Average Error: 14.7 → 8.2
Time: 22.9s
Precision: binary64
\[[M, D]=\mathsf{sort}([M, D])\]
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq 3.106650200411536 \cdot 10^{+293}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{1}{\frac{2 \cdot d}{M \cdot D}}\right)}^{2}}\\ \mathbf{elif}\;1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq \infty:\\ \;\;\;\;w0 \cdot \left(\sqrt{\left(\frac{h}{\ell} \cdot {\left(\frac{D}{d}\right)}^{2}\right) \cdot -0.25} \cdot \left(-M\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\left(h \cdot {\left(\left(\sqrt[3]{M} \cdot \sqrt[3]{M}\right) \cdot \left(\sqrt[3]{\frac{D}{d}} \cdot \sqrt[3]{\frac{D}{d}}\right)\right)}^{2}\right) \cdot {\left(\frac{\sqrt[3]{M}}{\frac{2}{\sqrt[3]{\frac{D}{d}}}}\right)}^{2}}{\ell}}\\ \end{array}\]
w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}
↓
\begin{array}{l}
\mathbf{if}\;1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq 3.106650200411536 \cdot 10^{+293}:\\
\;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{1}{\frac{2 \cdot d}{M \cdot D}}\right)}^{2}}\\

\mathbf{elif}\;1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq \infty:\\
\;\;\;\;w0 \cdot \left(\sqrt{\left(\frac{h}{\ell} \cdot {\left(\frac{D}{d}\right)}^{2}\right) \cdot -0.25} \cdot \left(-M\right)\right)\\

\mathbf{else}:\\
\;\;\;\;w0 \cdot \sqrt{1 - \frac{\left(h \cdot {\left(\left(\sqrt[3]{M} \cdot \sqrt[3]{M}\right) \cdot \left(\sqrt[3]{\frac{D}{d}} \cdot \sqrt[3]{\frac{D}{d}}\right)\right)}^{2}\right) \cdot {\left(\frac{\sqrt[3]{M}}{\frac{2}{\sqrt[3]{\frac{D}{d}}}}\right)}^{2}}{\ell}}\\

\end{array}
(FPCore (w0 M D h l d)
 :precision binary64
 (* w0 (sqrt (- 1.0 (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l))))))
↓
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<=
      (- 1.0 (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)))
      3.106650200411536e+293)
   (* w0 (sqrt (- 1.0 (* (/ h l) (pow (/ 1.0 (/ (* 2.0 d) (* M D))) 2.0)))))
   (if (<= (- 1.0 (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l))) INFINITY)
     (* w0 (* (sqrt (* (* (/ h l) (pow (/ D d) 2.0)) -0.25)) (- M)))
     (*
      w0
      (sqrt
       (-
        1.0
        (/
         (*
          (*
           h
           (pow
            (* (* (cbrt M) (cbrt M)) (* (cbrt (/ D d)) (cbrt (/ D d))))
            2.0))
          (pow (/ (cbrt M) (/ 2.0 (cbrt (/ D d)))) 2.0))
         l)))))))
double code(double w0, double M, double D, double h, double l, double d) {
	return w0 * sqrt(1.0 - (pow(((M * D) / (2.0 * d)), 2.0) * (h / l)));
}
↓
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((1.0 - (pow(((M * D) / (2.0 * d)), 2.0) * (h / l))) <= 3.106650200411536e+293) {
		tmp = w0 * sqrt(1.0 - ((h / l) * pow((1.0 / ((2.0 * d) / (M * D))), 2.0)));
	} else if ((1.0 - (pow(((M * D) / (2.0 * d)), 2.0) * (h / l))) <= ((double) INFINITY)) {
		tmp = w0 * (sqrt(((h / l) * pow((D / d), 2.0)) * -0.25) * -M);
	} else {
		tmp = w0 * sqrt(1.0 - (((h * pow(((cbrt(M) * cbrt(M)) * (cbrt(D / d) * cbrt(D / d))), 2.0)) * pow((cbrt(M) / (2.0 / cbrt(D / d))), 2.0)) / l));
	}
	return tmp;
}

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 (-.f64 1 (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l))) < 3.1066502004115359e293

    1. Initial program 0.2

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

    3. Applied clear-num_binary64_11000.2

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

    if 3.1066502004115359e293 < (-.f64 1 (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l))) < +inf.0

    1. Initial program 63.1

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

    2. Taylor expanded around -inf 56.5

      \[\leadsto w0 \cdot \color{blue}{\left(-1 \cdot \left(M \cdot \sqrt{-0.25 \cdot \frac{{D}^{2} \cdot h}{{d}^{2} \cdot \ell}}\right)\right)}\]

    3. Simplified47.4

      \[\leadsto w0 \cdot \color{blue}{\left(\sqrt{\left(\frac{h}{\ell} \cdot {\left(\frac{D}{d}\right)}^{2}\right) \cdot -0.25} \cdot \left(-M\right)\right)}\]

    if +inf.0 < (-.f64 1 (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l)))

    1. Initial program 64.0

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

    3. Applied associate-*r/_binary64_104326.3

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

    4. Simplified26.3

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

    6. Applied associate-/l*_binary64_104624.1

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

    7. Simplified24.1

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

    9. Applied add-cube-cbrt_binary64_113624.1

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

    10. Applied *-un-lft-identity_binary64_110124.1

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

    11. Applied times-frac_binary64_110724.1

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

    12. Applied add-cube-cbrt_binary64_113624.1

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

    13. Applied times-frac_binary64_110724.1

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

    14. Applied unpow-prod-down_binary64_118024.1

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

    15. Applied associate-*r*_binary64_104115.4

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

    16. Simplified15.4

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

  4. Final simplification8.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq 3.106650200411536 \cdot 10^{+293}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{1}{\frac{2 \cdot d}{M \cdot D}}\right)}^{2}}\\ \mathbf{elif}\;1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq \infty:\\ \;\;\;\;w0 \cdot \left(\sqrt{\left(\frac{h}{\ell} \cdot {\left(\frac{D}{d}\right)}^{2}\right) \cdot -0.25} \cdot \left(-M\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\left(h \cdot {\left(\left(\sqrt[3]{M} \cdot \sqrt[3]{M}\right) \cdot \left(\sqrt[3]{\frac{D}{d}} \cdot \sqrt[3]{\frac{D}{d}}\right)\right)}^{2}\right) \cdot {\left(\frac{\sqrt[3]{M}}{\frac{2}{\sqrt[3]{\frac{D}{d}}}}\right)}^{2}}{\ell}}\\ \end{array}\]

Reproduce

herbie shell --seed 2021098 
(FPCore (w0 M D h l d)
  :name "Henrywood and Agarwal, Equation (9a)"
  :precision binary64
  (* w0 (sqrt (- 1.0 (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l))))))