Henrywood and Agarwal, Equation (13)

Average Error: 59.5 → 19.1

Time: 27.8s

Precision: binary64

Math

\[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

↓

\[\begin{array}{l} t_0 := \sqrt[3]{\sqrt[3]{d} \cdot \sqrt[3]{d}}\\ t_1 := \frac{D}{\frac{\sqrt[3]{\sqrt[3]{d}}}{\frac{M}{\sqrt[3]{d}}}}\\ t_2 := \frac{c0}{2 \cdot w}\\ t_3 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ t_4 := t_2 \cdot \left(t_3 + \sqrt{t_3 \cdot t_3 - M \cdot M}\right)\\ t_5 := \frac{D}{\sqrt[3]{d}} \cdot \frac{h}{d}\\ t_6 := 0.25 \cdot \left(\left(t_5 \cdot \frac{M}{t_0}\right) \cdot t_1\right)\\ \mathbf{if}\;t_4 \leq -\infty:\\ \;\;\;\;t_6\\ \mathbf{elif}\;t_4 \leq -2.0809617348611996 \cdot 10^{-152}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;t_4 \leq 0:\\ \;\;\;\;0.25 \cdot \left(t_1 \cdot \left(t_5 \cdot \frac{1}{\frac{t_0}{M}}\right)\right)\\ \mathbf{elif}\;t_4 \leq 5.764322879322445 \cdot 10^{+250}:\\ \;\;\;\;t_2 \cdot \left(2 \cdot \frac{c0 \cdot {d}^{2}}{\left(w \cdot h\right) \cdot {D}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_6\\ \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (*
  (/ c0 (* 2.0 w))
  (+
   (/ (* c0 (* d d)) (* (* w h) (* D D)))
   (sqrt
    (-
     (*
      (/ (* c0 (* d d)) (* (* w h) (* D D)))
      (/ (* c0 (* d d)) (* (* w h) (* D D))))
     (* M M))))))

↓

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (cbrt (* (cbrt d) (cbrt d))))
        (t_1 (/ D (/ (cbrt (cbrt d)) (/ M (cbrt d)))))
        (t_2 (/ c0 (* 2.0 w)))
        (t_3 (/ (* c0 (* d d)) (* (* w h) (* D D))))
        (t_4 (* t_2 (+ t_3 (sqrt (- (* t_3 t_3) (* M M))))))
        (t_5 (* (/ D (cbrt d)) (/ h d)))
        (t_6 (* 0.25 (* (* t_5 (/ M t_0)) t_1))))
   (if (<= t_4 (- INFINITY))
     t_6
     (if (<= t_4 -2.0809617348611996e-152)
       t_4
       (if (<= t_4 0.0)
         (* 0.25 (* t_1 (* t_5 (/ 1.0 (/ t_0 M)))))
         (if (<= t_4 5.764322879322445e+250)
           (* t_2 (* 2.0 (/ (* c0 (pow d 2.0)) (* (* w h) (pow D 2.0)))))
           t_6))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	return (c0 / (2.0 * w)) * (((c0 * (d * d)) / ((w * h) * (D * D))) + sqrt((((c0 * (d * d)) / ((w * h) * (D * D))) * ((c0 * (d * d)) / ((w * h) * (D * D)))) - (M * M)));
}

↓

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = cbrt(cbrt(d) * cbrt(d));
	double t_1 = D / (cbrt(cbrt(d)) / (M / cbrt(d)));
	double t_2 = c0 / (2.0 * w);
	double t_3 = (c0 * (d * d)) / ((w * h) * (D * D));
	double t_4 = t_2 * (t_3 + sqrt((t_3 * t_3) - (M * M)));
	double t_5 = (D / cbrt(d)) * (h / d);
	double t_6 = 0.25 * ((t_5 * (M / t_0)) * t_1);
	double tmp;
	if (t_4 <= -((double) INFINITY)) {
		tmp = t_6;
	} else if (t_4 <= -2.0809617348611996e-152) {
		tmp = t_4;
	} else if (t_4 <= 0.0) {
		tmp = 0.25 * (t_1 * (t_5 * (1.0 / (t_0 / M))));
	} else if (t_4 <= 5.764322879322445e+250) {
		tmp = t_2 * (2.0 * ((c0 * pow(d, 2.0)) / ((w * h) * pow(D, 2.0))));
	} else {
		tmp = t_6;
	}
	return tmp;
}

