Average Error: 42.5 → 20.3
Time: 28.6s
Precision: 64
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;n \le -0.6272769565773454:\\ \;\;\;\;100 \cdot \left(\left(n + i \cdot \left(\frac{1}{2} \cdot n\right)\right) + \left(\left(i \cdot i\right) \cdot \frac{1}{6}\right) \cdot n\right)\\ \mathbf{elif}\;n \le 1.4389956636303172 \cdot 10^{-137}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(\left(n + i \cdot \left(\frac{1}{2} \cdot n\right)\right) + \left(\left(i \cdot i\right) \cdot \frac{1}{6}\right) \cdot n\right)\\ \end{array}\]
100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}
↓
\begin{array}{l}
\mathbf{if}\;n \le -0.6272769565773454:\\
\;\;\;\;100 \cdot \left(\left(n + i \cdot \left(\frac{1}{2} \cdot n\right)\right) + \left(\left(i \cdot i\right) \cdot \frac{1}{6}\right) \cdot n\right)\\

\mathbf{elif}\;n \le 1.4389956636303172 \cdot 10^{-137}:\\
\;\;\;\;0\\

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

\end{array}
double f(double i, double n) {
        double r6273603 = 100.0;
        double r6273604 = 1.0;
        double r6273605 = i;
        double r6273606 = n;
        double r6273607 = r6273605 / r6273606;
        double r6273608 = r6273604 + r6273607;
        double r6273609 = pow(r6273608, r6273606);
        double r6273610 = r6273609 - r6273604;
        double r6273611 = r6273610 / r6273607;
        double r6273612 = r6273603 * r6273611;
        return r6273612;
}
↓
double f(double i, double n) {
        double r6273613 = n;
        double r6273614 = -0.6272769565773454;
        bool r6273615 = r6273613 <= r6273614;
        double r6273616 = 100.0;
        double r6273617 = i;
        double r6273618 = 0.5;
        double r6273619 = r6273618 * r6273613;
        double r6273620 = r6273617 * r6273619;
        double r6273621 = r6273613 + r6273620;
        double r6273622 = r6273617 * r6273617;
        double r6273623 = 0.16666666666666666;
        double r6273624 = r6273622 * r6273623;
        double r6273625 = r6273624 * r6273613;
        double r6273626 = r6273621 + r6273625;
        double r6273627 = r6273616 * r6273626;
        double r6273628 = 1.4389956636303172e-137;
        bool r6273629 = r6273613 <= r6273628;
        double r6273630 = 0.0;
        double r6273631 = r6273629 ? r6273630 : r6273627;
        double r6273632 = r6273615 ? r6273627 : r6273631;
        return r6273632;
}

Error

Bits error versus i

Bits error versus n

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original42.5
Target42.6
Herbie20.3
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 2 regimes

  2. if n < -0.6272769565773454 or 1.4389956636303172e-137 < n

    1. Initial program 52.0

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

    2. Taylor expanded around 0 37.6

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

    3. Simplified37.6

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

    4. Taylor expanded around 0 21.6

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

    5. Simplified21.5

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

    if -0.6272769565773454 < n < 1.4389956636303172e-137

    1. Initial program 24.6

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

    2. Taylor expanded around 0 18.0

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

  4. Final simplification20.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -0.6272769565773454:\\ \;\;\;\;100 \cdot \left(\left(n + i \cdot \left(\frac{1}{2} \cdot n\right)\right) + \left(\left(i \cdot i\right) \cdot \frac{1}{6}\right) \cdot n\right)\\ \mathbf{elif}\;n \le 1.4389956636303172 \cdot 10^{-137}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(\left(n + i \cdot \left(\frac{1}{2} \cdot n\right)\right) + \left(\left(i \cdot i\right) \cdot \frac{1}{6}\right) \cdot n\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 
(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))))