Henrywood and Agarwal, Equation (9a)

Average Error: 13.8 → 7.8

Time: 10.9s

Precision: binary64

\[[M, D]=\mathsf{sort}([M, D])\]

Math

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

↓

\[\begin{array}{l} t_0 := \frac{M \cdot D}{2 \cdot d}\\ t_1 := 1 - {t_0}^{2} \cdot \frac{h}{\ell}\\ \mathbf{if}\;t_1 \leq 4.028325540402926 \cdot 10^{+299}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}}\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;\sqrt{\left(\frac{h}{\ell} \cdot {\left(\frac{D}{d}\right)}^{2}\right) \cdot -0.25} \cdot \left(w0 \cdot \left(-M\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{t_0 \cdot \left(h \cdot \left(\left(M \cdot D\right) \cdot \frac{0.5}{d}\right)\right)}{\ell}}\\ \end{array} \]

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
 (let* ((t_0 (/ (* M D) (* 2.0 d))) (t_1 (- 1.0 (* (pow t_0 2.0) (/ h l)))))
   (if (<= t_1 4.028325540402926e+299)
     (* w0 (sqrt (- 1.0 (* (/ h l) (pow (* (/ M 2.0) (/ D d)) 2.0)))))
     (if (<= t_1 INFINITY)
       (* (sqrt (* (* (/ h l) (pow (/ D d) 2.0)) -0.25)) (* w0 (- M)))
       (* w0 (sqrt (- 1.0 (/ (* t_0 (* h (* (* M D) (/ 0.5 d)))) l))))))))

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 t_0 = (M * D) / (2.0 * d);
	double t_1 = 1.0 - (pow(t_0, 2.0) * (h / l));
	double tmp;
	if (t_1 <= 4.028325540402926e+299) {
		tmp = w0 * sqrt(1.0 - ((h / l) * pow(((M / 2.0) * (D / d)), 2.0)));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = sqrt(((h / l) * pow((D / d), 2.0)) * -0.25) * (w0 * -M);
	} else {
		tmp = w0 * sqrt(1.0 - ((t_0 * (h * ((M * D) * (0.5 / d)))) / l));
	}
	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 3 regimes

2. if (-.f64 1 (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l))) < 4.02832554040292602e299

   1. Initial program 0.2

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

   3. Applied times-frac_binary64 0.8

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

   if 4.02832554040292602e299 < (-.f64 1 (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l))) < +inf.0

   1. Initial program 63.6

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

   2. Taylor expanded around -inf 55.3

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{-0.25 \cdot \frac{{D}^{2} \cdot h}{{d}^{2} \cdot \ell}} \cdot \left(w0 \cdot M\right)\right)} \]

   3. Simplified 46.6

      \[\leadsto \color{blue}{\sqrt{\left(\frac{h}{\ell} \cdot {\left(\frac{D}{d}\right)}^{2}\right) \cdot -0.25} \cdot \left(w0 \cdot \left(-M\right)\right)} \]

   if +inf.0 < (-.f64 1 (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l)))

   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/_binary64 26.6

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

   4. Simplified 26.6

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

   6. Applied sqr-pow_binary64 26.6

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

   7. Applied associate-*r*_binary64 13.9

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

   8. Simplified 13.9

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

   10. Applied div-inv_binary64 13.9

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

   11. Simplified 13.9

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

4. Final simplification 7.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq 4.028325540402926 \cdot 10^{+299}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}}\\ \mathbf{elif}\;1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq \infty:\\ \;\;\;\;\sqrt{\left(\frac{h}{\ell} \cdot {\left(\frac{D}{d}\right)}^{2}\right) \cdot -0.25} \cdot \left(w0 \cdot \left(-M\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\frac{M \cdot D}{2 \cdot d} \cdot \left(h \cdot \left(\left(M \cdot D\right) \cdot \frac{0.5}{d}\right)\right)}{\ell}}\\ \end{array} \]

Reproduce

herbie shell --seed 2021204 
(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))))))