Henrywood and Agarwal, Equation (12)

Average Error: 26.6 → 14.5

Time: 14.4s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;M \cdot D \leq 1.2154114847819588 \cdot 10^{-93}:\\ \;\;\;\;\left(\left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}}}\right) \cdot \left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}}\right)\right) \cdot \left(1 - \left(\left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \left(\sqrt[3]{h} \cdot \sqrt[3]{h}\right)\right) \cdot \frac{\sqrt[3]{h}}{\ell}\right)\\ \mathbf{elif}\;M \cdot D \leq 1.6110136472930255 \cdot 10^{+160}:\\ \;\;\;\;\left(\left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}}}\right) \cdot \left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}}\right)\right) \cdot \left(1 - \frac{\sqrt[3]{h}}{\ell} \cdot e^{\log 0.5 + 2 \cdot \log \left(\sqrt[3]{h} \cdot \frac{M \cdot D}{d \cdot 2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left|\sqrt[3]{d}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \end{array}\]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (*
  (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
  (- 1.0 (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l)))))

↓

(FPCore (d h l M D)
 :precision binary64
 (if (<= (* M D) 1.2154114847819588e-93)
   (*
    (*
     (* (fabs (/ (cbrt d) (cbrt h))) (sqrt (/ (cbrt d) (cbrt h))))
     (* (fabs (/ (cbrt d) (cbrt l))) (sqrt (/ (cbrt d) (cbrt l)))))
    (-
     1.0
     (*
      (* (* 0.5 (pow (/ (* M D) (* d 2.0)) 2.0)) (* (cbrt h) (cbrt h)))
      (/ (cbrt h) l))))
   (if (<= (* M D) 1.6110136472930255e+160)
     (*
      (*
       (* (fabs (/ (cbrt d) (cbrt h))) (sqrt (/ (cbrt d) (cbrt h))))
       (* (fabs (/ (cbrt d) (cbrt l))) (sqrt (/ (cbrt d) (cbrt l)))))
      (-
       1.0
       (*
        (/ (cbrt h) l)
        (exp (+ (log 0.5) (* 2.0 (log (* (cbrt h) (/ (* M D) (* d 2.0))))))))))
     (*
      (* (* (fabs (cbrt d)) (sqrt (/ (cbrt d) h))) (sqrt (/ d l)))
      (- 1.0 (* (* 0.5 (pow (/ (* M D) (* d 2.0)) 2.0)) (/ h l)))))))

C

double code(double d, double h, double l, double M, double D) {
	return (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
}

↓

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if ((M * D) <= 1.2154114847819588e-93) {
		tmp = ((fabs(cbrt(d) / cbrt(h)) * sqrt(cbrt(d) / cbrt(h))) * (fabs(cbrt(d) / cbrt(l)) * sqrt(cbrt(d) / cbrt(l)))) * (1.0 - (((0.5 * pow(((M * D) / (d * 2.0)), 2.0)) * (cbrt(h) * cbrt(h))) * (cbrt(h) / l)));
	} else if ((M * D) <= 1.6110136472930255e+160) {
		tmp = ((fabs(cbrt(d) / cbrt(h)) * sqrt(cbrt(d) / cbrt(h))) * (fabs(cbrt(d) / cbrt(l)) * sqrt(cbrt(d) / cbrt(l)))) * (1.0 - ((cbrt(h) / l) * exp(log(0.5) + (2.0 * log(cbrt(h) * ((M * D) / (d * 2.0)))))));
	} else {
		tmp = ((fabs(cbrt(d)) * sqrt(cbrt(d) / h)) * sqrt(d / l)) * (1.0 - ((0.5 * pow(((M * D) / (d * 2.0)), 2.0)) * (h / l)));
	}
	return tmp;
}

Error

Bits error versus d

Bits error versus h

Bits error versus l

Bits error versus M

Bits error versus D

Derivation

1. Split input into 3 regimes

2. if (*.f64 M D) < 1.2154114847819588e-93

   1. Initial program 24.9

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

   2. Simplified 24.9

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}\]
   3. Using strategy rm

   4. Applied add-cube-cbrt_binary64 25.2

      \[\leadsto \left(\sqrt{\frac{d}{\color{blue}{\left(\sqrt[3]{h} \cdot \sqrt[3]{h}\right) \cdot \sqrt[3]{h}}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]

   5. Applied add-cube-cbrt_binary64 25.3

      \[\leadsto \left(\sqrt{\frac{\color{blue}{\left(\sqrt[3]{d} \cdot \sqrt[3]{d}\right) \cdot \sqrt[3]{d}}}{\left(\sqrt[3]{h} \cdot \sqrt[3]{h}\right) \cdot \sqrt[3]{h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]

