Average Error: 42.3 → 16.2
Time: 27.2s
Precision: 64
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le -2.881224776443082 \cdot 10^{-05}:\\ \;\;\;\;\frac{\mathsf{fma}\left(100, e^{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n}, -100\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 0.06908503657888032:\\ \;\;\;\;\frac{\sqrt[3]{\mathsf{fma}\left(i, 100, \left(i \cdot i\right) \cdot \left(i \cdot \frac{50}{3} + 50\right)\right)}}{\frac{1}{n}} \cdot \left(\sqrt[3]{\frac{\sqrt[3]{\mathsf{fma}\left(i, 100, \left(i \cdot i\right) \cdot \left(i \cdot \frac{50}{3} + 50\right)\right)} \cdot \sqrt[3]{\mathsf{fma}\left(i, 100, \left(i \cdot i\right) \cdot \left(i \cdot \frac{50}{3} + 50\right)\right)}}{i}} \cdot \left(\sqrt[3]{\frac{\sqrt[3]{\mathsf{fma}\left(i, 100, \left(i \cdot i\right) \cdot \left(i \cdot \frac{50}{3} + 50\right)\right)} \cdot \sqrt[3]{\mathsf{fma}\left(i, 100, \left(i \cdot i\right) \cdot \left(i \cdot \frac{50}{3} + 50\right)\right)}}{i}} \cdot \sqrt[3]{\frac{\sqrt[3]{\mathsf{fma}\left(i, 100, \left(i \cdot i\right) \cdot \left(i \cdot \frac{50}{3} + 50\right)\right)} \cdot \sqrt[3]{\mathsf{fma}\left(i, 100, \left(i \cdot i\right) \cdot \left(i \cdot \frac{50}{3} + 50\right)\right)}}{i}}\right)\right)\\ \mathbf{elif}\;i \le 2.130741075012547 \cdot 10^{+204}:\\ \;\;\;\;\frac{1}{\frac{\frac{i}{n}}{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, -100\right)}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]
100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}
↓
\begin{array}{l}
\mathbf{if}\;i \le -2.881224776443082 \cdot 10^{-05}:\\
\;\;\;\;\frac{\mathsf{fma}\left(100, e^{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n}, -100\right)}{\frac{i}{n}}\\

\mathbf{elif}\;i \le 0.06908503657888032:\\
\;\;\;\;\frac{\sqrt[3]{\mathsf{fma}\left(i, 100, \left(i \cdot i\right) \cdot \left(i \cdot \frac{50}{3} + 50\right)\right)}}{\frac{1}{n}} \cdot \left(\sqrt[3]{\frac{\sqrt[3]{\mathsf{fma}\left(i, 100, \left(i \cdot i\right) \cdot \left(i \cdot \frac{50}{3} + 50\right)\right)} \cdot \sqrt[3]{\mathsf{fma}\left(i, 100, \left(i \cdot i\right) \cdot \left(i \cdot \frac{50}{3} + 50\right)\right)}}{i}} \cdot \left(\sqrt[3]{\frac{\sqrt[3]{\mathsf{fma}\left(i, 100, \left(i \cdot i\right) \cdot \left(i \cdot \frac{50}{3} + 50\right)\right)} \cdot \sqrt[3]{\mathsf{fma}\left(i, 100, \left(i \cdot i\right) \cdot \left(i \cdot \frac{50}{3} + 50\right)\right)}}{i}} \cdot \sqrt[3]{\frac{\sqrt[3]{\mathsf{fma}\left(i, 100, \left(i \cdot i\right) \cdot \left(i \cdot \frac{50}{3} + 50\right)\right)} \cdot \sqrt[3]{\mathsf{fma}\left(i, 100, \left(i \cdot i\right) \cdot \left(i \cdot \frac{50}{3} + 50\right)\right)}}{i}}\right)\right)\\

\mathbf{elif}\;i \le 2.130741075012547 \cdot 10^{+204}:\\
\;\;\;\;\frac{1}{\frac{\frac{i}{n}}{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, -100\right)}}\\

