Average Error: 26.9 → 16.6
Time: 25.3s
Precision: 64
\[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;d \le 1.47686370215769455 \cdot 10^{159}:\\ \;\;\;\;\left(\left(\left({\left(\frac{\sqrt[3]{d} \cdot \sqrt[3]{d}}{\sqrt[3]{h} \cdot \sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left({\left(\frac{1}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\sqrt[3]{\ell}}\right)}^{\left(\frac{1}{2}\right)}\right)\right) \cdot \left(\sqrt{1} + \sqrt{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right)\right) \cdot \left(\sqrt{1} - \sqrt{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left({\left({d}^{1}\right)}^{1} \cdot {\left(\frac{1}{{h}^{1} \cdot {\ell}^{1}}\right)}^{0.5}\right)\\ \end{array}\]
\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)
↓
\begin{array}{l}
\mathbf{if}\;d \le 1.47686370215769455 \cdot 10^{159}:\\
\;\;\;\;\left(\left(\left({\left(\frac{\sqrt[3]{d} \cdot \sqrt[3]{d}}{\sqrt[3]{h} \cdot \sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left({\left(\frac{1}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\sqrt[3]{\ell}}\right)}^{\left(\frac{1}{2}\right)}\right)\right) \cdot \left(\sqrt{1} + \sqrt{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right)\right) \cdot \left(\sqrt{1} - \sqrt{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right)\\

\mathbf{else}:\\
\;\;\;\;1 \cdot \left({\left({d}^{1}\right)}^{1} \cdot {\left(\frac{1}{{h}^{1} \cdot {\ell}^{1}}\right)}^{0.5}\right)\\

\end{array}
double f(double d, double h, double l, double M, double D) {
        double r275952 = d;
        double r275953 = h;
        double r275954 = r275952 / r275953;
        double r275955 = 1.0;
        double r275956 = 2.0;
        double r275957 = r275955 / r275956;
        double r275958 = pow(r275954, r275957);
        double r275959 = l;
        double r275960 = r275952 / r275959;
        double r275961 = pow(r275960, r275957);
        double r275962 = r275958 * r275961;
        double r275963 = M;
        double r275964 = D;
        double r275965 = r275963 * r275964;
        double r275966 = r275956 * r275952;
        double r275967 = r275965 / r275966;
        double r275968 = pow(r275967, r275956);
        double r275969 = r275957 * r275968;
        double r275970 = r275953 / r275959;
        double r275971 = r275969 * r275970;
        double r275972 = r275955 - r275971;
        double r275973 = r275962 * r275972;
        return r275973;
}
↓
double f(double d, double h, double l, double M, double D) {
        double r275974 = d;
        double r275975 = 1.4768637021576945e+159;
        bool r275976 = r275974 <= r275975;
        double r275977 = cbrt(r275974);
        double r275978 = r275977 * r275977;
        double r275979 = h;
        double r275980 = cbrt(r275979);
        double r275981 = r275980 * r275980;
        double r275982 = r275978 / r275981;
        double r275983 = 1.0;
        double r275984 = 2.0;
        double r275985 = r275983 / r275984;
        double r275986 = pow(r275982, r275985);
        double r275987 = r275977 / r275980;
        double r275988 = pow(r275987, r275985);
        double r275989 = r275986 * r275988;
        double r275990 = 1.0;
        double r275991 = l;
        double r275992 = cbrt(r275991);
        double r275993 = r275992 * r275992;
        double r275994 = r275990 / r275993;
        double r275995 = pow(r275994, r275985);
        double r275996 = r275974 / r275992;
        double r275997 = pow(r275996, r275985);
        double r275998 = r275995 * r275997;
        double r275999 = r275989 * r275998;
        double r276000 = sqrt(r275983);
        double r276001 = M;
        double r276002 = D;
        double r276003 = r276001 * r276002;
        double r276004 = r275984 * r275974;
        double r276005 = r276003 / r276004;
        double r276006 = pow(r276005, r275984);
        double r276007 = r275985 * r276006;
        double r276008 = r276007 * r275979;
        double r276009 = r276008 / r275991;
        double r276010 = sqrt(r276009);
        double r276011 = r276000 + r276010;
        double r276012 = r275999 * r276011;
        double r276013 = r276000 - r276010;
        double r276014 = r276012 * r276013;
        double r276015 = pow(r275974, r275983);
        double r276016 = pow(r276015, r275983);
        double r276017 = pow(r275979, r275983);
        double r276018 = pow(r275991, r275983);
        double r276019 = r276017 * r276018;
        double r276020 = r275990 / r276019;
        double r276021 = 0.5;
        double r276022 = pow(r276020, r276021);
        double r276023 = r276016 * r276022;
        double r276024 = r275983 * r276023;
        double r276025 = r275976 ? r276014 : r276024;
        return r276025;
}

Error

Bits error versus d

Bits error versus h

Bits error versus l

Bits error versus M

Bits error versus D

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if d < 1.4768637021576945e+159

    1. Initial program 26.7

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]
    2. Using strategy rm

    3. Applied add-cube-cbrt26.9

      \[\leadsto \left({\left(\frac{d}{\color{blue}{\left(\sqrt[3]{h} \cdot \sqrt[3]{h}\right) \cdot \sqrt[3]{h}}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]

    4. Applied add-cube-cbrt27.0

      \[\leadsto \left({\left(\frac{\color{blue}{\left(\sqrt[3]{d} \cdot \sqrt[3]{d}\right) \cdot \sqrt[3]{d}}}{\left(\sqrt[3]{h} \cdot \sqrt[3]{h}\right) \cdot \sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]

