Average Error: 42.8 → 22.3
Time: 27.5s
Precision: 64
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le -2.022117346357720003868253810797028791058 \cdot 10^{-5}:\\ \;\;\;\;100 \cdot \frac{{\left({\left(1 + \frac{i}{n}\right)}^{n}\right)}^{3} - {1}^{3}}{\mathsf{fma}\left(1, 1 + {\left(1 + \frac{i}{n}\right)}^{n}, {\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)}\right) \cdot \frac{i}{n}}\\ \mathbf{elif}\;i \le 187.3391238131879674710944527760148048401:\\ \;\;\;\;100 \cdot \left(\frac{\mathsf{fma}\left(1, i, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{i} \cdot n\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\mathsf{fma}\left(1, i, \mathsf{fma}\left(\log 1, n, 1\right)\right) - 1}{\frac{i}{n}}\\ \end{array}\]
100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}
↓
\begin{array}{l}
\mathbf{if}\;i \le -2.022117346357720003868253810797028791058 \cdot 10^{-5}:\\
\;\;\;\;100 \cdot \frac{{\left({\left(1 + \frac{i}{n}\right)}^{n}\right)}^{3} - {1}^{3}}{\mathsf{fma}\left(1, 1 + {\left(1 + \frac{i}{n}\right)}^{n}, {\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)}\right) \cdot \frac{i}{n}}\\

\mathbf{elif}\;i \le 187.3391238131879674710944527760148048401:\\
\;\;\;\;100 \cdot \left(\frac{\mathsf{fma}\left(1, i, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{i} \cdot n\right)\\

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

\end{array}
double f(double i, double n) {
        double r135824 = 100.0;
        double r135825 = 1.0;
        double r135826 = i;
        double r135827 = n;
        double r135828 = r135826 / r135827;
        double r135829 = r135825 + r135828;
        double r135830 = pow(r135829, r135827);
        double r135831 = r135830 - r135825;
        double r135832 = r135831 / r135828;
        double r135833 = r135824 * r135832;
        return r135833;
}
↓
double f(double i, double n) {
        double r135834 = i;
        double r135835 = -2.02211734635772e-05;
        bool r135836 = r135834 <= r135835;
        double r135837 = 100.0;
        double r135838 = 1.0;
        double r135839 = n;
        double r135840 = r135834 / r135839;
        double r135841 = r135838 + r135840;
        double r135842 = pow(r135841, r135839);
        double r135843 = 3.0;
        double r135844 = pow(r135842, r135843);
        double r135845 = pow(r135838, r135843);
        double r135846 = r135844 - r135845;
        double r135847 = r135838 + r135842;
        double r135848 = 2.0;
        double r135849 = r135848 * r135839;
        double r135850 = pow(r135841, r135849);
        double r135851 = fma(r135838, r135847, r135850);
        double r135852 = r135851 * r135840;
        double r135853 = r135846 / r135852;
        double r135854 = r135837 * r135853;
        double r135855 = 187.33912381318797;
        bool r135856 = r135834 <= r135855;
        double r135857 = 0.5;
        double r135858 = pow(r135834, r135848);
        double r135859 = log(r135838);
        double r135860 = r135859 * r135839;
        double r135861 = fma(r135857, r135858, r135860);
        double r135862 = fma(r135838, r135834, r135861);
        double r135863 = r135858 * r135859;
        double r135864 = r135857 * r135863;
        double r135865 = r135862 - r135864;
        double r135866 = r135865 / r135834;
        double r135867 = r135866 * r135839;
        double r135868 = r135837 * r135867;
        double r135869 = 1.0;
        double r135870 = fma(r135859, r135839, r135869);
        double r135871 = fma(r135838, r135834, r135870);
        double r135872 = r135871 - r135838;
        double r135873 = r135872 / r135840;
        double r135874 = r135837 * r135873;
        double r135875 = r135856 ? r135868 : r135874;
        double r135876 = r135836 ? r135854 : r135875;
        return r135876;
}

Error

Bits error versus i

Bits error versus n

Target

Original42.8
Target42.9
Herbie22.3
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 3 regimes

  2. if i < -2.02211734635772e-05

    1. Initial program 27.2

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

    3. Applied flip3--27.2

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

    4. Applied associate-/l/27.2

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

    5. Simplified27.2

      \[\leadsto 100 \cdot \frac{{\left({\left(1 + \frac{i}{n}\right)}^{n}\right)}^{3} - {1}^{3}}{\color{blue}{\mathsf{fma}\left(1, 1 + {\left(1 + \frac{i}{n}\right)}^{n}, {\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)}\right) \cdot \frac{i}{n}}}\]

    if -2.02211734635772e-05 < i < 187.33912381318797

    1. Initial program 50.6

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

    2. Taylor expanded around 0 34.4

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

    3. Simplified34.4

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

    5. Applied associate-/r/17.0

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

    if 187.33912381318797 < i

    1. Initial program 32.4

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

    2. Taylor expanded around 0 39.8

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

    3. Simplified39.8

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

  4. Final simplification22.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -2.022117346357720003868253810797028791058 \cdot 10^{-5}:\\ \;\;\;\;100 \cdot \frac{{\left({\left(1 + \frac{i}{n}\right)}^{n}\right)}^{3} - {1}^{3}}{\mathsf{fma}\left(1, 1 + {\left(1 + \frac{i}{n}\right)}^{n}, {\left(1 + \frac{i}{n}\right)}^{\left(2 \cdot n\right)}\right) \cdot \frac{i}{n}}\\ \mathbf{elif}\;i \le 187.3391238131879674710944527760148048401:\\ \;\;\;\;100 \cdot \left(\frac{\mathsf{fma}\left(1, i, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{i} \cdot n\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\mathsf{fma}\left(1, i, \mathsf{fma}\left(\log 1, n, 1\right)\right) - 1}{\frac{i}{n}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019305 +o rules:numerics
(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))))