Average Error: 43.0 → 18.8
Time: 32.7s
Precision: 64
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le -0.8489635077384908301567634225648362189531:\\ \;\;\;\;100 \cdot \frac{{\left(\frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 0.5476118163719528864064045592385809868574:\\ \;\;\;\;\frac{100}{i} \cdot \left(\left(\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right) \cdot n\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(\frac{1}{i} \cdot \left(\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right) \cdot n\right)\right)\\ \end{array}\]
100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}
↓
\begin{array}{l}
\mathbf{if}\;i \le -0.8489635077384908301567634225648362189531:\\
\;\;\;\;100 \cdot \frac{{\left(\frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\\

\mathbf{elif}\;i \le 0.5476118163719528864064045592385809868574:\\
\;\;\;\;\frac{100}{i} \cdot \left(\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)\\

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

\end{array}
double f(double i, double n) {
        double r268886 = 100.0;
        double r268887 = 1.0;
        double r268888 = i;
        double r268889 = n;
        double r268890 = r268888 / r268889;
        double r268891 = r268887 + r268890;
        double r268892 = pow(r268891, r268889);
        double r268893 = r268892 - r268887;
        double r268894 = r268893 / r268890;
        double r268895 = r268886 * r268894;
        return r268895;
}
↓
double f(double i, double n) {
        double r268896 = i;
        double r268897 = -0.8489635077384908;
        bool r268898 = r268896 <= r268897;
        double r268899 = 100.0;
        double r268900 = n;
        double r268901 = r268896 / r268900;
        double r268902 = pow(r268901, r268900);
        double r268903 = 1.0;
        double r268904 = r268902 - r268903;
        double r268905 = r268904 / r268901;
        double r268906 = r268899 * r268905;
        double r268907 = 0.5476118163719529;
        bool r268908 = r268896 <= r268907;
        double r268909 = r268899 / r268896;
        double r268910 = r268903 * r268896;
        double r268911 = 0.5;
        double r268912 = 2.0;
        double r268913 = pow(r268896, r268912);
        double r268914 = r268911 * r268913;
        double r268915 = log(r268903);
        double r268916 = r268915 * r268900;
        double r268917 = r268914 + r268916;
        double r268918 = r268910 + r268917;
        double r268919 = r268913 * r268915;
        double r268920 = r268911 * r268919;
        double r268921 = r268918 - r268920;
        double r268922 = r268921 * r268900;
        double r268923 = r268909 * r268922;
        double r268924 = 1.0;
        double r268925 = r268924 / r268896;
        double r268926 = r268903 + r268901;
        double r268927 = pow(r268926, r268900);
        double r268928 = r268927 - r268903;
        double r268929 = r268928 * r268900;
        double r268930 = r268925 * r268929;
        double r268931 = r268899 * r268930;
        double r268932 = r268908 ? r268923 : r268931;
        double r268933 = r268898 ? r268906 : r268932;
        return r268933;
}

Error

Bits error versus i

Bits error versus n

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original43.0
Target43.5
Herbie18.8
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 3 regimes

  2. if i < -0.8489635077384908

    1. Initial program 27.4

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

    2. Taylor expanded around inf 64.0

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

    3. Simplified18.2

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

    if -0.8489635077384908 < i < 0.5476118163719529

    1. Initial program 50.8

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

    2. Taylor expanded around 0 34.2

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

    4. Applied div-inv34.3

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

    5. Applied *-un-lft-identity34.3

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

    6. Applied times-frac15.8

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

    7. Simplified15.8

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

    9. Applied associate-*r*16.2

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

    10. Simplified16.1

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

    if 0.5476118163719529 < i

    1. Initial program 32.4

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

    3. Applied div-inv32.4

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

    4. Applied *-un-lft-identity32.4

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

    5. Applied times-frac32.4

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

    6. Simplified32.4

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

  4. Final simplification18.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -0.8489635077384908301567634225648362189531:\\ \;\;\;\;100 \cdot \frac{{\left(\frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 0.5476118163719528864064045592385809868574:\\ \;\;\;\;\frac{100}{i} \cdot \left(\left(\left(1 \cdot i + \left(0.5 \cdot {i}^{2} + \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right) \cdot n\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(\frac{1}{i} \cdot \left(\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right) \cdot n\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 
(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))))