Average Error: 43.0 → 23.1
Time: 26.9s
Precision: 64
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;n \le -2.260047491173430560879176932933542305034 \cdot 10^{95}:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{\left(1 \cdot i + 0.5 \cdot {i}^{2}\right) + \log 1 \cdot \left(n - 0.5 \cdot {i}^{2}\right)}{i}\right)\\ \mathbf{elif}\;n \le -3.040807945417772148751844957499095808404 \cdot 10^{-242}:\\ \;\;\;\;\sqrt{100} \cdot \left(\sqrt{100} \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\right)\\ \mathbf{elif}\;n \le 1.48872624183818144300366844690655908205 \cdot 10^{-106}:\\ \;\;\;\;100 \cdot \frac{\left(1 \cdot i + \left(\log 1 \cdot n + 1\right)\right) - 1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{\left(1 \cdot i + 0.5 \cdot {i}^{2}\right) + \log 1 \cdot \left(n - 0.5 \cdot {i}^{2}\right)}{i}\right)\\ \end{array}\]
100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}
↓
\begin{array}{l}
\mathbf{if}\;n \le -2.260047491173430560879176932933542305034 \cdot 10^{95}:\\
\;\;\;\;100 \cdot \left(n \cdot \frac{\left(1 \cdot i + 0.5 \cdot {i}^{2}\right) + \log 1 \cdot \left(n - 0.5 \cdot {i}^{2}\right)}{i}\right)\\

\mathbf{elif}\;n \le -3.040807945417772148751844957499095808404 \cdot 10^{-242}:\\
\;\;\;\;\sqrt{100} \cdot \left(\sqrt{100} \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\right)\\

\mathbf{elif}\;n \le 1.48872624183818144300366844690655908205 \cdot 10^{-106}:\\
\;\;\;\;100 \cdot \frac{\left(1 \cdot i + \left(\log 1 \cdot n + 1\right)\right) - 1}{\frac{i}{n}}\\

\mathbf{else}:\\
\;\;\;\;100 \cdot \left(n \cdot \frac{\left(1 \cdot i + 0.5 \cdot {i}^{2}\right) + \log 1 \cdot \left(n - 0.5 \cdot {i}^{2}\right)}{i}\right)\\

\end{array}
double f(double i, double n) {
        double r165769 = 100.0;
        double r165770 = 1.0;
        double r165771 = i;
        double r165772 = n;
        double r165773 = r165771 / r165772;
        double r165774 = r165770 + r165773;
        double r165775 = pow(r165774, r165772);
        double r165776 = r165775 - r165770;
        double r165777 = r165776 / r165773;
        double r165778 = r165769 * r165777;
        return r165778;
}
↓
double f(double i, double n) {
        double r165779 = n;
        double r165780 = -2.2600474911734306e+95;
        bool r165781 = r165779 <= r165780;
        double r165782 = 100.0;
        double r165783 = 1.0;
        double r165784 = i;
        double r165785 = r165783 * r165784;
        double r165786 = 0.5;
        double r165787 = 2.0;
        double r165788 = pow(r165784, r165787);
        double r165789 = r165786 * r165788;
        double r165790 = r165785 + r165789;
        double r165791 = log(r165783);
        double r165792 = r165779 - r165789;
        double r165793 = r165791 * r165792;
        double r165794 = r165790 + r165793;
        double r165795 = r165794 / r165784;
        double r165796 = r165779 * r165795;
        double r165797 = r165782 * r165796;
        double r165798 = -3.040807945417772e-242;
        bool r165799 = r165779 <= r165798;
        double r165800 = sqrt(r165782);
        double r165801 = r165784 / r165779;
        double r165802 = r165783 + r165801;
        double r165803 = pow(r165802, r165779);
        double r165804 = r165803 - r165783;
        double r165805 = r165804 / r165801;
        double r165806 = r165800 * r165805;
        double r165807 = r165800 * r165806;
        double r165808 = 1.4887262418381814e-106;
        bool r165809 = r165779 <= r165808;
        double r165810 = r165791 * r165779;
        double r165811 = 1.0;
        double r165812 = r165810 + r165811;
        double r165813 = r165785 + r165812;
        double r165814 = r165813 - r165783;
        double r165815 = r165814 / r165801;
        double r165816 = r165782 * r165815;
        double r165817 = r165809 ? r165816 : r165797;
        double r165818 = r165799 ? r165807 : r165817;
        double r165819 = r165781 ? r165797 : r165818;
        return r165819;
}

Error

Bits error versus i

Bits error versus n

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original43.0
Target43.2
Herbie23.1
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 3 regimes

  2. if n < -2.2600474911734306e+95 or 1.4887262418381814e-106 < n

    1. Initial program 55.9

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

    3. Applied div-inv55.9

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

    4. Applied *-un-lft-identity55.9

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

    5. Applied times-frac55.5

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

    6. Applied associate-*r*55.5

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

    7. Simplified55.5

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

    8. Taylor expanded around 0 21.7

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

    10. Applied div-inv21.8

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

    11. Applied associate-*l*21.4

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

    12. Simplified21.2

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

    14. Applied *-un-lft-identity21.2

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

    15. Applied times-frac20.1

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

    16. Simplified20.1

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

    17. Simplified20.1

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

    if -2.2600474911734306e+95 < n < -3.040807945417772e-242

    1. Initial program 24.4

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

    3. Applied add-sqr-sqrt24.4

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

    4. Applied associate-*l*24.4

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

    if -3.040807945417772e-242 < n < 1.4887262418381814e-106

    1. Initial program 35.6

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

    2. Taylor expanded around 0 29.6

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

  4. Final simplification23.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -2.260047491173430560879176932933542305034 \cdot 10^{95}:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{\left(1 \cdot i + 0.5 \cdot {i}^{2}\right) + \log 1 \cdot \left(n - 0.5 \cdot {i}^{2}\right)}{i}\right)\\ \mathbf{elif}\;n \le -3.040807945417772148751844957499095808404 \cdot 10^{-242}:\\ \;\;\;\;\sqrt{100} \cdot \left(\sqrt{100} \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\right)\\ \mathbf{elif}\;n \le 1.48872624183818144300366844690655908205 \cdot 10^{-106}:\\ \;\;\;\;100 \cdot \frac{\left(1 \cdot i + \left(\log 1 \cdot n + 1\right)\right) - 1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{\left(1 \cdot i + 0.5 \cdot {i}^{2}\right) + \log 1 \cdot \left(n - 0.5 \cdot {i}^{2}\right)}{i}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019208 
(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))))