Compound Interest

Average Error: 42.6 → 21.9

Time: 28.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;n \le -1.1051279060415901 \cdot 10^{+112}:\\ \;\;\;\;\left(n \cdot \frac{\left(\left(\log \left(\sqrt[3]{e^{\left(i \cdot \left(i \cdot i\right)\right) \cdot \frac{1}{6}}} \cdot \sqrt[3]{e^{\left(i \cdot \left(i \cdot i\right)\right) \cdot \frac{1}{6}}}\right) + \left(i \cdot i\right) \cdot \frac{1}{2}\right) + \log \left(\sqrt[3]{e^{\left(i \cdot \left(i \cdot i\right)\right) \cdot \frac{1}{6}}}\right)\right) + i}{i}\right) \cdot 100\\ \mathbf{elif}\;n \le -1.0933259526011947 \cdot 10^{+64}:\\ \;\;\;\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{1}{n}} \cdot \frac{100}{i}\\ \mathbf{elif}\;n \le -0.13295556128930017:\\ \;\;\;\;\left(n \cdot \frac{\left(\log \left(\sqrt{e^{\left(i \cdot \left(i \cdot i\right)\right) \cdot \frac{1}{6}}}\right) + \left(\left(i \cdot i\right) \cdot \frac{1}{2} + \log \left(\sqrt{e^{\left(i \cdot \left(i \cdot i\right)\right) \cdot \frac{1}{6}}}\right)\right)\right) + i}{i}\right) \cdot 100\\ \mathbf{elif}\;n \le 4.420752440477652 \cdot 10^{-89}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\left(n \cdot \frac{\left(\log \left(\sqrt{e^{\left(i \cdot \left(i \cdot i\right)\right) \cdot \frac{1}{6}}}\right) + \left(\left(i \cdot i\right) \cdot \frac{1}{2} + \log \left(\sqrt{e^{\left(i \cdot \left(i \cdot i\right)\right) \cdot \frac{1}{6}}}\right)\right)\right) + i}{i}\right) \cdot 100\\ \end{array}\]

C

double f(double i, double n) {
        double r2469670 = 100.0;
        double r2469671 = 1.0;
        double r2469672 = i;
        double r2469673 = n;
        double r2469674 = r2469672 / r2469673;
        double r2469675 = r2469671 + r2469674;
        double r2469676 = pow(r2469675, r2469673);
        double r2469677 = r2469676 - r2469671;
        double r2469678 = r2469677 / r2469674;
        double r2469679 = r2469670 * r2469678;
        return r2469679;
}

↓

double f(double i, double n) {
        double r2469680 = n;
        double r2469681 = -1.1051279060415901e+112;
        bool r2469682 = r2469680 <= r2469681;
        double r2469683 = i;
        double r2469684 = r2469683 * r2469683;
        double r2469685 = r2469683 * r2469684;
        double r2469686 = 0.16666666666666666;
        double r2469687 = r2469685 * r2469686;
        double r2469688 = exp(r2469687);
        double r2469689 = cbrt(r2469688);
        double r2469690 = r2469689 * r2469689;
        double r2469691 = log(r2469690);
        double r2469692 = 0.5;
        double r2469693 = r2469684 * r2469692;
        double r2469694 = r2469691 + r2469693;
        double r2469695 = log(r2469689);
        double r2469696 = r2469694 + r2469695;
        double r2469697 = r2469696 + r2469683;
        double r2469698 = r2469697 / r2469683;
        double r2469699 = r2469680 * r2469698;
        double r2469700 = 100.0;
        double r2469701 = r2469699 * r2469700;
        double r2469702 = -1.0933259526011947e+64;
        bool r2469703 = r2469680 <= r2469702;
        double r2469704 = 1.0;
        double r2469705 = r2469683 / r2469680;
        double r2469706 = r2469704 + r2469705;
        double r2469707 = pow(r2469706, r2469680);
        double r2469708 = r2469707 - r2469704;
        double r2469709 = r2469704 / r2469680;
        double r2469710 = r2469708 / r2469709;
        double r2469711 = r2469700 / r2469683;
        double r2469712 = r2469710 * r2469711;
        double r2469713 = -0.13295556128930017;
        bool r2469714 = r2469680 <= r2469713;
        double r2469715 = sqrt(r2469688);
        double r2469716 = log(r2469715);
        double r2469717 = r2469693 + r2469716;
        double r2469718 = r2469716 + r2469717;
        double r2469719 = r2469718 + r2469683;
        double r2469720 = r2469719 / r2469683;
        double r2469721 = r2469680 * r2469720;
        double r2469722 = r2469721 * r2469700;
        double r2469723 = 4.420752440477652e-89;
        bool r2469724 = r2469680 <= r2469723;
        double r2469725 = 0.0;
        double r2469726 = r2469724 ? r2469725 : r2469722;
        double r2469727 = r2469714 ? r2469722 : r2469726;
        double r2469728 = r2469703 ? r2469712 : r2469727;
        double r2469729 = r2469682 ? r2469701 : r2469728;
        return r2469729;
}

