Average Error: 42.6 → 20.6
Time: 29.7s
Precision: 64
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;n \le -0.7181130799545676:\\ \;\;\;\;100 \cdot n + \left(\left(\frac{1}{2} \cdot i + \frac{1}{6} \cdot \left(i \cdot i\right)\right) \cdot n\right) \cdot 100\\ \mathbf{elif}\;n \le 3.201668089989767 \cdot 10^{-118}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;100 \cdot n + \left(\left(\frac{1}{2} \cdot i + \frac{1}{6} \cdot \left(i \cdot i\right)\right) \cdot n\right) \cdot 100\\ \end{array}\]
100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}
↓
\begin{array}{l}
\mathbf{if}\;n \le -0.7181130799545676:\\
\;\;\;\;100 \cdot n + \left(\left(\frac{1}{2} \cdot i + \frac{1}{6} \cdot \left(i \cdot i\right)\right) \cdot n\right) \cdot 100\\

\mathbf{elif}\;n \le 3.201668089989767 \cdot 10^{-118}:\\
\;\;\;\;0\\

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

\end{array}
double f(double i, double n) {
        double r5245696 = 100.0;
        double r5245697 = 1.0;
        double r5245698 = i;
        double r5245699 = n;
        double r5245700 = r5245698 / r5245699;
        double r5245701 = r5245697 + r5245700;
        double r5245702 = pow(r5245701, r5245699);
        double r5245703 = r5245702 - r5245697;
        double r5245704 = r5245703 / r5245700;
        double r5245705 = r5245696 * r5245704;
        return r5245705;
}
↓
double f(double i, double n) {
        double r5245706 = n;
        double r5245707 = -0.7181130799545676;
        bool r5245708 = r5245706 <= r5245707;
        double r5245709 = 100.0;
        double r5245710 = r5245709 * r5245706;
        double r5245711 = 0.5;
        double r5245712 = i;
        double r5245713 = r5245711 * r5245712;
        double r5245714 = 0.16666666666666666;
        double r5245715 = r5245712 * r5245712;
        double r5245716 = r5245714 * r5245715;
        double r5245717 = r5245713 + r5245716;
        double r5245718 = r5245717 * r5245706;
        double r5245719 = r5245718 * r5245709;
        double r5245720 = r5245710 + r5245719;
        double r5245721 = 3.201668089989767e-118;
        bool r5245722 = r5245706 <= r5245721;
        double r5245723 = 0.0;
        double r5245724 = r5245722 ? r5245723 : r5245720;
        double r5245725 = r5245708 ? r5245720 : r5245724;
        return r5245725;
}

Error

Bits error versus i

Bits error versus n

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original42.6
Target42.2
Herbie20.6
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 2 regimes

  2. if n < -0.7181130799545676 or 3.201668089989767e-118 < n

    1. Initial program 51.9

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

    2. Taylor expanded around 0 37.1

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

    3. Simplified37.1

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

    4. Taylor expanded around inf 21.3

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

    5. Simplified21.3

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

    7. Applied distribute-rgt-in21.3

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

    8. Simplified21.1

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

    if -0.7181130799545676 < n < 3.201668089989767e-118

    1. Initial program 25.9

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

    2. Taylor expanded around 0 19.5

      \[\leadsto \color{blue}{0}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification20.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -0.7181130799545676:\\ \;\;\;\;100 \cdot n + \left(\left(\frac{1}{2} \cdot i + \frac{1}{6} \cdot \left(i \cdot i\right)\right) \cdot n\right) \cdot 100\\ \mathbf{elif}\;n \le 3.201668089989767 \cdot 10^{-118}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;100 \cdot n + \left(\left(\frac{1}{2} \cdot i + \frac{1}{6} \cdot \left(i \cdot i\right)\right) \cdot n\right) \cdot 100\\ \end{array}\]

Reproduce

herbie shell --seed 2019134 
(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))))