Compound Interest

Average Error: 42.3 → 32.9

Time: 17.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;i \le -4.726437827530884962234924984159079031087 \cdot 10^{-4}:\\ \;\;\;\;100 \cdot \frac{\frac{\frac{{\left({\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)}\right)}^{3} + {\left(-1 \cdot 1\right)}^{3}}{\mathsf{fma}\left(-1 \cdot 1, \left(-1 \cdot 1\right) - {\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)}, {\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot \left(2 \cdot n\right)\right)}\right)}}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{\frac{i}{n}}\\ \mathbf{elif}\;i \le -1.964684591502211290172320573095869134171 \cdot 10^{-206}:\\ \;\;\;\;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}\;i \le -2.882850856787981581879145472439421418808 \cdot 10^{-226}:\\ \;\;\;\;\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}}\\ \mathbf{elif}\;i \le -1.019093496626147386492788288826940719188 \cdot 10^{-241}:\\ \;\;\;\;100 \cdot \frac{\frac{\mathsf{fma}\left(i, 2, \mathsf{fma}\left(2, {i}^{2}, 2 \cdot \left(\log 1 \cdot n\right)\right) - 2 \cdot \left({i}^{2} \cdot \log 1\right)\right)}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{\frac{i}{n}}\\ \mathbf{elif}\;i \le -3.645341750186080323375566759305922898034 \cdot 10^{-293}:\\ \;\;\;\;\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}}\\ \mathbf{elif}\;i \le 9.27900448690603418810951552586629986763:\\ \;\;\;\;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{else}:\\ \;\;\;\;100 \cdot \frac{\frac{\frac{{\left({\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)}\right)}^{3} + {\left(-1 \cdot 1\right)}^{3}}{\mathsf{fma}\left(-1 \cdot 1, \left(-1 \cdot 1\right) - {\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)}, {\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot \left(2 \cdot n\right)\right)}\right)}}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{\frac{i}{n}}\\ \end{array}\]

C

double f(double i, double n) {
        double r160770 = 100.0;
        double r160771 = 1.0;
        double r160772 = i;
        double r160773 = n;
        double r160774 = r160772 / r160773;
        double r160775 = r160771 + r160774;
        double r160776 = pow(r160775, r160773);
        double r160777 = r160776 - r160771;
        double r160778 = r160777 / r160774;
        double r160779 = r160770 * r160778;
        return r160779;
}

↓

double f(double i, double n) {
        double r160780 = i;
        double r160781 = -0.0004726437827530885;
        bool r160782 = r160780 <= r160781;
        double r160783 = 100.0;
        double r160784 = 1.0;
        double r160785 = n;
        double r160786 = r160780 / r160785;
        double r160787 = r160784 + r160786;
        double r160788 = 2.0;
        double r160789 = r160788 * r160785;
        double r160790 = pow(r160787, r160789);
        double r160791 = 3.0;
        double r160792 = pow(r160790, r160791);
        double r160793 = r160784 * r160784;
        double r160794 = -r160793;
        double r160795 = pow(r160794, r160791);
        double r160796 = r160792 + r160795;
        double r160797 = r160794 - r160790;
        double r160798 = r160788 * r160789;
        double r160799 = pow(r160787, r160798);
        double r160800 = fma(r160794, r160797, r160799);
        double r160801 = r160796 / r160800;
        double r160802 = pow(r160787, r160785);
        double r160803 = r160802 + r160784;
        double r160804 = r160801 / r160803;
        double r160805 = r160804 / r160786;
        double r160806 = r160783 * r160805;
        double r160807 = -1.9646845915022113e-206;
        bool r160808 = r160780 <= r160807;
        double r160809 = 0.5;
        double r160810 = pow(r160780, r160788);
        double r160811 = log(r160784);
        double r160812 = r160811 * r160785;
        double r160813 = fma(r160809, r160810, r160812);
        double r160814 = r160810 * r160811;
        double r160815 = r160809 * r160814;
        double r160816 = r160813 - r160815;
        double r160817 = fma(r160780, r160784, r160816);
        double r160818 = r160817 / r160786;
        double r160819 = r160783 * r160818;
        double r160820 = -2.8828508567879816e-226;
        bool r160821 = r160780 <= r160820;
        double r160822 = r160802 - r160784;
        double r160823 = cbrt(r160822);
        double r160824 = r160823 * r160823;
        double r160825 = r160824 / r160780;
        double r160826 = r160783 * r160825;
        double r160827 = 1.0;
        double r160828 = r160827 / r160785;
        double r160829 = r160823 / r160828;
        double r160830 = r160826 * r160829;
        double r160831 = -1.0190934966261474e-241;
        bool r160832 = r160780 <= r160831;
        double r160833 = 2.0;
        double r160834 = r160788 * r160812;
        double r160835 = fma(r160833, r160810, r160834);
        double r160836 = r160833 * r160814;
        double r160837 = r160835 - r160836;
        double r160838 = fma(r160780, r160833, r160837);
        double r160839 = r160838 / r160803;
        double r160840 = r160839 / r160786;
        double r160841 = r160783 * r160840;
        double r160842 = -3.6453417501860803e-293;
        bool r160843 = r160780 <= r160842;
        double r160844 = 9.279004486906034;
        bool r160845 = r160780 <= r160844;
        double r160846 = r160845 ? r160819 : r160806;
        double r160847 = r160843 ? r160830 : r160846;
        double r160848 = r160832 ? r160841 : r160847;
        double r160849 = r160821 ? r160830 : r160848;
        double r160850 = r160808 ? r160819 : r160849;
        double r160851 = r160782 ? r160806 : r160850;
        return r160851;
}