    5. Applied times-frac27.0

      \[\leadsto \left({\color{blue}{\left(\frac{\sqrt[3]{d} \cdot \sqrt[3]{d}}{\sqrt[3]{h} \cdot \sqrt[3]{h}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]

    6. Applied unpow-prod-down22.7

      \[\leadsto \left(\color{blue}{\left({\left(\frac{\sqrt[3]{d} \cdot \sqrt[3]{d}}{\sqrt[3]{h} \cdot \sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]
    7. Using strategy rm

    8. Applied add-cube-cbrt22.8

      \[\leadsto \left(\left({\left(\frac{\sqrt[3]{d} \cdot \sqrt[3]{d}}{\sqrt[3]{h} \cdot \sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot {\left(\frac{d}{\color{blue}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}}}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]

    9. Applied *-un-lft-identity22.8

      \[\leadsto \left(\left({\left(\frac{\sqrt[3]{d} \cdot \sqrt[3]{d}}{\sqrt[3]{h} \cdot \sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot {\left(\frac{\color{blue}{1 \cdot d}}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]

    10. Applied times-frac22.8

      \[\leadsto \left(\left({\left(\frac{\sqrt[3]{d} \cdot \sqrt[3]{d}}{\sqrt[3]{h} \cdot \sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot {\color{blue}{\left(\frac{1}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \frac{d}{\sqrt[3]{\ell}}\right)}}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]

    11. Applied unpow-prod-down19.1

      \[\leadsto \left(\left({\left(\frac{\sqrt[3]{d} \cdot \sqrt[3]{d}}{\sqrt[3]{h} \cdot \sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\left({\left(\frac{1}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\sqrt[3]{\ell}}\right)}^{\left(\frac{1}{2}\right)}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]
    12. Using strategy rm

    13. Applied associate-*r/16.7

      \[\leadsto \left(\left({\left(\frac{\sqrt[3]{d} \cdot \sqrt[3]{d}}{\sqrt[3]{h} \cdot \sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left({\left(\frac{1}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\sqrt[3]{\ell}}\right)}^{\left(\frac{1}{2}\right)}\right)\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right)\]
    14. Using strategy rm

    15. Applied add-sqr-sqrt16.7

      \[\leadsto \left(\left({\left(\frac{\sqrt[3]{d} \cdot \sqrt[3]{d}}{\sqrt[3]{h} \cdot \sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left({\left(\frac{1}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\sqrt[3]{\ell}}\right)}^{\left(\frac{1}{2}\right)}\right)\right) \cdot \left(1 - \color{blue}{\sqrt{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}} \cdot \sqrt{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}}\right)\]

    16. Applied add-sqr-sqrt16.7

      \[\leadsto \left(\left({\left(\frac{\sqrt[3]{d} \cdot \sqrt[3]{d}}{\sqrt[3]{h} \cdot \sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left({\left(\frac{1}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\sqrt[3]{\ell}}\right)}^{\left(\frac{1}{2}\right)}\right)\right) \cdot \left(\color{blue}{\sqrt{1} \cdot \sqrt{1}} - \sqrt{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}} \cdot \sqrt{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right)\]

    17. Applied difference-of-squares16.7

      \[\leadsto \left(\left({\left(\frac{\sqrt[3]{d} \cdot \sqrt[3]{d}}{\sqrt[3]{h} \cdot \sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left({\left(\frac{1}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\sqrt[3]{\ell}}\right)}^{\left(\frac{1}{2}\right)}\right)\right) \cdot \color{blue}{\left(\left(\sqrt{1} + \sqrt{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right) \cdot \left(\sqrt{1} - \sqrt{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right)\right)}\]

    18. Applied associate-*r*16.7

      \[\leadsto \color{blue}{\left(\left(\left({\left(\frac{\sqrt[3]{d} \cdot \sqrt[3]{d}}{\sqrt[3]{h} \cdot \sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left({\left(\frac{1}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\sqrt[3]{\ell}}\right)}^{\left(\frac{1}{2}\right)}\right)\right) \cdot \left(\sqrt{1} + \sqrt{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right)\right) \cdot \left(\sqrt{1} - \sqrt{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right)}\]

    if 1.4768637021576945e+159 < d

    1. Initial program 28.9

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]

    2. Taylor expanded around 0 16.1

      \[\leadsto \color{blue}{1 \cdot \left({\left({d}^{1}\right)}^{1} \cdot {\left(\frac{1}{{h}^{1} \cdot {\ell}^{1}}\right)}^{0.5}\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification16.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;d \le 1.47686370215769455 \cdot 10^{159}:\\ \;\;\;\;\left(\left(\left({\left(\frac{\sqrt[3]{d} \cdot \sqrt[3]{d}}{\sqrt[3]{h} \cdot \sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left({\left(\frac{1}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\sqrt[3]{\ell}}\right)}^{\left(\frac{1}{2}\right)}\right)\right) \cdot \left(\sqrt{1} + \sqrt{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right)\right) \cdot \left(\sqrt{1} - \sqrt{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left({\left({d}^{1}\right)}^{1} \cdot {\left(\frac{1}{{h}^{1} \cdot {\ell}^{1}}\right)}^{0.5}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020064 +o rules:numerics
(FPCore (d h l M D)
  :name "Henrywood and Agarwal, Equation (12)"
  :precision binary64
  (* (* (pow (/ d h) (/ 1 2)) (pow (/ d l) (/ 1 2))) (- 1 (* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) (/ h l)))))