Average Error: 43.1 → 21.2
Time: 25.7s
Precision: 64
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le -0.1040807419505014597138625731531647033989:\\ \;\;\;\;100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\\ \mathbf{elif}\;i \le -1.235437068794871502671548956380542009081 \cdot 10^{-255}:\\ \;\;\;\;\frac{100}{i} \cdot \frac{\mathsf{fma}\left(1, i, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{\frac{1}{n}}\\ \mathbf{elif}\;i \le 3.912989910623956103641462485659015187104 \cdot 10^{-198}:\\ \;\;\;\;100 \cdot \left(\frac{\mathsf{fma}\left(1, i, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{i} \cdot n\right)\\ \mathbf{elif}\;i \le 18.6629372047199559858654538402333855629:\\ \;\;\;\;\frac{100}{i} \cdot \frac{\mathsf{fma}\left(1, i, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{\frac{1}{n}}\\ \mathbf{elif}\;i \le 5.064717226083425690234124091873168768786 \cdot 10^{144}:\\ \;\;\;\;\left(100 \cdot \frac{\sqrt{{\left(1 + \frac{i}{n}\right)}^{n}} + \sqrt{1}}{i}\right) \cdot \frac{\sqrt{{\left(1 + \frac{i}{n}\right)}^{n}} - \sqrt{1}}{\frac{1}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\mathsf{fma}\left(1, i, \mathsf{fma}\left(\log 1, n, 1\right)\right) - 1}{\frac{i}{n}}\\ \end{array}\]
100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}
↓
\begin{array}{l}
\mathbf{if}\;i \le -0.1040807419505014597138625731531647033989:\\
\;\;\;\;100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\\

\mathbf{elif}\;i \le -1.235437068794871502671548956380542009081 \cdot 10^{-255}:\\
\;\;\;\;\frac{100}{i} \cdot \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)}{\frac{1}{n}}\\

\mathbf{elif}\;i \le 3.912989910623956103641462485659015187104 \cdot 10^{-198}:\\
\;\;\;\;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{elif}\;i \le 18.6629372047199559858654538402333855629:\\
\;\;\;\;\frac{100}{i} \cdot \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)}{\frac{1}{n}}\\

\mathbf{elif}\;i \le 5.064717226083425690234124091873168768786 \cdot 10^{144}:\\
\;\;\;\;\left(100 \cdot \frac{\sqrt{{\left(1 + \frac{i}{n}\right)}^{n}} + \sqrt{1}}{i}\right) \cdot \frac{\sqrt{{\left(1 + \frac{i}{n}\right)}^{n}} - \sqrt{1}}{\frac{1}{n}}\\

\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 r94817 = 100.0;
        double r94818 = 1.0;
        double r94819 = i;
        double r94820 = n;
        double r94821 = r94819 / r94820;
        double r94822 = r94818 + r94821;
        double r94823 = pow(r94822, r94820);
        double r94824 = r94823 - r94818;
        double r94825 = r94824 / r94821;
        double r94826 = r94817 * r94825;
        return r94826;
}
↓
double f(double i, double n) {
        double r94827 = i;
        double r94828 = -0.10408074195050146;
        bool r94829 = r94827 <= r94828;
        double r94830 = 100.0;
        double r94831 = 1.0;
        double r94832 = n;
        double r94833 = r94827 / r94832;
        double r94834 = r94831 + r94833;
        double r94835 = pow(r94834, r94832);
        double r94836 = r94835 - r94831;
        double r94837 = r94836 / r94833;
        double r94838 = r94830 * r94837;
        double r94839 = -1.2354370687948715e-255;
        bool r94840 = r94827 <= r94839;
        double r94841 = r94830 / r94827;
        double r94842 = 0.5;
        double r94843 = 2.0;
        double r94844 = pow(r94827, r94843);
        double r94845 = log(r94831);
        double r94846 = r94845 * r94832;
        double r94847 = fma(r94842, r94844, r94846);
        double r94848 = fma(r94831, r94827, r94847);
        double r94849 = r94844 * r94845;
        double r94850 = r94842 * r94849;
        double r94851 = r94848 - r94850;
        double r94852 = 1.0;
        double r94853 = r94852 / r94832;
        double r94854 = r94851 / r94853;
        double r94855 = r94841 * r94854;
        double r94856 = 3.912989910623956e-198;
        bool r94857 = r94827 <= r94856;
        double r94858 = r94851 / r94827;
        double r94859 = r94858 * r94832;
        double r94860 = r94830 * r94859;
        double r94861 = 18.662937204719956;
        bool r94862 = r94827 <= r94861;
        double r94863 = 5.064717226083426e+144;
        bool r94864 = r94827 <= r94863;
        double r94865 = sqrt(r94835);
        double r94866 = sqrt(r94831);
        double r94867 = r94865 + r94866;
        double r94868 = r94867 / r94827;
        double r94869 = r94830 * r94868;
        double r94870 = r94865 - r94866;
        double r94871 = r94870 / r94853;
        double r94872 = r94869 * r94871;
        double r94873 = fma(r94845, r94832, r94852);
        double r94874 = fma(r94831, r94827, r94873);
        double r94875 = r94874 - r94831;
        double r94876 = r94875 / r94833;
        double r94877 = r94830 * r94876;
        double r94878 = r94864 ? r94872 : r94877;
        double r94879 = r94862 ? r94855 : r94878;
        double r94880 = r94857 ? r94860 : r94879;
        double r94881 = r94840 ? r94855 : r94880;
        double r94882 = r94829 ? r94838 : r94881;
        return r94882;
}