Error

Bits error versus i

Bits error versus n

Target

Original	42.3
Target	42.6
Herbie	32.9

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

Derivation

1. Split input into 4 regimes

2. if i < -0.0004726437827530885 or 9.279004486906034 < i

   1. Initial program 29.1

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

   3. Applied flip-- 29.1

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

   4. Simplified 29.1

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

   6. Applied flip3-+ 29.1

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

   7. Simplified 29.1

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

   if -0.0004726437827530885 < i < -1.9646845915022113e-206 or -3.6453417501860803e-293 < i < 9.279004486906034

   1. Initial program 50.8

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

   2. Taylor expanded around 0 33.1

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

   3. Simplified 33.1

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

   if -1.9646845915022113e-206 < i < -2.8828508567879816e-226 or -1.0190934966261474e-241 < i < -3.6453417501860803e-293

   1. Initial program 47.7

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

   3. Applied div-inv 47.7

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

   4. Applied add-cube-cbrt 47.7

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

   5. Applied times-frac 47.1

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

   6. Applied associate-*r* 47.1

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

   if -2.8828508567879816e-226 < i < -1.0190934966261474e-241

   1. Initial program 46.5

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

   3. Applied flip-- 46.5

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

   4. Simplified 46.5

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

   5. Taylor expanded around 0 42.2

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

   6. Simplified 42.2

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

4. Final simplification 32.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -4.726437827530884962234924984159079031087 \cdot 10^{-4}:\\ \;\;\;\;100 \cdot \frac{\frac{\frac{{\left({\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)}\right)}^{3} + {\left(-1 \cdot 1\right)}^{3}}{\mathsf{fma}\left(-1 \cdot 1, \left(-1 \cdot 1\right) - {\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)}, {\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot \left(2 \cdot n\right)\right)}\right)}}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{\frac{i}{n}}\\ \mathbf{elif}\;i \le -1.964684591502211290172320573095869134171 \cdot 10^{-206}:\\ \;\;\;\;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}\;i \le -2.882850856787981581879145472439421418808 \cdot 10^{-226}:\\ \;\;\;\;\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}}\\ \mathbf{elif}\;i \le -1.019093496626147386492788288826940719188 \cdot 10^{-241}:\\ \;\;\;\;100 \cdot \frac{\frac{\mathsf{fma}\left(i, 2, \mathsf{fma}\left(2, {i}^{2}, 2 \cdot \left(\log 1 \cdot n\right)\right) - 2 \cdot \left({i}^{2} \cdot \log 1\right)\right)}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{\frac{i}{n}}\\ \mathbf{elif}\;i \le -3.645341750186080323375566759305922898034 \cdot 10^{-293}:\\ \;\;\;\;\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}}\\ \mathbf{elif}\;i \le 9.27900448690603418810951552586629986763:\\ \;\;\;\;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{else}:\\ \;\;\;\;100 \cdot \frac{\frac{\frac{{\left({\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)}\right)}^{3} + {\left(-1 \cdot 1\right)}^{3}}{\mathsf{fma}\left(-1 \cdot 1, \left(-1 \cdot 1\right) - {\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)}, {\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot \left(2 \cdot n\right)\right)}\right)}}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{\frac{i}{n}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020002 +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))))