Average Error: 47.1 → 16.9
Time: 11.5s
Precision: 64
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le -0.0390445009989891442:\\ \;\;\;\;\frac{100}{i} \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{1}{n}}\\ \mathbf{elif}\;i \le 22.1533166385413587:\\ \;\;\;\;\left(\left(50 \cdot i + \left(100 \cdot \frac{\log 1 \cdot n}{i} + 100\right)\right) - 50 \cdot \left(i \cdot \log 1\right)\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\frac{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + \left(-1 \cdot 1\right)}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{\frac{i}{n}}\\ \end{array}\]
100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}
↓
\begin{array}{l}
\mathbf{if}\;i \le -0.0390445009989891442:\\
\;\;\;\;\frac{100}{i} \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{1}{n}}\\

\mathbf{elif}\;i \le 22.1533166385413587:\\
\;\;\;\;\left(\left(50 \cdot i + \left(100 \cdot \frac{\log 1 \cdot n}{i} + 100\right)\right) - 50 \cdot \left(i \cdot \log 1\right)\right) \cdot n\\

\mathbf{else}:\\
\;\;\;\;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}}\\

\end{array}
double f(double i, double n) {
        double r130911 = 100.0;
        double r130912 = 1.0;
        double r130913 = i;
        double r130914 = n;
        double r130915 = r130913 / r130914;
        double r130916 = r130912 + r130915;
        double r130917 = pow(r130916, r130914);
        double r130918 = r130917 - r130912;
        double r130919 = r130918 / r130915;
        double r130920 = r130911 * r130919;
        return r130920;
}
↓
double f(double i, double n) {
        double r130921 = i;
        double r130922 = -0.039044500998989144;
        bool r130923 = r130921 <= r130922;
        double r130924 = 100.0;
        double r130925 = r130924 / r130921;
        double r130926 = 1.0;
        double r130927 = n;
        double r130928 = r130921 / r130927;
        double r130929 = r130926 + r130928;
        double r130930 = pow(r130929, r130927);
        double r130931 = r130930 - r130926;
        double r130932 = 1.0;
        double r130933 = r130932 / r130927;
        double r130934 = r130931 / r130933;
        double r130935 = r130925 * r130934;
        double r130936 = 22.15331663854136;
        bool r130937 = r130921 <= r130936;
        double r130938 = 50.0;
        double r130939 = r130938 * r130921;
        double r130940 = log(r130926);
        double r130941 = r130940 * r130927;
        double r130942 = r130941 / r130921;
        double r130943 = r130924 * r130942;
        double r130944 = r130943 + r130924;
        double r130945 = r130939 + r130944;
        double r130946 = r130921 * r130940;
        double r130947 = r130938 * r130946;
        double r130948 = r130945 - r130947;
        double r130949 = r130948 * r130927;
        double r130950 = 2.0;
        double r130951 = r130950 * r130927;
        double r130952 = pow(r130929, r130951);
        double r130953 = r130926 * r130926;
        double r130954 = -r130953;
        double r130955 = r130952 + r130954;
        double r130956 = r130930 + r130926;
        double r130957 = r130955 / r130956;
        double r130958 = r130957 / r130928;
        double r130959 = r130924 * r130958;
        double r130960 = r130937 ? r130949 : r130959;
        double r130961 = r130923 ? r130935 : r130960;
        return r130961;
}

Error

Bits error versus i

Bits error versus n

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original47.1
Target46.8
Herbie16.9
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 3 regimes

  2. if i < -0.039044500998989144

    1. Initial program 27.3

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

    3. Applied div-inv27.3

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

    4. Applied *-un-lft-identity27.3

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

    5. Applied times-frac27.8

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

    6. Applied associate-*r*27.8

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

    7. Simplified27.8

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

    if -0.039044500998989144 < i < 22.15331663854136

    1. Initial program 57.8

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

    2. Taylor expanded around 0 26.8

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

    4. Applied associate-/r/9.8

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

    5. Applied associate-*r*9.8

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

    6. Taylor expanded around 0 9.8

      \[\leadsto \color{blue}{\left(\left(100 + \left(50 \cdot i + 100 \cdot \frac{\log 1 \cdot n}{i}\right)\right) - 50 \cdot \left(i \cdot \log 1\right)\right)} \cdot n\]

    7. Simplified9.8

      \[\leadsto \color{blue}{\left(\left(50 \cdot i + \left(100 \cdot \frac{\log 1 \cdot n}{i} + 100\right)\right) - 50 \cdot \left(i \cdot \log 1\right)\right)} \cdot n\]

    if 22.15331663854136 < i

    1. Initial program 31.6

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

    3. Applied flip--31.6

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

    4. Simplified31.6

      \[\leadsto 100 \cdot \frac{\frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + \left(-1 \cdot 1\right)}}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{\frac{i}{n}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification16.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -0.0390445009989891442:\\ \;\;\;\;\frac{100}{i} \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{1}{n}}\\ \mathbf{elif}\;i \le 22.1533166385413587:\\ \;\;\;\;\left(\left(50 \cdot i + \left(100 \cdot \frac{\log 1 \cdot n}{i} + 100\right)\right) - 50 \cdot \left(i \cdot \log 1\right)\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\frac{{\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)} + \left(-1 \cdot 1\right)}{{\left(1 + \frac{i}{n}\right)}^{n} + 1}}{\frac{i}{n}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020062 
(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))))