Henrywood and Agarwal, Equation (9a)

Average Error: 14.3 → 8.6

Time: 15.0s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} t_0 := {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\\ t_1 := 1 - t_0 \cdot \frac{h}{\ell}\\ \mathbf{if}\;t_1 \leq 1.2493150413033514 \cdot 10^{+238}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{1}{\frac{\ell}{t_0 \cdot h}}}\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;w0 \cdot \left(\sqrt{\left(\frac{h}{\ell} \cdot {\left(\frac{D}{d}\right)}^{2}\right) \cdot -0.25} \cdot \left(-M\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\begin{array}{l} t_2 := \frac{D}{\frac{2}{\frac{M}{d}}}\\ w0 \cdot \sqrt{1 - \frac{t_2 \cdot \left(h \cdot t_2\right)}{\ell}} \end{array}\\ \end{array} \]

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 (pow (/ (* M D) (* 2.0 d)) 2.0)) (t_1 (- 1.0 (* t_0 (/ h l)))))
   (if (<= t_1 1.2493150413033514e+238)
     (* w0 (sqrt (- 1.0 (/ 1.0 (/ l (* t_0 h))))))
     (if (<= t_1 INFINITY)
       (* w0 (* (sqrt (* (* (/ h l) (pow (/ D d) 2.0)) -0.25)) (- M)))
       (let* ((t_2 (/ D (/ 2.0 (/ M d)))))
         (* w0 (sqrt (- 1.0 (/ (* t_2 (* h t_2)) 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 = pow(((M * D) / (2.0 * d)), 2.0);
	double t_1 = 1.0 - (t_0 * (h / l));
	double tmp;
	if (t_1 <= 1.2493150413033514e+238) {
		tmp = w0 * sqrt(1.0 - (1.0 / (l / (t_0 * h))));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = w0 * (sqrt(((h / l) * pow((D / d), 2.0)) * -0.25) * -M);
	} else {
		double t_2 = D / (2.0 / (M / d));
		tmp = w0 * sqrt(1.0 - ((t_2 * (h * t_2)) / 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))) < 1.24931504130335139e238

   1. Initial program 0.3

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

   3. Applied associate-*r/_binary64 0.8

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

   4. Simplified 0.8

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

   6. Applied clear-num_binary64 0.8

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

   if 1.24931504130335139e238 < (-.f64 1 (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l))) < +inf.0

   1. Initial program 59.0

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

   2. Taylor expanded in M around -inf 55.7

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

   3. Simplified 48.2

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

   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 25.4

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

   4. Simplified 25.4

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

   6. Applied associate-/l*_binary64 23.2

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

   7. Simplified 23.2

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

   9. Applied unpow2_binary64 23.2

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

   10. Applied associate-*r*_binary64 12.5

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

4. Final simplification 8.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq 1.2493150413033514 \cdot 10^{+238}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{1}{\frac{\ell}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h}}}\\ \mathbf{elif}\;1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq \infty:\\ \;\;\;\;w0 \cdot \left(\sqrt{\left(\frac{h}{\ell} \cdot {\left(\frac{D}{d}\right)}^{2}\right) \cdot -0.25} \cdot \left(-M\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\frac{D}{\frac{2}{\frac{M}{d}}} \cdot \left(h \cdot \frac{D}{\frac{2}{\frac{M}{d}}}\right)}{\ell}}\\ \end{array} \]

Reproduce

herbie shell --seed 2021212 
(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))))))