Average Error: 43.0 → 24.3
Time: 29.8s
Precision: 64
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;n \le -5.464773096034009611019906036964894498925 \cdot 10^{59}:\\ \;\;\;\;\frac{\left(\log 1 \cdot \left(n - i \cdot \left(0.5 \cdot i\right)\right) + i \cdot \left(0.5 \cdot i\right)\right) + i \cdot 1}{i} \cdot \left(n \cdot 100\right)\\ \mathbf{elif}\;n \le -499596521052505172268102593940553728:\\ \;\;\;\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{1}{n}} \cdot \frac{100}{i}\\ \mathbf{elif}\;n \le -15240705750258725123260416:\\ \;\;\;\;\frac{\left(\log 1 \cdot \left(n - i \cdot \left(0.5 \cdot i\right)\right) + i \cdot \left(0.5 \cdot i\right)\right) + i \cdot 1}{i} \cdot \left(n \cdot 100\right)\\ \mathbf{elif}\;n \le 8.439481621089909706689165353901153669925 \cdot 10^{-297}:\\ \;\;\;\;\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \frac{1}{\frac{i}{n}}\right) \cdot 100\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\log 1 \cdot \left(n - i \cdot \left(0.5 \cdot i\right)\right) + i \cdot \left(0.5 \cdot i\right)\right) + i \cdot 1}{i} \cdot \left(n \cdot 100\right)\\ \end{array}\]
100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}
↓
\begin{array}{l}
\mathbf{if}\;n \le -5.464773096034009611019906036964894498925 \cdot 10^{59}:\\
\;\;\;\;\frac{\left(\log 1 \cdot \left(n - i \cdot \left(0.5 \cdot i\right)\right) + i \cdot \left(0.5 \cdot i\right)\right) + i \cdot 1}{i} \cdot \left(n \cdot 100\right)\\

\mathbf{elif}\;n \le -499596521052505172268102593940553728:\\
\;\;\;\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{1}{n}} \cdot \frac{100}{i}\\

\mathbf{elif}\;n \le -15240705750258725123260416:\\
\;\;\;\;\frac{\left(\log 1 \cdot \left(n - i \cdot \left(0.5 \cdot i\right)\right) + i \cdot \left(0.5 \cdot i\right)\right) + i \cdot 1}{i} \cdot \left(n \cdot 100\right)\\

\mathbf{elif}\;n \le 8.439481621089909706689165353901153669925 \cdot 10^{-297}:\\
\;\;\;\;\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \frac{1}{\frac{i}{n}}\right) \cdot 100\\

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

\end{array}
double f(double i, double n) {
        double r6170622 = 100.0;
        double r6170623 = 1.0;
        double r6170624 = i;
        double r6170625 = n;
        double r6170626 = r6170624 / r6170625;
        double r6170627 = r6170623 + r6170626;
        double r6170628 = pow(r6170627, r6170625);
        double r6170629 = r6170628 - r6170623;
        double r6170630 = r6170629 / r6170626;
        double r6170631 = r6170622 * r6170630;
        return r6170631;
}
↓
double f(double i, double n) {
        double r6170632 = n;
        double r6170633 = -5.46477309603401e+59;
        bool r6170634 = r6170632 <= r6170633;
        double r6170635 = 1.0;
        double r6170636 = log(r6170635);
        double r6170637 = i;
        double r6170638 = 0.5;
        double r6170639 = r6170638 * r6170637;
        double r6170640 = r6170637 * r6170639;
        double r6170641 = r6170632 - r6170640;
        double r6170642 = r6170636 * r6170641;
        double r6170643 = r6170642 + r6170640;
        double r6170644 = r6170637 * r6170635;
        double r6170645 = r6170643 + r6170644;
        double r6170646 = r6170645 / r6170637;
        double r6170647 = 100.0;
        double r6170648 = r6170632 * r6170647;
        double r6170649 = r6170646 * r6170648;
        double r6170650 = -4.995965210525052e+35;
        bool r6170651 = r6170632 <= r6170650;
        double r6170652 = r6170637 / r6170632;
        double r6170653 = r6170635 + r6170652;
        double r6170654 = pow(r6170653, r6170632);
        double r6170655 = r6170654 - r6170635;
        double r6170656 = 1.0;
        double r6170657 = r6170656 / r6170632;
        double r6170658 = r6170655 / r6170657;
        double r6170659 = r6170647 / r6170637;
        double r6170660 = r6170658 * r6170659;
        double r6170661 = -1.5240705750258725e+25;
        bool r6170662 = r6170632 <= r6170661;
        double r6170663 = 8.43948162108991e-297;
        bool r6170664 = r6170632 <= r6170663;
        double r6170665 = r6170654 / r6170652;
        double r6170666 = r6170635 / r6170652;
        double r6170667 = r6170665 - r6170666;
        double r6170668 = r6170667 * r6170647;
        double r6170669 = r6170664 ? r6170668 : r6170649;
        double r6170670 = r6170662 ? r6170649 : r6170669;
        double r6170671 = r6170651 ? r6170660 : r6170670;
        double r6170672 = r6170634 ? r6170649 : r6170671;
        return r6170672;
}

Error

Bits error versus i

Bits error versus n

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original43.0
Target42.8
Herbie24.3
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 3 regimes

  2. if n < -5.46477309603401e+59 or -4.995965210525052e+35 < n < -1.5240705750258725e+25 or 8.43948162108991e-297 < n

    1. Initial program 52.3

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

    2. Taylor expanded around 0 39.8

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

    3. Simplified39.8

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

    5. Applied *-un-lft-identity39.8

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

    6. Applied associate-*l*39.8

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

    7. Simplified25.8

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

    if -5.46477309603401e+59 < n < -4.995965210525052e+35

    1. Initial program 34.9

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

    3. Applied div-inv34.9

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

    4. Applied *-un-lft-identity34.9

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

    5. Applied times-frac34.8

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

    6. Applied associate-*r*34.8

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

    7. Simplified34.8

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

    if -1.5240705750258725e+25 < n < 8.43948162108991e-297

    1. Initial program 18.9

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

    3. Applied div-sub18.9

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

  4. Final simplification24.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -5.464773096034009611019906036964894498925 \cdot 10^{59}:\\ \;\;\;\;\frac{\left(\log 1 \cdot \left(n - i \cdot \left(0.5 \cdot i\right)\right) + i \cdot \left(0.5 \cdot i\right)\right) + i \cdot 1}{i} \cdot \left(n \cdot 100\right)\\ \mathbf{elif}\;n \le -499596521052505172268102593940553728:\\ \;\;\;\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{1}{n}} \cdot \frac{100}{i}\\ \mathbf{elif}\;n \le -15240705750258725123260416:\\ \;\;\;\;\frac{\left(\log 1 \cdot \left(n - i \cdot \left(0.5 \cdot i\right)\right) + i \cdot \left(0.5 \cdot i\right)\right) + i \cdot 1}{i} \cdot \left(n \cdot 100\right)\\ \mathbf{elif}\;n \le 8.439481621089909706689165353901153669925 \cdot 10^{-297}:\\ \;\;\;\;\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \frac{1}{\frac{i}{n}}\right) \cdot 100\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\log 1 \cdot \left(n - i \cdot \left(0.5 \cdot i\right)\right) + i \cdot \left(0.5 \cdot i\right)\right) + i \cdot 1}{i} \cdot \left(n \cdot 100\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 
(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))))