r125884;
        double r125886 = r125836 ? r125852 : r125885;
        return r125886;
}

Error

Bits error versus i

Bits error versus n

Target

Original	43.0
Target	42.6
Herbie	24.4

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

Derivation

1. Split input into 4 regimes

2. if n < -3.5588588888448117e+195

   1. Initial program 55.6

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

   3. Applied div-inv 55.6

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

   4. Applied *-un-lft-identity 55.6

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

   5. Applied times-frac 55.1

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

   6. Applied associate-*r* 55.1

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

   7. Simplified 55.1

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

   8. Taylor expanded around 0 24.7

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

   9. Simplified 24.7

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

   11. Applied associate-*l/ 24.6

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

   12. Simplified 24.6

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

   if -3.5588588888448117e+195 < n < 2.527653028095778e-300

   1. Initial program 27.3

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

   3. Applied div-inv 27.3

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

   4. Applied *-un-lft-identity 27.3

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

   5. Applied times-frac 27.6

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

   6. Applied associate-*r* 27.7

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

   7. Simplified 27.7

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

   9. Applied div-sub 27.7

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

   10. Simplified 30.4

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

   11. Simplified 27.7

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

   if 2.527653028095778e-300 < n < 1.0241992467778217e-176

   1. Initial program 41.3

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

   2. Taylor expanded around inf 20.7

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

   if 1.0241992467778217e-176 < n

   1. Initial program 58.8

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

   3. Applied div-inv 58.8

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

   4. Applied *-un-lft-identity 58.8

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

   5. Applied times-frac 58.5

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

   6. Applied associate-*r* 58.5

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

   7. Simplified 58.5

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

   8. Taylor expanded around 0 24.4

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

   9. Simplified 24.4

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

   11. Applied add-sqr-sqrt 24.6

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

   12. Applied add-cube-cbrt 25.1

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

   13. Applied times-frac 25.1

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

   14. Applied associate-*r* 21.2

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

4. Final simplification 24.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -3.55885888884481173 \cdot 10^{195}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(i, 1, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right) \cdot n\right) \cdot 100}{i}\\ \mathbf{elif}\;n \le 2.5276530280957779 \cdot 10^{-300}:\\ \;\;\;\;\frac{100}{i} \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} \cdot n - 1 \cdot n\right)\\ \mathbf{elif}\;n \le 1.0241992467778217 \cdot 10^{-176}:\\ \;\;\;\;100 \cdot \frac{\left(e^{\left(\log \left(\frac{1}{n}\right) - \log \left(\frac{1}{i}\right)\right) \cdot n} - 1\right) \cdot n}{i}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{100}{i} \cdot \frac{\sqrt[3]{\mathsf{fma}\left(i, 1, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right)} \cdot \sqrt[3]{\mathsf{fma}\left(i, 1, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right)}}{\sqrt{\frac{1}{n}}}\right) \cdot \frac{\sqrt[3]{\mathsf{fma}\left(i, 1, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right)}}{\sqrt{\frac{1}{n}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020089 +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))))