Error

Bits error versus c0

Bits error versus w

Bits error versus h

Bits error versus D

Bits error versus d

Bits error versus M

Derivation

1. Split input into 4 regimes

2. if (*.f64 (/.f64 c0 (*.f64 2 w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M))))) < -inf.0 or 5.76432287932244541e250 < (*.f64 (/.f64 c0 (*.f64 2 w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M)))))

   1. Initial program 63.9

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   2. Taylor expanded in c0 around -inf 35.5

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

   3. Applied associate-/l*_binary64 35.4

      \[\leadsto 0.25 \cdot \color{blue}{\frac{{D}^{2}}{\frac{{d}^{2}}{{M}^{2} \cdot h}}} \]

   4. Simplified 32.9

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

   5. Applied add-cube-cbrt_binary64 32.9

      \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{d}{\frac{h \cdot \left(M \cdot M\right)}{\color{blue}{\left(\sqrt[3]{d} \cdot \sqrt[3]{d}\right) \cdot \sqrt[3]{d}}}}} \]

   6. Applied times-frac_binary64 31.9

      \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{d}{\color{blue}{\frac{h}{\sqrt[3]{d} \cdot \sqrt[3]{d}} \cdot \frac{M \cdot M}{\sqrt[3]{d}}}}} \]

   7. Applied add-cube-cbrt_binary64 31.9

      \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\color{blue}{\left(\sqrt[3]{d} \cdot \sqrt[3]{d}\right) \cdot \sqrt[3]{d}}}{\frac{h}{\sqrt[3]{d} \cdot \sqrt[3]{d}} \cdot \frac{M \cdot M}{\sqrt[3]{d}}}} \]

   8. Applied times-frac_binary64 32.4

      \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\color{blue}{\frac{\sqrt[3]{d} \cdot \sqrt[3]{d}}{\frac{h}{\sqrt[3]{d} \cdot \sqrt[3]{d}}} \cdot \frac{\sqrt[3]{d}}{\frac{M \cdot M}{\sqrt[3]{d}}}}} \]

   9. Applied add-sqr-sqrt_binary64 48.3

      \[\leadsto 0.25 \cdot \frac{{\color{blue}{\left(\sqrt{D} \cdot \sqrt{D}\right)}}^{2}}{\frac{\sqrt[3]{d} \cdot \sqrt[3]{d}}{\frac{h}{\sqrt[3]{d} \cdot \sqrt[3]{d}}} \cdot \frac{\sqrt[3]{d}}{\frac{M \cdot M}{\sqrt[3]{d}}}} \]

   10. Applied unpow-prod-down_binary64 48.3

       \[\leadsto 0.25 \cdot \frac{\color{blue}{{\left(\sqrt{D}\right)}^{2} \cdot {\left(\sqrt{D}\right)}^{2}}}{\frac{\sqrt[3]{d} \cdot \sqrt[3]{d}}{\frac{h}{\sqrt[3]{d} \cdot \sqrt[3]{d}}} \cdot \frac{\sqrt[3]{d}}{\frac{M \cdot M}{\sqrt[3]{d}}}} \]

   11. Applied times-frac_binary64 46.0

       \[\leadsto 0.25 \cdot \color{blue}{\left(\frac{{\left(\sqrt{D}\right)}^{2}}{\frac{\sqrt[3]{d} \cdot \sqrt[3]{d}}{\frac{h}{\sqrt[3]{d} \cdot \sqrt[3]{d}}}} \cdot \frac{{\left(\sqrt{D}\right)}^{2}}{\frac{\sqrt[3]{d}}{\frac{M \cdot M}{\sqrt[3]{d}}}}\right)} \]

   12. Simplified 45.6

       \[\leadsto 0.25 \cdot \left(\color{blue}{\left(\frac{D}{\sqrt[3]{d}} \cdot \frac{h}{d}\right)} \cdot \frac{{\left(\sqrt{D}\right)}^{2}}{\frac{\sqrt[3]{d}}{\frac{M \cdot M}{\sqrt[3]{d}}}}\right) \]