\mathbf{else}:\\
\;\;\;\;0\\

\end{array}
double f(double i, double n) {
        double r4989871 = 100.0;
        double r4989872 = 1.0;
        double r4989873 = i;
        double r4989874 = n;
        double r4989875 = r4989873 / r4989874;
        double r4989876 = r4989872 + r4989875;
        double r4989877 = pow(r4989876, r4989874);
        double r4989878 = r4989877 - r4989872;
        double r4989879 = r4989878 / r4989875;
        double r4989880 = r4989871 * r4989879;
        return r4989880;
}
↓
double f(double i, double n) {
        double r4989881 = i;
        double r4989882 = -2.881224776443082e-05;
        bool r4989883 = r4989881 <= r4989882;
        double r4989884 = 100.0;
        double r4989885 = n;
        double r4989886 = r4989881 / r4989885;
        double r4989887 = log1p(r4989886);
        double r4989888 = r4989887 * r4989885;
        double r4989889 = exp(r4989888);
        double r4989890 = -100.0;
        double r4989891 = fma(r4989884, r4989889, r4989890);
        double r4989892 = r4989891 / r4989886;
        double r4989893 = 0.06908503657888032;
        bool r4989894 = r4989881 <= r4989893;
        double r4989895 = r4989881 * r4989881;
        double r4989896 = 16.666666666666668;
        double r4989897 = r4989881 * r4989896;
        double r4989898 = 50.0;
        double r4989899 = r4989897 + r4989898;
        double r4989900 = r4989895 * r4989899;
        double r4989901 = fma(r4989881, r4989884, r4989900);
        double r4989902 = cbrt(r4989901);
        double r4989903 = 1.0;
        double r4989904 = r4989903 / r4989885;
        double r4989905 = r4989902 / r4989904;
        double r4989906 = r4989902 * r4989902;
        double r4989907 = r4989906 / r4989881;
        double r4989908 = cbrt(r4989907);
        double r4989909 = r4989908 * r4989908;
        double r4989910 = r4989908 * r4989909;
        double r4989911 = r4989905 * r4989910;
        double r4989912 = 2.130741075012547e+204;
        bool r4989913 = r4989881 <= r4989912;
        double r4989914 = r4989903 + r4989886;
        double r4989915 = pow(r4989914, r4989885);
        double r4989916 = fma(r4989884, r4989915, r4989890);
        double r4989917 = r4989886 / r4989916;
        double r4989918 = r4989903 / r4989917;
        double r4989919 = 0.0;
        double r4989920 = r4989913 ? r4989918 : r4989919;
        double r4989921 = r4989894 ? r4989911 : r4989920;
        double r4989922 = r4989883 ? r4989892 : r4989921;
        return r4989922;
}

Error

Bits error versus i

Bits error versus n

Target

Original42.3
Target42.4
Herbie16.2
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 4 regimes

  2. if i < -2.881224776443082e-05

    1. Initial program 27.2

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

    2. Simplified27.2

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

    4. Applied add-exp-log27.2

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

    5. Applied pow-exp27.2

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

    6. Simplified5.9

      \[\leadsto \frac{\mathsf{fma}\left(100, e^{\color{blue}{n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)}}, -100\right)}{\frac{i}{n}}\]

    if -2.881224776443082e-05 < i < 0.06908503657888032

    1. Initial program 49.8

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

    2. Simplified49.8

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

    3. Taylor expanded around 0 32.8

      \[\leadsto \frac{\color{blue}{100 \cdot i + \left(50 \cdot {i}^{2} + \frac{50}{3} \cdot {i}^{3}\right)}}{\frac{i}{n}}\]

    4. Simplified32.8

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(i, 100, \left(i \cdot i\right) \cdot \left(50 + \frac{50}{3} \cdot i\right)\right)}}{\frac{i}{n}}\]
    5. Using strategy rm

    6. Applied div-inv32.8

      \[\leadsto \frac{\mathsf{fma}\left(i, 100, \left(i \cdot i\right) \cdot \left(50 + \frac{50}{3} \cdot i\right)\right)}{\color{blue}{i \cdot \frac{1}{n}}}\]

    7. Applied add-cube-cbrt33.3

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\mathsf{fma}\left(i, 100, \left(i \cdot i\right) \cdot \left(50 + \frac{50}{3} \cdot i\right)\right)} \cdot \sqrt[3]{\mathsf{fma}\left(i, 100, \left(i \cdot i\right) \cdot \left(50 + \frac{50}{3} \cdot i\right)\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(i, 100, \left(i \cdot i\right) \cdot \left(50 + \frac{50}{3} \cdot i\right)\right)}}}{i \cdot \frac{1}{n}}\]

