Average Error: 14.4 → 8.7
Time: 34.1s
Precision: 64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;M \cdot D \le 257758865443741.375:\\ \;\;\;\;\sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(\frac{1}{\ell} \cdot \left(h \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}\right)\right)} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left({\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(h \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{\left(\frac{2}{2}\right)}\right)\right) \cdot \frac{1}{\ell}}\\ \end{array}\]
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 257758865443741.375:\\
\;\;\;\;\sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(\frac{1}{\ell} \cdot \left(h \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}\right)\right)} \cdot w0\\

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

\end{array}
double f(double w0, double M, double D, double h, double l, double d) {
        double r6103434 = w0;
        double r6103435 = 1.0;
        double r6103436 = M;
        double r6103437 = D;
        double r6103438 = r6103436 * r6103437;
        double r6103439 = 2.0;
        double r6103440 = d;
        double r6103441 = r6103439 * r6103440;
        double r6103442 = r6103438 / r6103441;
        double r6103443 = pow(r6103442, r6103439);
        double r6103444 = h;
        double r6103445 = l;
        double r6103446 = r6103444 / r6103445;
        double r6103447 = r6103443 * r6103446;
        double r6103448 = r6103435 - r6103447;
        double r6103449 = sqrt(r6103448);
        double r6103450 = r6103434 * r6103449;
        return r6103450;
}
↓
double f(double w0, double M, double D, double h, double l, double d) {
        double r6103451 = M;
        double r6103452 = D;
        double r6103453 = r6103451 * r6103452;
        double r6103454 = 257758865443741.38;
        bool r6103455 = r6103453 <= r6103454;
        double r6103456 = 1.0;
        double r6103457 = 2.0;
        double r6103458 = d;
        double r6103459 = r6103457 * r6103458;
        double r6103460 = r6103453 / r6103459;
        double r6103461 = 2.0;
        double r6103462 = r6103457 / r6103461;
        double r6103463 = pow(r6103460, r6103462);
        double r6103464 = 1.0;
        double r6103465 = l;
        double r6103466 = r6103464 / r6103465;
        double r6103467 = h;
        double r6103468 = r6103467 * r6103463;
        double r6103469 = r6103466 * r6103468;
        double r6103470 = r6103463 * r6103469;
        double r6103471 = r6103456 - r6103470;
        double r6103472 = sqrt(r6103471);
        double r6103473 = w0;
        double r6103474 = r6103472 * r6103473;
        double r6103475 = r6103452 / r6103458;
        double r6103476 = r6103451 / r6103457;
        double r6103477 = r6103475 * r6103476;
        double r6103478 = pow(r6103477, r6103462);
        double r6103479 = r6103467 * r6103478;
        double r6103480 = r6103478 * r6103479;
        double r6103481 = r6103480 * r6103466;
        double r6103482 = r6103456 - r6103481;
        double r6103483 = sqrt(r6103482);
        double r6103484 = r6103473 * r6103483;
        double r6103485 = r6103455 ? r6103474 : r6103484;
        return r6103485;
}

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) < 257758865443741.38

    1. Initial program 12.1

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

    3. Applied div-inv12.1

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

    4. Applied associate-*r*8.1

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

    6. Applied sqr-pow8.1

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

    7. Applied associate-*l*6.8

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

    9. Applied associate-*l*6.4

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

    if 257758865443741.38 < (* M D)

    1. Initial program 26.9

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

    3. Applied div-inv26.9

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

    4. Applied associate-*r*26.1

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

    6. Applied sqr-pow26.1

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

    7. Applied associate-*l*22.6

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

    9. Applied *-un-lft-identity22.6

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

    10. Applied *-un-lft-identity22.6

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

    11. Applied times-frac22.6

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

    12. Applied associate-*r*22.6

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

    13. Simplified21.5

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

  4. Final simplification8.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;M \cdot D \le 257758865443741.375:\\ \;\;\;\;\sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(\frac{1}{\ell} \cdot \left(h \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}\right)\right)} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left({\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(h \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{\left(\frac{2}{2}\right)}\right)\right) \cdot \frac{1}{\ell}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 +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))))))