Henrywood and Agarwal, Equation (9a)

Average Error: 13.9 → 7.5

Time: 12.9s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -1.6705765303034724 \cdot 10^{+290} \lor \neg \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq 0.7782603050346516\right):\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(h \cdot \frac{M}{\frac{2}{\frac{D}{d}}}\right) \cdot \frac{\frac{M}{\frac{2}{\frac{D}{d}}}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\left(M \cdot D\right) \cdot \frac{0.5}{d}\right)}^{2}}\\ \end{array}\]

FPCore

(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 (or (<=
          (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l))
          -1.6705765303034724e+290)
         (not
          (<= (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)) 0.7782603050346516)))
   (*
    w0
    (sqrt (- 1.0 (* (* h (/ M (/ 2.0 (/ D d)))) (/ (/ M (/ 2.0 (/ D d))) l)))))
   (* w0 (sqrt (- 1.0 (* (/ h l) (pow (* (* M D) (/ 0.5 d)) 2.0)))))))

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 tmp;
	if (((pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -1.6705765303034724e+290) || !((pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= 0.7782603050346516)) {
		tmp = w0 * sqrt(1.0 - ((h * (M / (2.0 / (D / d)))) * ((M / (2.0 / (D / d))) / l)));
	} else {
		tmp = w0 * sqrt(1.0 - ((h / l) * pow(((M * D) * (0.5 / d)), 2.0)));
	}
	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

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l)) < -1.67057653030347238e290 or 0.7782603050346516 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l))

   1. Initial program 63.2

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

   3. Applied associate-*r/_binary64_702 45.5

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

   4. Simplified 45.5

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

   6. Applied unpow2_binary64_825 45.5

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

   7. Applied associate-*r*_binary64_700 38.4

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

   9. Applied associate-/l*_binary64_705 39.9

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

   10. Simplified 40.0

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

   12. Applied *-un-lft-identity_binary64_760 40.0

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

   13. Applied times-frac_binary64_766 38.4

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

   14. Simplified 33.9

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

   if -1.67057653030347238e290 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l)) < 0.7782603050346516

   1. Initial program 0.1

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

   3. Applied div-inv_binary64_757 0.1

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

   4. Simplified 0.1

      \[\leadsto w0 \cdot \sqrt{1 - {\left(\left(M \cdot D\right) \cdot \color{blue}{\frac{0.5}{d}}\right)}^{2} \cdot \frac{h}{\ell}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 7.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -1.6705765303034724 \cdot 10^{+290} \lor \neg \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq 0.7782603050346516\right):\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(h \cdot \frac{M}{\frac{2}{\frac{D}{d}}}\right) \cdot \frac{\frac{M}{\frac{2}{\frac{D}{d}}}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\left(M \cdot D\right) \cdot \frac{0.5}{d}\right)}^{2}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020354 
(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))))))