Henrywood and Agarwal, Equation (9a)

Average Error: 14.1 → 7.8

Time: 10.9s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -\infty:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(h \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \frac{{\left(\frac{D}{d \cdot \sqrt{2}} \cdot \frac{M}{\sqrt{2}}\right)}^{\left(\frac{2}{2}\right)}}{\ell}}\\ \mathbf{elif}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -0:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(h \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \frac{{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{\left(\frac{2}{2}\right)}}{\ell}}\\ \end{array}\]

C

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 ((((double) (((double) pow((((double) (M * D)) / ((double) (2.0 * d))), 2.0)) * (h / l))) <= ((double) -(((double) INFINITY))))) {
		VAR = ((double) (w0 * ((double) sqrt(((double) (1.0 - ((double) (((double) (h * ((double) pow(((double) ((D / d) * (M / 2.0))), (2.0 / 2.0))))) * (((double) pow(((double) ((D / ((double) (d * ((double) sqrt(2.0))))) * (M / ((double) sqrt(2.0))))), (2.0 / 2.0))) / l)))))))));
	} else {
		double VAR_1;
		if ((((double) (((double) pow((((double) (M * D)) / ((double) (2.0 * d))), 2.0)) * (h / l))) <= -0.0)) {
			VAR_1 = ((double) (w0 * ((double) sqrt(((double) (1.0 - ((double) (((double) pow((((double) (M * D)) / ((double) (2.0 * d))), 2.0)) * (h / l)))))))));
		} else {
			VAR_1 = ((double) (w0 * ((double) sqrt(((double) (1.0 - ((double) (((double) (h * ((double) pow(((double) ((D / d) * (M / 2.0))), (2.0 / 2.0))))) * (((double) pow(((double) ((D / d) * (M / 2.0))), (2.0 / 2.0))) / l)))))))));
		}
		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

Derivation

1. Split input into 3 regimes

2. if (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)) < -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 associate-*r/ 59.5

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

   4. Simplified 58.0

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

   6. Applied sqr-pow 58.0

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

   7. Applied associate-*r* 55.1

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

   9. Applied *-un-lft-identity 55.1

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

   10. Applied times-frac 52.6

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

   11. Simplified 52.6

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

   13. Applied add-sqr-sqrt 52.7

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

   14. Applied *-un-lft-identity 52.7

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

   15. Applied times-frac 52.7

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

   16. Applied associate-*r* 52.7

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

   17. Simplified 52.7

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

   if -inf.0 < (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)) < -0.0

   1. Initial program 0.1

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

   if -0.0 < (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l))

   1. Initial program 32.7

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

   3. Applied associate-*r/ 13.3

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

   4. Simplified 12.1

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

   6. Applied sqr-pow 12.1

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

   7. Applied associate-*r* 6.4

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

   9. Applied *-un-lft-identity 6.4

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

   10. Applied times-frac 5.5

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

   11. Simplified 5.5

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

4. Final simplification 7.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -\infty:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(h \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \frac{{\left(\frac{D}{d \cdot \sqrt{2}} \cdot \frac{M}{\sqrt{2}}\right)}^{\left(\frac{2}{2}\right)}}{\ell}}\\ \mathbf{elif}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -0:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(h \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \frac{{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{\left(\frac{2}{2}\right)}}{\ell}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020199 
(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))))))