Average Error: 42.7 → 17.8
Time: 27.0s
Precision: 64
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le -1.8795028361709332:\\ \;\;\;\;100 \cdot \left(\left(\left(\frac{{\left(\frac{i}{n} + 1\right)}^{n}}{\frac{i}{n}}\right)\right) - \frac{1}{\frac{i}{n}}\right)\\ \mathbf{elif}\;i \le 0.0004866774544189746:\\ \;\;\;\;100 \cdot \left(\left(n + \left(n \cdot i\right) \cdot \frac{1}{2}\right) + n \cdot \left(\frac{1}{6} \cdot \left(i \cdot i\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(n \cdot \left(\frac{{\left(\frac{i}{n} + 1\right)}^{n}}{i} - \frac{1}{i}\right)\right) \cdot 100\\ \end{array}\]
100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}
↓
\begin{array}{l}
\mathbf{if}\;i \le -1.8795028361709332:\\
\;\;\;\;100 \cdot \left(\left(\left(\frac{{\left(\frac{i}{n} + 1\right)}^{n}}{\frac{i}{n}}\right)\right) - \frac{1}{\frac{i}{n}}\right)\\

\mathbf{elif}\;i \le 0.0004866774544189746:\\
\;\;\;\;100 \cdot \left(\left(n + \left(n \cdot i\right) \cdot \frac{1}{2}\right) + n \cdot \left(\frac{1}{6} \cdot \left(i \cdot i\right)\right)\right)\\

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

\end{array}
double f(double i, double n) {
        double r7017867 = 100.0;
        double r7017868 = 1.0;
        double r7017869 = i;
        double r7017870 = n;
        double r7017871 = r7017869 / r7017870;
        double r7017872 = r7017868 + r7017871;
        double r7017873 = pow(r7017872, r7017870);
        double r7017874 = r7017873 - r7017868;
        double r7017875 = r7017874 / r7017871;
        double r7017876 = r7017867 * r7017875;
        return r7017876;
}
↓
double f(double i, double n) {
        double r7017877 = i;
        double r7017878 = -1.8795028361709332;
        bool r7017879 = r7017877 <= r7017878;
        double r7017880 = 100.0;
        double r7017881 = n;
        double r7017882 = r7017877 / r7017881;
        double r7017883 = 1.0;
        double r7017884 = r7017882 + r7017883;
        double r7017885 = pow(r7017884, r7017881);
        double r7017886 = r7017885 / r7017882;
        double r7017887 = /* ERROR: no posit support in C */;
        double r7017888 = /* ERROR: no posit support in C */;
        double r7017889 = r7017883 / r7017882;
        double r7017890 = r7017888 - r7017889;
        double r7017891 = r7017880 * r7017890;
        double r7017892 = 0.0004866774544189746;
        bool r7017893 = r7017877 <= r7017892;
        double r7017894 = r7017881 * r7017877;
        double r7017895 = 0.5;
        double r7017896 = r7017894 * r7017895;
        double r7017897 = r7017881 + r7017896;
        double r7017898 = 0.16666666666666666;
        double r7017899 = r7017877 * r7017877;
        double r7017900 = r7017898 * r7017899;
        double r7017901 = r7017881 * r7017900;
        double r7017902 = r7017897 + r7017901;
        double r7017903 = r7017880 * r7017902;
        double r7017904 = r7017885 / r7017877;
        double r7017905 = r7017883 / r7017877;
        double r7017906 = r7017904 - r7017905;
        double r7017907 = r7017881 * r7017906;
        double r7017908 = r7017907 * r7017880;
        double r7017909 = r7017893 ? r7017903 : r7017908;
        double r7017910 = r7017879 ? r7017891 : r7017909;
        return r7017910;
}

Error

Bits error versus i

Bits error versus n

Target

Original42.7
Target42.2
Herbie17.8
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 3 regimes

  2. if i < -1.8795028361709332

    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-sub28.6

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

    5. Applied insert-posit1612.7

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

    if -1.8795028361709332 < i < 0.0004866774544189746

    1. Initial program 50.1

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

    3. Applied div-sub50.1

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

    4. Taylor expanded around 0 16.7

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

    5. Simplified16.7

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

    if 0.0004866774544189746 < i

    1. Initial program 31.4

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

    3. Applied div-sub31.4

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

    5. Applied associate-/r/34.4

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

    6. Applied associate-/r/31.4

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

    7. Applied distribute-rgt-out--31.4

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

  4. Final simplification17.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -1.8795028361709332:\\ \;\;\;\;100 \cdot \left(\left(\left(\frac{{\left(\frac{i}{n} + 1\right)}^{n}}{\frac{i}{n}}\right)\right) - \frac{1}{\frac{i}{n}}\right)\\ \mathbf{elif}\;i \le 0.0004866774544189746:\\ \;\;\;\;100 \cdot \left(\left(n + \left(n \cdot i\right) \cdot \frac{1}{2}\right) + n \cdot \left(\frac{1}{6} \cdot \left(i \cdot i\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(n \cdot \left(\frac{{\left(\frac{i}{n} + 1\right)}^{n}}{i} - \frac{1}{i}\right)\right) \cdot 100\\ \end{array}\]

Reproduce

herbie shell --seed 2019163 
(FPCore (i n)
  :name "Compound Interest"

  :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))))