Compound Interest

Average Error: 41.3 → 24.2

Time: 14.8s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;i \le -1.62626015699271742 \cdot 10^{-16}:\\ \;\;\;\;\frac{100}{i} \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} \cdot n - 1 \cdot n\right)\\ \mathbf{elif}\;i \le 622281.331476159976:\\ \;\;\;\;\frac{100 \cdot \left(\sqrt[3]{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)} \cdot \sqrt[3]{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}\right)}{i} \cdot \frac{1 \cdot \sqrt[3]{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}}{\frac{\sqrt{1}}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{100}{i} \cdot \frac{\frac{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + \left(-1 \cdot 1\right)}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{\frac{1}{n}}\\ \end{array}\]

C

double f(double i, double n) {
        double r126663 = 100.0;
        double r126664 = 1.0;
        double r126665 = i;
        double r126666 = n;
        double r126667 = r126665 / r126666;
        double r126668 = r126664 + r126667;
        double r126669 = pow(r126668, r126666);
        double r126670 = r126669 - r126664;
        double r126671 = r126670 / r126667;
        double r126672 = r126663 * r126671;
        return r126672;
}

↓

double f(double i, double n) {
        double r126673 = i;
        double r126674 = -1.6262601569927174e-16;
        bool r126675 = r126673 <= r126674;
        double r126676 = 100.0;
        double r126677 = r126676 / r126673;
        double r126678 = 1.0;
        double r126679 = n;
        double r126680 = r126673 / r126679;
        double r126681 = r126678 + r126680;
        double r126682 = pow(r126681, r126679);
        double r126683 = r126682 * r126679;
        double r126684 = r126678 * r126679;
        double r126685 = r126683 - r126684;
        double r126686 = r126677 * r126685;
        double r126687 = 622281.33147616;
        bool r126688 = r126673 <= r126687;
        double r126689 = r126678 * r126673;
        double r126690 = 0.5;
        double r126691 = 2.0;
        double r126692 = pow(r126673, r126691);
        double r126693 = r126690 * r126692;
        double r126694 = log(r126678);
        double r126695 = r126694 * r126679;
        double r126696 = r126693 + r126695;
        double r126697 = r126689 + r126696;
        double r126698 = r126692 * r126694;
        double r126699 = r126690 * r126698;
        double r126700 = r126697 - r126699;
        double r126701 = cbrt(r126700);
        double r126702 = r126701 * r126701;
        double r126703 = r126676 * r126702;
        double r126704 = r126703 / r126673;
        double r126705 = 1.0;
        double r126706 = r126705 * r126701;
        double r126707 = sqrt(r126705);
        double r126708 = r126707 / r126679;
        double r126709 = r126706 / r126708;
        double r126710 = r126704 * r126709;
        double r126711 = r126691 * r126679;
        double r126712 = pow(r126681, r126711);
        double r126713 = r126678 * r126678;
        double r126714 = -r126713;
        double r126715 = r126712 + r126714;
        double r126716 = r126682 + r126678;
        double r126717 = r126715 / r126716;
        double r126718 = r126705 / r126679;
        double r126719 = r126717 / r126718;
        double r126720 = r126677 * r126719;
        double r126721 = r126688 ? r126710 : r126720;
        double r126722 = r126675 ? r126686 : r126721;
        return r126722;
}

Error

Bits error versus i

Bits error versus n

Target

Original	41.3
Target	41.0
Herbie	24.2

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

Derivation

1. Split input into 3 regimes

2. if i < -1.6262601569927174e-16

   1. Initial program 29.7

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

   3. Applied div-inv 29.7

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

   4. Applied *-un-lft-identity 29.7

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

   5. Applied times-frac 30.3

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

   6. Applied associate-*r* 30.4

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

   7. Simplified 30.3

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

   9. Applied div-sub 30.3

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

   10. Simplified 31.0

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

   11. Simplified 30.3

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

   if -1.6262601569927174e-16 < i < 622281.33147616

   1. Initial program 48.5

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

   3. Applied div-inv 48.5

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

   4. Applied *-un-lft-identity 48.5

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

   5. Applied times-frac 48.2

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

   6. Applied associate-*r* 48.3

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

   7. Simplified 48.3

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

   8. Taylor expanded around 0 18.5

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

   10. Applied *-un-lft-identity 18.5

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

   11. Applied add-sqr-sqrt 18.5

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

   12. Applied times-frac 18.5

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

   13. Applied add-cube-cbrt 19.1

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

   14. Applied times-frac 19.1

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

   15. Applied associate-*r* 19.8

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

   16. Simplified 19.5

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

   18. Applied *-un-lft-identity 19.5

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

   if 622281.33147616 < i

   1. Initial program 32.4

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

   3. Applied div-inv 32.5

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

   4. Applied *-un-lft-identity 32.5

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

   5. Applied times-frac 32.4

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

   6. Applied associate-*r* 32.4

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

   7. Simplified 32.4

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

   9. Applied flip-- 32.5

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

   10. Simplified 32.5

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

4. Final simplification 24.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -1.62626015699271742 \cdot 10^{-16}:\\ \;\;\;\;\frac{100}{i} \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} \cdot n - 1 \cdot n\right)\\ \mathbf{elif}\;i \le 622281.331476159976:\\ \;\;\;\;\frac{100 \cdot \left(\sqrt[3]{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)} \cdot \sqrt[3]{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}\right)}{i} \cdot \frac{1 \cdot \sqrt[3]{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}}{\frac{\sqrt{1}}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{100}{i} \cdot \frac{\frac{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + \left(-1 \cdot 1\right)}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{\frac{1}{n}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020021 
(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))))