Average Error: 14.2 → 9.2
Time: 11.7s
Precision: 64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[w0 \cdot \sqrt{1 - \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{1}{\frac{2 \cdot d}{M \cdot D}}\right)}^{\left(\frac{2}{2}\right)} \cdot h\right)\right) \cdot \frac{1}{\ell}}\]
w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}
↓
w0 \cdot \sqrt{1 - \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{1}{\frac{2 \cdot d}{M \cdot D}}\right)}^{\left(\frac{2}{2}\right)} \cdot h\right)\right) \cdot \frac{1}{\ell}}
double f(double w0, double M, double D, double h, double l, double d) {
        double r211573 = w0;
        double r211574 = 1.0;
        double r211575 = M;
        double r211576 = D;
        double r211577 = r211575 * r211576;
        double r211578 = 2.0;
        double r211579 = d;
        double r211580 = r211578 * r211579;
        double r211581 = r211577 / r211580;
        double r211582 = pow(r211581, r211578);
        double r211583 = h;
        double r211584 = l;
        double r211585 = r211583 / r211584;
        double r211586 = r211582 * r211585;
        double r211587 = r211574 - r211586;
        double r211588 = sqrt(r211587);
        double r211589 = r211573 * r211588;
        return r211589;
}
↓
double f(double w0, double M, double D, double h, double l, double d) {
        double r211590 = w0;
        double r211591 = 1.0;
        double r211592 = M;
        double r211593 = D;
        double r211594 = r211592 * r211593;
        double r211595 = 2.0;
        double r211596 = d;
        double r211597 = r211595 * r211596;
        double r211598 = r211594 / r211597;
        double r211599 = 2.0;
        double r211600 = r211595 / r211599;
        double r211601 = pow(r211598, r211600);
        double r211602 = 1.0;
        double r211603 = r211597 / r211594;
        double r211604 = r211602 / r211603;
        double r211605 = pow(r211604, r211600);
        double r211606 = h;
        double r211607 = r211605 * r211606;
        double r211608 = r211601 * r211607;
        double r211609 = l;
        double r211610 = r211602 / r211609;
        double r211611 = r211608 * r211610;
        double r211612 = r211591 - r211611;
        double r211613 = sqrt(r211612);
        double r211614 = r211590 * r211613;
        return r211614;
}

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. Initial program 14.2

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

  3. Applied div-inv14.2

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

  4. Applied associate-*r*10.7

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

  6. Applied sqr-pow10.7

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

  7. Applied associate-*l*9.2

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

  9. Applied clear-num9.2

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

  10. Final simplification9.2

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

Reproduce

herbie shell --seed 2020021 
(FPCore (w0 M D h l d)
  :name "Henrywood and Agarwal, Equation (9a)"
  :precision binary64
  (* w0 (sqrt (- 1 (* (pow (/ (* M D) (* 2 d)) 2) (/ h l))))))