Average Error: 14.4 → 8.6
Time: 9.4s
Precision: binary64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -1.2596240831703367 \cdot 10^{+249}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(h \cdot {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \frac{{\left(M \cdot \frac{D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}}{\ell}}\\ \mathbf{elif}\;\frac{h}{\ell} \leq -1.0384090607706485 \cdot 10^{-291}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(\frac{h}{\ell} \cdot {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1}\\ \end{array}\]
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} \leq -1.2596240831703367 \cdot 10^{+249}:\\
\;\;\;\;w0 \cdot \sqrt{1 - \left(h \cdot {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \frac{{\left(M \cdot \frac{D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}}{\ell}}\\

\mathbf{elif}\;\frac{h}{\ell} \leq -1.0384090607706485 \cdot 10^{-291}:\\
\;\;\;\;w0 \cdot \sqrt{1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(\frac{h}{\ell} \cdot {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}\right)}\\

\mathbf{else}:\\
\;\;\;\;w0 \cdot \sqrt{1}\\

\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 (<= (/ h l) -1.2596240831703367e+249)
   (*
    w0
    (sqrt
     (-
      1.0
      (*
       (* h (pow (* M (/ D (* 2.0 d))) (/ 2.0 2.0)))
       (/ (pow (* M (/ D (* 2.0 d))) (/ 2.0 2.0)) l)))))
   (if (<= (/ h l) -1.0384090607706485e-291)
     (*
      w0
      (sqrt
       (-
        1.0
        (*
         (pow (* M (/ D (* 2.0 d))) (/ 2.0 2.0))
         (* (/ h l) (pow (* M (/ D (* 2.0 d))) (/ 2.0 2.0)))))))
     (* w0 (sqrt 1.0)))))
double code(double w0, double M, double D, double h, double l, double d) {
	return ((double) (w0 * ((double) sqrt(((double) (1.0 - ((double) (((double) pow((((double) (M * D)) / ((double) (2.0 * d))), 2.0)) * (h / l)))))))));
}
↓
double code(double w0, double M, double D, double h, double l, double d) {
	double VAR;
	if (((h / l) <= -1.2596240831703367e+249)) {
		VAR = ((double) (w0 * ((double) sqrt(((double) (1.0 - ((double) (((double) (h * ((double) pow(((double) (M * (D / ((double) (2.0 * d))))), (2.0 / 2.0))))) * (((double) pow(((double) (M * (D / ((double) (2.0 * d))))), (2.0 / 2.0))) / l)))))))));
	} else {
		double VAR_1;
		if (((h / l) <= -1.0384090607706485e-291)) {
			VAR_1 = ((double) (w0 * ((double) sqrt(((double) (1.0 - ((double) (((double) pow(((double) (M * (D / ((double) (2.0 * d))))), (2.0 / 2.0))) * ((double) ((h / l) * ((double) pow(((double) (M * (D / ((double) (2.0 * d))))), (2.0 / 2.0)))))))))))));
		} else {
			VAR_1 = ((double) (w0 * ((double) sqrt(1.0))));
		}
		VAR = VAR_1;
	}
	return VAR;
}

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) < -1.25962408317033673e249

    1. Initial program 49.5

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

    2. Simplified49.6

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

    4. Applied associate-*r/23.2

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

    5. Simplified23.2

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

    7. Applied sqr-pow23.2

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

    8. Applied associate-*r*21.1

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

    10. Applied *-un-lft-identity21.1

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

    11. Applied times-frac21.1

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

    12. Simplified21.1

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

    if -1.25962408317033673e249 < (/ h l) < -1.0384090607706485e-291

    1. Initial program 14.1

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

    2. Simplified14.0

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

    4. Applied sqr-pow14.0

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

    5. Applied associate-*l*12.0

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

    6. Simplified12.0

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

    if -1.0384090607706485e-291 < (/ h l)

    1. Initial program 8.0

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

    2. Simplified8.0

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

    3. Taylor expanded around 0 3.3

      \[\leadsto w0 \cdot \color{blue}{\sqrt{1}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification8.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -1.2596240831703367 \cdot 10^{+249}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(h \cdot {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \frac{{\left(M \cdot \frac{D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}}{\ell}}\\ \mathbf{elif}\;\frac{h}{\ell} \leq -1.0384090607706485 \cdot 10^{-291}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(\frac{h}{\ell} \cdot {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020198 
(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))))))