Error

Bits error versus i

Bits error versus n

Target

Original	42.6
Target	42.3
Herbie	21.9

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

Derivation

1. Split input into 4 regimes

2. if n < -1.1051279060415901e+112

   1. Initial program 51.0

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

   2. Taylor expanded around 0 48.3

      \[\leadsto 100 \cdot \frac{\color{blue}{i + \left(\frac{1}{2} \cdot {i}^{2} + \frac{1}{6} \cdot {i}^{3}\right)}}{\frac{i}{n}}\]

   3. Simplified 48.3

      \[\leadsto 100 \cdot \frac{\color{blue}{\left(\left(i \cdot i\right) \cdot \frac{1}{2} + \frac{1}{6} \cdot \left(i \cdot \left(i \cdot i\right)\right)\right) + i}}{\frac{i}{n}}\]
   4. Using strategy rm

   5. Applied associate-/r/ 25.2

      \[\leadsto 100 \cdot \color{blue}{\left(\frac{\left(\left(i \cdot i\right) \cdot \frac{1}{2} + \frac{1}{6} \cdot \left(i \cdot \left(i \cdot i\right)\right)\right) + i}{i} \cdot n\right)}\]
   6. Using strategy rm

   7. Applied add-log-exp 25.3

      \[\leadsto 100 \cdot \left(\frac{\left(\left(i \cdot i\right) \cdot \frac{1}{2} + \color{blue}{\log \left(e^{\frac{1}{6} \cdot \left(i \cdot \left(i \cdot i\right)\right)}\right)}\right) + i}{i} \cdot n\right)\]
   8. Using strategy rm

   9. Applied add-cube-cbrt 25.3

      \[\leadsto 100 \cdot \left(\frac{\left(\left(i \cdot i\right) \cdot \frac{1}{2} + \log \color{blue}{\left(\left(\sqrt[3]{e^{\frac{1}{6} \cdot \left(i \cdot \left(i \cdot i\right)\right)}} \cdot \sqrt[3]{e^{\frac{1}{6} \cdot \left(i \cdot \left(i \cdot i\right)\right)}}\right) \cdot \sqrt[3]{e^{\frac{1}{6} \cdot \left(i \cdot \left(i \cdot i\right)\right)}}\right)}\right) + i}{i} \cdot n\right)\]

   10. Applied log-prod 25.3

       \[\leadsto 100 \cdot \left(\frac{\left(\left(i \cdot i\right) \cdot \frac{1}{2} + \color{blue}{\left(\log \left(\sqrt[3]{e^{\frac{1}{6} \cdot \left(i \cdot \left(i \cdot i\right)\right)}} \cdot \sqrt[3]{e^{\frac{1}{6} \cdot \left(i \cdot \left(i \cdot i\right)\right)}}\right) + \log \left(\sqrt[3]{e^{\frac{1}{6} \cdot \left(i \cdot \left(i \cdot i\right)\right)}}\right)\right)}\right) + i}{i} \cdot n\right)\]

   11. Applied associate-+r+ 25.3

       \[\leadsto 100 \cdot \left(\frac{\color{blue}{\left(\left(\left(i \cdot i\right) \cdot \frac{1}{2} + \log \left(\sqrt[3]{e^{\frac{1}{6} \cdot \left(i \cdot \left(i \cdot i\right)\right)}} \cdot \sqrt[3]{e^{\frac{1}{6} \cdot \left(i \cdot \left(i \cdot i\right)\right)}}\right)\right) + \log \left(\sqrt[3]{e^{\frac{1}{6} \cdot \left(i \cdot \left(i \cdot i\right)\right)}}\right)\right)} + i}{i} \cdot n\right)\]

   if -1.1051279060415901e+112 < n < -1.0933259526011947e+64

   1. Initial program 35.7

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

   3. Applied div-inv 35.7

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

   4. Applied *-un-lft-identity 35.7

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

   5. Applied *-un-lft-identity 35.7

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

   6. Applied distribute-lft-out-- 35.7

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

   7. Applied times-frac 35.5

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

   8. Applied associate-*r* 35.6

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

   9. Simplified 35.5

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

   if -1.0933259526011947e+64 < n < -0.13295556128930017 or 4.420752440477652e-89 < n

   1. Initial program 54.4

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

   2. Taylor expanded around 0 32.3

      \[\leadsto 100 \cdot \frac{\color{blue}{i + \left(\frac{1}{2} \cdot {i}^{2} + \frac{1}{6} \cdot {i}^{3}\right)}}{\frac{i}{n}}\]

