Average Error: 42.1 → 22.3
Time: 33.3s
Precision: 64
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;n \le -5.858969519248875 \cdot 10^{+150}:\\ \;\;\;\;100 \cdot \left(\left(i \cdot n\right) \cdot \frac{1}{2} + \left(n + \left(i \cdot i\right) \cdot \left(\frac{1}{6} \cdot n\right)\right)\right)\\ \mathbf{elif}\;n \le -4.0699167252326846 \cdot 10^{+112}:\\ \;\;\;\;\frac{\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right) \cdot 100}{\frac{i}{n}}\\ \mathbf{elif}\;n \le -8.426499588148776 \cdot 10^{+62}:\\ \;\;\;\;100 \cdot \left(\left(i \cdot n\right) \cdot \frac{1}{2} + \left(n + \left(i \cdot i\right) \cdot \left(\frac{1}{6} \cdot n\right)\right)\right)\\ \mathbf{elif}\;n \le -1.9721475649731032 \cdot 10^{+48}:\\ \;\;\;\;100 \cdot \frac{\frac{{\left(1 + \frac{i}{n}\right)}^{n} \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} \cdot {\left(1 + \frac{i}{n}\right)}^{n}\right) - 1}{{\left(1 + \frac{i}{n}\right)}^{n} \cdot {\left(1 + \frac{i}{n}\right)}^{n} + \left(1 + {\left(1 + \frac{i}{n}\right)}^{n}\right)}}{\frac{i}{n}}\\ \mathbf{elif}\;n \le -8051382600516.531:\\ \;\;\;\;100 \cdot \left(\left(i \cdot n\right) \cdot \frac{1}{2} + \left(n + \left(i \cdot i\right) \cdot \left(\frac{1}{6} \cdot n\right)\right)\right)\\ \mathbf{elif}\;n \le 1.0148966282182346 \cdot 10^{-203}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(\left(i \cdot n\right) \cdot \frac{1}{2} + \left(n + \left(i \cdot i\right) \cdot \left(\frac{1}{6} \cdot n\right)\right)\right)\\ \end{array}\]
100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}
↓
\begin{array}{l}
\mathbf{if}\;n \le -5.858969519248875 \cdot 10^{+150}:\\
\;\;\;\;100 \cdot \left(\left(i \cdot n\right) \cdot \frac{1}{2} + \left(n + \left(i \cdot i\right) \cdot \left(\frac{1}{6} \cdot n\right)\right)\right)\\

\mathbf{elif}\;n \le -4.0699167252326846 \cdot 10^{+112}:\\
\;\;\;\;\frac{\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right) \cdot 100}{\frac{i}{n}}\\

\mathbf{elif}\;n \le -8.426499588148776 \cdot 10^{+62}:\\
\;\;\;\;100 \cdot \left(\left(i \cdot n\right) \cdot \frac{1}{2} + \left(n + \left(i \cdot i\right) \cdot \left(\frac{1}{6} \cdot n\right)\right)\right)\\

\mathbf{elif}\;n \le -1.9721475649731032 \cdot 10^{+48}:\\
\;\;\;\;100 \cdot \frac{\frac{{\left(1 + \frac{i}{n}\right)}^{n} \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} \cdot {\left(1 + \frac{i}{n}\right)}^{n}\right) - 1}{{\left(1 + \frac{i}{n}\right)}^{n} \cdot {\left(1 + \frac{i}{n}\right)}^{n} + \left(1 + {\left(1 + \frac{i}{n}\right)}^{n}\right)}}{\frac{i}{n}}\\

\mathbf{elif}\;n \le -8051382600516.531:\\
\;\;\;\;100 \cdot \left(\left(i \cdot n\right) \cdot \frac{1}{2} + \left(n + \left(i \cdot i\right) \cdot \left(\frac{1}{6} \cdot n\right)\right)\right)\\

