Average Error: 42.1 → 19.9
Time: 3.8m
Precision: 64
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le -93263.25339554752:\\ \;\;\;\;\left(\frac{n}{i} \cdot \left({\left(\frac{1}{\frac{n}{i}}\right)}^{n} - 1\right)\right) \cdot 100\\ \mathbf{elif}\;i \le -1.312300068674317 \cdot 10^{-227}:\\ \;\;\;\;\frac{100}{i} \cdot \frac{\left(i \cdot i\right) \cdot \left(i \cdot \frac{1}{6} + \frac{1}{2}\right) + i}{\frac{1}{n}}\\ \mathbf{elif}\;i \le 2.1570816796635502 \cdot 10^{-262}:\\ \;\;\;\;100 \cdot \left(\log \left(e^{\left(n \cdot i\right) \cdot \left(i \cdot \frac{1}{6} + \frac{1}{2}\right)}\right) + n\right)\\ \mathbf{elif}\;i \le 332694488255.6885:\\ \;\;\;\;\frac{100}{i} \cdot \frac{\left(i \cdot i\right) \cdot \left(i \cdot \frac{1}{6} + \frac{1}{2}\right) + i}{\frac{1}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{i}}{\frac{1}{n}}\\ \end{array}\]
100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}
↓
\begin{array}{l}
\mathbf{if}\;i \le -93263.25339554752:\\
\;\;\;\;\left(\frac{n}{i} \cdot \left({\left(\frac{1}{\frac{n}{i}}\right)}^{n} - 1\right)\right) \cdot 100\\

\mathbf{elif}\;i \le -1.312300068674317 \cdot 10^{-227}:\\
\;\;\;\;\frac{100}{i} \cdot \frac{\left(i \cdot i\right) \cdot \left(i \cdot \frac{1}{6} + \frac{1}{2}\right) + i}{\frac{1}{n}}\\

\mathbf{elif}\;i \le 2.1570816796635502 \cdot 10^{-262}:\\
\;\;\;\;100 \cdot \left(\log \left(e^{\left(n \cdot i\right) \cdot \left(i \cdot \frac{1}{6} + \frac{1}{2}\right)}\right) + n\right)\\

\mathbf{elif}\;i \le 332694488255.6885:\\
\;\;\;\;\frac{100}{i} \cdot \frac{\left(i \cdot i\right) \cdot \left(i \cdot \frac{1}{6} + \frac{1}{2}\right) + i}{\frac{1}{n}}\\

\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 r48422558 = 100.0;
        double r48422559 = 1.0;
        double r48422560 = i;
        double r48422561 = n;
        double r48422562 = r48422560 / r48422561;
        double r48422563 = r48422559 + r48422562;
        double r48422564 = pow(r48422563, r48422561);
        double r48422565 = r48422564 - r48422559;
        double r48422566 = r48422565 / r48422562;
        double r48422567 = r48422558 * r48422566;
        return r48422567;
}
↓
double f(double i, double n) {
        double r48422568 = i;
        double r48422569 = -93263.25339554752;
        bool r48422570 = r48422568 <= r48422569;
        double r48422571 = n;
        double r48422572 = r48422571 / r48422568;
        double r48422573 = 1.0;
        double r48422574 = r48422573 / r48422572;
        double r48422575 = pow(r48422574, r48422571);
        double r48422576 = r48422575 - r48422573;
        double r48422577 = r48422572 * r48422576;
        double r48422578 = 100.0;
        double r48422579 = r48422577 * r48422578;
        double r48422580 = -1.312300068674317e-227;
        bool r48422581 = r48422568 <= r48422580;
        double r48422582 = r48422578 / r48422568;
        double r48422583 = r48422568 * r48422568;
        double r48422584 = 0.16666666666666666;
        double r48422585 = r48422568 * r48422584;
        double r48422586 = 0.5;
        double r48422587 = r48422585 + r48422586;
        double r48422588 = r48422583 * r48422587;
        double r48422589 = r48422588 + r48422568;
        double r48422590 = r48422573 / r48422571;
        double r48422591 = r48422589 / r48422590;
        double r48422592 = r48422582 * r48422591;
        double r48422593 = 2.1570816796635502e-262;
        bool r48422594 = r48422568 <= r48422593;
        double r48422595 = r48422571 * r48422568;
        double r48422596 = r48422595 * r48422587;
        double r48422597 = exp(r48422596);
        double r48422598 = log(r48422597);
        double r48422599 = r48422598 + r48422571;
        double r48422600 = r48422578 * r48422599;
        double r48422601 = 332694488255.6885;
        bool r48422602 = r48422568 <= r48422601;
        double r48422603 = r48422568 / r48422571;
        double r48422604 = r48422603 + r48422573;
        double r48422605 = pow(r48422604, r48422571);
        double r48422606 = r48422605 - r48422573;
        double r48422607 = r48422606 / r48422568;
        double r48422608 = r48422607 / r48422590;
        double r48422609 = r48422578 * r48422608;
        double r48422610 = r48422602 ? r48422592 : r48422609;
        double r48422611 = r48422594 ? r48422600 : r48422610;
        double r48422612 = r48422581 ? r48422592 : r48422611;
        double r48422613 = r48422570 ? r48422579 : r48422612;
        return r48422613;
}