Error

Bits error versus i

Bits error versus n

Target

Original43.1
Target42.7
Herbie21.2
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 5 regimes

  2. if i < -0.10408074195050146

    1. Initial program 29.8

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

    if -0.10408074195050146 < i < -1.2354370687948715e-255 or 3.912989910623956e-198 < i < 18.662937204719956

    1. Initial program 51.0

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

    2. Taylor expanded around 0 30.6

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

    3. Simplified30.6

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

    5. Applied div-inv30.7

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

    6. Applied *-un-lft-identity30.7

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

    7. Applied times-frac15.6

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

    8. Applied associate-*r*15.7

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

    9. Simplified15.7

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

    if -1.2354370687948715e-255 < i < 3.912989910623956e-198

    1. Initial program 49.3

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

    2. Taylor expanded around 0 40.8

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

    3. Simplified40.8

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

    5. Applied associate-/r/14.4

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

    if 18.662937204719956 < i < 5.064717226083426e+144

    1. Initial program 30.0

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

    3. Applied div-inv30.0

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

    4. Applied add-sqr-sqrt30.0

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

    5. Applied add-sqr-sqrt30.0

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

    6. Applied difference-of-squares30.0

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

    7. Applied times-frac30.1

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

    8. Applied associate-*r*30.1

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

    if 5.064717226083426e+144 < i

    1. Initial program 30.1

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

    2. Taylor expanded around 0 36.9

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

    3. Simplified36.9

      \[\leadsto 100 \cdot \frac{\color{blue}{\mathsf{fma}\left(1, i, \mathsf{fma}\left(\log 1, n, 1\right)\right)} - 1}{\frac{i}{n}}\]
  3. Recombined 5 regimes into one program.

  4. Final simplification21.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -0.1040807419505014597138625731531647033989:\\ \;\;\;\;100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\\ \mathbf{elif}\;i \le -1.235437068794871502671548956380542009081 \cdot 10^{-255}:\\ \;\;\;\;\frac{100}{i} \cdot \frac{\mathsf{fma}\left(1, i, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{\frac{1}{n}}\\ \mathbf{elif}\;i \le 3.912989910623956103641462485659015187104 \cdot 10^{-198}:\\ \;\;\;\;100 \cdot \left(\frac{\mathsf{fma}\left(1, i, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{i} \cdot n\right)\\ \mathbf{elif}\;i \le 18.6629372047199559858654538402333855629:\\ \;\;\;\;\frac{100}{i} \cdot \frac{\mathsf{fma}\left(1, i, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)}{\frac{1}{n}}\\ \mathbf{elif}\;i \le 5.064717226083425690234124091873168768786 \cdot 10^{144}:\\ \;\;\;\;\left(100 \cdot \frac{\sqrt{{\left(1 + \frac{i}{n}\right)}^{n}} + \sqrt{1}}{i}\right) \cdot \frac{\sqrt{{\left(1 + \frac{i}{n}\right)}^{n}} - \sqrt{1}}{\frac{1}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\mathsf{fma}\left(1, i, \mathsf{fma}\left(\log 1, n, 1\right)\right) - 1}{\frac{i}{n}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019235 +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))))