Average Error: 42.8 → 21.0
Time: 25.8s
Precision: 64
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;n \le -2.018924830568215247325901083553769937794 \cdot 10^{68}:\\ \;\;\;\;\left(100 \cdot \left(\left(0.5 \cdot i + \left(\frac{\log 1 \cdot n}{i} + 1\right)\right) - 0.5 \cdot \left(i \cdot \log 1\right)\right)\right) \cdot n\\ \mathbf{elif}\;n \le 5.743050826285918365465354232866532417819 \cdot 10^{-309}:\\ \;\;\;\;\left(100 \cdot \frac{\frac{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + \left(-1 \cdot 1\right)}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{i}\right) \cdot n\\ \mathbf{elif}\;n \le 1.657016025576251879264718591748443204092 \cdot 10^{-183}:\\ \;\;\;\;\frac{100 \cdot e^{\left(\log \left(\frac{1}{n}\right) - \log \left(\frac{1}{i}\right)\right) \cdot n} - 100}{i} \cdot n\\ \mathbf{else}:\\ \;\;\;\;\left(100 \cdot \left(\left(0.5 \cdot i + \left(\frac{\log 1 \cdot n}{i} + 1\right)\right) - 0.5 \cdot \left(i \cdot \log 1\right)\right)\right) \cdot n\\ \end{array}\]
100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}
↓
\begin{array}{l}
\mathbf{if}\;n \le -2.018924830568215247325901083553769937794 \cdot 10^{68}:\\
\;\;\;\;\left(100 \cdot \left(\left(0.5 \cdot i + \left(\frac{\log 1 \cdot n}{i} + 1\right)\right) - 0.5 \cdot \left(i \cdot \log 1\right)\right)\right) \cdot n\\

\mathbf{elif}\;n \le 5.743050826285918365465354232866532417819 \cdot 10^{-309}:\\
\;\;\;\;\left(100 \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}}{i}\right) \cdot n\\

\mathbf{elif}\;n \le 1.657016025576251879264718591748443204092 \cdot 10^{-183}:\\
\;\;\;\;\frac{100 \cdot e^{\left(\log \left(\frac{1}{n}\right) - \log \left(\frac{1}{i}\right)\right) \cdot n} - 100}{i} \cdot n\\

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

\end{array}
double f(double i, double n) {
        double r125671 = 100.0;
        double r125672 = 1.0;
        double r125673 = i;
        double r125674 = n;
        double r125675 = r125673 / r125674;
        double r125676 = r125672 + r125675;
        double r125677 = pow(r125676, r125674);
        double r125678 = r125677 - r125672;
        double r125679 = r125678 / r125675;
        double r125680 = r125671 * r125679;
        return r125680;
}
↓
double f(double i, double n) {
        double r125681 = n;
        double r125682 = -2.0189248305682152e+68;
        bool r125683 = r125681 <= r125682;
        double r125684 = 100.0;
        double r125685 = 0.5;
        double r125686 = i;
        double r125687 = r125685 * r125686;
        double r125688 = 1.0;
        double r125689 = log(r125688);
        double r125690 = r125689 * r125681;
        double r125691 = r125690 / r125686;
        double r125692 = r125691 + r125688;
        double r125693 = r125687 + r125692;
        double r125694 = r125686 * r125689;
        double r125695 = r125685 * r125694;
        double r125696 = r125693 - r125695;
        double r125697 = r125684 * r125696;
        double r125698 = r125697 * r125681;
        double r125699 = 5.74305082628592e-309;
        bool r125700 = r125681 <= r125699;
        double r125701 = r125686 / r125681;
        double r125702 = r125688 + r125701;
        double r125703 = 2.0;
        double r125704 = r125703 * r125681;
        double r125705 = pow(r125702, r125704);
        double r125706 = r125688 * r125688;
        double r125707 = -r125706;
        double r125708 = r125705 + r125707;
        double r125709 = pow(r125702, r125681);
        double r125710 = r125709 + r125688;
        double r125711 = r125708 / r125710;
        double r125712 = r125711 / r125686;
        double r125713 = r125684 * r125712;
        double r125714 = r125713 * r125681;
        double r125715 = 1.6570160255762519e-183;
        bool r125716 = r125681 <= r125715;
        double r125717 = 1.0;
        double r125718 = r125717 / r125681;
        double r125719 = log(r125718);
        double r125720 = r125717 / r125686;
        double r125721 = log(r125720);
        double r125722 = r125719 - r125721;
        double r125723 = r125722 * r125681;
        double r125724 = exp(r125723);
        double r125725 = r125684 * r125724;
        double r125726 = r125725 - r125684;
        double r125727 = r125726 / r125686;
        double r125728 = r125727 * r125681;
        double r125729 = r125716 ? r125728 : r125698;
        double r125730 = r125700 ? r125714 : r125729;
        double r125731 = r125683 ? r125698 : r125730;
        return r125731;
}

Error

Bits error versus i

Bits error versus n

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original42.8
Target42.7
Herbie21.0
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 3 regimes

  2. if n < -2.0189248305682152e+68 or 1.6570160255762519e-183 < n

    1. Initial program 55.0

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

    3. Applied associate-/r/54.7

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

    4. Applied associate-*r*54.6

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

    5. Taylor expanded around 0 21.9

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

    6. Taylor expanded around 0 21.8

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

    7. Simplified21.8

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

    if -2.0189248305682152e+68 < n < 5.74305082628592e-309

    1. Initial program 19.3

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

    3. Applied associate-/r/19.8

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

    4. Applied associate-*r*19.8

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

    6. Applied flip--19.8

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

    7. Simplified19.8

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

    if 5.74305082628592e-309 < n < 1.6570160255762519e-183

    1. Initial program 41.0

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

    3. Applied associate-/r/41.0

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

    4. Applied associate-*r*41.0

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

    5. Taylor expanded around inf 19.7

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

  4. Final simplification21.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -2.018924830568215247325901083553769937794 \cdot 10^{68}:\\ \;\;\;\;\left(100 \cdot \left(\left(0.5 \cdot i + \left(\frac{\log 1 \cdot n}{i} + 1\right)\right) - 0.5 \cdot \left(i \cdot \log 1\right)\right)\right) \cdot n\\ \mathbf{elif}\;n \le 5.743050826285918365465354232866532417819 \cdot 10^{-309}:\\ \;\;\;\;\left(100 \cdot \frac{\frac{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + \left(-1 \cdot 1\right)}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{i}\right) \cdot n\\ \mathbf{elif}\;n \le 1.657016025576251879264718591748443204092 \cdot 10^{-183}:\\ \;\;\;\;\frac{100 \cdot e^{\left(\log \left(\frac{1}{n}\right) - \log \left(\frac{1}{i}\right)\right) \cdot n} - 100}{i} \cdot n\\ \mathbf{else}:\\ \;\;\;\;\left(100 \cdot \left(\left(0.5 \cdot i + \left(\frac{\log 1 \cdot n}{i} + 1\right)\right) - 0.5 \cdot \left(i \cdot \log 1\right)\right)\right) \cdot n\\ \end{array}\]

Reproduce

herbie shell --seed 2019298 
(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))))