Average Error: 42.1 → 21.0
Time: 26.5s
Precision: 64
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;n \le -6.470021336482934 \cdot 10^{+94}:\\ \;\;\;\;\left(n + n \cdot \left(\frac{1}{2} \cdot i + \frac{1}{6} \cdot \left(i \cdot i\right)\right)\right) \cdot 100\\ \mathbf{elif}\;n \le -6.429664273656172 \cdot 10^{+68}:\\ \;\;\;\;n \cdot \left(\frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{i} \cdot 100\right)\\ \mathbf{elif}\;n \le -0.574841385015252:\\ \;\;\;\;\left(n + n \cdot \left(\frac{1}{2} \cdot i + \frac{1}{6} \cdot \left(i \cdot i\right)\right)\right) \cdot 100\\ \mathbf{elif}\;n \le 1.2083195710773826 \cdot 10^{-138}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\left(n + n \cdot \left(\frac{1}{2} \cdot i + \frac{1}{6} \cdot \left(i \cdot i\right)\right)\right) \cdot 100\\ \end{array}\]
100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}
↓
\begin{array}{l}
\mathbf{if}\;n \le -6.470021336482934 \cdot 10^{+94}:\\
\;\;\;\;\left(n + n \cdot \left(\frac{1}{2} \cdot i + \frac{1}{6} \cdot \left(i \cdot i\right)\right)\right) \cdot 100\\

\mathbf{elif}\;n \le -6.429664273656172 \cdot 10^{+68}:\\
\;\;\;\;n \cdot \left(\frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{i} \cdot 100\right)\\

\mathbf{elif}\;n \le -0.574841385015252:\\
\;\;\;\;\left(n + n \cdot \left(\frac{1}{2} \cdot i + \frac{1}{6} \cdot \left(i \cdot i\right)\right)\right) \cdot 100\\

\mathbf{elif}\;n \le 1.2083195710773826 \cdot 10^{-138}:\\
\;\;\;\;0\\

\mathbf{else}:\\
\;\;\;\;\left(n + n \cdot \left(\frac{1}{2} \cdot i + \frac{1}{6} \cdot \left(i \cdot i\right)\right)\right) \cdot 100\\

\end{array}
double f(double i, double n) {
        double r5069820 = 100.0;
        double r5069821 = 1.0;
        double r5069822 = i;
        double r5069823 = n;
        double r5069824 = r5069822 / r5069823;
        double r5069825 = r5069821 + r5069824;
        double r5069826 = pow(r5069825, r5069823);
        double r5069827 = r5069826 - r5069821;
        double r5069828 = r5069827 / r5069824;
        double r5069829 = r5069820 * r5069828;
        return r5069829;
}
↓
double f(double i, double n) {
        double r5069830 = n;
        double r5069831 = -6.470021336482934e+94;
        bool r5069832 = r5069830 <= r5069831;
        double r5069833 = 0.5;
        double r5069834 = i;
        double r5069835 = r5069833 * r5069834;
        double r5069836 = 0.16666666666666666;
        double r5069837 = r5069834 * r5069834;
        double r5069838 = r5069836 * r5069837;
        double r5069839 = r5069835 + r5069838;
        double r5069840 = r5069830 * r5069839;
        double r5069841 = r5069830 + r5069840;
        double r5069842 = 100.0;
        double r5069843 = r5069841 * r5069842;
        double r5069844 = -6.429664273656172e+68;
        bool r5069845 = r5069830 <= r5069844;
        double r5069846 = r5069834 / r5069830;
        double r5069847 = 1.0;
        double r5069848 = r5069846 + r5069847;
        double r5069849 = pow(r5069848, r5069830);
        double r5069850 = r5069849 - r5069847;
        double r5069851 = r5069850 / r5069834;
        double r5069852 = r5069851 * r5069842;
        double r5069853 = r5069830 * r5069852;
        double r5069854 = -0.574841385015252;
        bool r5069855 = r5069830 <= r5069854;
        double r5069856 = 1.2083195710773826e-138;
        bool r5069857 = r5069830 <= r5069856;
        double r5069858 = 0.0;
        double r5069859 = r5069857 ? r5069858 : r5069843;
        double r5069860 = r5069855 ? r5069843 : r5069859;
        double r5069861 = r5069845 ? r5069853 : r5069860;
        double r5069862 = r5069832 ? r5069843 : r5069861;
        return r5069862;
}

Error

Bits error versus i

Bits error versus n

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original42.1
Target41.8
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 < -6.470021336482934e+94 or -6.429664273656172e+68 < n < -0.574841385015252 or 1.2083195710773826e-138 < n

    1. Initial program 52.4

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

    2. Taylor expanded around 0 37.5

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

    3. Simplified37.5

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

    5. Applied associate-/r/21.9

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

    6. Simplified21.9

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

    7. Taylor expanded around 0 22.0

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

    8. Simplified21.9

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

    if -6.470021336482934e+94 < n < -6.429664273656172e+68

    1. Initial program 33.9

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

    3. Applied associate-/r/33.8

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

    4. Applied associate-*r*33.7

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

    if -0.574841385015252 < n < 1.2083195710773826e-138

    1. Initial program 24.4

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

    2. Taylor expanded around 0 18.5

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

  4. Final simplification21.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -6.470021336482934 \cdot 10^{+94}:\\ \;\;\;\;\left(n + n \cdot \left(\frac{1}{2} \cdot i + \frac{1}{6} \cdot \left(i \cdot i\right)\right)\right) \cdot 100\\ \mathbf{elif}\;n \le -6.429664273656172 \cdot 10^{+68}:\\ \;\;\;\;n \cdot \left(\frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{i} \cdot 100\right)\\ \mathbf{elif}\;n \le -0.574841385015252:\\ \;\;\;\;\left(n + n \cdot \left(\frac{1}{2} \cdot i + \frac{1}{6} \cdot \left(i \cdot i\right)\right)\right) \cdot 100\\ \mathbf{elif}\;n \le 1.2083195710773826 \cdot 10^{-138}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\left(n + n \cdot \left(\frac{1}{2} \cdot i + \frac{1}{6} \cdot \left(i \cdot i\right)\right)\right) \cdot 100\\ \end{array}\]

Reproduce

herbie shell --seed 2019162 
(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))))