Average Error: 42.0 → 21.3
Time: 25.8s
Precision: 64
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le -1.9324505473046543 \cdot 10^{-07}:\\ \;\;\;\;100 \cdot \left(\frac{{\left(\frac{i}{n} + 1\right)}^{n}}{\frac{i}{n}} - \frac{1}{\frac{i}{n}}\right)\\ \mathbf{elif}\;i \le 4.5457902375689345:\\ \;\;\;\;100 \cdot \left(n \cdot \left(\left(i \cdot \left(i \cdot \frac{1}{6}\right) + 1\right) + \frac{1}{2} \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{100}{i} \cdot \frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{\frac{1}{n}}\\ \end{array}\]
100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}
↓
\begin{array}{l}
\mathbf{if}\;i \le -1.9324505473046543 \cdot 10^{-07}:\\
\;\;\;\;100 \cdot \left(\frac{{\left(\frac{i}{n} + 1\right)}^{n}}{\frac{i}{n}} - \frac{1}{\frac{i}{n}}\right)\\

\mathbf{elif}\;i \le 4.5457902375689345:\\
\;\;\;\;100 \cdot \left(n \cdot \left(\left(i \cdot \left(i \cdot \frac{1}{6}\right) + 1\right) + \frac{1}{2} \cdot i\right)\right)\\

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

\end{array}
double f(double i, double n) {
        double r2309406 = 100.0;
        double r2309407 = 1.0;
        double r2309408 = i;
        double r2309409 = n;
        double r2309410 = r2309408 / r2309409;
        double r2309411 = r2309407 + r2309410;
        double r2309412 = pow(r2309411, r2309409);
        double r2309413 = r2309412 - r2309407;
        double r2309414 = r2309413 / r2309410;
        double r2309415 = r2309406 * r2309414;
        return r2309415;
}
↓
double f(double i, double n) {
        double r2309416 = i;
        double r2309417 = -1.9324505473046543e-07;
        bool r2309418 = r2309416 <= r2309417;
        double r2309419 = 100.0;
        double r2309420 = n;
        double r2309421 = r2309416 / r2309420;
        double r2309422 = 1.0;
        double r2309423 = r2309421 + r2309422;
        double r2309424 = pow(r2309423, r2309420);
        double r2309425 = r2309424 / r2309421;
        double r2309426 = r2309422 / r2309421;
        double r2309427 = r2309425 - r2309426;
        double r2309428 = r2309419 * r2309427;
        double r2309429 = 4.5457902375689345;
        bool r2309430 = r2309416 <= r2309429;
        double r2309431 = 0.16666666666666666;
        double r2309432 = r2309416 * r2309431;
        double r2309433 = r2309416 * r2309432;
        double r2309434 = r2309433 + r2309422;
        double r2309435 = 0.5;
        double r2309436 = r2309435 * r2309416;
        double r2309437 = r2309434 + r2309436;
        double r2309438 = r2309420 * r2309437;
        double r2309439 = r2309419 * r2309438;
        double r2309440 = r2309419 / r2309416;
        double r2309441 = r2309424 - r2309422;
        double r2309442 = r2309422 / r2309420;
        double r2309443 = r2309441 / r2309442;
        double r2309444 = r2309440 * r2309443;
        double r2309445 = r2309430 ? r2309439 : r2309444;
        double r2309446 = r2309418 ? r2309428 : r2309445;
        return r2309446;
}

Error

Bits error versus i

Bits error versus n

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original42.0
Target42.3
Herbie21.3
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 3 regimes

  2. if i < -1.9324505473046543e-07

    1. Initial program 27.4

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

    3. Applied div-sub27.4

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

    if -1.9324505473046543e-07 < i < 4.5457902375689345

    1. Initial program 50.1

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

    2. Taylor expanded around 0 33.0

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

    3. Simplified33.0

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

    5. Applied associate-/r/16.9

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

    6. Taylor expanded around 0 16.9

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

    7. Simplified16.9

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

    if 4.5457902375689345 < i

    1. Initial program 31.0

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

    3. Applied div-inv31.1

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

    4. Applied *-un-lft-identity31.1

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

    5. Applied times-frac31.1

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

    6. Applied associate-*r*31.1

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

    7. Simplified31.0

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

  4. Final simplification21.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -1.9324505473046543 \cdot 10^{-07}:\\ \;\;\;\;100 \cdot \left(\frac{{\left(\frac{i}{n} + 1\right)}^{n}}{\frac{i}{n}} - \frac{1}{\frac{i}{n}}\right)\\ \mathbf{elif}\;i \le 4.5457902375689345:\\ \;\;\;\;100 \cdot \left(n \cdot \left(\left(i \cdot \left(i \cdot \frac{1}{6}\right) + 1\right) + \frac{1}{2} \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{100}{i} \cdot \frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{\frac{1}{n}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019152 
(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))))