Average Error: 42.9 → 21.9
Time: 36.6s
Precision: 64
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;n \le -3.094835530129789 \cdot 10^{+133}:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{i + \left(\log \left(\sqrt[3]{e^{i \cdot \left(i \cdot \left(\frac{1}{2} + i \cdot \frac{1}{6}\right)\right)}} \cdot \sqrt[3]{e^{i \cdot \left(i \cdot \left(\frac{1}{2} + i \cdot \frac{1}{6}\right)\right)}}\right) + \log \left(\sqrt[3]{e^{i \cdot \left(i \cdot \left(\frac{1}{2} + i \cdot \frac{1}{6}\right)\right)}}\right)\right)}{i}\right)\\ \mathbf{elif}\;n \le -3.866080760544941 \cdot 10^{+49}:\\ \;\;\;\;\left(\frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{i} \cdot 100\right) \cdot n\\ \mathbf{elif}\;n \le -1.4249003372371525:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{i + \left(\log \left(\sqrt[3]{e^{i \cdot \left(i \cdot \left(\frac{1}{2} + i \cdot \frac{1}{6}\right)\right)}} \cdot \sqrt[3]{e^{i \cdot \left(i \cdot \left(\frac{1}{2} + i \cdot \frac{1}{6}\right)\right)}}\right) + \log \left(\sqrt[3]{e^{i \cdot \left(i \cdot \left(\frac{1}{2} + i \cdot \frac{1}{6}\right)\right)}}\right)\right)}{i}\right)\\ \mathbf{elif}\;n \le 3.760450716247655 \cdot 10^{-168}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{i + \left(\log \left(\sqrt[3]{e^{i \cdot \left(i \cdot \left(\frac{1}{2} + i \cdot \frac{1}{6}\right)\right)}} \cdot \sqrt[3]{e^{i \cdot \left(i \cdot \left(\frac{1}{2} + i \cdot \frac{1}{6}\right)\right)}}\right) + \log \left(\sqrt[3]{e^{i \cdot \left(i \cdot \left(\frac{1}{2} + i \cdot \frac{1}{6}\right)\right)}}\right)\right)}{i}\right)\\ \end{array}\]
100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}
↓
\begin{array}{l}
\mathbf{if}\;n \le -3.094835530129789 \cdot 10^{+133}:\\
\;\;\;\;100 \cdot \left(n \cdot \frac{i + \left(\log \left(\sqrt[3]{e^{i \cdot \left(i \cdot \left(\frac{1}{2} + i \cdot \frac{1}{6}\right)\right)}} \cdot \sqrt[3]{e^{i \cdot \left(i \cdot \left(\frac{1}{2} + i \cdot \frac{1}{6}\right)\right)}}\right) + \log \left(\sqrt[3]{e^{i \cdot \left(i \cdot \left(\frac{1}{2} + i \cdot \frac{1}{6}\right)\right)}}\right)\right)}{i}\right)\\

\mathbf{elif}\;n \le -3.866080760544941 \cdot 10^{+49}:\\
\;\;\;\;\left(\frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{i} \cdot 100\right) \cdot n\\

\mathbf{elif}\;n \le -1.4249003372371525:\\
\;\;\;\;100 \cdot \left(n \cdot \frac{i + \left(\log \left(\sqrt[3]{e^{i \cdot \left(i \cdot \left(\frac{1}{2} + i \cdot \frac{1}{6}\right)\right)}} \cdot \sqrt[3]{e^{i \cdot \left(i \cdot \left(\frac{1}{2} + i \cdot \frac{1}{6}\right)\right)}}\right) + \log \left(\sqrt[3]{e^{i \cdot \left(i \cdot \left(\frac{1}{2} + i \cdot \frac{1}{6}\right)\right)}}\right)\right)}{i}\right)\\

\mathbf{elif}\;n \le 3.760450716247655 \cdot 10^{-168}:\\
\;\;\;\;0\\

\mathbf{else}:\\
\;\;\;\;100 \cdot \left(n \cdot \frac{i + \left(\log \left(\sqrt[3]{e^{i \cdot \left(i \cdot \left(\frac{1}{2} + i \cdot \frac{1}{6}\right)\right)}} \cdot \sqrt[3]{e^{i \cdot \left(i \cdot \left(\frac{1}{2} + i \cdot \frac{1}{6}\right)\right)}}\right) + \log \left(\sqrt[3]{e^{i \cdot \left(i \cdot \left(\frac{1}{2} + i \cdot \frac{1}{6}\right)\right)}}\right)\right)}{i}\right)\\

