Henrywood and Agarwal, Equation (9a)

Average Error: 13.7 → 8.4

Time: 22.6s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \leq 1.1623878408255412 \cdot 10^{+299}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(h \cdot \frac{\frac{M \cdot D}{2 \cdot d}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}\right) \cdot \frac{\frac{M \cdot D}{2 \cdot d}}{\sqrt[3]{\ell}}}\\ \mathbf{elif}\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \leq \infty:\\ \;\;\;\;w0 \cdot \left(\sqrt{\frac{h \cdot \left(D \cdot D\right)}{d \cdot \left(d \cdot \ell\right)} \cdot -0.25} \cdot \left(-M\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(\frac{M \cdot D}{2 \cdot d} \cdot h\right) \cdot \frac{\frac{1}{\frac{2 \cdot d}{M \cdot D}}}{\ell}}\\ \end{array}\]

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
 (if (<=
      (* w0 (sqrt (- 1.0 (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)))))
      1.1623878408255412e+299)
   (*
    w0
    (sqrt
     (-
      1.0
      (*
       (* h (/ (/ (* M D) (* 2.0 d)) (* (cbrt l) (cbrt l))))
       (/ (/ (* M D) (* 2.0 d)) (cbrt l))))))
   (if (<=
        (* w0 (sqrt (- 1.0 (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)))))
        INFINITY)
     (* w0 (* (sqrt (* (/ (* h (* D D)) (* d (* d l))) -0.25)) (- M)))
     (*
      w0
      (sqrt
       (-
        1.0
        (*
         (* (/ (* M D) (* 2.0 d)) h)
         (/ (/ 1.0 (/ (* 2.0 d) (* M 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 tmp;
	if ((w0 * sqrt(1.0 - (pow(((M * D) / (2.0 * d)), 2.0) * (h / l)))) <= 1.1623878408255412e+299) {
		tmp = w0 * sqrt(1.0 - ((h * (((M * D) / (2.0 * d)) / (cbrt(l) * cbrt(l)))) * (((M * D) / (2.0 * d)) / cbrt(l))));
	} else if ((w0 * sqrt(1.0 - (pow(((M * D) / (2.0 * d)), 2.0) * (h / l)))) <= ((double) INFINITY)) {
		tmp = w0 * (sqrt(((h * (D * D)) / (d * (d * l))) * -0.25) * -M);
	} else {
		tmp = w0 * sqrt(1.0 - ((((M * D) / (2.0 * d)) * h) * ((1.0 / ((2.0 * d) / (M * 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 w0 (sqrt.f64 (-.f64 1 (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l))))) < 1.16238784082554121e299

   1. Initial program 5.0

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

   3. Applied div-inv_binary64_757 5.0

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

   4. Applied associate-*r*_binary64_700 5.1

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

   5. Simplified 5.1

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

   7. Applied associate-*l*_binary64_701 5.3

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

   8. Simplified 5.3

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

   10. Applied add-cube-cbrt_binary64_795 5.4

       \[\leadsto w0 \cdot \sqrt{1 - h \cdot \frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}{\color{blue}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}}}}\]

   11. Applied unpow2_binary64_825 5.4

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

   12. Applied times-frac_binary64_766 5.0

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

   13. Applied associate-*r*_binary64_700 4.2

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

   if 1.16238784082554121e299 < (*.f64 w0 (sqrt.f64 (-.f64 1 (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l))))) < +inf.0

   1. Initial program 58.4

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

   3. Applied div-inv_binary64_757 58.4

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

   4. Applied associate-*r*_binary64_700 53.7

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

   5. Simplified 53.7

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

   7. Applied associate-*l*_binary64_701 53.9

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

   8. Simplified 53.9

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

   10. Applied *-un-lft-identity_binary64_760 53.9

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

   11. Applied unpow2_binary64_825 53.9

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

   12. Applied times-frac_binary64_766 50.4

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

   13. Applied associate-*r*_binary64_700 48.1

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

   14. Simplified 48.1

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

   15. Taylor expanded around -inf 57.9

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

   16. Simplified 55.9

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

   if +inf.0 < (*.f64 w0 (sqrt.f64 (-.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 div-inv_binary64_757 64.0

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

   4. Applied associate-*r*_binary64_700 26.3

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

   5. Simplified 26.3

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

   7. Applied associate-*l*_binary64_701 26.3

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

   8. Simplified 26.3

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

   10. Applied *-un-lft-identity_binary64_760 26.3

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

   11. Applied unpow2_binary64_825 26.3

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

   12. Applied times-frac_binary64_766 14.3

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

   13. Applied associate-*r*_binary64_700 12.7

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

   14. Simplified 12.7

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

   16. Applied clear-num_binary64_759 12.7

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

4. Final simplification 8.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \leq 1.1623878408255412 \cdot 10^{+299}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(h \cdot \frac{\frac{M \cdot D}{2 \cdot d}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}\right) \cdot \frac{\frac{M \cdot D}{2 \cdot d}}{\sqrt[3]{\ell}}}\\ \mathbf{elif}\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \leq \infty:\\ \;\;\;\;w0 \cdot \left(\sqrt{\frac{h \cdot \left(D \cdot D\right)}{d \cdot \left(d \cdot \ell\right)} \cdot -0.25} \cdot \left(-M\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(\frac{M \cdot D}{2 \cdot d} \cdot h\right) \cdot \frac{\frac{1}{\frac{2 \cdot d}{M \cdot D}}}{\ell}}\\ \end{array}\]

Reproduce

herbie shell --seed 2021096 
(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))))))