   6. Applied times-frac_binary64 25.3

      \[\leadsto \left(\sqrt{\color{blue}{\frac{\sqrt[3]{d} \cdot \sqrt[3]{d}}{\sqrt[3]{h} \cdot \sqrt[3]{h}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{h}}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]

   7. Applied sqrt-prod_binary64 20.4

      \[\leadsto \left(\color{blue}{\left(\sqrt{\frac{\sqrt[3]{d} \cdot \sqrt[3]{d}}{\sqrt[3]{h} \cdot \sqrt[3]{h}}} \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}}}\right)} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]

   8. Simplified 19.8

      \[\leadsto \left(\left(\color{blue}{\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right|} \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}}}\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]
   9. Using strategy rm

   10. Applied add-cube-cbrt_binary64 19.9

       \[\leadsto \left(\left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}}}\right) \cdot \sqrt{\frac{d}{\color{blue}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}}}}\right) \cdot \left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]

   11. Applied add-cube-cbrt_binary64 20.0

       \[\leadsto \left(\left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}}}\right) \cdot \sqrt{\frac{\color{blue}{\left(\sqrt[3]{d} \cdot \sqrt[3]{d}\right) \cdot \sqrt[3]{d}}}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}}}\right) \cdot \left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]

   12. Applied times-frac_binary64 20.0

       \[\leadsto \left(\left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}}}\right) \cdot \sqrt{\color{blue}{\frac{\sqrt[3]{d} \cdot \sqrt[3]{d}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}}}\right) \cdot \left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]

   13. Applied sqrt-prod_binary64 15.5

       \[\leadsto \left(\left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}}}\right) \cdot \color{blue}{\left(\sqrt{\frac{\sqrt[3]{d} \cdot \sqrt[3]{d}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}} \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}}\right)}\right) \cdot \left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]

   14. Simplified 15.3

       \[\leadsto \left(\left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}}}\right) \cdot \left(\color{blue}{\left|\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right|} \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}}\right)\right) \cdot \left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]
   15. Using strategy rm

   16. Applied *-un-lft-identity_binary64 15.3

       \[\leadsto \left(\left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}}}\right) \cdot \left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}}\right)\right) \cdot \left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\color{blue}{1 \cdot \ell}}\right)\]

   17. Applied add-cube-cbrt_binary64 15.3

       \[\leadsto \left(\left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}}}\right) \cdot \left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}}\right)\right) \cdot \left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{\color{blue}{\left(\sqrt[3]{h} \cdot \sqrt[3]{h}\right) \cdot \sqrt[3]{h}}}{1 \cdot \ell}\right)\]

   18. Applied times-frac_binary64 15.3

       \[\leadsto \left(\left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}}}\right) \cdot \left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}}\right)\right) \cdot \left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \color{blue}{\left(\frac{\sqrt[3]{h} \cdot \sqrt[3]{h}}{1} \cdot \frac{\sqrt[3]{h}}{\ell}\right)}\right)\]

   19. Applied associate-*r*_binary64 12.4

       \[\leadsto \left(\left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}}}\right) \cdot \left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}}\right)\right) \cdot \left(1 - \color{blue}{\left(\left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{\sqrt[3]{h} \cdot \sqrt[3]{h}}{1}\right) \cdot \frac{\sqrt[3]{h}}{\ell}}\right)\]

   20. Simplified 12.4

       \[\leadsto \left(\left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}}}\right) \cdot \left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}}\right)\right) \cdot \left(1 - \color{blue}{\left(\left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \left(\sqrt[3]{h} \cdot \sqrt[3]{h}\right)\right)} \cdot \frac{\sqrt[3]{h}}{\ell}\right)\]

   if 1.2154114847819588e-93 < (*.f64 M D) < 1.61101364729302546e160

   1. Initial program 29.6

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

   2. Simplified 29.6

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}\]
   3. Using strategy rm

   4. Applied add-cube-cbrt_binary64 29.9

      \[\leadsto \left(\sqrt{\frac{d}{\color{blue}{\left(\sqrt[3]{h} \cdot \sqrt[3]{h}\right) \cdot \sqrt[3]{h}}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]

   5. Applied add-cube-cbrt_binary64 30.0

      \[\leadsto \left(\sqrt{\frac{\color{blue}{\left(\sqrt[3]{d} \cdot \sqrt[3]{d}\right) \cdot \sqrt[3]{d}}}{\left(\sqrt[3]{h} \cdot \sqrt[3]{h}\right) \cdot \sqrt[3]{h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]

   6. Applied times-frac_binary64 30.0

      \[\leadsto \left(\sqrt{\color{blue}{\frac{\sqrt[3]{d} \cdot \sqrt[3]{d}}{\sqrt[3]{h} \cdot \sqrt[3]{h}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{h}}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]

