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

\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 r7714435 = w0;
        double r7714436 = 1.0;
        double r7714437 = M;
        double r7714438 = D;
        double r7714439 = r7714437 * r7714438;
        double r7714440 = 2.0;
        double r7714441 = d;
        double r7714442 = r7714440 * r7714441;
        double r7714443 = r7714439 / r7714442;
        double r7714444 = pow(r7714443, r7714440);
        double r7714445 = h;
        double r7714446 = l;
        double r7714447 = r7714445 / r7714446;
        double r7714448 = r7714444 * r7714447;
        double r7714449 = r7714436 - r7714448;
        double r7714450 = sqrt(r7714449);
        double r7714451 = r7714435 * r7714450;
        return r7714451;
}
↓
double f(double w0, double M, double D, double h, double l, double d) {
        double r7714452 = M;
        double r7714453 = D;
        double r7714454 = r7714452 * r7714453;
        double r7714455 = -4.7692322344315233e+219;
        bool r7714456 = r7714454 <= r7714455;
        double r7714457 = w0;
        double r7714458 = 1.0;
        double r7714459 = sqrt(r7714458);
        double r7714460 = r7714457 * r7714459;
        double r7714461 = 2.0;
        double r7714462 = d;
        double r7714463 = r7714461 * r7714462;
        double r7714464 = r7714454 / r7714463;
        double r7714465 = 2.0;
        double r7714466 = r7714461 / r7714465;
        double r7714467 = pow(r7714464, r7714466);
        double r7714468 = 1.0;
        double r7714469 = l;
        double r7714470 = r7714468 / r7714469;
        double r7714471 = h;
        double r7714472 = r7714467 * r7714471;
        double r7714473 = r7714470 * r7714472;
        double r7714474 = r7714467 * r7714473;
        double r7714475 = r7714458 - r7714474;
        double r7714476 = sqrt(r7714475);
        double r7714477 = r7714476 * r7714457;
        double r7714478 = r7714456 ? r7714460 : r7714477;
        return r7714478;
}

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. Taylor expanded around 0 44.9

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

    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

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

Reproduce

herbie shell --seed 2019168 +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))))))