Error

Bits error versus i

Bits error versus n

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original42.1
Target41.6
Herbie19.9
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 4 regimes

  2. if i < -93263.25339554752

    1. Initial program 27.4

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

    2. Taylor expanded around inf 63.0

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

    3. Simplified19.3

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

    if -93263.25339554752 < i < -1.312300068674317e-227 or 2.1570816796635502e-262 < i < 332694488255.6885

    1. Initial program 49.7

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

    2. Taylor expanded around 0 31.8

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

    3. Simplified31.8

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

    5. Applied div-inv31.8

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

    6. Applied *-un-lft-identity31.8

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

    7. Applied times-frac17.0

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

    8. Applied associate-*r*17.0

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

    9. Simplified17.0

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

    if -1.312300068674317e-227 < i < 2.1570816796635502e-262

    1. Initial program 47.2

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

    2. Taylor expanded around 0 42.4

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

    3. Simplified42.4

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

    4. Taylor expanded around inf 16.0

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

    5. Simplified16.0

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

    7. Applied add-log-exp19.0

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

    if 332694488255.6885 < i

    1. Initial program 33.0

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

    3. Applied div-inv33.1

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

    4. Applied associate-/r*33.1

      \[\leadsto 100 \cdot \color{blue}{\frac{\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}}{\frac{1}{n}}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification19.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -93263.25339554752:\\ \;\;\;\;\left(\frac{n}{i} \cdot \left({\left(\frac{1}{\frac{n}{i}}\right)}^{n} - 1\right)\right) \cdot 100\\ \mathbf{elif}\;i \le -1.312300068674317 \cdot 10^{-227}:\\ \;\;\;\;\frac{100}{i} \cdot \frac{\left(i \cdot i\right) \cdot \left(i \cdot \frac{1}{6} + \frac{1}{2}\right) + i}{\frac{1}{n}}\\ \mathbf{elif}\;i \le 2.1570816796635502 \cdot 10^{-262}:\\ \;\;\;\;100 \cdot \left(\log \left(e^{\left(n \cdot i\right) \cdot \left(i \cdot \frac{1}{6} + \frac{1}{2}\right)}\right) + n\right)\\ \mathbf{elif}\;i \le 332694488255.6885:\\ \;\;\;\;\frac{100}{i} \cdot \frac{\left(i \cdot i\right) \cdot \left(i \cdot \frac{1}{6} + \frac{1}{2}\right) + i}{\frac{1}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{i}}{\frac{1}{n}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019125 
(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))))