Average Error: 42.4 → 22.4
Time: 16.8s
Precision: 64
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;n \le -2.71552055838372837 \cdot 10^{82}:\\ \;\;\;\;100 \cdot \left(\frac{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{i} \cdot n\right)\\ \mathbf{elif}\;n \le -3.9974032830534776 \cdot 10^{-251}:\\ \;\;\;\;100 \cdot \frac{{\left({\left(1 + \frac{i}{n}\right)}^{n}\right)}^{3} - {1}^{3}}{\left(1 \cdot \left(1 + {\left(1 + \frac{i}{n}\right)}^{n}\right) + {\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)}\right) \cdot \frac{i}{n}}\\ \mathbf{elif}\;n \le 7.18594828335009282 \cdot 10^{-152}:\\ \;\;\;\;100 \cdot \frac{\left(1 \cdot i + \left(\log 1 \cdot n + 1\right)\right) - 1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(\frac{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{i} \cdot n\right)\\ \end{array}\]
100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}
↓
\begin{array}{l}
\mathbf{if}\;n \le -2.71552055838372837 \cdot 10^{82}:\\
\;\;\;\;100 \cdot \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)\\

\mathbf{elif}\;n \le -3.9974032830534776 \cdot 10^{-251}:\\
\;\;\;\;100 \cdot \frac{{\left({\left(1 + \frac{i}{n}\right)}^{n}\right)}^{3} - {1}^{3}}{\left(1 \cdot \left(1 + {\left(1 + \frac{i}{n}\right)}^{n}\right) + {\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)}\right) \cdot \frac{i}{n}}\\

\mathbf{elif}\;n \le 7.18594828335009282 \cdot 10^{-152}:\\
\;\;\;\;100 \cdot \frac{\left(1 \cdot i + \left(\log 1 \cdot n + 1\right)\right) - 1}{\frac{i}{n}}\\

\mathbf{else}:\\
\;\;\;\;100 \cdot \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)\\

\end{array}
double f(double i, double n) {
        double r159605 = 100.0;
        double r159606 = 1.0;
        double r159607 = i;
        double r159608 = n;
        double r159609 = r159607 / r159608;
        double r159610 = r159606 + r159609;
        double r159611 = pow(r159610, r159608);
        double r159612 = r159611 - r159606;
        double r159613 = r159612 / r159609;
        double r159614 = r159605 * r159613;
        return r159614;
}
↓
double f(double i, double n) {
        double r159615 = n;
        double r159616 = -2.7155205583837284e+82;
        bool r159617 = r159615 <= r159616;
        double r159618 = 100.0;
        double r159619 = 1.0;
        double r159620 = i;
        double r159621 = r159619 * r159620;
        double r159622 = 0.5;
        double r159623 = 2.0;
        double r159624 = pow(r159620, r159623);
        double r159625 = r159622 * r159624;
        double r159626 = log(r159619);
        double r159627 = r159626 * r159615;
        double r159628 = r159625 + r159627;
        double r159629 = r159621 + r159628;
        double r159630 = r159624 * r159626;
        double r159631 = r159622 * r159630;
        double r159632 = r159629 - r159631;
        double r159633 = r159632 / r159620;
        double r159634 = r159633 * r159615;
        double r159635 = r159618 * r159634;
        double r159636 = -3.9974032830534776e-251;
        bool r159637 = r159615 <= r159636;
        double r159638 = r159620 / r159615;
        double r159639 = r159619 + r159638;
        double r159640 = pow(r159639, r159615);
        double r159641 = 3.0;
        double r159642 = pow(r159640, r159641);
        double r159643 = pow(r159619, r159641);
        double r159644 = r159642 - r159643;
        double r159645 = r159619 + r159640;
        double r159646 = r159619 * r159645;
        double r159647 = r159623 * r159615;
        double r159648 = pow(r159639, r159647);
        double r159649 = r159646 + r159648;
        double r159650 = r159649 * r159638;
        double r159651 = r159644 / r159650;
        double r159652 = r159618 * r159651;
        double r159653 = 7.185948283350093e-152;
        bool r159654 = r159615 <= r159653;
        double r159655 = 1.0;
        double r159656 = r159627 + r159655;
        double r159657 = r159621 + r159656;
        double r159658 = r159657 - r159619;
        double r159659 = r159658 / r159638;
        double r159660 = r159618 * r159659;
        double r159661 = r159654 ? r159660 : r159635;
        double r159662 = r159637 ? r159652 : r159661;
        double r159663 = r159617 ? r159635 : r159662;
        return r159663;
}

Error

Bits error versus i

Bits error versus n

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original42.4
Target42.3
Herbie22.4
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 3 regimes

  2. if n < -2.7155205583837284e+82 or 7.185948283350093e-152 < n

    1. Initial program 54.6

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

    2. Taylor expanded around 0 40.4

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

    4. Applied associate-/r/22.3

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

    if -2.7155205583837284e+82 < n < -3.9974032830534776e-251

    1. Initial program 22.0

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

    3. Applied flip3--22.0

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

    4. Applied associate-/l/22.0

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

    5. Simplified22.0

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

    if -3.9974032830534776e-251 < n < 7.185948283350093e-152

    1. Initial program 33.4

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

    2. Taylor expanded around 0 24.1

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

  4. Final simplification22.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -2.71552055838372837 \cdot 10^{82}:\\ \;\;\;\;100 \cdot \left(\frac{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{i} \cdot n\right)\\ \mathbf{elif}\;n \le -3.9974032830534776 \cdot 10^{-251}:\\ \;\;\;\;100 \cdot \frac{{\left({\left(1 + \frac{i}{n}\right)}^{n}\right)}^{3} - {1}^{3}}{\left(1 \cdot \left(1 + {\left(1 + \frac{i}{n}\right)}^{n}\right) + {\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)}\right) \cdot \frac{i}{n}}\\ \mathbf{elif}\;n \le 7.18594828335009282 \cdot 10^{-152}:\\ \;\;\;\;100 \cdot \frac{\left(1 \cdot i + \left(\log 1 \cdot n + 1\right)\right) - 1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(\frac{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{i} \cdot n\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020081 
(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))))