Average Error: 43.0 → 24.3
Time: 30.3s
Precision: 64
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;n \le -5.464773096034009611019906036964894498925 \cdot 10^{59}:\\ \;\;\;\;n \cdot \left(100 \cdot \frac{\left(\left(i \cdot i\right) \cdot 0.5 - \left(\left(i \cdot i\right) \cdot 0.5\right) \cdot \log 1\right) + \left(i \cdot 1 + \log 1 \cdot n\right)}{i}\right)\\ \mathbf{elif}\;n \le -499596521052505172268102593940553728:\\ \;\;\;\;n \cdot \left(100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}\right)\\ \mathbf{elif}\;n \le -15240705750258725123260416:\\ \;\;\;\;n \cdot \left(100 \cdot \frac{\left(\left(i \cdot i\right) \cdot 0.5 - \left(\left(i \cdot i\right) \cdot 0.5\right) \cdot \log 1\right) + \left(i \cdot 1 + \log 1 \cdot n\right)}{i}\right)\\ \mathbf{elif}\;n \le 8.439481621089909706689165353901153669925 \cdot 10^{-297}:\\ \;\;\;\;\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \frac{1}{\frac{i}{n}}\right) \cdot 100\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 \cdot \frac{\left(\left(i \cdot i\right) \cdot 0.5 - \left(\left(i \cdot i\right) \cdot 0.5\right) \cdot \log 1\right) + \left(i \cdot 1 + \log 1 \cdot n\right)}{i}\right)\\ \end{array}\]
100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}
↓
\begin{array}{l}
\mathbf{if}\;n \le -5.464773096034009611019906036964894498925 \cdot 10^{59}:\\
\;\;\;\;n \cdot \left(100 \cdot \frac{\left(\left(i \cdot i\right) \cdot 0.5 - \left(\left(i \cdot i\right) \cdot 0.5\right) \cdot \log 1\right) + \left(i \cdot 1 + \log 1 \cdot n\right)}{i}\right)\\

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

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

\end{array}
double f(double i, double n) {
        double r6316965 = 100.0;
        double r6316966 = 1.0;
        double r6316967 = i;
        double r6316968 = n;
        double r6316969 = r6316967 / r6316968;
        double r6316970 = r6316966 + r6316969;
        double r6316971 = pow(r6316970, r6316968);
        double r6316972 = r6316971 - r6316966;
        double r6316973 = r6316972 / r6316969;
        double r6316974 = r6316965 * r6316973;
        return r6316974;
}
↓
double f(double i, double n) {
        double r6316975 = n;
        double r6316976 = -5.46477309603401e+59;
        bool r6316977 = r6316975 <= r6316976;
        double r6316978 = 100.0;
        double r6316979 = i;
        double r6316980 = r6316979 * r6316979;
        double r6316981 = 0.5;
        double r6316982 = r6316980 * r6316981;
        double r6316983 = 1.0;
        double r6316984 = log(r6316983);
        double r6316985 = r6316982 * r6316984;
        double r6316986 = r6316982 - r6316985;
        double r6316987 = r6316979 * r6316983;
        double r6316988 = r6316984 * r6316975;
        double r6316989 = r6316987 + r6316988;
        double r6316990 = r6316986 + r6316989;
        double r6316991 = r6316990 / r6316979;
        double r6316992 = r6316978 * r6316991;
        double r6316993 = r6316975 * r6316992;
        double r6316994 = -4.995965210525052e+35;
        bool r6316995 = r6316975 <= r6316994;
        double r6316996 = r6316979 / r6316975;
        double r6316997 = r6316983 + r6316996;
        double r6316998 = pow(r6316997, r6316975);
        double r6316999 = r6316998 - r6316983;
        double r6317000 = r6316999 / r6316979;
        double r6317001 = r6316978 * r6317000;
        double r6317002 = r6316975 * r6317001;
        double r6317003 = -1.5240705750258725e+25;
        bool r6317004 = r6316975 <= r6317003;
        double r6317005 = 8.43948162108991e-297;
        bool r6317006 = r6316975 <= r6317005;
        double r6317007 = r6316998 / r6316996;
        double r6317008 = r6316983 / r6316996;
        double r6317009 = r6317007 - r6317008;
        double r6317010 = r6317009 * r6316978;
        double r6317011 = r6317006 ? r6317010 : r6316993;
        double r6317012 = r6317004 ? r6316993 : r6317011;
        double r6317013 = r6316995 ? r6317002 : r6317012;
        double r6317014 = r6316977 ? r6316993 : r6317013;
        return r6317014;
}

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

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

    5. Applied associate-/r/25.8

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

    6. Applied associate-*r*25.8

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

    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 associate-/r/34.8

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

    4. Applied associate-*r*34.8

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

    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

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

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))))