\mathbf{elif}\;n \le 1.0148966282182346 \cdot 10^{-203}:\\
\;\;\;\;0\\

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

\end{array}
double f(double i, double n) {
        double r5340161 = 100.0;
        double r5340162 = 1.0;
        double r5340163 = i;
        double r5340164 = n;
        double r5340165 = r5340163 / r5340164;
        double r5340166 = r5340162 + r5340165;
        double r5340167 = pow(r5340166, r5340164);
        double r5340168 = r5340167 - r5340162;
        double r5340169 = r5340168 / r5340165;
        double r5340170 = r5340161 * r5340169;
        return r5340170;
}
↓
double f(double i, double n) {
        double r5340171 = n;
        double r5340172 = -5.858969519248875e+150;
        bool r5340173 = r5340171 <= r5340172;
        double r5340174 = 100.0;
        double r5340175 = i;
        double r5340176 = r5340175 * r5340171;
        double r5340177 = 0.5;
        double r5340178 = r5340176 * r5340177;
        double r5340179 = r5340175 * r5340175;
        double r5340180 = 0.16666666666666666;
        double r5340181 = r5340180 * r5340171;
        double r5340182 = r5340179 * r5340181;
        double r5340183 = r5340171 + r5340182;
        double r5340184 = r5340178 + r5340183;
        double r5340185 = r5340174 * r5340184;
        double r5340186 = -4.0699167252326846e+112;
        bool r5340187 = r5340171 <= r5340186;
        double r5340188 = 1.0;
        double r5340189 = r5340175 / r5340171;
        double r5340190 = r5340188 + r5340189;
        double r5340191 = pow(r5340190, r5340171);
        double r5340192 = r5340191 - r5340188;
        double r5340193 = r5340192 * r5340174;
        double r5340194 = r5340193 / r5340189;
        double r5340195 = -8.426499588148776e+62;
        bool r5340196 = r5340171 <= r5340195;
        double r5340197 = -1.9721475649731032e+48;
        bool r5340198 = r5340171 <= r5340197;
        double r5340199 = r5340191 * r5340191;
        double r5340200 = r5340191 * r5340199;
        double r5340201 = r5340200 - r5340188;
        double r5340202 = r5340188 + r5340191;
        double r5340203 = r5340199 + r5340202;
        double r5340204 = r5340201 / r5340203;
        double r5340205 = r5340204 / r5340189;
        double r5340206 = r5340174 * r5340205;
        double r5340207 = -8051382600516.531;
        bool r5340208 = r5340171 <= r5340207;
        double r5340209 = 1.0148966282182346e-203;
        bool r5340210 = r5340171 <= r5340209;
        double r5340211 = 0.0;
        double r5340212 = r5340210 ? r5340211 : r5340185;
        double r5340213 = r5340208 ? r5340185 : r5340212;
        double r5340214 = r5340198 ? r5340206 : r5340213;
        double r5340215 = r5340196 ? r5340185 : r5340214;
        double r5340216 = r5340187 ? r5340194 : r5340215;
        double r5340217 = r5340173 ? r5340185 : r5340216;
        return r5340217;
}

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
Target42.1
Herbie22.3
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 4 regimes

  2. if n < -5.858969519248875e+150 or -4.0699167252326846e+112 < n < -8.426499588148776e+62 or -1.9721475649731032e+48 < n < -8051382600516.531 or 1.0148966282182346e-203 < n

    1. Initial program 52.8

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

    2. Taylor expanded around 0 37.8

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

    3. Simplified37.8

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

    4. Taylor expanded around inf 22.8

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

    5. Simplified22.8

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

    if -5.858969519248875e+150 < n < -4.0699167252326846e+112

    1. Initial program 39.7

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

    3. Applied associate-*r/39.7

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

    if -8.426499588148776e+62 < n < -1.9721475649731032e+48

    1. Initial program 31.9

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

    3. Applied flip3--31.9

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

    4. Simplified31.9

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

    5. Simplified31.9

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

    if -8051382600516.531 < n < 1.0148966282182346e-203

    1. Initial program 21.5

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

    2. Taylor expanded around 0 18.4

      \[\leadsto \color{blue}{0}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification22.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -5.858969519248875 \cdot 10^{+150}:\\ \;\;\;\;100 \cdot \left(\left(i \cdot n\right) \cdot \frac{1}{2} + \left(n + \left(i \cdot i\right) \cdot \left(\frac{1}{6} \cdot n\right)\right)\right)\\ \mathbf{elif}\;n \le -4.0699167252326846 \cdot 10^{+112}:\\ \;\;\;\;\frac{\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right) \cdot 100}{\frac{i}{n}}\\ \mathbf{elif}\;n \le -8.426499588148776 \cdot 10^{+62}:\\ \;\;\;\;100 \cdot \left(\left(i \cdot n\right) \cdot \frac{1}{2} + \left(n + \left(i \cdot i\right) \cdot \left(\frac{1}{6} \cdot n\right)\right)\right)\\ \mathbf{elif}\;n \le -1.9721475649731032 \cdot 10^{+48}:\\ \;\;\;\;100 \cdot \frac{\frac{{\left(1 + \frac{i}{n}\right)}^{n} \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} \cdot {\left(1 + \frac{i}{n}\right)}^{n}\right) - 1}{{\left(1 + \frac{i}{n}\right)}^{n} \cdot {\left(1 + \frac{i}{n}\right)}^{n} + \left(1 + {\left(1 + \frac{i}{n}\right)}^{n}\right)}}{\frac{i}{n}}\\ \mathbf{elif}\;n \le -8051382600516.531:\\ \;\;\;\;100 \cdot \left(\left(i \cdot n\right) \cdot \frac{1}{2} + \left(n + \left(i \cdot i\right) \cdot \left(\frac{1}{6} \cdot n\right)\right)\right)\\ \mathbf{elif}\;n \le 1.0148966282182346 \cdot 10^{-203}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(\left(i \cdot n\right) \cdot \frac{1}{2} + \left(n + \left(i \cdot i\right) \cdot \left(\frac{1}{6} \cdot n\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019135 
(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))))