   7. Applied sqrt-prod_binary64 24.6

      \[\leadsto \left(\color{blue}{\left(\sqrt{\frac{\sqrt[3]{d} \cdot \sqrt[3]{d}}{\sqrt[3]{h} \cdot \sqrt[3]{h}}} \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}}}\right)} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]

   8. Simplified 23.5

      \[\leadsto \left(\left(\color{blue}{\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right|} \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}}}\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]
   9. Using strategy rm

   10. Applied add-cube-cbrt_binary64 23.6

       \[\leadsto \left(\left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}}}\right) \cdot \sqrt{\frac{d}{\color{blue}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}}}}\right) \cdot \left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]

   11. Applied add-cube-cbrt_binary64 23.8

       \[\leadsto \left(\left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}}}\right) \cdot \sqrt{\frac{\color{blue}{\left(\sqrt[3]{d} \cdot \sqrt[3]{d}\right) \cdot \sqrt[3]{d}}}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}}}\right) \cdot \left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]

   12. Applied times-frac_binary64 23.8

       \[\leadsto \left(\left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}}}\right) \cdot \sqrt{\color{blue}{\frac{\sqrt[3]{d} \cdot \sqrt[3]{d}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}}}\right) \cdot \left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]

   13. Applied sqrt-prod_binary64 20.6

       \[\leadsto \left(\left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}}}\right) \cdot \color{blue}{\left(\sqrt{\frac{\sqrt[3]{d} \cdot \sqrt[3]{d}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}} \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}}\right)}\right) \cdot \left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]

   14. Simplified 20.6

       \[\leadsto \left(\left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}}}\right) \cdot \left(\color{blue}{\left|\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right|} \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}}\right)\right) \cdot \left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]
   15. Using strategy rm

   16. Applied *-un-lft-identity_binary64 20.6

       \[\leadsto \left(\left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}}}\right) \cdot \left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}}\right)\right) \cdot \left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\color{blue}{1 \cdot \ell}}\right)\]

   17. Applied add-cube-cbrt_binary64 20.6

       \[\leadsto \left(\left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}}}\right) \cdot \left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}}\right)\right) \cdot \left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{\color{blue}{\left(\sqrt[3]{h} \cdot \sqrt[3]{h}\right) \cdot \sqrt[3]{h}}}{1 \cdot \ell}\right)\]

   18. Applied times-frac_binary64 20.6

       \[\leadsto \left(\left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}}}\right) \cdot \left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}}\right)\right) \cdot \left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \color{blue}{\left(\frac{\sqrt[3]{h} \cdot \sqrt[3]{h}}{1} \cdot \frac{\sqrt[3]{h}}{\ell}\right)}\right)\]

   19. Applied associate-*r*_binary64 19.3

       \[\leadsto \left(\left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}}}\right) \cdot \left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}}\right)\right) \cdot \left(1 - \color{blue}{\left(\left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{\sqrt[3]{h} \cdot \sqrt[3]{h}}{1}\right) \cdot \frac{\sqrt[3]{h}}{\ell}}\right)\]

   20. Simplified 19.3

       \[\leadsto \left(\left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}}}\right) \cdot \left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}}\right)\right) \cdot \left(1 - \color{blue}{\left(\left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \left(\sqrt[3]{h} \cdot \sqrt[3]{h}\right)\right)} \cdot \frac{\sqrt[3]{h}}{\ell}\right)\]
   21. Using strategy rm

   22. Applied add-exp-log_binary64 41.8

       \[\leadsto \left(\left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}}}\right) \cdot \left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}}\right)\right) \cdot \left(1 - \left(\left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \left(\sqrt[3]{h} \cdot \color{blue}{e^{\log \left(\sqrt[3]{h}\right)}}\right)\right) \cdot \frac{\sqrt[3]{h}}{\ell}\right)\]

   23. Applied add-exp-log_binary64 41.8

       \[\leadsto \left(\left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}}}\right) \cdot \left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}}\right)\right) \cdot \left(1 - \left(\left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \left(\color{blue}{e^{\log \left(\sqrt[3]{h}\right)}} \cdot e^{\log \left(\sqrt[3]{h}\right)}\right)\right) \cdot \frac{\sqrt[3]{h}}{\ell}\right)\]

   24. Applied prod-exp_binary64 41.8

       \[\leadsto \left(\left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}}}\right) \cdot \left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}}\right)\right) \cdot \left(1 - \left(\left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \color{blue}{e^{\log \left(\sqrt[3]{h}\right) + \log \left(\sqrt[3]{h}\right)}}\right) \cdot \frac{\sqrt[3]{h}}{\ell}\right)\]

