Average Error: 47.1 → 16.9
Time: 11.9s
Precision: 64
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le -0.0390445009989891442:\\ \;\;\;\;\frac{100}{i} \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{1}{n}}\\ \mathbf{elif}\;i \le 22.1533166385413587:\\ \;\;\;\;\left(\left(50 \cdot i + \left(100 \cdot \frac{\log 1 \cdot n}{i} + 100\right)\right) - 50 \cdot \left(i \cdot \log 1\right)\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\frac{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + \left(-1 \cdot 1\right)}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{\frac{i}{n}}\\ \end{array}\]
100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}
↓
\begin{array}{l}
\mathbf{if}\;i \le -0.0390445009989891442:\\
\;\;\;\;\frac{100}{i} \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{1}{n}}\\

\mathbf{elif}\;i \le 22.1533166385413587:\\
\;\;\;\;\left(\left(50 \cdot i + \left(100 \cdot \frac{\log 1 \cdot n}{i} + 100\right)\right) - 50 \cdot \left(i \cdot \log 1\right)\right) \cdot n\\

\mathbf{else}:\\
\;\;\;\;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}}{\frac{i}{n}}\\

\end{array}
double f(double i, double n) {
        double r135725 = 100.0;
        double r135726 = 1.0;
        double r135727 = i;
        double r135728 = n;
        double r135729 = r135727 / r135728;
        double r135730 = r135726 + r135729;
        double r135731 = pow(r135730, r135728);
        double r135732 = r135731 - r135726;
        double r135733 = r135732 / r135729;
        double r135734 = r135725 * r135733;
        return r135734;
}
↓
double f(double i, double n) {
        double r135735 = i;
        double r135736 = -0.039044500998989144;
        bool r135737 = r135735 <= r135736;
        double r135738 = 100.0;
        double r135739 = r135738 / r135735;
        double r135740 = 1.0;
        double r135741 = n;
        double r135742 = r135735 / r135741;
        double r135743 = r135740 + r135742;
        double r135744 = pow(r135743, r135741);
        double r135745 = r135744 - r135740;
        double r135746 = 1.0;
        double r135747 = r135746 / r135741;
        double r135748 = r135745 / r135747;
        double r135749 = r135739 * r135748;
        double r135750 = 22.15331663854136;
        bool r135751 = r135735 <= r135750;
        double r135752 = 50.0;
        double r135753 = r135752 * r135735;
        double r135754 = log(r135740);
        double r135755 = r135754 * r135741;
        double r135756 = r135755 / r135735;
        double r135757 = r135738 * r135756;
        double r135758 = r135757 + r135738;
        double r135759 = r135753 + r135758;
        double r135760 = r135735 * r135754;
        double r135761 = r135752 * r135760;
        double r135762 = r135759 - r135761;
        double r135763 = r135762 * r135741;
        double r135764 = 2.0;
        double r135765 = r135764 * r135741;
        double r135766 = pow(r135743, r135765);
        double r135767 = r135740 * r135740;
        double r135768 = -r135767;
        double r135769 = r135766 + r135768;
        double r135770 = r135744 + r135740;
        double r135771 = r135769 / r135770;
        double r135772 = r135771 / r135742;
        double r135773 = r135738 * r135772;
        double r135774 = r135751 ? r135763 : r135773;
        double r135775 = r135737 ? r135749 : r135774;
        return r135775;
}

Error

Bits error versus i

Bits error versus n

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original47.1
Target46.8
Herbie16.9
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 3 regimes

  2. if i < -0.039044500998989144

    1. Initial program 27.3

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

    3. Applied div-inv27.3

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

    4. Applied *-un-lft-identity27.3

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

    5. Applied times-frac27.8

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

    6. Applied associate-*r*27.8

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

    7. Simplified27.8

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

    if -0.039044500998989144 < i < 22.15331663854136

    1. Initial program 57.8

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

    2. Taylor expanded around 0 26.8

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

    4. Applied associate-/r/9.8

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

    5. Applied associate-*r*9.8

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

    6. Taylor expanded around 0 9.8

      \[\leadsto \color{blue}{\left(\left(100 + \left(50 \cdot i + 100 \cdot \frac{\log 1 \cdot n}{i}\right)\right) - 50 \cdot \left(i \cdot \log 1\right)\right)} \cdot n\]

    7. Simplified9.8

      \[\leadsto \color{blue}{\left(\left(50 \cdot i + \left(100 \cdot \frac{\log 1 \cdot n}{i} + 100\right)\right) - 50 \cdot \left(i \cdot \log 1\right)\right)} \cdot n\]

    if 22.15331663854136 < i

    1. Initial program 31.6

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

    3. Applied flip--31.6

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

    4. Simplified31.6

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

  4. Final simplification16.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -0.0390445009989891442:\\ \;\;\;\;\frac{100}{i} \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{1}{n}}\\ \mathbf{elif}\;i \le 22.1533166385413587:\\ \;\;\;\;\left(\left(50 \cdot i + \left(100 \cdot \frac{\log 1 \cdot n}{i} + 100\right)\right) - 50 \cdot \left(i \cdot \log 1\right)\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\frac{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + \left(-1 \cdot 1\right)}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{\frac{i}{n}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020062 
(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))))