   13. Simplified 27.2

       \[\leadsto 0.25 \cdot \left(\left(\frac{D}{\sqrt[3]{d}} \cdot \frac{h}{d}\right) \cdot \color{blue}{\frac{D}{\frac{\sqrt[3]{d}}{\frac{M \cdot M}{\sqrt[3]{d}}}}}\right) \]

   14. Applied *-un-lft-identity_binary64 27.2

       \[\leadsto 0.25 \cdot \left(\left(\frac{D}{\sqrt[3]{d}} \cdot \frac{h}{d}\right) \cdot \frac{D}{\frac{\sqrt[3]{d}}{\frac{M \cdot M}{\sqrt[3]{\color{blue}{1 \cdot d}}}}}\right) \]

   15. Applied cbrt-prod_binary64 27.2

       \[\leadsto 0.25 \cdot \left(\left(\frac{D}{\sqrt[3]{d}} \cdot \frac{h}{d}\right) \cdot \frac{D}{\frac{\sqrt[3]{d}}{\frac{M \cdot M}{\color{blue}{\sqrt[3]{1} \cdot \sqrt[3]{d}}}}}\right) \]

   16. Applied times-frac_binary64 26.1

       \[\leadsto 0.25 \cdot \left(\left(\frac{D}{\sqrt[3]{d}} \cdot \frac{h}{d}\right) \cdot \frac{D}{\frac{\sqrt[3]{d}}{\color{blue}{\frac{M}{\sqrt[3]{1}} \cdot \frac{M}{\sqrt[3]{d}}}}}\right) \]

   17. Applied add-cube-cbrt_binary64 26.1

       \[\leadsto 0.25 \cdot \left(\left(\frac{D}{\sqrt[3]{d}} \cdot \frac{h}{d}\right) \cdot \frac{D}{\frac{\sqrt[3]{\color{blue}{\left(\sqrt[3]{d} \cdot \sqrt[3]{d}\right) \cdot \sqrt[3]{d}}}}{\frac{M}{\sqrt[3]{1}} \cdot \frac{M}{\sqrt[3]{d}}}}\right) \]

   18. Applied cbrt-prod_binary64 26.1

       \[\leadsto 0.25 \cdot \left(\left(\frac{D}{\sqrt[3]{d}} \cdot \frac{h}{d}\right) \cdot \frac{D}{\frac{\color{blue}{\sqrt[3]{\sqrt[3]{d} \cdot \sqrt[3]{d}} \cdot \sqrt[3]{\sqrt[3]{d}}}}{\frac{M}{\sqrt[3]{1}} \cdot \frac{M}{\sqrt[3]{d}}}}\right) \]

   19. Applied times-frac_binary64 24.9

       \[\leadsto 0.25 \cdot \left(\left(\frac{D}{\sqrt[3]{d}} \cdot \frac{h}{d}\right) \cdot \frac{D}{\color{blue}{\frac{\sqrt[3]{\sqrt[3]{d} \cdot \sqrt[3]{d}}}{\frac{M}{\sqrt[3]{1}}} \cdot \frac{\sqrt[3]{\sqrt[3]{d}}}{\frac{M}{\sqrt[3]{d}}}}}\right) \]

   20. Applied *-un-lft-identity_binary64 24.9

       \[\leadsto 0.25 \cdot \left(\left(\frac{D}{\sqrt[3]{d}} \cdot \frac{h}{d}\right) \cdot \frac{\color{blue}{1 \cdot D}}{\frac{\sqrt[3]{\sqrt[3]{d} \cdot \sqrt[3]{d}}}{\frac{M}{\sqrt[3]{1}}} \cdot \frac{\sqrt[3]{\sqrt[3]{d}}}{\frac{M}{\sqrt[3]{d}}}}\right) \]

   21. Applied times-frac_binary64 21.7

       \[\leadsto 0.25 \cdot \left(\left(\frac{D}{\sqrt[3]{d}} \cdot \frac{h}{d}\right) \cdot \color{blue}{\left(\frac{1}{\frac{\sqrt[3]{\sqrt[3]{d} \cdot \sqrt[3]{d}}}{\frac{M}{\sqrt[3]{1}}}} \cdot \frac{D}{\frac{\sqrt[3]{\sqrt[3]{d}}}{\frac{M}{\sqrt[3]{d}}}}\right)}\right) \]

