Average Error: 42.9 → 29.8
Time: 14.8s
Precision: 64
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;n \le -2.51457813685989335 \cdot 10^{153}:\\ \;\;\;\;100 \cdot \frac{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{\frac{i}{n}}\\ \mathbf{elif}\;n \le -854986621658212610:\\ \;\;\;\;\left(100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}\right) \cdot n\\ \mathbf{elif}\;n \le -29.9716390321119768:\\ \;\;\;\;100 \cdot \frac{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{\frac{i}{n}}\\ \mathbf{elif}\;n \le -7.70416426394191 \cdot 10^{-311}:\\ \;\;\;\;100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \frac{1}{\frac{i}{n}}\right)\\ \mathbf{elif}\;n \le 1.86219659931192698 \cdot 10^{-104}:\\ \;\;\;\;\frac{100 \cdot \left(\left(\left(\left(\frac{1}{2} \cdot \left({\left(\log i\right)}^{2} \cdot {n}^{2} + {n}^{2} \cdot {\left(\log n\right)}^{2}\right) + \left(\frac{1}{6} \cdot \left({\left(\log i\right)}^{3} \cdot {n}^{3}\right) + \log i \cdot n\right)\right) + \frac{1}{2} \cdot \left(\log i \cdot \left({n}^{3} \cdot {\left(\log n\right)}^{2}\right)\right)\right) - \log n \cdot \left(\log i \cdot {n}^{2} + n\right)\right) - \left(\frac{1}{2} \cdot \left({\left(\log i\right)}^{2} \cdot \left({n}^{3} \cdot \log n\right)\right) + \frac{1}{6} \cdot \left({n}^{3} \cdot {\left(\log n\right)}^{3}\right)\right)\right)}{\frac{i}{n}}\\ \mathbf{elif}\;n \le 4.7215330661381895 \cdot 10^{175}:\\ \;\;\;\;100 \cdot \frac{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\left(100 \cdot \left({\left(\frac{i}{n}\right)}^{n} - 1\right)\right) \cdot \frac{n}{i}\\ \end{array}\]
100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}
↓
\begin{array}{l}
\mathbf{if}\;n \le -2.51457813685989335 \cdot 10^{153}:\\
\;\;\;\;100 \cdot \frac{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{\frac{i}{n}}\\

\mathbf{elif}\;n \le -854986621658212610:\\
\;\;\;\;\left(100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}\right) \cdot n\\

\mathbf{elif}\;n \le -29.9716390321119768:\\
\;\;\;\;100 \cdot \frac{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{\frac{i}{n}}\\

\mathbf{elif}\;n \le -7.70416426394191 \cdot 10^{-311}:\\
\;\;\;\;100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \frac{1}{\frac{i}{n}}\right)\\

\mathbf{elif}\;n \le 1.86219659931192698 \cdot 10^{-104}:\\
\;\;\;\;\frac{100 \cdot \left(\left(\left(\left(\frac{1}{2} \cdot \left({\left(\log i\right)}^{2} \cdot {n}^{2} + {n}^{2} \cdot {\left(\log n\right)}^{2}\right) + \left(\frac{1}{6} \cdot \left({\left(\log i\right)}^{3} \cdot {n}^{3}\right) + \log i \cdot n\right)\right) + \frac{1}{2} \cdot \left(\log i \cdot \left({n}^{3} \cdot {\left(\log n\right)}^{2}\right)\right)\right) - \log n \cdot \left(\log i \cdot {n}^{2} + n\right)\right) - \left(\frac{1}{2} \cdot \left({\left(\log i\right)}^{2} \cdot \left({n}^{3} \cdot \log n\right)\right) + \frac{1}{6} \cdot \left({n}^{3} \cdot {\left(\log n\right)}^{3}\right)\right)\right)}{\frac{i}{n}}\\

\mathbf{elif}\;n \le 4.7215330661381895 \cdot 10^{175}:\\
\;\;\;\;100 \cdot \frac{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{\frac{i}{n}}\\

