Average Error: 42.8 → 23.5
Time: 22.9s
Precision: 64
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;n \le -6.1428874105226178 \cdot 10^{122}:\\ \;\;\;\;\left(100 \cdot \frac{\left(\log 1 \cdot n + \left(1 \cdot i + 0.5 \cdot {i}^{2}\right)\right) - 0.5 \cdot \left(\log 1 \cdot {i}^{2}\right)}{i}\right) \cdot n\\ \mathbf{elif}\;n \le -8.23875326441091999 \cdot 10^{106}:\\ \;\;\;\;100 \cdot \frac{\frac{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} - 1 \cdot 1}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{\frac{i}{n}}\\ \mathbf{elif}\;n \le -1.8594290872480363 \lor \neg \left(n \le 1.41832978451163636 \cdot 10^{-154}\right):\\ \;\;\;\;\left(100 \cdot \frac{\left(\log 1 \cdot n + \left(1 \cdot i + 0.5 \cdot {i}^{2}\right)\right) - 0.5 \cdot \left(\log 1 \cdot {i}^{2}\right)}{i}\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\left(\log 1 \cdot n + \left(1 \cdot i + 1\right)\right) - 1}{\frac{i}{n}}\\ \end{array}\]
100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}
↓
\begin{array}{l}
\mathbf{if}\;n \le -6.1428874105226178 \cdot 10^{122}:\\
\;\;\;\;\left(100 \cdot \frac{\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)}{i}\right) \cdot n\\

\mathbf{elif}\;n \le -8.23875326441091999 \cdot 10^{106}:\\
\;\;\;\;100 \cdot \frac{\frac{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} - 1 \cdot 1}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{\frac{i}{n}}\\

\mathbf{elif}\;n \le -1.8594290872480363 \lor \neg \left(n \le 1.41832978451163636 \cdot 10^{-154}\right):\\
\;\;\;\;\left(100 \cdot \frac{\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)}{i}\right) \cdot n\\

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

\end{array}
double f(double i, double n) {
        double r142103 = 100.0;
        double r142104 = 1.0;
        double r142105 = i;
        double r142106 = n;
        double r142107 = r142105 / r142106;
        double r142108 = r142104 + r142107;
        double r142109 = pow(r142108, r142106);
        double r142110 = r142109 - r142104;
        double r142111 = r142110 / r142107;
        double r142112 = r142103 * r142111;
        return r142112;
}
↓
double f(double i, double n) {
        double r142113 = n;
        double r142114 = -6.142887410522618e+122;
        bool r142115 = r142113 <= r142114;
        double r142116 = 100.0;
        double r142117 = 1.0;
        double r142118 = log(r142117);
        double r142119 = r142118 * r142113;
        double r142120 = i;
        double r142121 = r142117 * r142120;
        double r142122 = 0.5;
        double r142123 = 2.0;
        double r142124 = pow(r142120, r142123);
        double r142125 = r142122 * r142124;
        double r142126 = r142121 + r142125;
        double r142127 = r142119 + r142126;
        double r142128 = r142118 * r142124;
        double r142129 = r142122 * r142128;
        double r142130 = r142127 - r142129;
        double r142131 = r142130 / r142120;
        double r142132 = r142116 * r142131;
        double r142133 = r142132 * r142113;
        double r142134 = -8.23875326441092e+106;
        bool r142135 = r142113 <= r142134;
        double r142136 = r142120 / r142113;
        double r142137 = r142117 + r142136;
        double r142138 = r142123 * r142113;
        double r142139 = pow(r142137, r142138);
        double r142140 = r142117 * r142117;
        double r142141 = r142139 - r142140;
        double r142142 = pow(r142137, r142113);
        double r142143 = r142142 + r142117;
        double r142144 = r142141 / r142143;
        double r142145 = r142144 / r142136;
        double r142146 = r142116 * r142145;
        double r142147 = -1.8594290872480363;
        bool r142148 = r142113 <= r142147;
        double r142149 = 1.4183297845116364e-154;
        bool r142150 = r142113 <= r142149;
        double r142151 = !r142150;
        bool r142152 = r142148 || r142151;
        double r142153 = 1.0;
        double r142154 = r142121 + r142153;
        double r142155 = r142119 + r142154;
        double r142156 = r142155 - r142117;
        double r142157 = r142156 / r142136;
        double r142158 = r142116 * r142157;
        double r142159 = r142152 ? r142133 : r142158;
        double r142160 = r142135 ? r142146 : r142159;
        double r142161 = r142115 ? r142133 : r142160;
        return r142161;
}

Error

Bits error versus i

Bits error versus n

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original42.8
Target42.8
Herbie23.5
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 3 regimes

  2. if n < -6.142887410522618e+122 or -8.23875326441092e+106 < n < -1.8594290872480363 or 1.4183297845116364e-154 < 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 39.5

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

    4. Applied associate-/r/22.5

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

    5. Applied associate-*r*22.5

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

    if -6.142887410522618e+122 < n < -8.23875326441092e+106

    1. Initial program 40.4

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

    3. Applied flip--40.4

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

    4. Simplified40.4

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

    if -1.8594290872480363 < n < 1.4183297845116364e-154

    1. Initial program 23.6

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

    2. Taylor expanded around 0 24.6

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

  4. Final simplification23.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -6.1428874105226178 \cdot 10^{122}:\\ \;\;\;\;\left(100 \cdot \frac{\left(\log 1 \cdot n + \left(1 \cdot i + 0.5 \cdot {i}^{2}\right)\right) - 0.5 \cdot \left(\log 1 \cdot {i}^{2}\right)}{i}\right) \cdot n\\ \mathbf{elif}\;n \le -8.23875326441091999 \cdot 10^{106}:\\ \;\;\;\;100 \cdot \frac{\frac{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} - 1 \cdot 1}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{\frac{i}{n}}\\ \mathbf{elif}\;n \le -1.8594290872480363 \lor \neg \left(n \le 1.41832978451163636 \cdot 10^{-154}\right):\\ \;\;\;\;\left(100 \cdot \frac{\left(\log 1 \cdot n + \left(1 \cdot i + 0.5 \cdot {i}^{2}\right)\right) - 0.5 \cdot \left(\log 1 \cdot {i}^{2}\right)}{i}\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\left(\log 1 \cdot n + \left(1 \cdot i + 1\right)\right) - 1}{\frac{i}{n}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019195 
(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))))