Henrywood and Agarwal, Equation (9a)

Average Error: 13.9 → 8.7

Time: 14.7s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;M \le -9.8638740699774658 \cdot 10^{-221}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(\frac{{\left(\sqrt[3]{\frac{D}{d}} \cdot \left(\frac{M}{2} \cdot \left(\sqrt[3]{\frac{D}{d}} \cdot \sqrt[3]{\frac{D}{d}}\right)\right)\right)}^{\left(\frac{2}{2}\right)}}{\sqrt[3]{\ell}} \cdot \frac{h}{\sqrt[3]{\ell}}\right) \cdot \frac{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)}}{\sqrt[3]{\ell}}}\\ \mathbf{elif}\;M \le 1.4172298698894145 \cdot 10^{-163}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(\left(\sqrt[3]{h} \cdot \sqrt[3]{h}\right) \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)} \cdot \sqrt[3]{h}\right)\right)}{\ell}}\\ \end{array}\]

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) (((double) (M * D)) / ((double) (2.0 * d)))), 2.0)) * ((double) (h / l))))))))));
}

↓

double code(double w0, double M, double D, double h, double l, double d) {
	double VAR;
	if ((M <= -9.863874069977466e-221)) {
		VAR = ((double) (w0 * ((double) sqrt(((double) (1.0 - ((double) (((double) (((double) (((double) pow(((double) (((double) cbrt(((double) (D / d)))) * ((double) (((double) (M / 2.0)) * ((double) (((double) cbrt(((double) (D / d)))) * ((double) cbrt(((double) (D / d)))))))))), ((double) (2.0 / 2.0)))) / ((double) cbrt(l)))) * ((double) (h / ((double) cbrt(l)))))) * ((double) (((double) pow(((double) (((double) (M / 2.0)) * ((double) (D / d)))), ((double) (2.0 / 2.0)))) / ((double) cbrt(l))))))))))));
	} else {
		double VAR_1;
		if ((M <= 1.4172298698894145e-163)) {
			VAR_1 = ((double) (w0 * ((double) sqrt(((double) (1.0 - ((double) (((double) (h * ((double) pow(((double) (((double) (M * D)) / ((double) (2.0 * d)))), 2.0)))) / l))))))));
		} else {
			VAR_1 = ((double) (w0 * ((double) sqrt(((double) (1.0 - ((double) (((double) (((double) pow(((double) (((double) (M / 2.0)) * ((double) (D / d)))), ((double) (2.0 / 2.0)))) * ((double) (((double) (((double) cbrt(h)) * ((double) cbrt(h)))) * ((double) (((double) pow(((double) (((double) (M / 2.0)) * ((double) (D / d)))), ((double) (2.0 / 2.0)))) * ((double) cbrt(h)))))))) / 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 M < -9.8638740699774658e-221

   1. Initial program 14.8

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

   3. Applied associate-*r/ 11.9

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

   4. Simplified 11.4

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

   6. Applied sqr-pow 11.4

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

   7. Applied associate-*r* 9.6

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

   8. Simplified 9.6

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

   10. Applied add-cube-cbrt 9.7

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

   11. Applied times-frac 8.6

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

   12. Simplified 9.4

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

   13. Simplified 9.4

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

   15. Applied add-cube-cbrt 9.4

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

   16. Applied associate-*r* 9.4

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

   if -9.8638740699774658e-221 < M < 1.4172298698894145e-163

   1. Initial program 8.1

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

   3. Applied associate-*r/ 3.1

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

   4. Simplified 4.4

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

   6. Applied frac-times 3.1

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

   if 1.4172298698894145e-163 < M

   1. Initial program 17.2

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

   3. Applied associate-*r/ 14.6

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

   4. Simplified 14.2

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

   6. Applied sqr-pow 14.2

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

   7. Applied associate-*r* 11.9

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

   8. Simplified 11.9

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

   10. Applied add-cube-cbrt 11.9

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

   11. Applied associate-*l* 11.9

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

   12. Simplified 11.9

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

4. Final simplification 8.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \le -9.8638740699774658 \cdot 10^{-221}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(\frac{{\left(\sqrt[3]{\frac{D}{d}} \cdot \left(\frac{M}{2} \cdot \left(\sqrt[3]{\frac{D}{d}} \cdot \sqrt[3]{\frac{D}{d}}\right)\right)\right)}^{\left(\frac{2}{2}\right)}}{\sqrt[3]{\ell}} \cdot \frac{h}{\sqrt[3]{\ell}}\right) \cdot \frac{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)}}{\sqrt[3]{\ell}}}\\ \mathbf{elif}\;M \le 1.4172298698894145 \cdot 10^{-163}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(\left(\sqrt[3]{h} \cdot \sqrt[3]{h}\right) \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)} \cdot \sqrt[3]{h}\right)\right)}{\ell}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020181 
(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))))))