    8. Applied times-frac17.0

      \[\leadsto \color{blue}{\frac{\sqrt[3]{\mathsf{fma}\left(i, 100, \left(i \cdot i\right) \cdot \left(50 + \frac{50}{3} \cdot i\right)\right)} \cdot \sqrt[3]{\mathsf{fma}\left(i, 100, \left(i \cdot i\right) \cdot \left(50 + \frac{50}{3} \cdot i\right)\right)}}{i} \cdot \frac{\sqrt[3]{\mathsf{fma}\left(i, 100, \left(i \cdot i\right) \cdot \left(50 + \frac{50}{3} \cdot i\right)\right)}}{\frac{1}{n}}}\]
    9. Using strategy rm

    10. Applied add-cube-cbrt17.1

      \[\leadsto \color{blue}{\left(\left(\sqrt[3]{\frac{\sqrt[3]{\mathsf{fma}\left(i, 100, \left(i \cdot i\right) \cdot \left(50 + \frac{50}{3} \cdot i\right)\right)} \cdot \sqrt[3]{\mathsf{fma}\left(i, 100, \left(i \cdot i\right) \cdot \left(50 + \frac{50}{3} \cdot i\right)\right)}}{i}} \cdot \sqrt[3]{\frac{\sqrt[3]{\mathsf{fma}\left(i, 100, \left(i \cdot i\right) \cdot \left(50 + \frac{50}{3} \cdot i\right)\right)} \cdot \sqrt[3]{\mathsf{fma}\left(i, 100, \left(i \cdot i\right) \cdot \left(50 + \frac{50}{3} \cdot i\right)\right)}}{i}}\right) \cdot \sqrt[3]{\frac{\sqrt[3]{\mathsf{fma}\left(i, 100, \left(i \cdot i\right) \cdot \left(50 + \frac{50}{3} \cdot i\right)\right)} \cdot \sqrt[3]{\mathsf{fma}\left(i, 100, \left(i \cdot i\right) \cdot \left(50 + \frac{50}{3} \cdot i\right)\right)}}{i}}\right)} \cdot \frac{\sqrt[3]{\mathsf{fma}\left(i, 100, \left(i \cdot i\right) \cdot \left(50 + \frac{50}{3} \cdot i\right)\right)}}{\frac{1}{n}}\]

    if 0.06908503657888032 < i < 2.130741075012547e+204

    1. Initial program 29.7

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

    2. Simplified29.7

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

    4. Applied clear-num29.7

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

    if 2.130741075012547e+204 < i

    1. Initial program 30.9

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

    2. Simplified30.9

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

    3. Taylor expanded around 0 30.4

      \[\leadsto \color{blue}{0}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification16.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -2.881224776443082 \cdot 10^{-05}:\\ \;\;\;\;\frac{\mathsf{fma}\left(100, e^{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n}, -100\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 0.06908503657888032:\\ \;\;\;\;\frac{\sqrt[3]{\mathsf{fma}\left(i, 100, \left(i \cdot i\right) \cdot \left(i \cdot \frac{50}{3} + 50\right)\right)}}{\frac{1}{n}} \cdot \left(\sqrt[3]{\frac{\sqrt[3]{\mathsf{fma}\left(i, 100, \left(i \cdot i\right) \cdot \left(i \cdot \frac{50}{3} + 50\right)\right)} \cdot \sqrt[3]{\mathsf{fma}\left(i, 100, \left(i \cdot i\right) \cdot \left(i \cdot \frac{50}{3} + 50\right)\right)}}{i}} \cdot \left(\sqrt[3]{\frac{\sqrt[3]{\mathsf{fma}\left(i, 100, \left(i \cdot i\right) \cdot \left(i \cdot \frac{50}{3} + 50\right)\right)} \cdot \sqrt[3]{\mathsf{fma}\left(i, 100, \left(i \cdot i\right) \cdot \left(i \cdot \frac{50}{3} + 50\right)\right)}}{i}} \cdot \sqrt[3]{\frac{\sqrt[3]{\mathsf{fma}\left(i, 100, \left(i \cdot i\right) \cdot \left(i \cdot \frac{50}{3} + 50\right)\right)} \cdot \sqrt[3]{\mathsf{fma}\left(i, 100, \left(i \cdot i\right) \cdot \left(i \cdot \frac{50}{3} + 50\right)\right)}}{i}}\right)\right)\\ \mathbf{elif}\;i \le 2.130741075012547 \cdot 10^{+204}:\\ \;\;\;\;\frac{1}{\frac{\frac{i}{n}}{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, -100\right)}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Reproduce

herbie shell --seed 2019158 +o rules:numerics
(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))))