\end{array}
double f(double i, double n) {
        double r3583466 = 100.0;
        double r3583467 = 1.0;
        double r3583468 = i;
        double r3583469 = n;
        double r3583470 = r3583468 / r3583469;
        double r3583471 = r3583467 + r3583470;
        double r3583472 = pow(r3583471, r3583469);
        double r3583473 = r3583472 - r3583467;
        double r3583474 = r3583473 / r3583470;
        double r3583475 = r3583466 * r3583474;
        return r3583475;
}
↓
double f(double i, double n) {
        double r3583476 = n;
        double r3583477 = -3.094835530129789e+133;
        bool r3583478 = r3583476 <= r3583477;
        double r3583479 = 100.0;
        double r3583480 = i;
        double r3583481 = 0.5;
        double r3583482 = 0.16666666666666666;
        double r3583483 = r3583480 * r3583482;
        double r3583484 = r3583481 + r3583483;
        double r3583485 = r3583480 * r3583484;
        double r3583486 = r3583480 * r3583485;
        double r3583487 = exp(r3583486);
        double r3583488 = cbrt(r3583487);
        double r3583489 = r3583488 * r3583488;
        double r3583490 = log(r3583489);
        double r3583491 = log(r3583488);
        double r3583492 = r3583490 + r3583491;
        double r3583493 = r3583480 + r3583492;
        double r3583494 = r3583493 / r3583480;
        double r3583495 = r3583476 * r3583494;
        double r3583496 = r3583479 * r3583495;
        double r3583497 = -3.866080760544941e+49;
        bool r3583498 = r3583476 <= r3583497;
        double r3583499 = r3583480 / r3583476;
        double r3583500 = 1.0;
        double r3583501 = r3583499 + r3583500;
        double r3583502 = pow(r3583501, r3583476);
        double r3583503 = r3583502 - r3583500;
        double r3583504 = r3583503 / r3583480;
        double r3583505 = r3583504 * r3583479;
        double r3583506 = r3583505 * r3583476;
        double r3583507 = -1.4249003372371525;
        bool r3583508 = r3583476 <= r3583507;
        double r3583509 = 3.760450716247655e-168;
        bool r3583510 = r3583476 <= r3583509;
        double r3583511 = 0.0;
        double r3583512 = r3583510 ? r3583511 : r3583496;
        double r3583513 = r3583508 ? r3583496 : r3583512;
        double r3583514 = r3583498 ? r3583506 : r3583513;
        double r3583515 = r3583478 ? r3583496 : r3583514;
        return r3583515;
}

Error

Bits error versus i

Bits error versus n

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original42.9
Target42.2
Herbie21.9
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 3 regimes

  2. if n < -3.094835530129789e+133 or -3.866080760544941e+49 < n < -1.4249003372371525 or 3.760450716247655e-168 < n

    1. Initial program 54.0

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

    2. Taylor expanded around 0 37.5

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

    3. Simplified37.5

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

    5. Applied associate-/r/21.7

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

    7. Applied add-log-exp21.9

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

    8. Applied add-log-exp22.2

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

    9. Applied sum-log22.2

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

    10. Simplified22.0

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

    12. Applied add-cube-cbrt22.0

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

    13. Applied log-prod22.0

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

    if -3.094835530129789e+133 < n < -3.866080760544941e+49

    1. Initial program 37.5

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

    3. Applied associate-/r/37.3

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

    4. Applied associate-*r*37.3

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

    if -1.4249003372371525 < n < 3.760450716247655e-168

    1. Initial program 23.6

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

    2. Taylor expanded around 0 17.4

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

  4. Final simplification21.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -3.094835530129789 \cdot 10^{+133}:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{i + \left(\log \left(\sqrt[3]{e^{i \cdot \left(i \cdot \left(\frac{1}{2} + i \cdot \frac{1}{6}\right)\right)}} \cdot \sqrt[3]{e^{i \cdot \left(i \cdot \left(\frac{1}{2} + i \cdot \frac{1}{6}\right)\right)}}\right) + \log \left(\sqrt[3]{e^{i \cdot \left(i \cdot \left(\frac{1}{2} + i \cdot \frac{1}{6}\right)\right)}}\right)\right)}{i}\right)\\ \mathbf{elif}\;n \le -3.866080760544941 \cdot 10^{+49}:\\ \;\;\;\;\left(\frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{i} \cdot 100\right) \cdot n\\ \mathbf{elif}\;n \le -1.4249003372371525:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{i + \left(\log \left(\sqrt[3]{e^{i \cdot \left(i \cdot \left(\frac{1}{2} + i \cdot \frac{1}{6}\right)\right)}} \cdot \sqrt[3]{e^{i \cdot \left(i \cdot \left(\frac{1}{2} + i \cdot \frac{1}{6}\right)\right)}}\right) + \log \left(\sqrt[3]{e^{i \cdot \left(i \cdot \left(\frac{1}{2} + i \cdot \frac{1}{6}\right)\right)}}\right)\right)}{i}\right)\\ \mathbf{elif}\;n \le 3.760450716247655 \cdot 10^{-168}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{i + \left(\log \left(\sqrt[3]{e^{i \cdot \left(i \cdot \left(\frac{1}{2} + i \cdot \frac{1}{6}\right)\right)}} \cdot \sqrt[3]{e^{i \cdot \left(i \cdot \left(\frac{1}{2} + i \cdot \frac{1}{6}\right)\right)}}\right) + \log \left(\sqrt[3]{e^{i \cdot \left(i \cdot \left(\frac{1}{2} + i \cdot \frac{1}{6}\right)\right)}}\right)\right)}{i}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019146 
(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))))