Average Error: 43.0 → 20.1
Time: 1.3m
Precision: 64
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le -1.237131456279114563109935683102053260005 \cdot 10^{-18}:\\ \;\;\;\;\frac{100 \cdot {\left(\frac{1}{n} \cdot i\right)}^{n} - 100}{i} \cdot n\\ \mathbf{elif}\;i \le 3222196569.804427623748779296875:\\ \;\;\;\;\left(100 \cdot \frac{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{i}\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\frac{n \cdot \left({\left(\frac{i}{n}\right)}^{\left(2 \cdot n\right)} - 1\right)}{{\left(\frac{i}{n}\right)}^{n} + 1}}{i}\\ \end{array}\]
100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}
↓
\begin{array}{l}
\mathbf{if}\;i \le -1.237131456279114563109935683102053260005 \cdot 10^{-18}:\\
\;\;\;\;\frac{100 \cdot {\left(\frac{1}{n} \cdot i\right)}^{n} - 100}{i} \cdot n\\

\mathbf{elif}\;i \le 3222196569.804427623748779296875:\\
\;\;\;\;\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\\

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

\end{array}
double f(double i, double n) {
        double r362694 = 100.0;
        double r362695 = 1.0;
        double r362696 = i;
        double r362697 = n;
        double r362698 = r362696 / r362697;
        double r362699 = r362695 + r362698;
        double r362700 = pow(r362699, r362697);
        double r362701 = r362700 - r362695;
        double r362702 = r362701 / r362698;
        double r362703 = r362694 * r362702;
        return r362703;
}
↓
double f(double i, double n) {
        double r362704 = i;
        double r362705 = -1.2371314562791146e-18;
        bool r362706 = r362704 <= r362705;
        double r362707 = 100.0;
        double r362708 = 1.0;
        double r362709 = n;
        double r362710 = r362708 / r362709;
        double r362711 = r362710 * r362704;
        double r362712 = pow(r362711, r362709);
        double r362713 = r362707 * r362712;
        double r362714 = r362713 - r362707;
        double r362715 = r362714 / r362704;
        double r362716 = r362715 * r362709;
        double r362717 = 3222196569.8044276;
        bool r362718 = r362704 <= r362717;
        double r362719 = 1.0;
        double r362720 = r362719 * r362704;
        double r362721 = 0.5;
        double r362722 = 2.0;
        double r362723 = pow(r362704, r362722);
        double r362724 = r362721 * r362723;
        double r362725 = log(r362719);
        double r362726 = r362725 * r362709;
        double r362727 = r362724 + r362726;
        double r362728 = r362720 + r362727;
        double r362729 = r362723 * r362725;
        double r362730 = r362721 * r362729;
        double r362731 = r362728 - r362730;
        double r362732 = r362731 / r362704;
        double r362733 = r362707 * r362732;
        double r362734 = r362733 * r362709;
        double r362735 = r362704 / r362709;
        double r362736 = r362722 * r362709;
        double r362737 = pow(r362735, r362736);
        double r362738 = r362737 - r362719;
        double r362739 = r362709 * r362738;
        double r362740 = pow(r362735, r362709);
        double r362741 = r362740 + r362719;
        double r362742 = r362739 / r362741;
        double r362743 = r362742 / r362704;
        double r362744 = r362707 * r362743;
        double r362745 = r362718 ? r362734 : r362744;
        double r362746 = r362706 ? r362716 : r362745;
        return r362746;
}

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
Target42.7
Herbie20.1
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 3 regimes

  2. if i < -1.2371314562791146e-18

    1. Initial program 29.3

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

    3. Applied associate-/r/29.9

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

    4. Applied associate-*r*29.9

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

    5. Taylor expanded around inf 64.0

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

    6. Simplified21.6

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

    if -1.2371314562791146e-18 < i < 3222196569.8044276

    1. Initial program 50.8

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

    3. Applied associate-/r/50.4

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

    4. Applied associate-*r*50.4

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

    5. Taylor expanded around 0 16.9

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

    if 3222196569.8044276 < i

    1. Initial program 32.3

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

    3. Applied flip--32.3

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

    4. Applied associate-/l/32.3

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

    5. Taylor expanded around inf 28.4

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

    6. Simplified32.2

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

  4. Final simplification20.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -1.237131456279114563109935683102053260005 \cdot 10^{-18}:\\ \;\;\;\;\frac{100 \cdot {\left(\frac{1}{n} \cdot i\right)}^{n} - 100}{i} \cdot n\\ \mathbf{elif}\;i \le 3222196569.804427623748779296875:\\ \;\;\;\;\left(100 \cdot \frac{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{i}\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\frac{n \cdot \left({\left(\frac{i}{n}\right)}^{\left(2 \cdot n\right)} - 1\right)}{{\left(\frac{i}{n}\right)}^{n} + 1}}{i}\\ \end{array}\]

Reproduce

herbie shell --seed 2019212 
(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))))