Average Error: 43.0 → 24.3
Time: 31.1s
Precision: 64
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;n \le -5.464773096034009611019906036964894498925 \cdot 10^{59}:\\ \;\;\;\;n \cdot \left(100 \cdot \frac{\left(\left(i \cdot i\right) \cdot 0.5 - \left(\left(i \cdot i\right) \cdot 0.5\right) \cdot \log 1\right) + \left(i \cdot 1 + \log 1 \cdot n\right)}{i}\right)\\ \mathbf{elif}\;n \le -499596521052505172268102593940553728:\\ \;\;\;\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{1}{n}} \cdot \frac{100}{i}\\ \mathbf{elif}\;n \le -15240705750258725123260416:\\ \;\;\;\;n \cdot \left(100 \cdot \frac{\left(\left(i \cdot i\right) \cdot 0.5 - \left(\left(i \cdot i\right) \cdot 0.5\right) \cdot \log 1\right) + \left(i \cdot 1 + \log 1 \cdot n\right)}{i}\right)\\ \mathbf{elif}\;n \le 8.439481621089909706689165353901153669925 \cdot 10^{-297}:\\ \;\;\;\;100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \frac{1}{\frac{i}{n}}\right)\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 \cdot \frac{\left(\left(i \cdot i\right) \cdot 0.5 - \left(\left(i \cdot i\right) \cdot 0.5\right) \cdot \log 1\right) + \left(i \cdot 1 + \log 1 \cdot n\right)}{i}\right)\\ \end{array}\]
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:\\
\;\;\;\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{1}{n}} \cdot \frac{100}{i}\\

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

\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 r6687167 = 100.0;
        double r6687168 = 1.0;
        double r6687169 = i;
        double r6687170 = n;
        double r6687171 = r6687169 / r6687170;
        double r6687172 = r6687168 + r6687171;
        double r6687173 = pow(r6687172, r6687170);
        double r6687174 = r6687173 - r6687168;
        double r6687175 = r6687174 / r6687171;
        double r6687176 = r6687167 * r6687175;
        return r6687176;
}
↓
double f(double i, double n) {
        double r6687177 = n;
        double r6687178 = -5.46477309603401e+59;
        bool r6687179 = r6687177 <= r6687178;
        double r6687180 = 100.0;
        double r6687181 = i;
        double r6687182 = r6687181 * r6687181;
        double r6687183 = 0.5;
        double r6687184 = r6687182 * r6687183;
        double r6687185 = 1.0;
        double r6687186 = log(r6687185);
        double r6687187 = r6687184 * r6687186;
        double r6687188 = r6687184 - r6687187;
        double r6687189 = r6687181 * r6687185;
        double r6687190 = r6687186 * r6687177;
        double r6687191 = r6687189 + r6687190;
        double r6687192 = r6687188 + r6687191;
        double r6687193 = r6687192 / r6687181;
        double r6687194 = r6687180 * r6687193;
        double r6687195 = r6687177 * r6687194;
        double r6687196 = -4.995965210525052e+35;
        bool r6687197 = r6687177 <= r6687196;
        double r6687198 = r6687181 / r6687177;
        double r6687199 = r6687185 + r6687198;
        double r6687200 = pow(r6687199, r6687177);
        double r6687201 = r6687200 - r6687185;
        double r6687202 = 1.0;
        double r6687203 = r6687202 / r6687177;
        double r6687204 = r6687201 / r6687203;
        double r6687205 = r6687180 / r6687181;
        double r6687206 = r6687204 * r6687205;
        double r6687207 = -1.5240705750258725e+25;
        bool r6687208 = r6687177 <= r6687207;
        double r6687209 = 8.43948162108991e-297;
        bool r6687210 = r6687177 <= r6687209;
        double r6687211 = r6687200 / r6687198;
        double r6687212 = r6687185 / r6687198;
        double r6687213 = r6687211 - r6687212;
        double r6687214 = r6687180 * r6687213;
        double r6687215 = r6687210 ? r6687214 : r6687195;
        double r6687216 = r6687208 ? r6687195 : r6687215;
        double r6687217 = r6687197 ? r6687206 : r6687216;
        double r6687218 = r6687179 ? r6687195 : r6687217;
        return r6687218;
}

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 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

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

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))))