Average Error: 43.0 → 24.3
Time: 28.7s
Precision: 64
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;n \le -5.464773096034009611019906036964894498925 \cdot 10^{59}:\\ \;\;\;\;n \cdot \left(\frac{\mathsf{fma}\left(n, \log 1, \mathsf{fma}\left(i \cdot i, 0.5, i \cdot 1\right)\right) - \left(0.5 \cdot \log 1\right) \cdot \left(i \cdot i\right)}{i} \cdot 100\right)\\ \mathbf{elif}\;n \le -499596521052505172268102593940553728:\\ \;\;\;\;n \cdot \left(\frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{i} \cdot 100\right)\\ \mathbf{elif}\;n \le -15240705750258725123260416:\\ \;\;\;\;n \cdot \left(\frac{\mathsf{fma}\left(n, \log 1, \mathsf{fma}\left(i \cdot i, 0.5, i \cdot 1\right)\right) - \left(0.5 \cdot \log 1\right) \cdot \left(i \cdot i\right)}{i} \cdot 100\right)\\ \mathbf{elif}\;n \le 8.439481621089909706689165353901153669925 \cdot 10^{-297}:\\ \;\;\;\;100 \cdot \left(\frac{{\left(\frac{i}{n} + 1\right)}^{n}}{\frac{i}{n}} - \frac{1}{\frac{i}{n}}\right)\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(\frac{\mathsf{fma}\left(n, \log 1, \mathsf{fma}\left(i \cdot i, 0.5, i \cdot 1\right)\right) - \left(0.5 \cdot \log 1\right) \cdot \left(i \cdot i\right)}{i} \cdot 100\right)\\ \end{array}\]
100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}
↓
\begin{array}{l}
\mathbf{if}\;n \le -5.464773096034009611019906036964894498925 \cdot 10^{59}:\\
\;\;\;\;n \cdot \left(\frac{\mathsf{fma}\left(n, \log 1, \mathsf{fma}\left(i \cdot i, 0.5, i \cdot 1\right)\right) - \left(0.5 \cdot \log 1\right) \cdot \left(i \cdot i\right)}{i} \cdot 100\right)\\

\mathbf{elif}\;n \le -499596521052505172268102593940553728:\\
\;\;\;\;n \cdot \left(\frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{i} \cdot 100\right)\\

\mathbf{elif}\;n \le -15240705750258725123260416:\\
\;\;\;\;n \cdot \left(\frac{\mathsf{fma}\left(n, \log 1, \mathsf{fma}\left(i \cdot i, 0.5, i \cdot 1\right)\right) - \left(0.5 \cdot \log 1\right) \cdot \left(i \cdot i\right)}{i} \cdot 100\right)\\

\mathbf{elif}\;n \le 8.439481621089909706689165353901153669925 \cdot 10^{-297}:\\
\;\;\;\;100 \cdot \left(\frac{{\left(\frac{i}{n} + 1\right)}^{n}}{\frac{i}{n}} - \frac{1}{\frac{i}{n}}\right)\\

\mathbf{else}:\\
\;\;\;\;n \cdot \left(\frac{\mathsf{fma}\left(n, \log 1, \mathsf{fma}\left(i \cdot i, 0.5, i \cdot 1\right)\right) - \left(0.5 \cdot \log 1\right) \cdot \left(i \cdot i\right)}{i} \cdot 100\right)\\

