Compound Interest

Average Error: 43.0 → 18.8

Time: 32.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;i \le -0.8489635077384908301567634225648362189531:\\ \;\;\;\;100 \cdot \frac{{\left(\frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 0.5476118163719528864064045592385809868574:\\ \;\;\;\;\frac{100}{i} \cdot \left(\left(\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right) \cdot n\right)\\ \mathbf{else}:\\ \;\;\;\;\left(100 \cdot \frac{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} - 1} \cdot \sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} - 1}}{i}\right) \cdot \frac{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} - 1}}{\frac{1}{n}}\\ \end{array}\]

C

double f(double i, double n) {
        double r110804 = 100.0;
        double r110805 = 1.0;
        double r110806 = i;
        double r110807 = n;
        double r110808 = r110806 / r110807;
        double r110809 = r110805 + r110808;
        double r110810 = pow(r110809, r110807);
        double r110811 = r110810 - r110805;
        double r110812 = r110811 / r110808;
        double r110813 = r110804 * r110812;
        return r110813;
}

↓

double f(double i, double n) {
        double r110814 = i;
        double r110815 = -0.8489635077384908;
        bool r110816 = r110814 <= r110815;
        double r110817 = 100.0;
        double r110818 = n;
        double r110819 = r110814 / r110818;
        double r110820 = pow(r110819, r110818);
        double r110821 = 1.0;
        double r110822 = r110820 - r110821;
        double r110823 = r110822 / r110819;
        double r110824 = r110817 * r110823;
        double r110825 = 0.5476118163719529;
        bool r110826 = r110814 <= r110825;
        double r110827 = r110817 / r110814;
        double r110828 = r110821 * r110814;
        double r110829 = 0.5;
        double r110830 = 2.0;
        double r110831 = pow(r110814, r110830);
        double r110832 = r110829 * r110831;
        double r110833 = log(r110821);
        double r110834 = r110833 * r110818;
        double r110835 = r110832 + r110834;
        double r110836 = r110828 + r110835;
        double r110837 = r110831 * r110833;
        double r110838 = r110829 * r110837;
        double r110839 = r110836 - r110838;
        double r110840 = r110839 * r110818;
        double r110841 = r110827 * r110840;
        double r110842 = r110821 + r110819;
        double r110843 = pow(r110842, r110818);
        double r110844 = r110843 - r110821;
        double r110845 = cbrt(r110844);
        double r110846 = r110845 * r110845;
        double r110847 = r110846 / r110814;
        double r110848 = r110817 * r110847;
        double r110849 = 1.0;
        double r110850 = r110849 / r110818;
        double r110851 = r110845 / r110850;
        double r110852 = r110848 * r110851;
        double r110853 = r110826 ? r110841 : r110852;
        double r110854 = r110816 ? r110824 : r110853;
        return r110854;
}

Error

Bits error versus i

Bits error versus n

Target

Original	43.0
Target	43.5
Herbie	18.8

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

Derivation

1. Split input into 3 regimes

2. if i < -0.8489635077384908

   1. Initial program 27.4

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

   2. Taylor expanded around inf 64.0

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

   3. Simplified 18.2

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

   if -0.8489635077384908 < i < 0.5476118163719529

   1. Initial program 50.8

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

   2. Taylor expanded around 0 34.2

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

   4. Applied div-inv 34.3

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

   5. Applied *-un-lft-identity 34.3

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

   6. Applied times-frac 15.8

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

   7. Simplified 15.8

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

   9. Applied associate-*r* 16.2

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

   10. Simplified 16.1

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

   if 0.5476118163719529 < 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 32.4

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

   4. Applied add-cube-cbrt 32.4

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

   5. Applied times-frac 32.4

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

   6. Applied associate-*r* 32.4

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

4. Final simplification 18.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -0.8489635077384908301567634225648362189531:\\ \;\;\;\;100 \cdot \frac{{\left(\frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 0.5476118163719528864064045592385809868574:\\ \;\;\;\;\frac{100}{i} \cdot \left(\left(\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right) \cdot n\right)\\ \mathbf{else}:\\ \;\;\;\;\left(100 \cdot \frac{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} - 1} \cdot \sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} - 1}}{i}\right) \cdot \frac{\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n} - 1}}{\frac{1}{n}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 
(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))))