   22. Applied associate-*r*_binary64 19.9

       \[\leadsto 0.25 \cdot \color{blue}{\left(\left(\left(\frac{D}{\sqrt[3]{d}} \cdot \frac{h}{d}\right) \cdot \frac{1}{\frac{\sqrt[3]{\sqrt[3]{d} \cdot \sqrt[3]{d}}}{\frac{M}{\sqrt[3]{1}}}}\right) \cdot \frac{D}{\frac{\sqrt[3]{\sqrt[3]{d}}}{\frac{M}{\sqrt[3]{d}}}}\right)} \]

   23. Simplified 19.9

       \[\leadsto 0.25 \cdot \left(\color{blue}{\left(\left(\frac{D}{\sqrt[3]{d}} \cdot \frac{h}{d}\right) \cdot \frac{M}{\sqrt[3]{\sqrt[3]{d} \cdot \sqrt[3]{d}}}\right)} \cdot \frac{D}{\frac{\sqrt[3]{\sqrt[3]{d}}}{\frac{M}{\sqrt[3]{d}}}}\right) \]

   if -inf.0 < (*.f64 (/.f64 c0 (*.f64 2 w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M))))) < -2.08096173486119964e-152

   1. Initial program 5.3

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   if -2.08096173486119964e-152 < (*.f64 (/.f64 c0 (*.f64 2 w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M))))) < 0.0

   1. Initial program 28.6

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   2. Taylor expanded in c0 around -inf 26.3

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

   3. Applied associate-/l*_binary64 26.1

      \[\leadsto 0.25 \cdot \color{blue}{\frac{{D}^{2}}{\frac{{d}^{2}}{{M}^{2} \cdot h}}} \]

   4. Simplified 23.7

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

   5. Applied add-cube-cbrt_binary64 23.8

      \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{d}{\frac{h \cdot \left(M \cdot M\right)}{\color{blue}{\left(\sqrt[3]{d} \cdot \sqrt[3]{d}\right) \cdot \sqrt[3]{d}}}}} \]

   6. Applied times-frac_binary64 23.2

      \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{d}{\color{blue}{\frac{h}{\sqrt[3]{d} \cdot \sqrt[3]{d}} \cdot \frac{M \cdot M}{\sqrt[3]{d}}}}} \]

   7. Applied add-cube-cbrt_binary64 23.2

      \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\frac{\color{blue}{\left(\sqrt[3]{d} \cdot \sqrt[3]{d}\right) \cdot \sqrt[3]{d}}}{\frac{h}{\sqrt[3]{d} \cdot \sqrt[3]{d}} \cdot \frac{M \cdot M}{\sqrt[3]{d}}}} \]

   8. Applied times-frac_binary64 24.0

      \[\leadsto 0.25 \cdot \frac{{D}^{2}}{\color{blue}{\frac{\sqrt[3]{d} \cdot \sqrt[3]{d}}{\frac{h}{\sqrt[3]{d} \cdot \sqrt[3]{d}}} \cdot \frac{\sqrt[3]{d}}{\frac{M \cdot M}{\sqrt[3]{d}}}}} \]

   9. Applied add-sqr-sqrt_binary64 45.0

      \[\leadsto 0.25 \cdot \frac{{\color{blue}{\left(\sqrt{D} \cdot \sqrt{D}\right)}}^{2}}{\frac{\sqrt[3]{d} \cdot \sqrt[3]{d}}{\frac{h}{\sqrt[3]{d} \cdot \sqrt[3]{d}}} \cdot \frac{\sqrt[3]{d}}{\frac{M \cdot M}{\sqrt[3]{d}}}} \]

   10. Applied unpow-prod-down_binary64 45.0

       \[\leadsto 0.25 \cdot \frac{\color{blue}{{\left(\sqrt{D}\right)}^{2} \cdot {\left(\sqrt{D}\right)}^{2}}}{\frac{\sqrt[3]{d} \cdot \sqrt[3]{d}}{\frac{h}{\sqrt[3]{d} \cdot \sqrt[3]{d}}} \cdot \frac{\sqrt[3]{d}}{\frac{M \cdot M}{\sqrt[3]{d}}}} \]

