Average Error: 42.8 → 24.0
Time: 29.7s
Precision: 64
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;n \le -4.12718938658690470334225094125126747311 \cdot 10^{214}:\\ \;\;\;\;n \cdot \left(\frac{\mathsf{fma}\left(\log 1, n, \mathsf{fma}\left(i, 1, \mathsf{fma}\left(0.5, i \cdot i, \left(i \cdot i\right) \cdot \left(-0.5 \cdot \log 1\right)\right)\right)\right)}{i} \cdot 100\right)\\ \mathbf{elif}\;n \le -1.767552428992892694746701860906965261931 \cdot 10^{78}:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{i}\right)\\ \mathbf{elif}\;n \le -7.520459214308253406879712470442768842157 \cdot 10^{45}:\\ \;\;\;\;n \cdot \left(\frac{\mathsf{fma}\left(\log 1, n, \mathsf{fma}\left(i, 1, \mathsf{fma}\left(0.5, i \cdot i, \left(i \cdot i\right) \cdot \left(-0.5 \cdot \log 1\right)\right)\right)\right)}{i} \cdot 100\right)\\ \mathbf{elif}\;n \le -3.513107364644464039347190573730070991037 \cdot 10^{-256}:\\ \;\;\;\;\frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{\sqrt[3]{\frac{i}{n}}} \cdot \frac{100}{\sqrt[3]{\frac{i}{n}} \cdot \sqrt[3]{\frac{i}{n}}}\\ \mathbf{elif}\;n \le 4.674985157503464314300604414735053267466 \cdot 10^{-152}:\\ \;\;\;\;100 \cdot \frac{\mathsf{fma}\left(n, \log 1, \mathsf{fma}\left(i, 1, 1\right)\right) - 1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(\frac{\mathsf{fma}\left(\log 1, n, \mathsf{fma}\left(i, 1, \mathsf{fma}\left(0.5, i \cdot i, \left(i \cdot i\right) \cdot \left(-0.5 \cdot \log 1\right)\right)\right)\right)}{i} \cdot 100\right)\\ \end{array}\]
100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}
↓
\begin{array}{l}
\mathbf{if}\;n \le -4.12718938658690470334225094125126747311 \cdot 10^{214}:\\
\;\;\;\;n \cdot \left(\frac{\mathsf{fma}\left(\log 1, n, \mathsf{fma}\left(i, 1, \mathsf{fma}\left(0.5, i \cdot i, \left(i \cdot i\right) \cdot \left(-0.5 \cdot \log 1\right)\right)\right)\right)}{i} \cdot 100\right)\\

\mathbf{elif}\;n \le -1.767552428992892694746701860906965261931 \cdot 10^{78}:\\
\;\;\;\;100 \cdot \left(n \cdot \frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{i}\right)\\

\mathbf{elif}\;n \le -7.520459214308253406879712470442768842157 \cdot 10^{45}:\\
\;\;\;\;n \cdot \left(\frac{\mathsf{fma}\left(\log 1, n, \mathsf{fma}\left(i, 1, \mathsf{fma}\left(0.5, i \cdot i, \left(i \cdot i\right) \cdot \left(-0.5 \cdot \log 1\right)\right)\right)\right)}{i} \cdot 100\right)\\

\mathbf{elif}\;n \le -3.513107364644464039347190573730070991037 \cdot 10^{-256}:\\
\;\;\;\;\frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{\sqrt[3]{\frac{i}{n}}} \cdot \frac{100}{\sqrt[3]{\frac{i}{n}} \cdot \sqrt[3]{\frac{i}{n}}}\\

\mathbf{elif}\;n \le 4.674985157503464314300604414735053267466 \cdot 10^{-152}:\\
\;\;\;\;100 \cdot \frac{\mathsf{fma}\left(n, \log 1, \mathsf{fma}\left(i, 1, 1\right)\right) - 1}{\frac{i}{n}}\\