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

\end{array}
double f(double i, double n) {
        double r127048 = 100.0;
        double r127049 = 1.0;
        double r127050 = i;
        double r127051 = n;
        double r127052 = r127050 / r127051;
        double r127053 = r127049 + r127052;
        double r127054 = pow(r127053, r127051);
        double r127055 = r127054 - r127049;
        double r127056 = r127055 / r127052;
        double r127057 = r127048 * r127056;
        return r127057;
}
↓
double f(double i, double n) {
        double r127058 = n;
        double r127059 = -2.5145781368598933e+153;
        bool r127060 = r127058 <= r127059;
        double r127061 = 100.0;
        double r127062 = 1.0;
        double r127063 = i;
        double r127064 = r127062 * r127063;
        double r127065 = 0.5;
        double r127066 = 2.0;
        double r127067 = pow(r127063, r127066);
        double r127068 = r127065 * r127067;
        double r127069 = log(r127062);
        double r127070 = r127069 * r127058;
        double r127071 = r127068 + r127070;
        double r127072 = r127064 + r127071;
        double r127073 = r127067 * r127069;
        double r127074 = r127065 * r127073;
        double r127075 = r127072 - r127074;
        double r127076 = r127063 / r127058;
        double r127077 = r127075 / r127076;
        double r127078 = r127061 * r127077;
        double r127079 = -8.549866216582126e+17;
        bool r127080 = r127058 <= r127079;
        double r127081 = r127062 + r127076;
        double r127082 = pow(r127081, r127058);
        double r127083 = r127082 - r127062;
        double r127084 = r127083 / r127063;
        double r127085 = r127061 * r127084;
        double r127086 = r127085 * r127058;
        double r127087 = -29.971639032111977;
        bool r127088 = r127058 <= r127087;
        double r127089 = -7.7041642639419e-311;
        bool r127090 = r127058 <= r127089;
        double r127091 = r127082 / r127076;
        double r127092 = r127062 / r127076;
        double r127093 = r127091 - r127092;
        double r127094 = r127061 * r127093;
        double r127095 = 1.862196599311927e-104;
        bool r127096 = r127058 <= r127095;
        double r127097 = 0.5;
        double r127098 = log(r127063);
        double r127099 = pow(r127098, r127066);
        double r127100 = pow(r127058, r127066);
        double r127101 = r127099 * r127100;
        double r127102 = log(r127058);
        double r127103 = pow(r127102, r127066);
        double r127104 = r127100 * r127103;
        double r127105 = r127101 + r127104;
        double r127106 = r127097 * r127105;
        double r127107 = 0.16666666666666666;
        double r127108 = 3.0;
        double r127109 = pow(r127098, r127108);
        double r127110 = pow(r127058, r127108);
        double r127111 = r127109 * r127110;
        double r127112 = r127107 * r127111;
        double r127113 = r127098 * r127058;
        double r127114 = r127112 + r127113;
        double r127115 = r127106 + r127114;
        double r127116 = r127110 * r127103;
        double r127117 = r127098 * r127116;
        double r127118 = r127097 * r127117;
        double r127119 = r127115 + r127118;
        double r127120 = r127098 * r127100;
        double r127121 = r127120 + r127058;
        double r127122 = r127102 * r127121;
        double r127123 = r127119 - r127122;
        double r127124 = r127110 * r127102;
        double r127125 = r127099 * r127124;
        double r127126 = r127097 * r127125;
        double r127127 = pow(r127102, r127108);
        double r127128 = r127110 * r127127;
        double r127129 = r127107 * r127128;
        double r127130 = r127126 + r127129;
        double r127131 = r127123 - r127130;
        double r127132 = r127061 * r127131;
        double r127133 = r127132 / r127076;
        double r127134 = 4.7215330661381895e+175;
        bool r127135 = r127058 <= r127134;
        double r127136 = pow(r127076, r127058);
        double r127137 = r127136 - r127062;
        double r127138 = r127061 * r127137;
        double r127139 = r127058 / r127063;
        double r127140 = r127138 * r127139;
        double r127141 = r127135 ? r127078 : r127140;
        double r127142 = r127096 ? r127133 : r127141;
        double r127143 = r127090 ? r127094 : r127142;
        double r127144 = r127088 ? r127078 : r127143;
        double r127145 = r127080 ? r127086 : r127144;
        double r127146 = r127060 ? r127078 : r127145;
        return r127146;
}