\end{array}
double f(double i, double n) {
        double r5029833 = 100.0;
        double r5029834 = 1.0;
        double r5029835 = i;
        double r5029836 = n;
        double r5029837 = r5029835 / r5029836;
        double r5029838 = r5029834 + r5029837;
        double r5029839 = pow(r5029838, r5029836);
        double r5029840 = r5029839 - r5029834;
        double r5029841 = r5029840 / r5029837;
        double r5029842 = r5029833 * r5029841;
        return r5029842;
}
↓
double f(double i, double n) {
        double r5029843 = n;
        double r5029844 = -5.46477309603401e+59;
        bool r5029845 = r5029843 <= r5029844;
        double r5029846 = 1.0;
        double r5029847 = log(r5029846);
        double r5029848 = i;
        double r5029849 = r5029848 * r5029848;
        double r5029850 = 0.5;
        double r5029851 = r5029848 * r5029846;
        double r5029852 = fma(r5029849, r5029850, r5029851);
        double r5029853 = fma(r5029843, r5029847, r5029852);
        double r5029854 = r5029850 * r5029847;
        double r5029855 = r5029854 * r5029849;
        double r5029856 = r5029853 - r5029855;
        double r5029857 = r5029856 / r5029848;
        double r5029858 = 100.0;
        double r5029859 = r5029857 * r5029858;
        double r5029860 = r5029843 * r5029859;
        double r5029861 = -4.995965210525052e+35;
        bool r5029862 = r5029843 <= r5029861;
        double r5029863 = r5029848 / r5029843;
        double r5029864 = r5029863 + r5029846;
        double r5029865 = pow(r5029864, r5029843);
        double r5029866 = r5029865 - r5029846;
        double r5029867 = r5029866 / r5029848;
        double r5029868 = r5029867 * r5029858;
        double r5029869 = r5029843 * r5029868;
        double r5029870 = -1.5240705750258725e+25;
        bool r5029871 = r5029843 <= r5029870;
        double r5029872 = 8.43948162108991e-297;
        bool r5029873 = r5029843 <= r5029872;
        double r5029874 = r5029865 / r5029863;
        double r5029875 = r5029846 / r5029863;
        double r5029876 = r5029874 - r5029875;
        double r5029877 = r5029858 * r5029876;
        double r5029878 = r5029873 ? r5029877 : r5029860;
        double r5029879 = r5029871 ? r5029860 : r5029878;
        double r5029880 = r5029862 ? r5029869 : r5029879;
        double r5029881 = r5029845 ? r5029860 : r5029880;
        return r5029881;
}

Error

Bits error versus i

Bits error versus n

Target

Original43.0
Target42.8
Herbie24.3
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 3 regimes

  2. if n < -5.46477309603401e+59 or -4.995965210525052e+35 < n < -1.5240705750258725e+25 or 8.43948162108991e-297 < n

    1. Initial program 52.3

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

    2. Taylor expanded around 0 39.8

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

    3. Simplified39.8

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

    5. Applied associate-/r/25.8

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

    6. Applied associate-*r*25.8

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

    if -5.46477309603401e+59 < n < -4.995965210525052e+35

    1. Initial program 34.9

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

    3. Applied associate-/r/34.8

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

    4. Applied associate-*r*34.8

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

    if -1.5240705750258725e+25 < n < 8.43948162108991e-297

    1. Initial program 18.9

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

    3. Applied div-sub18.9

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

  4. Final simplification24.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -5.464773096034009611019906036964894498925 \cdot 10^{59}:\\ \;\;\;\;n \cdot \left(\frac{\mathsf{fma}\left(n, \log 1, \mathsf{fma}\left(i \cdot i, 0.5, i \cdot 1\right)\right) - \left(0.5 \cdot \log 1\right) \cdot \left(i \cdot i\right)}{i} \cdot 100\right)\\ \mathbf{elif}\;n \le -499596521052505172268102593940553728:\\ \;\;\;\;n \cdot \left(\frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{i} \cdot 100\right)\\ \mathbf{elif}\;n \le -15240705750258725123260416:\\ \;\;\;\;n \cdot \left(\frac{\mathsf{fma}\left(n, \log 1, \mathsf{fma}\left(i \cdot i, 0.5, i \cdot 1\right)\right) - \left(0.5 \cdot \log 1\right) \cdot \left(i \cdot i\right)}{i} \cdot 100\right)\\ \mathbf{elif}\;n \le 8.439481621089909706689165353901153669925 \cdot 10^{-297}:\\ \;\;\;\;100 \cdot \left(\frac{{\left(\frac{i}{n} + 1\right)}^{n}}{\frac{i}{n}} - \frac{1}{\frac{i}{n}}\right)\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(\frac{\mathsf{fma}\left(n, \log 1, \mathsf{fma}\left(i \cdot i, 0.5, i \cdot 1\right)\right) - \left(0.5 \cdot \log 1\right) \cdot \left(i \cdot i\right)}{i} \cdot 100\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(FPCore (i n)
  :name "Compound Interest"

  :herbie-target
  (* 100.0 (/ (- (exp (* n (if (== (+ 1.0 (/ i n)) 1.0) (/ i n) (/ (* (/ i n) (log (+ 1.0 (/ i n)))) (- (+ (/ i n) 1.0) 1.0))))) 1.0) (/ i n)))

  (* 100.0 (/ (- (pow (+ 1.0 (/ i n)) n) 1.0) (/ i n))))