Average Error: 42.6 → 20.1
Time: 32.1s
Precision: 64
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le -0.09568466101685804692245085334434406831861:\\ \;\;\;\;\frac{100 \cdot \left({\left(\frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 6.433403710498657594142250154269082661313 \cdot 10^{46}:\\ \;\;\;\;\left(\left(\frac{100}{i} \cdot \sqrt[3]{\left(\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right) \cdot n}\right) \cdot \sqrt[3]{\left(\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right) \cdot n}\right) \cdot \sqrt[3]{\frac{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{\frac{1}{n}}}\\ \mathbf{elif}\;i \le 1.353418716849568981322833055613312096611 \cdot 10^{183}:\\ \;\;\;\;100 \cdot \frac{n \cdot \left({\left(\frac{i}{n}\right)}^{n} - 1\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\left(1 \cdot i + \left(\log 1 \cdot n + 1\right)\right) - 1}{\frac{i}{n}}\\ \end{array}\]
100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}
↓
\begin{array}{l}
\mathbf{if}\;i \le -0.09568466101685804692245085334434406831861:\\
\;\;\;\;\frac{100 \cdot \left({\left(\frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}\\

\mathbf{elif}\;i \le 6.433403710498657594142250154269082661313 \cdot 10^{46}:\\
\;\;\;\;\left(\left(\frac{100}{i} \cdot \sqrt[3]{\left(\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)\right) \cdot n}\right) \cdot \sqrt[3]{\left(\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)\right) \cdot n}\right) \cdot \sqrt[3]{\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{1}{n}}}\\

\mathbf{elif}\;i \le 1.353418716849568981322833055613312096611 \cdot 10^{183}:\\
\;\;\;\;100 \cdot \frac{n \cdot \left({\left(\frac{i}{n}\right)}^{n} - 1\right)}{i}\\

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

\end{array}
double f(double i, double n) {
        double r127146 = 100.0;
        double r127147 = 1.0;
        double r127148 = i;
        double r127149 = n;
        double r127150 = r127148 / r127149;
        double r127151 = r127147 + r127150;
        double r127152 = pow(r127151, r127149);
        double r127153 = r127152 - r127147;
        double r127154 = r127153 / r127150;
        double r127155 = r127146 * r127154;
        return r127155;
}
↓
double f(double i, double n) {
        double r127156 = i;
        double r127157 = -0.09568466101685805;
        bool r127158 = r127156 <= r127157;
        double r127159 = 100.0;
        double r127160 = n;
        double r127161 = r127156 / r127160;
        double r127162 = pow(r127161, r127160);
        double r127163 = 1.0;
        double r127164 = r127162 - r127163;
        double r127165 = r127159 * r127164;
        double r127166 = r127165 / r127161;
        double r127167 = 6.433403710498658e+46;
        bool r127168 = r127156 <= r127167;
        double r127169 = r127159 / r127156;
        double r127170 = r127163 * r127156;
        double r127171 = 0.5;
        double r127172 = 2.0;
        double r127173 = pow(r127156, r127172);
        double r127174 = r127171 * r127173;
        double r127175 = log(r127163);
        double r127176 = r127175 * r127160;
        double r127177 = r127174 + r127176;
        double r127178 = r127170 + r127177;
        double r127179 = r127173 * r127175;
        double r127180 = r127171 * r127179;
        double r127181 = r127178 - r127180;
        double r127182 = r127181 * r127160;
        double r127183 = cbrt(r127182);
        double r127184 = r127169 * r127183;
        double r127185 = r127184 * r127183;
        double r127186 = 1.0;
        double r127187 = r127186 / r127160;
        double r127188 = r127181 / r127187;
        double r127189 = cbrt(r127188);
        double r127190 = r127185 * r127189;
        double r127191 = 1.353418716849569e+183;
        bool r127192 = r127156 <= r127191;
        double r127193 = r127160 * r127164;
        double r127194 = r127193 / r127156;
        double r127195 = r127159 * r127194;
        double r127196 = r127176 + r127186;
        double r127197 = r127170 + r127196;
        double r127198 = r127197 - r127163;
        double r127199 = r127198 / r127161;
        double r127200 = r127159 * r127199;
        double r127201 = r127192 ? r127195 : r127200;
        double r127202 = r127168 ? r127190 : r127201;
        double r127203 = r127158 ? r127166 : r127202;
        return r127203;
}

Error

Bits error versus i

Bits error versus n

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original42.6
Target42.1
Herbie20.1
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 4 regimes

  2. if i < -0.09568466101685805

    1. Initial program 28.6

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

    3. Applied div-inv28.6

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

    4. Applied *-un-lft-identity28.6

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

    5. Applied times-frac29.3

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

    6. Applied associate-*r*29.3

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

    7. Simplified29.3

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

    8. Taylor expanded around inf 64.0

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

    9. Simplified18.7

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

    if -0.09568466101685805 < i < 6.433403710498658e+46

    1. Initial program 49.5

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

    3. Applied div-inv49.5

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

    4. Applied *-un-lft-identity49.5

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

    5. Applied times-frac49.2

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

    6. Applied associate-*r*49.2

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

    7. Simplified49.2

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

    8. Taylor expanded around 0 17.8

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

    10. Applied add-cube-cbrt18.4

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

    11. Applied associate-*r*18.4

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

    12. Simplified18.4

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

    if 6.433403710498658e+46 < i < 1.353418716849569e+183

    1. Initial program 32.0

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

    2. Taylor expanded around inf 29.2

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

    3. Simplified32.0

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

    if 1.353418716849569e+183 < i

    1. Initial program 33.1

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

    2. Taylor expanded around 0 32.9

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

  4. Final simplification20.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -0.09568466101685804692245085334434406831861:\\ \;\;\;\;\frac{100 \cdot \left({\left(\frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 6.433403710498657594142250154269082661313 \cdot 10^{46}:\\ \;\;\;\;\left(\left(\frac{100}{i} \cdot \sqrt[3]{\left(\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right) \cdot n}\right) \cdot \sqrt[3]{\left(\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right) \cdot n}\right) \cdot \sqrt[3]{\frac{\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{\frac{1}{n}}}\\ \mathbf{elif}\;i \le 1.353418716849568981322833055613312096611 \cdot 10^{183}:\\ \;\;\;\;100 \cdot \frac{n \cdot \left({\left(\frac{i}{n}\right)}^{n} - 1\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\left(1 \cdot i + \left(\log 1 \cdot n + 1\right)\right) - 1}{\frac{i}{n}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019326 
(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))))