Henrywood and Agarwal, Equation (9a)

Average Error: 14.2 → 8.4

Time: 10.1s

Precision: 64

Math

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

↓

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

C

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 VAR;
	if ((((M * D) / (2.0 * d)) <= -inf.0)) {
		VAR = (w0 * sqrt((1.0 - (pow((D / ((2.0 * d) / M)), 2.0) * (h / l)))));
	} else {
		VAR = (w0 * sqrt((1.0 - (pow(((M * D) / (2.0 * d)), (2.0 / 2.0)) / (1.0 / ((pow(((M * D) / (2.0 * d)), (2.0 / 2.0)) * h) / l))))));
	}
	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

Derivation

1. Split input into 2 regimes

2. if (/ (* M D) (* 2.0 d)) < -inf.0

   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 *-commutative 64.0

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

   4. Applied associate-/l* 52.9

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

   if -inf.0 < (/ (* M D) (* 2.0 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 associate-*r/ 9.2

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

   5. Applied sqr-pow 9.2

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

   6. Applied associate-*l* 7.6

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

   7. Applied associate-/l* 6.9

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

   9. Applied clear-num 6.9

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

4. Final simplification 8.4

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

Reproduce

herbie shell --seed 2020071 +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))))))