Henrywood and Agarwal, Equation (9a)

Average Error: 14.2 → 8.5

Time: 13.9s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \frac{M}{\frac{2 \cdot d}{D}}\\ t_1 := {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\\ t_2 := w0 \cdot \sqrt{1 - \frac{t_0 \cdot \left(t_0 \cdot h\right)}{\ell}}\\ \mathbf{if}\;t_1 \leq 4.005006849645771 \cdot 10^{-69}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq 6.109391534461873 \cdot 10^{+302}:\\ \;\;\;\;w0 \cdot \sqrt{1 - t_1 \cdot \frac{h}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

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
 (let* ((t_0 (/ M (/ (* 2.0 d) D)))
        (t_1 (pow (/ (* M D) (* 2.0 d)) 2.0))
        (t_2 (* w0 (sqrt (- 1.0 (/ (* t_0 (* t_0 h)) l))))))
   (if (<= t_1 4.005006849645771e-69)
     t_2
     (if (<= t_1 6.109391534461873e+302)
       (* w0 (sqrt (- 1.0 (* t_1 (/ h l)))))
       t_2))))

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 / ((2.0 * d) / D);
	double t_1 = pow(((M * D) / (2.0 * d)), 2.0);
	double t_2 = w0 * sqrt(1.0 - ((t_0 * (t_0 * h)) / l));
	double tmp;
	if (t_1 <= 4.005006849645771e-69) {
		tmp = t_2;
	} else if (t_1 <= 6.109391534461873e+302) {
		tmp = w0 * sqrt(1.0 - (t_1 * (h / l)));
	} else {
		tmp = t_2;
	}
	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 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) < 4.00500684964577099e-69 or 6.1093915344618732e302 < (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)

   1. Initial program 15.6

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

   2. Applied associate-*r/_binary64 10.9

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

   3. Applied associate-/l*_binary64 10.5

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

   4. Applied unpow2_binary64 10.5

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

   5. Applied associate-*l*_binary64 8.5

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

   if 4.00500684964577099e-69 < (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) < 6.1093915344618732e302

   1. Initial program 8.3

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

4. Final simplification 8.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \leq 4.005006849645771 \cdot 10^{-69}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\frac{M}{\frac{2 \cdot d}{D}} \cdot \left(\frac{M}{\frac{2 \cdot d}{D}} \cdot h\right)}{\ell}}\\ \mathbf{elif}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \leq 6.109391534461873 \cdot 10^{+302}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\frac{M}{\frac{2 \cdot d}{D}} \cdot \left(\frac{M}{\frac{2 \cdot d}{D}} \cdot h\right)}{\ell}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022100 
(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))))))