Compound Interest

Average Error: 43.1 → 22.8

Time: 22.7s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;i \le -5.4348495206706917 \cdot 10^{-14}:\\ \;\;\;\;\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \le -3.6505909100430136 \cdot 10^{-289}:\\ \;\;\;\;\frac{100 \cdot \left(\left(\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right) \cdot n\right)}{i}\\ \mathbf{elif}\;i \le 1.10371457373729495 \cdot 10^{-38}:\\ \;\;\;\;100 \cdot \left(\frac{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{i} \cdot n\right)\\ \mathbf{elif}\;i \le 8.04120890918656311 \cdot 10^{241}:\\ \;\;\;\;\frac{\frac{\left({\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} - 1 \cdot 1\right) \cdot n}{i} \cdot 100}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}\\ \mathbf{elif}\;i \le 4.24623267635107831 \cdot 10^{289}:\\ \;\;\;\;100 \cdot \frac{\left(1 \cdot i + \left(\log 1 \cdot n + 1\right)\right) - 1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\frac{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} - 1 \cdot 1}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{\frac{i}{n}}\\ \end{array}\]

C

double f(double i, double n) {
        double r115762 = 100.0;
        double r115763 = 1.0;
        double r115764 = i;
        double r115765 = n;
        double r115766 = r115764 / r115765;
        double r115767 = r115763 + r115766;
        double r115768 = pow(r115767, r115765);
        double r115769 = r115768 - r115763;
        double r115770 = r115769 / r115766;
        double r115771 = r115762 * r115770;
        return r115771;
}

↓

double f(double i, double n) {
        double r115772 = i;
        double r115773 = -5.434849520670692e-14;
        bool r115774 = r115772 <= r115773;
        double r115775 = 100.0;
        double r115776 = 1.0;
        double r115777 = n;
        double r115778 = r115772 / r115777;
        double r115779 = r115776 + r115778;
        double r115780 = pow(r115779, r115777);
        double r115781 = r115780 - r115776;
        double r115782 = r115775 * r115781;
        double r115783 = r115782 / r115778;
        double r115784 = -3.6505909100430136e-289;
        bool r115785 = r115772 <= r115784;
        double r115786 = r115776 * r115772;
        double r115787 = 0.5;
        double r115788 = 2.0;
        double r115789 = pow(r115772, r115788);
        double r115790 = r115787 * r115789;
        double r115791 = log(r115776);
        double r115792 = r115791 * r115777;
        double r115793 = r115790 + r115792;
        double r115794 = r115786 + r115793;
        double r115795 = r115789 * r115791;
        double r115796 = r115787 * r115795;
        double r115797 = r115794 - r115796;
        double r115798 = r115797 * r115777;
        double r115799 = r115775 * r115798;
        double r115800 = r115799 / r115772;
        double r115801 = 1.103714573737295e-38;
        bool r115802 = r115772 <= r115801;
        double r115803 = r115797 / r115772;
        double r115804 = r115803 * r115777;
        double r115805 = r115775 * r115804;
        double r115806 = 8.041208909186563e+241;
        bool r115807 = r115772 <= r115806;
        double r115808 = r115788 * r115777;
        double r115809 = pow(r115779, r115808);
        double r115810 = r115776 * r115776;
        double r115811 = r115809 - r115810;
        double r115812 = r115811 * r115777;
        double r115813 = r115812 / r115772;
        double r115814 = r115813 * r115775;
        double r115815 = r115780 + r115776;
        double r115816 = r115814 / r115815;
        double r115817 = 4.246232676351078e+289;
        bool r115818 = r115772 <= r115817;
        double r115819 = 1.0;
        double r115820 = r115792 + r115819;
        double r115821 = r115786 + r115820;
        double r115822 = r115821 - r115776;
        double r115823 = r115822 / r115778;
        double r115824 = r115775 * r115823;
        double r115825 = r115811 / r115815;
        double r115826 = r115825 / r115778;
        double r115827 = r115775 * r115826;
        double r115828 = r115818 ? r115824 : r115827;
        double r115829 = r115807 ? r115816 : r115828;
        double r115830 = r115802 ? r115805 : r115829;
        double r115831 = r115785 ? r115800 : r115830;
        double r115832 = r115774 ? r115783 : r115831;
        return r115832;
}

Error

Bits error versus i

Bits error versus n

Target

Original	43.1
Target	43.0
Herbie	22.8

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

Derivation

1. Split input into 6 regimes

2. if i < -5.434849520670692e-14

   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

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

   if -5.434849520670692e-14 < i < -3.6505909100430136e-289

   1. Initial program 51.3

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

   3. Applied div-inv 51.3

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

   4. Applied *-un-lft-identity 51.3

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

   5. Applied times-frac 51.0

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

   6. Simplified 51.0

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

   7. Taylor expanded around 0 15.0

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

   9. Applied associate-*l/ 14.9

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

   10. Applied associate-*r/ 15.0

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

   11. Simplified 15.0

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

   if -3.6505909100430136e-289 < i < 1.103714573737295e-38

   1. Initial program 50.3

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

   3. Applied div-inv 50.3

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

   4. Applied *-un-lft-identity 50.3

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

   5. Applied times-frac 49.9

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

   6. Simplified 49.9

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

   7. Taylor expanded around 0 15.9

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

   9. Applied associate-*r* 16.6

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

   10. Simplified 16.5

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

   if 1.103714573737295e-38 < i < 8.041208909186563e+241

   1. Initial program 37.1

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

   3. Applied div-inv 37.1

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

   4. Applied *-un-lft-identity 37.1

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

   5. Applied times-frac 37.1

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

   6. Simplified 37.1

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

   8. Applied flip-- 37.1

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

   9. Applied associate-*l/ 37.1

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

   10. Applied associate-*r/ 37.1

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

   11. Applied associate-*r/ 37.1

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

   12. Simplified 37.0

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

   if 8.041208909186563e+241 < i < 4.246232676351078e+289

   1. Initial program 28.6

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

   2. Taylor expanded around 0 36.4

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

   if 4.246232676351078e+289 < i

   1. Initial program 35.5

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

   3. Applied flip-- 35.4

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

   4. Simplified 35.4

      \[\leadsto 100 \cdot \frac{\frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} - 1 \cdot 1}}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{\frac{i}{n}}\]
3. Recombined 6 regimes into one program.

4. Final simplification 22.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -5.4348495206706917 \cdot 10^{-14}:\\ \;\;\;\;\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \le -3.6505909100430136 \cdot 10^{-289}:\\ \;\;\;\;\frac{100 \cdot \left(\left(\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right) \cdot n\right)}{i}\\ \mathbf{elif}\;i \le 1.10371457373729495 \cdot 10^{-38}:\\ \;\;\;\;100 \cdot \left(\frac{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{i} \cdot n\right)\\ \mathbf{elif}\;i \le 8.04120890918656311 \cdot 10^{241}:\\ \;\;\;\;\frac{\frac{\left({\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} - 1 \cdot 1\right) \cdot n}{i} \cdot 100}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}\\ \mathbf{elif}\;i \le 4.24623267635107831 \cdot 10^{289}:\\ \;\;\;\;100 \cdot \frac{\left(1 \cdot i + \left(\log 1 \cdot n + 1\right)\right) - 1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\frac{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} - 1 \cdot 1}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{\frac{i}{n}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 
(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))))