Average Error: 14.1 → 8.7
Time: 33.1s
Precision: 64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot h}{\ell}}\]
w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}
↓
w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot h}{\ell}}
double f(double w0, double M, double D, double h, double l, double d) {
        double r7108561 = w0;
        double r7108562 = 1.0;
        double r7108563 = M;
        double r7108564 = D;
        double r7108565 = r7108563 * r7108564;
        double r7108566 = 2.0;
        double r7108567 = d;
        double r7108568 = r7108566 * r7108567;
        double r7108569 = r7108565 / r7108568;
        double r7108570 = pow(r7108569, r7108566);
        double r7108571 = h;
        double r7108572 = l;
        double r7108573 = r7108571 / r7108572;
        double r7108574 = r7108570 * r7108573;
        double r7108575 = r7108562 - r7108574;
        double r7108576 = sqrt(r7108575);
        double r7108577 = r7108561 * r7108576;
        return r7108577;
}
↓
double f(double w0, double M, double D, double h, double l, double d) {
        double r7108578 = w0;
        double r7108579 = 1.0;
        double r7108580 = M;
        double r7108581 = D;
        double r7108582 = r7108580 * r7108581;
        double r7108583 = 2.0;
        double r7108584 = d;
        double r7108585 = r7108583 * r7108584;
        double r7108586 = r7108582 / r7108585;
        double r7108587 = 2.0;
        double r7108588 = r7108583 / r7108587;
        double r7108589 = pow(r7108586, r7108588);
        double r7108590 = h;
        double r7108591 = r7108589 * r7108590;
        double r7108592 = l;
        double r7108593 = r7108591 / r7108592;
        double r7108594 = r7108589 * r7108593;
        double r7108595 = r7108579 - r7108594;
        double r7108596 = sqrt(r7108595);
        double r7108597 = r7108578 * r7108596;
        return r7108597;
}

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 14.1

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

  3. Applied div-inv14.1

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

  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*9.4

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

  9. Applied associate-*l*8.7

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

  10. Simplified8.7

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

  11. Final simplification8.7

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

Reproduce

herbie shell --seed 2019171 +o rules:numerics
(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))))))