Average Error: 42.8 → 21.7
Time: 30.3s
Precision: 64
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le -3.076493361831821548083157134309908748576 \cdot 10^{-11}:\\ \;\;\;\;100 \cdot \frac{{\left({\left(\frac{i}{n} + 1\right)}^{n}\right)}^{3} - {1}^{3}}{\frac{i}{n} \cdot \left({\left(\frac{i}{n} + 1\right)}^{n} \cdot {\left(\frac{i}{n} + 1\right)}^{n} + \left(1 \cdot 1 + 1 \cdot {\left(\frac{i}{n} + 1\right)}^{n}\right)\right)}\\ \mathbf{elif}\;i \le 7.743283777926976085836940910667181015015:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{\left(\left(n - i \cdot \left(i \cdot 0.5\right)\right) \cdot \log 1 + i \cdot \left(i \cdot 0.5\right)\right) + 1 \cdot i}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{i}}{\frac{1}{n}}\\ \end{array}\]
100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}
↓
\begin{array}{l}
\mathbf{if}\;i \le -3.076493361831821548083157134309908748576 \cdot 10^{-11}:\\
\;\;\;\;100 \cdot \frac{{\left({\left(\frac{i}{n} + 1\right)}^{n}\right)}^{3} - {1}^{3}}{\frac{i}{n} \cdot \left({\left(\frac{i}{n} + 1\right)}^{n} \cdot {\left(\frac{i}{n} + 1\right)}^{n} + \left(1 \cdot 1 + 1 \cdot {\left(\frac{i}{n} + 1\right)}^{n}\right)\right)}\\

\mathbf{elif}\;i \le 7.743283777926976085836940910667181015015:\\
\;\;\;\;100 \cdot \left(n \cdot \frac{\left(\left(n - i \cdot \left(i \cdot 0.5\right)\right) \cdot \log 1 + i \cdot \left(i \cdot 0.5\right)\right) + 1 \cdot i}{i}\right)\\

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

\end{array}
double f(double i, double n) {
        double r7237590 = 100.0;
        double r7237591 = 1.0;
        double r7237592 = i;
        double r7237593 = n;
        double r7237594 = r7237592 / r7237593;
        double r7237595 = r7237591 + r7237594;
        double r7237596 = pow(r7237595, r7237593);
        double r7237597 = r7237596 - r7237591;
        double r7237598 = r7237597 / r7237594;
        double r7237599 = r7237590 * r7237598;
        return r7237599;
}
↓
double f(double i, double n) {
        double r7237600 = i;
        double r7237601 = -3.0764933618318215e-11;
        bool r7237602 = r7237600 <= r7237601;
        double r7237603 = 100.0;
        double r7237604 = n;
        double r7237605 = r7237600 / r7237604;
        double r7237606 = 1.0;
        double r7237607 = r7237605 + r7237606;
        double r7237608 = pow(r7237607, r7237604);
        double r7237609 = 3.0;
        double r7237610 = pow(r7237608, r7237609);
        double r7237611 = pow(r7237606, r7237609);
        double r7237612 = r7237610 - r7237611;
        double r7237613 = r7237608 * r7237608;
        double r7237614 = r7237606 * r7237606;
        double r7237615 = r7237606 * r7237608;
        double r7237616 = r7237614 + r7237615;
        double r7237617 = r7237613 + r7237616;
        double r7237618 = r7237605 * r7237617;
        double r7237619 = r7237612 / r7237618;
        double r7237620 = r7237603 * r7237619;
        double r7237621 = 7.743283777926976;
        bool r7237622 = r7237600 <= r7237621;
        double r7237623 = 0.5;
        double r7237624 = r7237600 * r7237623;
        double r7237625 = r7237600 * r7237624;
        double r7237626 = r7237604 - r7237625;
        double r7237627 = log(r7237606);
        double r7237628 = r7237626 * r7237627;
        double r7237629 = r7237628 + r7237625;
        double r7237630 = r7237606 * r7237600;
        double r7237631 = r7237629 + r7237630;
        double r7237632 = r7237631 / r7237600;
        double r7237633 = r7237604 * r7237632;
        double r7237634 = r7237603 * r7237633;
        double r7237635 = r7237608 - r7237606;
        double r7237636 = r7237635 / r7237600;
        double r7237637 = 1.0;
        double r7237638 = r7237637 / r7237604;
        double r7237639 = r7237636 / r7237638;
        double r7237640 = r7237603 * r7237639;
        double r7237641 = r7237622 ? r7237634 : r7237640;
        double r7237642 = r7237602 ? r7237620 : r7237641;
        return r7237642;
}

Error

Bits error versus i

Bits error versus n

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original42.8
Target42.7
Herbie21.7
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 3 regimes

  2. if i < -3.0764933618318215e-11

    1. Initial program 28.6

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

    3. Applied flip3--28.6

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

    4. Applied associate-/l/28.6

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

    if -3.0764933618318215e-11 < i < 7.743283777926976

    1. Initial program 50.7

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

    2. Taylor expanded around 0 34.0

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

    3. Simplified34.0

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

    5. Applied associate-/r/17.0

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

    6. Simplified17.0

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

    if 7.743283777926976 < i

    1. Initial program 31.3

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

    3. Applied div-inv31.4

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

    4. Applied associate-/r*31.4

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

  4. Final simplification21.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -3.076493361831821548083157134309908748576 \cdot 10^{-11}:\\ \;\;\;\;100 \cdot \frac{{\left({\left(\frac{i}{n} + 1\right)}^{n}\right)}^{3} - {1}^{3}}{\frac{i}{n} \cdot \left({\left(\frac{i}{n} + 1\right)}^{n} \cdot {\left(\frac{i}{n} + 1\right)}^{n} + \left(1 \cdot 1 + 1 \cdot {\left(\frac{i}{n} + 1\right)}^{n}\right)\right)}\\ \mathbf{elif}\;i \le 7.743283777926976085836940910667181015015:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{\left(\left(n - i \cdot \left(i \cdot 0.5\right)\right) \cdot \log 1 + i \cdot \left(i \cdot 0.5\right)\right) + 1 \cdot i}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{i}}{\frac{1}{n}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019179 
(FPCore (i n)
  :name "Compound Interest"

  :herbie-target
  (* 100.0 (/ (- (exp (* n (if (== (+ 1.0 (/ i n)) 1.0) (/ i n) (/ (* (/ i n) (log (+ 1.0 (/ i n)))) (- (+ (/ i n) 1.0) 1.0))))) 1.0) (/ i n)))

  (* 100.0 (/ (- (pow (+ 1.0 (/ i n)) n) 1.0) (/ i n))))