   11. Applied times-frac_binary64 43.9

       \[\leadsto 0.25 \cdot \color{blue}{\left(\frac{{\left(\sqrt{D}\right)}^{2}}{\frac{\sqrt[3]{d} \cdot \sqrt[3]{d}}{\frac{h}{\sqrt[3]{d} \cdot \sqrt[3]{d}}}} \cdot \frac{{\left(\sqrt{D}\right)}^{2}}{\frac{\sqrt[3]{d}}{\frac{M \cdot M}{\sqrt[3]{d}}}}\right)} \]

   12. Simplified 43.7

       \[\leadsto 0.25 \cdot \left(\color{blue}{\left(\frac{D}{\sqrt[3]{d}} \cdot \frac{h}{d}\right)} \cdot \frac{{\left(\sqrt{D}\right)}^{2}}{\frac{\sqrt[3]{d}}{\frac{M \cdot M}{\sqrt[3]{d}}}}\right) \]

   13. Simplified 20.9

       \[\leadsto 0.25 \cdot \left(\left(\frac{D}{\sqrt[3]{d}} \cdot \frac{h}{d}\right) \cdot \color{blue}{\frac{D}{\frac{\sqrt[3]{d}}{\frac{M \cdot M}{\sqrt[3]{d}}}}}\right) \]

   14. Applied *-un-lft-identity_binary64 20.9

       \[\leadsto 0.25 \cdot \left(\left(\frac{D}{\sqrt[3]{d}} \cdot \frac{h}{d}\right) \cdot \frac{D}{\frac{\sqrt[3]{d}}{\frac{M \cdot M}{\sqrt[3]{\color{blue}{1 \cdot d}}}}}\right) \]

   15. Applied cbrt-prod_binary64 20.9

       \[\leadsto 0.25 \cdot \left(\left(\frac{D}{\sqrt[3]{d}} \cdot \frac{h}{d}\right) \cdot \frac{D}{\frac{\sqrt[3]{d}}{\frac{M \cdot M}{\color{blue}{\sqrt[3]{1} \cdot \sqrt[3]{d}}}}}\right) \]

   16. Applied times-frac_binary64 20.5

       \[\leadsto 0.25 \cdot \left(\left(\frac{D}{\sqrt[3]{d}} \cdot \frac{h}{d}\right) \cdot \frac{D}{\frac{\sqrt[3]{d}}{\color{blue}{\frac{M}{\sqrt[3]{1}} \cdot \frac{M}{\sqrt[3]{d}}}}}\right) \]

   17. Applied add-cube-cbrt_binary64 20.5

       \[\leadsto 0.25 \cdot \left(\left(\frac{D}{\sqrt[3]{d}} \cdot \frac{h}{d}\right) \cdot \frac{D}{\frac{\sqrt[3]{\color{blue}{\left(\sqrt[3]{d} \cdot \sqrt[3]{d}\right) \cdot \sqrt[3]{d}}}}{\frac{M}{\sqrt[3]{1}} \cdot \frac{M}{\sqrt[3]{d}}}}\right) \]

   18. Applied cbrt-prod_binary64 20.5

       \[\leadsto 0.25 \cdot \left(\left(\frac{D}{\sqrt[3]{d}} \cdot \frac{h}{d}\right) \cdot \frac{D}{\frac{\color{blue}{\sqrt[3]{\sqrt[3]{d} \cdot \sqrt[3]{d}} \cdot \sqrt[3]{\sqrt[3]{d}}}}{\frac{M}{\sqrt[3]{1}} \cdot \frac{M}{\sqrt[3]{d}}}}\right) \]

   19. Applied times-frac_binary64 20.0

       \[\leadsto 0.25 \cdot \left(\left(\frac{D}{\sqrt[3]{d}} \cdot \frac{h}{d}\right) \cdot \frac{D}{\color{blue}{\frac{\sqrt[3]{\sqrt[3]{d} \cdot \sqrt[3]{d}}}{\frac{M}{\sqrt[3]{1}}} \cdot \frac{\sqrt[3]{\sqrt[3]{d}}}{\frac{M}{\sqrt[3]{d}}}}}\right) \]