   25. Applied pow-to-exp_binary64 41.9

       \[\leadsto \left(\left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}}}\right) \cdot \left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}}\right)\right) \cdot \left(1 - \left(\left(0.5 \cdot \color{blue}{e^{\log \left(\frac{M \cdot D}{d \cdot 2}\right) \cdot 2}}\right) \cdot e^{\log \left(\sqrt[3]{h}\right) + \log \left(\sqrt[3]{h}\right)}\right) \cdot \frac{\sqrt[3]{h}}{\ell}\right)\]

   26. Applied add-exp-log_binary64 41.9

       \[\leadsto \left(\left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}}}\right) \cdot \left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}}\right)\right) \cdot \left(1 - \left(\left(\color{blue}{e^{\log 0.5}} \cdot e^{\log \left(\frac{M \cdot D}{d \cdot 2}\right) \cdot 2}\right) \cdot e^{\log \left(\sqrt[3]{h}\right) + \log \left(\sqrt[3]{h}\right)}\right) \cdot \frac{\sqrt[3]{h}}{\ell}\right)\]

   27. Applied prod-exp_binary64 42.0

       \[\leadsto \left(\left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}}}\right) \cdot \left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}}\right)\right) \cdot \left(1 - \left(\color{blue}{e^{\log 0.5 + \log \left(\frac{M \cdot D}{d \cdot 2}\right) \cdot 2}} \cdot e^{\log \left(\sqrt[3]{h}\right) + \log \left(\sqrt[3]{h}\right)}\right) \cdot \frac{\sqrt[3]{h}}{\ell}\right)\]

   28. Applied prod-exp_binary64 40.7

       \[\leadsto \left(\left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}}}\right) \cdot \left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}}\right)\right) \cdot \left(1 - \color{blue}{e^{\left(\log 0.5 + \log \left(\frac{M \cdot D}{d \cdot 2}\right) \cdot 2\right) + \left(\log \left(\sqrt[3]{h}\right) + \log \left(\sqrt[3]{h}\right)\right)}} \cdot \frac{\sqrt[3]{h}}{\ell}\right)\]

   29. Simplified 17.4

       \[\leadsto \left(\left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}}}\right) \cdot \left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}}\right)\right) \cdot \left(1 - e^{\color{blue}{\log 0.5 + 2 \cdot \log \left(\frac{M \cdot D}{d \cdot 2} \cdot \sqrt[3]{h}\right)}} \cdot \frac{\sqrt[3]{h}}{\ell}\right)\]

   if 1.61101364729302546e160 < (*.f64 M D)

   1. Initial program 42.9

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

   2. Simplified 42.9

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}\]
   3. Using strategy rm

   4. Applied *-un-lft-identity_binary64 42.9

      \[\leadsto \left(\sqrt{\frac{d}{\color{blue}{1 \cdot h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]

   5. Applied add-cube-cbrt_binary64 43.0

      \[\leadsto \left(\sqrt{\frac{\color{blue}{\left(\sqrt[3]{d} \cdot \sqrt[3]{d}\right) \cdot \sqrt[3]{d}}}{1 \cdot h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]

   6. Applied times-frac_binary64 43.0

      \[\leadsto \left(\sqrt{\color{blue}{\frac{\sqrt[3]{d} \cdot \sqrt[3]{d}}{1} \cdot \frac{\sqrt[3]{d}}{h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]

   7. Applied sqrt-prod_binary64 38.2

      \[\leadsto \left(\color{blue}{\left(\sqrt{\frac{\sqrt[3]{d} \cdot \sqrt[3]{d}}{1}} \cdot \sqrt{\frac{\sqrt[3]{d}}{h}}\right)} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]

   8. Simplified 38.2

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

4. Final simplification 14.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \cdot D \leq 1.2154114847819588 \cdot 10^{-93}:\\ \;\;\;\;\left(\left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}}}\right) \cdot \left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}}\right)\right) \cdot \left(1 - \left(\left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \left(\sqrt[3]{h} \cdot \sqrt[3]{h}\right)\right) \cdot \frac{\sqrt[3]{h}}{\ell}\right)\\ \mathbf{elif}\;M \cdot D \leq 1.6110136472930255 \cdot 10^{+160}:\\ \;\;\;\;\left(\left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}}}\right) \cdot \left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}}\right)\right) \cdot \left(1 - \frac{\sqrt[3]{h}}{\ell} \cdot e^{\log 0.5 + 2 \cdot \log \left(\sqrt[3]{h} \cdot \frac{M \cdot D}{d \cdot 2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left|\sqrt[3]{d}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020232 
(FPCore (d h l M D)
  :name "Henrywood and Agarwal, Equation (12)"
  :precision binary64
  (* (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0))) (- 1.0 (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l)))))