Average Error: 14.1 → 8.6
Time: 1.2m
Precision: 64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;M \cdot D \le -4.769232234431523337483911707392693863577 \cdot 10^{219}:\\ \;\;\;\;\sqrt{1} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(\frac{1}{\ell} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot h\right)\right)} \cdot w0\\ \end{array}\]
w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}
↓
\begin{array}{l}
\mathbf{if}\;M \cdot D \le -4.769232234431523337483911707392693863577 \cdot 10^{219}:\\
\;\;\;\;\sqrt{1} \cdot w0\\

\mathbf{else}:\\
\;\;\;\;\sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(\frac{1}{\ell} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot h\right)\right)} \cdot w0\\

\end{array}
double f(double w0, double M, double D, double h, double l, double d) {
        double r7166689 = w0;
        double r7166690 = 1.0;
        double r7166691 = M;
        double r7166692 = D;
        double r7166693 = r7166691 * r7166692;
        double r7166694 = 2.0;
        double r7166695 = d;
        double r7166696 = r7166694 * r7166695;
        double r7166697 = r7166693 / r7166696;
        double r7166698 = pow(r7166697, r7166694);
        double r7166699 = h;
        double r7166700 = l;
        double r7166701 = r7166699 / r7166700;
        double r7166702 = r7166698 * r7166701;
        double r7166703 = r7166690 - r7166702;
        double r7166704 = sqrt(r7166703);
        double r7166705 = r7166689 * r7166704;
        return r7166705;
}
↓
double f(double w0, double M, double D, double h, double l, double d) {
        double r7166706 = M;
        double r7166707 = D;
        double r7166708 = r7166706 * r7166707;
        double r7166709 = -4.7692322344315233e+219;
        bool r7166710 = r7166708 <= r7166709;
        double r7166711 = 1.0;
        double r7166712 = sqrt(r7166711);
        double r7166713 = w0;
        double r7166714 = r7166712 * r7166713;
        double r7166715 = 2.0;
        double r7166716 = d;
        double r7166717 = r7166715 * r7166716;
        double r7166718 = r7166708 / r7166717;
        double r7166719 = 2.0;
        double r7166720 = r7166715 / r7166719;
        double r7166721 = pow(r7166718, r7166720);
        double r7166722 = 1.0;
        double r7166723 = l;
        double r7166724 = r7166722 / r7166723;
        double r7166725 = h;
        double r7166726 = r7166721 * r7166725;
        double r7166727 = r7166724 * r7166726;
        double r7166728 = r7166721 * r7166727;
        double r7166729 = r7166711 - r7166728;
        double r7166730 = sqrt(r7166729);
        double r7166731 = r7166730 * r7166713;
        double r7166732 = r7166710 ? r7166714 : r7166731;
        return r7166732;
}

Error

Bits error versus w0

Bits error versus M

Bits error versus D

Bits error versus h

Bits error versus l

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 (* M D) < -4.7692322344315233e+219

    1. Initial program 48.3

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
    2. Using strategy rm

    3. Applied div-inv48.3

      \[\leadsto w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \color{blue}{\left(h \cdot \frac{1}{\ell}\right)}}\]

    4. Applied associate-*r*48.3

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h\right) \cdot \frac{1}{\ell}}}\]

    5. Taylor expanded around 0 44.9

      \[\leadsto w0 \cdot \color{blue}{\sqrt{1}}\]

    if -4.7692322344315233e+219 < (* M D)

    1. Initial program 12.5

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
    2. Using strategy rm

    3. Applied div-inv12.5

      \[\leadsto w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \color{blue}{\left(h \cdot \frac{1}{\ell}\right)}}\]

    4. Applied associate-*r*8.9

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h\right) \cdot \frac{1}{\ell}}}\]
    5. Using strategy rm

    6. Applied sqr-pow8.9

      \[\leadsto w0 \cdot \sqrt{1 - \left(\color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}\right)} \cdot h\right) \cdot \frac{1}{\ell}}\]

    7. Applied associate-*l*7.5

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot h\right)\right)} \cdot \frac{1}{\ell}}\]
    8. Using strategy rm

    9. Applied associate-*l*6.9

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot h\right) \cdot \frac{1}{\ell}\right)}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification8.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;M \cdot D \le -4.769232234431523337483911707392693863577 \cdot 10^{219}:\\ \;\;\;\;\sqrt{1} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(\frac{1}{\ell} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot h\right)\right)} \cdot w0\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 
(FPCore (w0 M D h l d)
  :name "Henrywood and Agarwal, Equation (9a)"
  (* w0 (sqrt (- 1.0 (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l))))))