Error

Bits error versus i

Bits error versus n

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original42.9
Target42.2
Herbie29.8
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 5 regimes

  2. if n < -2.5145781368598933e+153 or -8.549866216582126e+17 < n < -29.971639032111977 or 1.862196599311927e-104 < n < 4.7215330661381895e+175

    1. Initial program 55.7

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

    2. Taylor expanded around 0 36.1

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

    if -2.5145781368598933e+153 < n < -8.549866216582126e+17

    1. Initial program 37.9

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

    3. Applied associate-/r/37.7

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

    4. Applied associate-*r*37.7

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

    if -29.971639032111977 < n < -7.7041642639419e-311

    1. Initial program 15.7

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

    3. Applied div-sub15.8

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

    if -7.7041642639419e-311 < n < 1.862196599311927e-104

    1. Initial program 45.1

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

    3. Applied associate-*r/45.1

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

    4. Taylor expanded around inf 26.7

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

    5. Simplified45.6

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

    6. Taylor expanded around 0 18.1

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

    7. Simplified18.1

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

    if 4.7215330661381895e+175 < n

    1. Initial program 62.3

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

    3. Applied associate-*r/62.3

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

    4. Taylor expanded around inf 64.0

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

    5. Simplified44.2

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

    7. Applied div-inv44.2

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

    8. Simplified44.2

      \[\leadsto \left(100 \cdot \left({\left(\frac{i}{n}\right)}^{n} - 1\right)\right) \cdot \color{blue}{\frac{n}{i}}\]
  3. Recombined 5 regimes into one program.

  4. Final simplification29.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -2.51457813685989335 \cdot 10^{153}:\\ \;\;\;\;100 \cdot \frac{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{\frac{i}{n}}\\ \mathbf{elif}\;n \le -854986621658212610:\\ \;\;\;\;\left(100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}\right) \cdot n\\ \mathbf{elif}\;n \le -29.9716390321119768:\\ \;\;\;\;100 \cdot \frac{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{\frac{i}{n}}\\ \mathbf{elif}\;n \le -7.70416426394191 \cdot 10^{-311}:\\ \;\;\;\;100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \frac{1}{\frac{i}{n}}\right)\\ \mathbf{elif}\;n \le 1.86219659931192698 \cdot 10^{-104}:\\ \;\;\;\;\frac{100 \cdot \left(\left(\left(\left(\frac{1}{2} \cdot \left({\left(\log i\right)}^{2} \cdot {n}^{2} + {n}^{2} \cdot {\left(\log n\right)}^{2}\right) + \left(\frac{1}{6} \cdot \left({\left(\log i\right)}^{3} \cdot {n}^{3}\right) + \log i \cdot n\right)\right) + \frac{1}{2} \cdot \left(\log i \cdot \left({n}^{3} \cdot {\left(\log n\right)}^{2}\right)\right)\right) - \log n \cdot \left(\log i \cdot {n}^{2} + n\right)\right) - \left(\frac{1}{2} \cdot \left({\left(\log i\right)}^{2} \cdot \left({n}^{3} \cdot \log n\right)\right) + \frac{1}{6} \cdot \left({n}^{3} \cdot {\left(\log n\right)}^{3}\right)\right)\right)}{\frac{i}{n}}\\ \mathbf{elif}\;n \le 4.7215330661381895 \cdot 10^{175}:\\ \;\;\;\;100 \cdot \frac{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\left(100 \cdot \left({\left(\frac{i}{n}\right)}^{n} - 1\right)\right) \cdot \frac{n}{i}\\ \end{array}\]

Reproduce

herbie shell --seed 2020035 
(FPCore (i n)
  :name "Compound Interest"
  :precision binary64

  :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))))