Henrywood and Agarwal, Equation (9a)

Average Error: 14.5 → 7.7

Time: 11.5s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} t_0 := \frac{M \cdot D}{2 \cdot d}\\ t_1 := {t_0}^{2} \cdot \frac{h}{\ell}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;\sqrt{\left(\frac{h}{\ell} \cdot {\left(\frac{D}{d}\right)}^{2}\right) \cdot -0.25} \cdot \left(w0 \cdot \left(-M\right)\right)\\ \mathbf{elif}\;t_1 \leq -5.687910821836587 \cdot 10^{+122}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{1}{\frac{2 \cdot d}{M \cdot D}}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(t_0 \cdot h\right) \cdot \frac{\left(M \cdot D\right) \cdot \frac{0.5}{d}}{\ell}}\\ \end{array} \]

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 (* (pow t_0 2.0) (/ h l))))
   (if (<= t_1 (- INFINITY))
     (* (sqrt (* (* (/ h l) (pow (/ D d) 2.0)) -0.25)) (* w0 (- M)))
     (if (<= t_1 -5.687910821836587e+122)
       (*
        w0
        (sqrt (- 1.0 (* (/ h l) (pow (/ 1.0 (/ (* 2.0 d) (* M D))) 2.0)))))
       (* 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 = pow(t_0, 2.0) * (h / l);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = sqrt(((h / l) * pow((D / d), 2.0)) * -0.25) * (w0 * -M);
	} else if (t_1 <= -5.687910821836587e+122) {
		tmp = w0 * sqrt(1.0 - ((h / l) * pow((1.0 / ((2.0 * d) / (M * D))), 2.0)));
	} 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 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l)) < -inf.0

   1. Initial program 64.0

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

   2. Taylor expanded around -inf 56.4

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

   3. Simplified 46.4

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

   if -inf.0 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l)) < -5.6879108218365872e122

   1. Initial program 0.5

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

   3. Applied clear-num_binary64 0.5

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

   if -5.6879108218365872e122 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l))

   1. Initial program 6.7

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

   3. Applied div-inv_binary64 6.8

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

   4. Applied associate-*r*_binary64 3.0

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

   5. Simplified 3.0

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

   7. Applied unpow2_binary64 3.0

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

   8. Applied associate-*r*_binary64 1.9

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

   9. Simplified 1.9

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

   11. Applied associate-*l*_binary64 1.5

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

   12. Simplified 1.4

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

   14. Applied div-inv_binary64 1.4

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

   15. Simplified 1.4

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

4. Final simplification 7.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -\infty:\\ \;\;\;\;\sqrt{\left(\frac{h}{\ell} \cdot {\left(\frac{D}{d}\right)}^{2}\right) \cdot -0.25} \cdot \left(w0 \cdot \left(-M\right)\right)\\ \mathbf{elif}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -5.687910821836587 \cdot 10^{+122}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{1}{\frac{2 \cdot d}{M \cdot D}}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(\frac{M \cdot D}{2 \cdot d} \cdot h\right) \cdot \frac{\left(M \cdot D\right) \cdot \frac{0.5}{d}}{\ell}}\\ \end{array} \]

Reproduce

herbie shell --seed 2021206 
(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))))))