Average Error: 14.6 → 9.3
Time: 9.3s
Precision: 64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \le -4.434769257142831 \cdot 10^{294} \lor \neg \left(\frac{h}{\ell} \le -3.0353158251 \cdot 10^{-314}\right):\\ \;\;\;\;w0 \cdot \sqrt{1}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(\sqrt{\sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{h}{\ell}\right)}} \cdot \sqrt{\sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{h}{\ell}\right)}}\right)\\ \end{array}\]
w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}
↓
\begin{array}{l}
\mathbf{if}\;\frac{h}{\ell} \le -4.434769257142831 \cdot 10^{294} \lor \neg \left(\frac{h}{\ell} \le -3.0353158251 \cdot 10^{-314}\right):\\
\;\;\;\;w0 \cdot \sqrt{1}\\

\mathbf{else}:\\
\;\;\;\;w0 \cdot \left(\sqrt{\sqrt{1 - {\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 \frac{h}{\ell}\right)}} \cdot \sqrt{\sqrt{1 - {\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 \frac{h}{\ell}\right)}}\right)\\

\end{array}
double f(double w0, double M, double D, double h, double l, double d) {
        double r174181 = w0;
        double r174182 = 1.0;
        double r174183 = M;
        double r174184 = D;
        double r174185 = r174183 * r174184;
        double r174186 = 2.0;
        double r174187 = d;
        double r174188 = r174186 * r174187;
        double r174189 = r174185 / r174188;
        double r174190 = pow(r174189, r174186);
        double r174191 = h;
        double r174192 = l;
        double r174193 = r174191 / r174192;
        double r174194 = r174190 * r174193;
        double r174195 = r174182 - r174194;
        double r174196 = sqrt(r174195);
        double r174197 = r174181 * r174196;
        return r174197;
}
↓
double f(double w0, double M, double D, double h, double l, double d) {
        double r174198 = h;
        double r174199 = l;
        double r174200 = r174198 / r174199;
        double r174201 = -4.434769257142831e+294;
        bool r174202 = r174200 <= r174201;
        double r174203 = -3.0353158251021e-314;
        bool r174204 = r174200 <= r174203;
        double r174205 = !r174204;
        bool r174206 = r174202 || r174205;
        double r174207 = w0;
        double r174208 = 1.0;
        double r174209 = sqrt(r174208);
        double r174210 = r174207 * r174209;
        double r174211 = M;
        double r174212 = D;
        double r174213 = r174211 * r174212;
        double r174214 = 2.0;
        double r174215 = d;
        double r174216 = r174214 * r174215;
        double r174217 = r174213 / r174216;
        double r174218 = 2.0;
        double r174219 = r174214 / r174218;
        double r174220 = pow(r174217, r174219);
        double r174221 = r174220 * r174200;
        double r174222 = r174220 * r174221;
        double r174223 = r174208 - r174222;
        double r174224 = sqrt(r174223);
        double r174225 = sqrt(r174224);
        double r174226 = r174225 * r174225;
        double r174227 = r174207 * r174226;
        double r174228 = r174206 ? r174210 : r174227;
        return r174228;
}

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 (/ h l) < -4.434769257142831e+294 or -3.0353158251021e-314 < (/ h l)

    1. Initial program 14.5

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

    2. Taylor expanded around 0 6.5

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

    if -4.434769257142831e+294 < (/ h l) < -3.0353158251021e-314

    1. Initial program 14.6

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

    3. Applied sqr-pow14.6

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

    4. Applied associate-*l*12.6

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

    6. Applied add-sqr-sqrt12.6

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

    7. Applied sqrt-prod12.7

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

  4. Final simplification9.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \le -4.434769257142831 \cdot 10^{294} \lor \neg \left(\frac{h}{\ell} \le -3.0353158251 \cdot 10^{-314}\right):\\ \;\;\;\;w0 \cdot \sqrt{1}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(\sqrt{\sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{h}{\ell}\right)}} \cdot \sqrt{\sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{h}{\ell}\right)}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 +o rules:numerics
(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))))))