Average Error: 43.1 → 21.7
Time: 34.1s
Precision: 64
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le -0.008529206764181104793998144941724603995681:\\ \;\;\;\;\frac{100}{i} \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{1}{n}}\\ \mathbf{elif}\;i \le 4.039962480132392563803023222135379910469:\\ \;\;\;\;100 \cdot \left(\frac{\log 1 \cdot n + \left(\left(\left(i \cdot i\right) \cdot 0.5 + 1 \cdot i\right) - \left(0.5 \cdot \log 1\right) \cdot \left(i \cdot i\right)\right)}{i} \cdot n\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{100}{i} \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{1}{n}}\\ \end{array}\]
100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}
↓
\begin{array}{l}
\mathbf{if}\;i \le -0.008529206764181104793998144941724603995681:\\
\;\;\;\;\frac{100}{i} \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{1}{n}}\\

\mathbf{elif}\;i \le 4.039962480132392563803023222135379910469:\\
\;\;\;\;100 \cdot \left(\frac{\log 1 \cdot n + \left(\left(\left(i \cdot i\right) \cdot 0.5 + 1 \cdot i\right) - \left(0.5 \cdot \log 1\right) \cdot \left(i \cdot i\right)\right)}{i} \cdot n\right)\\

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

\end{array}
double f(double i, double n) {
        double r9639606 = 100.0;
        double r9639607 = 1.0;
        double r9639608 = i;
        double r9639609 = n;
        double r9639610 = r9639608 / r9639609;
        double r9639611 = r9639607 + r9639610;
        double r9639612 = pow(r9639611, r9639609);
        double r9639613 = r9639612 - r9639607;
        double r9639614 = r9639613 / r9639610;
        double r9639615 = r9639606 * r9639614;
        return r9639615;
}
↓
double f(double i, double n) {
        double r9639616 = i;
        double r9639617 = -0.008529206764181105;
        bool r9639618 = r9639616 <= r9639617;
        double r9639619 = 100.0;
        double r9639620 = r9639619 / r9639616;
        double r9639621 = 1.0;
        double r9639622 = n;
        double r9639623 = r9639616 / r9639622;
        double r9639624 = r9639621 + r9639623;
        double r9639625 = pow(r9639624, r9639622);
        double r9639626 = r9639625 - r9639621;
        double r9639627 = 1.0;
        double r9639628 = r9639627 / r9639622;
        double r9639629 = r9639626 / r9639628;
        double r9639630 = r9639620 * r9639629;
        double r9639631 = 4.039962480132393;
        bool r9639632 = r9639616 <= r9639631;
        double r9639633 = log(r9639621);
        double r9639634 = r9639633 * r9639622;
        double r9639635 = r9639616 * r9639616;
        double r9639636 = 0.5;
        double r9639637 = r9639635 * r9639636;
        double r9639638 = r9639621 * r9639616;
        double r9639639 = r9639637 + r9639638;
        double r9639640 = r9639636 * r9639633;
        double r9639641 = r9639640 * r9639635;
        double r9639642 = r9639639 - r9639641;
        double r9639643 = r9639634 + r9639642;
        double r9639644 = r9639643 / r9639616;
        double r9639645 = r9639644 * r9639622;
        double r9639646 = r9639619 * r9639645;
        double r9639647 = r9639632 ? r9639646 : r9639630;
        double r9639648 = r9639618 ? r9639630 : r9639647;
        return r9639648;
}

Error

Bits error versus i

Bits error versus n

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original43.1
Target42.9
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 2 regimes

  2. if i < -0.008529206764181105 or 4.039962480132393 < i

    1. Initial program 30.1

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

    3. Applied div-inv30.1

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

    4. Applied *-un-lft-identity30.1

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

    5. Applied times-frac30.6

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

    6. Applied associate-*r*30.6

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

    7. Simplified30.6

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

    if -0.008529206764181105 < i < 4.039962480132393

    1. Initial program 50.8

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

    2. Taylor expanded around 0 33.6

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

    3. Simplified33.6

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

    5. Applied associate-/r/16.4

      \[\leadsto 100 \cdot \color{blue}{\left(\frac{\log 1 \cdot n + \left(\left(\left(i \cdot i\right) \cdot 0.5 + 1 \cdot i\right) - \left(0.5 \cdot \log 1\right) \cdot \left(i \cdot i\right)\right)}{i} \cdot n\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification21.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -0.008529206764181104793998144941724603995681:\\ \;\;\;\;\frac{100}{i} \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{1}{n}}\\ \mathbf{elif}\;i \le 4.039962480132392563803023222135379910469:\\ \;\;\;\;100 \cdot \left(\frac{\log 1 \cdot n + \left(\left(\left(i \cdot i\right) \cdot 0.5 + 1 \cdot i\right) - \left(0.5 \cdot \log 1\right) \cdot \left(i \cdot i\right)\right)}{i} \cdot n\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{100}{i} \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{1}{n}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 
(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))))