Average Error: 43.2 → 23.8
Time: 14.9s
Precision: 64
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;n \le -5.5136808217910137 \cdot 10^{209}:\\ \;\;\;\;\left(100 \cdot \frac{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{i}\right) \cdot n\\ \mathbf{elif}\;n \le -4.20006178668092851 \cdot 10^{124}:\\ \;\;\;\;\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}\\ \mathbf{elif}\;n \le -1.98325517621712044:\\ \;\;\;\;\left(100 \cdot \frac{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{i}\right) \cdot n\\ \mathbf{elif}\;n \le -3.081068569055166 \cdot 10^{-191}:\\ \;\;\;\;100 \cdot \frac{\frac{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + \left(-1 \cdot 1\right)}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{\frac{i}{n}}\\ \mathbf{elif}\;n \le 3.21498631270168259 \cdot 10^{-193}:\\ \;\;\;\;100 \cdot \frac{\left(1 \cdot i + \left(\log 1 \cdot n + 1\right)\right) - 1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\left(100 \cdot \frac{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{i}\right) \cdot n\\ \end{array}\]
100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}
↓
\begin{array}{l}
\mathbf{if}\;n \le -5.5136808217910137 \cdot 10^{209}:\\
\;\;\;\;\left(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)}{i}\right) \cdot n\\

\mathbf{elif}\;n \le -4.20006178668092851 \cdot 10^{124}:\\
\;\;\;\;\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}\\

\mathbf{elif}\;n \le -1.98325517621712044:\\
\;\;\;\;\left(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)}{i}\right) \cdot n\\

\mathbf{elif}\;n \le -3.081068569055166 \cdot 10^{-191}:\\
\;\;\;\;100 \cdot \frac{\frac{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + \left(-1 \cdot 1\right)}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{\frac{i}{n}}\\

\mathbf{elif}\;n \le 3.21498631270168259 \cdot 10^{-193}:\\
\;\;\;\;100 \cdot \frac{\left(1 \cdot i + \left(\log 1 \cdot n + 1\right)\right) - 1}{\frac{i}{n}}\\

\mathbf{else}:\\
\;\;\;\;\left(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)}{i}\right) \cdot n\\

\end{array}
double f(double i, double n) {
        double r167066 = 100.0;
        double r167067 = 1.0;
        double r167068 = i;
        double r167069 = n;
        double r167070 = r167068 / r167069;
        double r167071 = r167067 + r167070;
        double r167072 = pow(r167071, r167069);
        double r167073 = r167072 - r167067;
        double r167074 = r167073 / r167070;
        double r167075 = r167066 * r167074;
        return r167075;
}
↓
double f(double i, double n) {
        double r167076 = n;
        double r167077 = -5.513680821791014e+209;
        bool r167078 = r167076 <= r167077;
        double r167079 = 100.0;
        double r167080 = 1.0;
        double r167081 = i;
        double r167082 = r167080 * r167081;
        double r167083 = 0.5;
        double r167084 = 2.0;
        double r167085 = pow(r167081, r167084);
        double r167086 = r167083 * r167085;
        double r167087 = log(r167080);
        double r167088 = r167087 * r167076;
        double r167089 = r167086 + r167088;
        double r167090 = r167082 + r167089;
        double r167091 = r167085 * r167087;
        double r167092 = r167083 * r167091;
        double r167093 = r167090 - r167092;
        double r167094 = r167093 / r167081;
        double r167095 = r167079 * r167094;
        double r167096 = r167095 * r167076;
        double r167097 = -4.2000617866809285e+124;
        bool r167098 = r167076 <= r167097;
        double r167099 = r167081 / r167076;
        double r167100 = r167080 + r167099;
        double r167101 = pow(r167100, r167076);
        double r167102 = r167101 - r167080;
        double r167103 = r167079 * r167102;
        double r167104 = r167103 / r167099;
        double r167105 = -1.9832551762171204;
        bool r167106 = r167076 <= r167105;
        double r167107 = -3.0810685690551663e-191;
        bool r167108 = r167076 <= r167107;
        double r167109 = r167084 * r167076;
        double r167110 = pow(r167100, r167109);
        double r167111 = r167080 * r167080;
        double r167112 = -r167111;
        double r167113 = r167110 + r167112;
        double r167114 = r167101 + r167080;
        double r167115 = r167113 / r167114;
        double r167116 = r167115 / r167099;
        double r167117 = r167079 * r167116;
        double r167118 = 3.2149863127016826e-193;
        bool r167119 = r167076 <= r167118;
        double r167120 = 1.0;
        double r167121 = r167088 + r167120;
        double r167122 = r167082 + r167121;
        double r167123 = r167122 - r167080;
        double r167124 = r167123 / r167099;
        double r167125 = r167079 * r167124;
        double r167126 = r167119 ? r167125 : r167096;
        double r167127 = r167108 ? r167117 : r167126;
        double r167128 = r167106 ? r167096 : r167127;
        double r167129 = r167098 ? r167104 : r167128;
        double r167130 = r167078 ? r167096 : r167129;
        return r167130;
}

Error

Bits error versus i

Bits error versus n

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original43.2
Target42.4
Herbie23.8
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 4 regimes

  2. if n < -5.513680821791014e+209 or -4.2000617866809285e+124 < n < -1.9832551762171204 or 3.2149863127016826e-193 < n

    1. Initial program 53.2

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

    2. Taylor expanded around 0 38.3

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

    4. Applied associate-/r/23.3

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

    5. Applied associate-*r*23.3

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

    if -5.513680821791014e+209 < n < -4.2000617866809285e+124

    1. Initial program 45.4

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

    3. Applied associate-*r/45.4

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

    if -1.9832551762171204 < n < -3.0810685690551663e-191

    1. Initial program 20.1

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

    3. Applied flip--20.1

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

    4. Simplified20.1

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

    if -3.0810685690551663e-191 < n < 3.2149863127016826e-193

    1. Initial program 26.1

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

    2. Taylor expanded around 0 17.7

      \[\leadsto 100 \cdot \frac{\color{blue}{\left(1 \cdot i + \left(\log 1 \cdot n + 1\right)\right)} - 1}{\frac{i}{n}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification23.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -5.5136808217910137 \cdot 10^{209}:\\ \;\;\;\;\left(100 \cdot \frac{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{i}\right) \cdot n\\ \mathbf{elif}\;n \le -4.20006178668092851 \cdot 10^{124}:\\ \;\;\;\;\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}\\ \mathbf{elif}\;n \le -1.98325517621712044:\\ \;\;\;\;\left(100 \cdot \frac{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{i}\right) \cdot n\\ \mathbf{elif}\;n \le -3.081068569055166 \cdot 10^{-191}:\\ \;\;\;\;100 \cdot \frac{\frac{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + \left(-1 \cdot 1\right)}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{\frac{i}{n}}\\ \mathbf{elif}\;n \le 3.21498631270168259 \cdot 10^{-193}:\\ \;\;\;\;100 \cdot \frac{\left(1 \cdot i + \left(\log 1 \cdot n + 1\right)\right) - 1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\left(100 \cdot \frac{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{i}\right) \cdot n\\ \end{array}\]

Reproduce

herbie shell --seed 2020036 
(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))))