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

\end{array}
double f(double i, double n) {
        double r4580811 = 100.0;
        double r4580812 = 1.0;
        double r4580813 = i;
        double r4580814 = n;
        double r4580815 = r4580813 / r4580814;
        double r4580816 = r4580812 + r4580815;
        double r4580817 = pow(r4580816, r4580814);
        double r4580818 = r4580817 - r4580812;
        double r4580819 = r4580818 / r4580815;
        double r4580820 = r4580811 * r4580819;
        return r4580820;
}
↓
double f(double i, double n) {
        double r4580821 = n;
        double r4580822 = -4.1271893865869047e+214;
        bool r4580823 = r4580821 <= r4580822;
        double r4580824 = 1.0;
        double r4580825 = log(r4580824);
        double r4580826 = i;
        double r4580827 = 0.5;
        double r4580828 = r4580826 * r4580826;
        double r4580829 = r4580827 * r4580825;
        double r4580830 = -r4580829;
        double r4580831 = r4580828 * r4580830;
        double r4580832 = fma(r4580827, r4580828, r4580831);
        double r4580833 = fma(r4580826, r4580824, r4580832);
        double r4580834 = fma(r4580825, r4580821, r4580833);
        double r4580835 = r4580834 / r4580826;
        double r4580836 = 100.0;
        double r4580837 = r4580835 * r4580836;
        double r4580838 = r4580821 * r4580837;
        double r4580839 = -1.7675524289928927e+78;
        bool r4580840 = r4580821 <= r4580839;
        double r4580841 = r4580826 / r4580821;
        double r4580842 = r4580841 + r4580824;
        double r4580843 = pow(r4580842, r4580821);
        double r4580844 = r4580843 - r4580824;
        double r4580845 = r4580844 / r4580826;
        double r4580846 = r4580821 * r4580845;
        double r4580847 = r4580836 * r4580846;
        double r4580848 = -7.520459214308253e+45;
        bool r4580849 = r4580821 <= r4580848;
        double r4580850 = -3.513107364644464e-256;
        bool r4580851 = r4580821 <= r4580850;
        double r4580852 = cbrt(r4580841);
        double r4580853 = r4580844 / r4580852;
        double r4580854 = r4580852 * r4580852;
        double r4580855 = r4580836 / r4580854;
        double r4580856 = r4580853 * r4580855;
        double r4580857 = 4.674985157503464e-152;
        bool r4580858 = r4580821 <= r4580857;
        double r4580859 = 1.0;
        double r4580860 = fma(r4580826, r4580824, r4580859);
        double r4580861 = fma(r4580821, r4580825, r4580860);
        double r4580862 = r4580861 - r4580824;
        double r4580863 = r4580862 / r4580841;
        double r4580864 = r4580836 * r4580863;
        double r4580865 = r4580858 ? r4580864 : r4580838;
        double r4580866 = r4580851 ? r4580856 : r4580865;
        double r4580867 = r4580849 ? r4580838 : r4580866;
        double r4580868 = r4580840 ? r4580847 : r4580867;
        double r4580869 = r4580823 ? r4580838 : r4580868;
        return r4580869;
}

Error

Bits error versus i

Bits error versus n

Target

Original42.8
Target42.5
Herbie24.0
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 4 regimes

  2. if n < -4.1271893865869047e+214 or -1.7675524289928927e+78 < n < -7.520459214308253e+45 or 4.674985157503464e-152 < n

    1. Initial program 56.8

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

    3. Applied associate-/r/56.5

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

    4. Applied associate-*r*56.5

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

    5. Taylor expanded around 0 20.7

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

    6. Simplified20.7

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

    if -4.1271893865869047e+214 < n < -1.7675524289928927e+78

    1. Initial program 42.3

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

    3. Applied associate-/r/42.0

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

    if -7.520459214308253e+45 < n < -3.513107364644464e-256

    1. Initial program 20.6

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

    3. Applied add-cube-cbrt20.7

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

    4. Applied *-un-lft-identity20.7

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

    5. Applied times-frac20.7

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

    6. Applied associate-*r*20.7

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

    7. Simplified20.7

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

    if -3.513107364644464e-256 < n < 4.674985157503464e-152

    1. Initial program 33.8

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

    2. Taylor expanded around 0 23.8

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

    3. Simplified23.8

      \[\leadsto 100 \cdot \frac{\color{blue}{\mathsf{fma}\left(n, \log 1, \mathsf{fma}\left(i, 1, 1\right)\right)} - 1}{\frac{i}{n}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification24.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -4.12718938658690470334225094125126747311 \cdot 10^{214}:\\ \;\;\;\;n \cdot \left(\frac{\mathsf{fma}\left(\log 1, n, \mathsf{fma}\left(i, 1, \mathsf{fma}\left(0.5, i \cdot i, \left(i \cdot i\right) \cdot \left(-0.5 \cdot \log 1\right)\right)\right)\right)}{i} \cdot 100\right)\\ \mathbf{elif}\;n \le -1.767552428992892694746701860906965261931 \cdot 10^{78}:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{i}\right)\\ \mathbf{elif}\;n \le -7.520459214308253406879712470442768842157 \cdot 10^{45}:\\ \;\;\;\;n \cdot \left(\frac{\mathsf{fma}\left(\log 1, n, \mathsf{fma}\left(i, 1, \mathsf{fma}\left(0.5, i \cdot i, \left(i \cdot i\right) \cdot \left(-0.5 \cdot \log 1\right)\right)\right)\right)}{i} \cdot 100\right)\\ \mathbf{elif}\;n \le -3.513107364644464039347190573730070991037 \cdot 10^{-256}:\\ \;\;\;\;\frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{\sqrt[3]{\frac{i}{n}}} \cdot \frac{100}{\sqrt[3]{\frac{i}{n}} \cdot \sqrt[3]{\frac{i}{n}}}\\ \mathbf{elif}\;n \le 4.674985157503464314300604414735053267466 \cdot 10^{-152}:\\ \;\;\;\;100 \cdot \frac{\mathsf{fma}\left(n, \log 1, \mathsf{fma}\left(i, 1, 1\right)\right) - 1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(\frac{\mathsf{fma}\left(\log 1, n, \mathsf{fma}\left(i, 1, \mathsf{fma}\left(0.5, i \cdot i, \left(i \cdot i\right) \cdot \left(-0.5 \cdot \log 1\right)\right)\right)\right)}{i} \cdot 100\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019192 +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))))