   3. Simplified 32.3

      \[\leadsto 100 \cdot \frac{\color{blue}{\left(\left(i \cdot i\right) \cdot \frac{1}{2} + \frac{1}{6} \cdot \left(i \cdot \left(i \cdot i\right)\right)\right) + i}}{\frac{i}{n}}\]
   4. Using strategy rm

   5. Applied associate-/r/ 17.9

      \[\leadsto 100 \cdot \color{blue}{\left(\frac{\left(\left(i \cdot i\right) \cdot \frac{1}{2} + \frac{1}{6} \cdot \left(i \cdot \left(i \cdot i\right)\right)\right) + i}{i} \cdot n\right)}\]
   6. Using strategy rm

   7. Applied add-log-exp 18.1

      \[\leadsto 100 \cdot \left(\frac{\left(\left(i \cdot i\right) \cdot \frac{1}{2} + \color{blue}{\log \left(e^{\frac{1}{6} \cdot \left(i \cdot \left(i \cdot i\right)\right)}\right)}\right) + i}{i} \cdot n\right)\]
   8. Using strategy rm

   9. Applied add-sqr-sqrt 18.1

      \[\leadsto 100 \cdot \left(\frac{\left(\left(i \cdot i\right) \cdot \frac{1}{2} + \log \color{blue}{\left(\sqrt{e^{\frac{1}{6} \cdot \left(i \cdot \left(i \cdot i\right)\right)}} \cdot \sqrt{e^{\frac{1}{6} \cdot \left(i \cdot \left(i \cdot i\right)\right)}}\right)}\right) + i}{i} \cdot n\right)\]

   10. Applied log-prod 18.1

       \[\leadsto 100 \cdot \left(\frac{\left(\left(i \cdot i\right) \cdot \frac{1}{2} + \color{blue}{\left(\log \left(\sqrt{e^{\frac{1}{6} \cdot \left(i \cdot \left(i \cdot i\right)\right)}}\right) + \log \left(\sqrt{e^{\frac{1}{6} \cdot \left(i \cdot \left(i \cdot i\right)\right)}}\right)\right)}\right) + i}{i} \cdot n\right)\]

   11. Applied associate-+r+ 18.1

       \[\leadsto 100 \cdot \left(\frac{\color{blue}{\left(\left(\left(i \cdot i\right) \cdot \frac{1}{2} + \log \left(\sqrt{e^{\frac{1}{6} \cdot \left(i \cdot \left(i \cdot i\right)\right)}}\right)\right) + \log \left(\sqrt{e^{\frac{1}{6} \cdot \left(i \cdot \left(i \cdot i\right)\right)}}\right)\right)} + i}{i} \cdot n\right)\]

   if -0.13295556128930017 < n < 4.420752440477652e-89

   1. Initial program 28.2

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

   2. Taylor expanded around 0 21.7

      \[\leadsto \color{blue}{0}\]
3. Recombined 4 regimes into one program.

4. Final simplification 21.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -1.1051279060415901 \cdot 10^{+112}:\\ \;\;\;\;\left(n \cdot \frac{\left(\left(\log \left(\sqrt[3]{e^{\left(i \cdot \left(i \cdot i\right)\right) \cdot \frac{1}{6}}} \cdot \sqrt[3]{e^{\left(i \cdot \left(i \cdot i\right)\right) \cdot \frac{1}{6}}}\right) + \left(i \cdot i\right) \cdot \frac{1}{2}\right) + \log \left(\sqrt[3]{e^{\left(i \cdot \left(i \cdot i\right)\right) \cdot \frac{1}{6}}}\right)\right) + i}{i}\right) \cdot 100\\ \mathbf{elif}\;n \le -1.0933259526011947 \cdot 10^{+64}:\\ \;\;\;\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{1}{n}} \cdot \frac{100}{i}\\ \mathbf{elif}\;n \le -0.13295556128930017:\\ \;\;\;\;\left(n \cdot \frac{\left(\log \left(\sqrt{e^{\left(i \cdot \left(i \cdot i\right)\right) \cdot \frac{1}{6}}}\right) + \left(\left(i \cdot i\right) \cdot \frac{1}{2} + \log \left(\sqrt{e^{\left(i \cdot \left(i \cdot i\right)\right) \cdot \frac{1}{6}}}\right)\right)\right) + i}{i}\right) \cdot 100\\ \mathbf{elif}\;n \le 4.420752440477652 \cdot 10^{-89}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\left(n \cdot \frac{\left(\log \left(\sqrt{e^{\left(i \cdot \left(i \cdot i\right)\right) \cdot \frac{1}{6}}}\right) + \left(\left(i \cdot i\right) \cdot \frac{1}{2} + \log \left(\sqrt{e^{\left(i \cdot \left(i \cdot i\right)\right) \cdot \frac{1}{6}}}\right)\right)\right) + i}{i}\right) \cdot 100\\ \end{array}\]

Reproduce

herbie shell --seed 2019153 
(FPCore (i n)
  :name "Compound Interest"

  :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))))