   20. Applied *-un-lft-identity_binary64 20.0

       \[\leadsto 0.25 \cdot \left(\left(\frac{D}{\sqrt[3]{d}} \cdot \frac{h}{d}\right) \cdot \frac{\color{blue}{1 \cdot D}}{\frac{\sqrt[3]{\sqrt[3]{d} \cdot \sqrt[3]{d}}}{\frac{M}{\sqrt[3]{1}}} \cdot \frac{\sqrt[3]{\sqrt[3]{d}}}{\frac{M}{\sqrt[3]{d}}}}\right) \]

   21. Applied times-frac_binary64 18.3

       \[\leadsto 0.25 \cdot \left(\left(\frac{D}{\sqrt[3]{d}} \cdot \frac{h}{d}\right) \cdot \color{blue}{\left(\frac{1}{\frac{\sqrt[3]{\sqrt[3]{d} \cdot \sqrt[3]{d}}}{\frac{M}{\sqrt[3]{1}}}} \cdot \frac{D}{\frac{\sqrt[3]{\sqrt[3]{d}}}{\frac{M}{\sqrt[3]{d}}}}\right)}\right) \]

   22. Applied associate-*r*_binary64 15.1

       \[\leadsto 0.25 \cdot \color{blue}{\left(\left(\left(\frac{D}{\sqrt[3]{d}} \cdot \frac{h}{d}\right) \cdot \frac{1}{\frac{\sqrt[3]{\sqrt[3]{d} \cdot \sqrt[3]{d}}}{\frac{M}{\sqrt[3]{1}}}}\right) \cdot \frac{D}{\frac{\sqrt[3]{\sqrt[3]{d}}}{\frac{M}{\sqrt[3]{d}}}}\right)} \]

   if 0.0 < (*.f64 (/.f64 c0 (*.f64 2 w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M))))) < 5.76432287932244541e250

   1. Initial program 8.9

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   2. Taylor expanded in c0 around inf 7.5

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(2 \cdot \frac{{d}^{2} \cdot c0}{{D}^{2} \cdot \left(w \cdot h\right)}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 19.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq -\infty:\\ \;\;\;\;0.25 \cdot \left(\left(\left(\frac{D}{\sqrt[3]{d}} \cdot \frac{h}{d}\right) \cdot \frac{M}{\sqrt[3]{\sqrt[3]{d} \cdot \sqrt[3]{d}}}\right) \cdot \frac{D}{\frac{\sqrt[3]{\sqrt[3]{d}}}{\frac{M}{\sqrt[3]{d}}}}\right)\\ \mathbf{elif}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq -2.0809617348611996 \cdot 10^{-152}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\\ \mathbf{elif}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq 0:\\ \;\;\;\;0.25 \cdot \left(\frac{D}{\frac{\sqrt[3]{\sqrt[3]{d}}}{\frac{M}{\sqrt[3]{d}}}} \cdot \left(\left(\frac{D}{\sqrt[3]{d}} \cdot \frac{h}{d}\right) \cdot \frac{1}{\frac{\sqrt[3]{\sqrt[3]{d} \cdot \sqrt[3]{d}}}{M}}\right)\right)\\ \mathbf{elif}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq 5.764322879322445 \cdot 10^{+250}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{c0 \cdot {d}^{2}}{\left(w \cdot h\right) \cdot {D}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\left(\left(\frac{D}{\sqrt[3]{d}} \cdot \frac{h}{d}\right) \cdot \frac{M}{\sqrt[3]{\sqrt[3]{d} \cdot \sqrt[3]{d}}}\right) \cdot \frac{D}{\frac{\sqrt[3]{\sqrt[3]{d}}}{\frac{M}{\sqrt[3]{d}}}}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022077 
(FPCore (c0 w h D d M)
  :name "Henrywood and Agarwal, Equation (13)"
  :precision binary64
  (* (/ c0 (* 2.0 w)) (+ (/ (* c0 (* d d)) (* (* w h) (* D D))) (sqrt (- (* (/ (* c0 (* d d)) (* (* w h) (* D D))) (/ (* c0 (* d d)) (* (* w h) (* D D)))) (* M M))))))