Average Error: 14.1 → 8.5
Time: 55.3s
Precision: 64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \le -2.282180062770275197908078530250940945557 \cdot 10^{149}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{1}{\ell} \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot h\right)}\\ \mathbf{elif}\;\frac{h}{\ell} \le -2.53200167457589704840317226177229055452 \cdot 10^{-63}:\\ \;\;\;\;\left(w0 \cdot \sqrt{\sqrt{1 - {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\ell}}}\right) \cdot \sqrt{\sqrt{1 - {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\ell}}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(\left(h \cdot {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \frac{1}{\ell}\right) \cdot {\left(\left(D \cdot M\right) \cdot \frac{1}{d \cdot 2}\right)}^{\left(\frac{2}{2}\right)}}\\ \end{array}\]
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 -2.282180062770275197908078530250940945557 \cdot 10^{149}:\\
\;\;\;\;w0 \cdot \sqrt{1 - \frac{1}{\ell} \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot h\right)}\\

\mathbf{elif}\;\frac{h}{\ell} \le -2.53200167457589704840317226177229055452 \cdot 10^{-63}:\\
\;\;\;\;\left(w0 \cdot \sqrt{\sqrt{1 - {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\ell}}}\right) \cdot \sqrt{\sqrt{1 - {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\ell}}}\\

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

\end{array}
double f(double w0, double M, double D, double h, double l, double d) {
        double r5089196 = w0;
        double r5089197 = 1.0;
        double r5089198 = M;
        double r5089199 = D;
        double r5089200 = r5089198 * r5089199;
        double r5089201 = 2.0;
        double r5089202 = d;
        double r5089203 = r5089201 * r5089202;
        double r5089204 = r5089200 / r5089203;
        double r5089205 = pow(r5089204, r5089201);
        double r5089206 = h;
        double r5089207 = l;
        double r5089208 = r5089206 / r5089207;
        double r5089209 = r5089205 * r5089208;
        double r5089210 = r5089197 - r5089209;
        double r5089211 = sqrt(r5089210);
        double r5089212 = r5089196 * r5089211;
        return r5089212;
}
↓
double f(double w0, double M, double D, double h, double l, double d) {
        double r5089213 = h;
        double r5089214 = l;
        double r5089215 = r5089213 / r5089214;
        double r5089216 = -2.282180062770275e+149;
        bool r5089217 = r5089215 <= r5089216;
        double r5089218 = w0;
        double r5089219 = 1.0;
        double r5089220 = 1.0;
        double r5089221 = r5089220 / r5089214;
        double r5089222 = M;
        double r5089223 = 2.0;
        double r5089224 = r5089222 / r5089223;
        double r5089225 = D;
        double r5089226 = d;
        double r5089227 = r5089225 / r5089226;
        double r5089228 = r5089224 * r5089227;
        double r5089229 = pow(r5089228, r5089223);
        double r5089230 = r5089229 * r5089213;
        double r5089231 = r5089221 * r5089230;
        double r5089232 = r5089219 - r5089231;
        double r5089233 = sqrt(r5089232);
        double r5089234 = r5089218 * r5089233;
        double r5089235 = -2.532001674575897e-63;
        bool r5089236 = r5089215 <= r5089235;
        double r5089237 = r5089225 * r5089222;
        double r5089238 = r5089226 * r5089223;
        double r5089239 = r5089237 / r5089238;
        double r5089240 = pow(r5089239, r5089223);
        double r5089241 = r5089240 * r5089215;
        double r5089242 = r5089219 - r5089241;
        double r5089243 = sqrt(r5089242);
        double r5089244 = sqrt(r5089243);
        double r5089245 = r5089218 * r5089244;
        double r5089246 = r5089245 * r5089244;
        double r5089247 = 2.0;
        double r5089248 = r5089223 / r5089247;
        double r5089249 = pow(r5089239, r5089248);
        double r5089250 = r5089213 * r5089249;
        double r5089251 = r5089250 * r5089221;
        double r5089252 = r5089220 / r5089238;
        double r5089253 = r5089237 * r5089252;
        double r5089254 = pow(r5089253, r5089248);
        double r5089255 = r5089251 * r5089254;
        double r5089256 = r5089219 - r5089255;
        double r5089257 = sqrt(r5089256);
        double r5089258 = r5089218 * r5089257;
        double r5089259 = r5089236 ? r5089246 : r5089258;
        double r5089260 = r5089217 ? r5089234 : r5089259;
        return r5089260;
}

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 3 regimes

  2. if (/ h l) < -2.282180062770275e+149

    1. Initial program 37.0

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

    3. Applied div-inv37.0

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

    4. Applied associate-*r*21.8

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

    6. Applied times-frac21.9

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

    if -2.282180062770275e+149 < (/ h l) < -2.532001674575897e-63

    1. Initial program 13.0

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

    3. Applied add-sqr-sqrt13.0

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

    4. Applied sqrt-prod13.1

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

    5. Applied associate-*r*13.1

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

    if -2.532001674575897e-63 < (/ h l)

    1. Initial program 9.6

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

    3. Applied div-inv9.6

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

    4. Applied associate-*r*7.3

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

    6. Applied sqr-pow7.3

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

    7. Applied associate-*l*5.5

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

    9. Applied associate-*l*4.4

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

    11. Applied div-inv4.4

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

  4. Final simplification8.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \le -2.282180062770275197908078530250940945557 \cdot 10^{149}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{1}{\ell} \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot h\right)}\\ \mathbf{elif}\;\frac{h}{\ell} \le -2.53200167457589704840317226177229055452 \cdot 10^{-63}:\\ \;\;\;\;\left(w0 \cdot \sqrt{\sqrt{1 - {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\ell}}}\right) \cdot \sqrt{\sqrt{1 - {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\ell}}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(\left(h \cdot {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \frac{1}{\ell}\right) \cdot {\left(\left(D \cdot M\right) \cdot \frac{1}{d \cdot 2}\right)}^{\left(\frac{2}{2}\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019200 +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))))))