Henrywood and Agarwal, Equation (12)

Average Error: 26.5 → 11.1

Time: 24.1s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \frac{M \cdot D}{d \cdot 2}\\ t_1 := 1 - \left(0.5 \cdot \left(t_0 \cdot \left(h \cdot t_0\right)\right)\right) \cdot \frac{1}{\ell}\\ t_2 := {\left(\frac{d}{h}\right)}^{0.5}\\ t_3 := \frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\\ t_4 := \left|t_3\right| \cdot \sqrt{t_3}\\ t_5 := 1 - \left(0.5 \cdot {t_0}^{2}\right) \cdot \frac{h}{\ell}\\ t_6 := \left(t_2 \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot t_5\\ \mathbf{if}\;t_6 \leq -1.2609594260553415 \cdot 10^{+155}:\\ \;\;\;\;\left(t_2 \cdot t_4\right) \cdot t_1\\ \mathbf{elif}\;t_6 \leq \infty:\\ \;\;\;\;\begin{array}{l} t_7 := \frac{\sqrt[3]{d}}{\sqrt[3]{h}}\\ t_5 \cdot \left(t_4 \cdot \left(\left|t_7\right| \cdot \sqrt{t_7}\right)\right) \end{array}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(t_4 \cdot \left(\left|\frac{1}{\sqrt[3]{h}}\right| \cdot \sqrt{\frac{d}{\sqrt[3]{h}}}\right)\right)\\ \end{array} \]

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
 (let* ((t_0 (/ (* M D) (* d 2.0)))
        (t_1 (- 1.0 (* (* 0.5 (* t_0 (* h t_0))) (/ 1.0 l))))
        (t_2 (pow (/ d h) 0.5))
        (t_3 (/ (cbrt d) (cbrt l)))
        (t_4 (* (fabs t_3) (sqrt t_3)))
        (t_5 (- 1.0 (* (* 0.5 (pow t_0 2.0)) (/ h l))))
        (t_6 (* (* t_2 (pow (/ d l) 0.5)) t_5)))
   (if (<= t_6 -1.2609594260553415e+155)
     (* (* t_2 t_4) t_1)
     (if (<= t_6 INFINITY)
       (let* ((t_7 (/ (cbrt d) (cbrt h))))
         (* t_5 (* t_4 (* (fabs t_7) (sqrt t_7)))))
       (* t_1 (* t_4 (* (fabs (/ 1.0 (cbrt h))) (sqrt (/ d (cbrt h))))))))))

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 t_0 = (M * D) / (d * 2.0);
	double t_1 = 1.0 - ((0.5 * (t_0 * (h * t_0))) * (1.0 / l));
	double t_2 = pow((d / h), 0.5);
	double t_3 = cbrt(d) / cbrt(l);
	double t_4 = fabs(t_3) * sqrt(t_3);
	double t_5 = 1.0 - ((0.5 * pow(t_0, 2.0)) * (h / l));
	double t_6 = (t_2 * pow((d / l), 0.5)) * t_5;
	double tmp;
	if (t_6 <= -1.2609594260553415e+155) {
		tmp = (t_2 * t_4) * t_1;
	} else if (t_6 <= ((double) INFINITY)) {
		double t_7 = cbrt(d) / cbrt(h);
		tmp = t_5 * (t_4 * (fabs(t_7) * sqrt(t_7)));
	} else {
		tmp = t_1 * (t_4 * (fabs(1.0 / cbrt(h)) * sqrt(d / cbrt(h))));
	}
	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 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (-.f64 1 (*.f64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)) (/.f64 h l)))) < -1.2609594260553415e155

   1. Initial program 44.6

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

   2. Applied add-cube-cbrt_binary64 44.7

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

   3. Applied add-cube-cbrt_binary64 44.8

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

   4. Applied times-frac_binary64 44.7

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

   5. Applied unpow-prod-down_binary64 44.1

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

   6. Simplified 44.1

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

   7. Simplified 44.1

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

   8. Applied div-inv_binary64 44.1

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

   9. Applied associate-*r*_binary64 38.9

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

   10. Simplified 38.9

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

   11. Applied unpow2_binary64 38.9

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

   12. Applied associate-*r*_binary64 31.4

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

   if -1.2609594260553415e155 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (-.f64 1 (*.f64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)) (/.f64 h l)))) < +inf.0

   1. Initial program 15.6

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

   2. Applied add-cube-cbrt_binary64 15.9

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

   3. Applied add-cube-cbrt_binary64 16.1

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

   4. Applied times-frac_binary64 16.1

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

   5. Applied unpow-prod-down_binary64 10.7

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

   6. Simplified 10.4

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

   7. Simplified 10.4

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

   8. Applied add-cube-cbrt_binary64 10.5

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

   9. Applied add-cube-cbrt_binary64 10.7

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

   10. Applied times-frac_binary64 10.7

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

   11. Applied unpow-prod-down_binary64 3.6

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

   12. Simplified 2.6

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

   13. Simplified 2.6

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

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

   1. Initial program 64.0

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

   2. Applied add-cube-cbrt_binary64 64.0

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

   3. Applied add-cube-cbrt_binary64 64.0

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

   4. Applied times-frac_binary64 64.0

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

   5. Applied unpow-prod-down_binary64 64.0

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

   6. Simplified 64.0

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

   7. Simplified 64.0

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

   8. Applied div-inv_binary64 64.0

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

   9. Applied associate-*r*_binary64 47.5

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

   10. Simplified 47.5

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

   11. Applied add-cube-cbrt_binary64 47.5

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

   12. Applied *-un-lft-identity_binary64 47.5

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

   13. Applied times-frac_binary64 47.5

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

   14. Applied unpow-prod-down_binary64 44.8

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

   15. Simplified 44.8

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

   16. Simplified 44.8

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

   17. Applied unpow2_binary64 44.8

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

   18. Applied associate-*r*_binary64 35.8

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

   19. Simplified 35.8

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

4. Final simplification 11.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -1.2609594260553415 \cdot 10^{+155}:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot \left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}}\right)\right) \cdot \left(1 - \left(0.5 \cdot \left(\frac{M \cdot D}{d \cdot 2} \cdot \left(h \cdot \frac{M \cdot D}{d \cdot 2}\right)\right)\right) \cdot \frac{1}{\ell}\right)\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq \infty:\\ \;\;\;\;\left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left(\left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}}\right) \cdot \left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \left(0.5 \cdot \left(\frac{M \cdot D}{d \cdot 2} \cdot \left(h \cdot \frac{M \cdot D}{d \cdot 2}\right)\right)\right) \cdot \frac{1}{\ell}\right) \cdot \left(\left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}}\right) \cdot \left(\left|\frac{1}{\sqrt[3]{h}}\right| \cdot \sqrt{\frac{d}{\sqrt[3]{h}}}\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2021313 
(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)))))