Average Error: 43.0 → 20.7
Time: 31.2s
Precision: 64
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le -1.74625455624762593309640124061843380332:\\ \;\;\;\;100 \cdot \frac{\mathsf{fma}\left(\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n}} \cdot \sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n}}, \sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n}}, -1\right) + \left(\left(-1\right) + 1\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 0.5476118163719528864064045592385809868574:\\ \;\;\;\;\frac{100}{\frac{i}{\left(\mathsf{fma}\left(1, i, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right) \cdot n}}\\ \mathbf{else}:\\ \;\;\;\;\left(100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}\right) \cdot n\\ \end{array}\]
100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}
↓
\begin{array}{l}
\mathbf{if}\;i \le -1.74625455624762593309640124061843380332:\\
\;\;\;\;100 \cdot \frac{\mathsf{fma}\left(\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n}} \cdot \sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n}}, \sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n}}, -1\right) + \left(\left(-1\right) + 1\right)}{\frac{i}{n}}\\

\mathbf{elif}\;i \le 0.5476118163719528864064045592385809868574:\\
\;\;\;\;\frac{100}{\frac{i}{\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) \cdot n}}\\

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

\end{array}
double f(double i, double n) {
        double r99863 = 100.0;
        double r99864 = 1.0;
        double r99865 = i;
        double r99866 = n;
        double r99867 = r99865 / r99866;
        double r99868 = r99864 + r99867;
        double r99869 = pow(r99868, r99866);
        double r99870 = r99869 - r99864;
        double r99871 = r99870 / r99867;
        double r99872 = r99863 * r99871;
        return r99872;
}
↓
double f(double i, double n) {
        double r99873 = i;
        double r99874 = -1.746254556247626;
        bool r99875 = r99873 <= r99874;
        double r99876 = 100.0;
        double r99877 = 1.0;
        double r99878 = n;
        double r99879 = r99873 / r99878;
        double r99880 = r99877 + r99879;
        double r99881 = pow(r99880, r99878);
        double r99882 = cbrt(r99881);
        double r99883 = r99882 * r99882;
        double r99884 = -r99877;
        double r99885 = fma(r99883, r99882, r99884);
        double r99886 = r99884 + r99877;
        double r99887 = r99885 + r99886;
        double r99888 = r99887 / r99879;
        double r99889 = r99876 * r99888;
        double r99890 = 0.5476118163719529;
        bool r99891 = r99873 <= r99890;
        double r99892 = 0.5;
        double r99893 = 2.0;
        double r99894 = pow(r99873, r99893);
        double r99895 = log(r99877);
        double r99896 = r99895 * r99878;
        double r99897 = fma(r99892, r99894, r99896);
        double r99898 = fma(r99877, r99873, r99897);
        double r99899 = r99894 * r99895;
        double r99900 = r99892 * r99899;
        double r99901 = r99898 - r99900;
        double r99902 = r99901 * r99878;
        double r99903 = r99873 / r99902;
        double r99904 = r99876 / r99903;
        double r99905 = r99881 - r99877;
        double r99906 = r99905 / r99873;
        double r99907 = r99876 * r99906;
        double r99908 = r99907 * r99878;
        double r99909 = r99891 ? r99904 : r99908;
        double r99910 = r99875 ? r99889 : r99909;
        return r99910;
}

Error

Bits error versus i

Bits error versus n

Target

Original43.0
Target43.5
Herbie20.7
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 3 regimes

  2. if i < -1.746254556247626

    1. Initial program 27.3

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

    3. Applied add-cube-cbrt27.3

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

    4. Applied add-cube-cbrt27.4

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

    5. Applied prod-diff27.4

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

    6. Simplified27.4

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

    7. Simplified27.4

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

    if -1.746254556247626 < i < 0.5476118163719529

    1. Initial program 50.8

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

    2. Taylor expanded around 0 34.2

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

    3. Simplified34.2

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

    5. Applied div-inv34.3

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

    6. Applied *-un-lft-identity34.3

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

    7. Applied times-frac15.9

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

    8. Simplified15.8

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

    10. Applied associate-*l/15.7

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

    11. Applied associate-*r/15.8

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

    12. Simplified15.8

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

    14. Applied associate-/l*15.8

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

    if 0.5476118163719529 < i

    1. Initial program 32.4

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

    3. Applied associate-/r/32.4

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

    4. Applied associate-*r*32.4

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

  4. Final simplification20.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -1.74625455624762593309640124061843380332:\\ \;\;\;\;100 \cdot \frac{\mathsf{fma}\left(\sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n}} \cdot \sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n}}, \sqrt[3]{{\left(1 + \frac{i}{n}\right)}^{n}}, -1\right) + \left(\left(-1\right) + 1\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \le 0.5476118163719528864064045592385809868574:\\ \;\;\;\;\frac{100}{\frac{i}{\left(\mathsf{fma}\left(1, i, \mathsf{fma}\left(0.5, {i}^{2}, \log 1 \cdot n\right)\right) - 0.5 \cdot \left({i}^{2} \cdot \log 1\right)\right) \cdot n}}\\ \mathbf{else}:\\ \;\;\;\;\left(100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}\right) \cdot n\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 +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))))