Henrywood and Agarwal, Equation (12)

Average Error: 21.2 → 11.4

Time: 19.0s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} t_0 := {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\\ t_1 := \sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\\ t_2 := \sqrt{\sqrt[3]{h} \cdot \sqrt[3]{h}} \cdot \sqrt{t_1}\\ t_3 := \sqrt{\frac{d}{\sqrt[3]{h}}}\\ \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \left(0.5 \cdot t_0\right) \cdot \frac{h}{\ell}\right) \leq \infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(t_0, \frac{h}{\ell} \cdot -0.5, 1\right) \cdot \left(t_3 \cdot \left(\sqrt{\frac{1}{\sqrt[3]{t_1}}} \cdot \sqrt{\frac{d}{\sqrt[3]{\sqrt[3]{\ell}}}}\right)\right)}{t_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)}{\ell \cdot \left(d \cdot d\right)}, -0.125, 1\right) \cdot \left(t_3 \cdot \sqrt{\frac{d}{\sqrt[3]{\ell}}}\right)}{t_2}\\ \end{array} \]

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 (pow (/ (* M D) (* d 2.0)) 2.0))
        (t_1 (* (cbrt l) (cbrt l)))
        (t_2 (* (sqrt (* (cbrt h) (cbrt h))) (sqrt t_1)))
        (t_3 (sqrt (/ d (cbrt h)))))
   (if (<=
        (*
         (* (pow (/ d h) 0.5) (pow (/ d l) 0.5))
         (- 1.0 (* (* 0.5 t_0) (/ h l))))
        INFINITY)
     (/
      (*
       (fma t_0 (* (/ h l) -0.5) 1.0)
       (* t_3 (* (sqrt (/ 1.0 (cbrt t_1))) (sqrt (/ d (cbrt (cbrt l)))))))
      t_2)
     (/
      (*
       (fma (/ (* (* D D) (* h (* M M))) (* l (* d d))) -0.125 1.0)
       (* t_3 (sqrt (/ d (cbrt l)))))
      t_2))))

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 = pow(((M * D) / (d * 2.0)), 2.0);
	double t_1 = cbrt(l) * cbrt(l);
	double t_2 = sqrt(cbrt(h) * cbrt(h)) * sqrt(t_1);
	double t_3 = sqrt(d / cbrt(h));
	double tmp;
	if (((pow((d / h), 0.5) * pow((d / l), 0.5)) * (1.0 - ((0.5 * t_0) * (h / l)))) <= ((double) INFINITY)) {
		tmp = (fma(t_0, ((h / l) * -0.5), 1.0) * (t_3 * (sqrt(1.0 / cbrt(t_1)) * sqrt(d / cbrt(cbrt(l)))))) / t_2;
	} else {
		tmp = (fma((((D * D) * (h * (M * M))) / (l * (d * d))), -0.125, 1.0) * (t_3 * sqrt(d / cbrt(l)))) / t_2;
	}
	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 2 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)))) < +inf.0

   1. Initial program 12.0

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

   2. Simplified 12.0

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

   3. Applied add-cube-cbrt_binary64 12.2

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

   4. Applied *-un-lft-identity_binary64 12.2

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

   5. Applied times-frac_binary64 12.2

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

   6. Applied sqrt-prod_binary64 8.8

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

   7. Applied add-cube-cbrt_binary64 8.9

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

   8. Applied *-un-lft-identity_binary64 8.9

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

   9. Applied times-frac_binary64 8.9

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

   10. Applied sqrt-prod_binary64 6.0

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

   11. Applied sqrt-div_binary64 6.0

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

   12. Applied associate-*l/_binary64 5.9

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

   13. Applied sqrt-div_binary64 5.9

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

   14. Applied associate-*l/_binary64 5.9

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

   15. Applied frac-times_binary64 5.9

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

   16. Applied associate-*l/_binary64 5.6

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

   17. Simplified 5.6

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

   18. Applied add-cube-cbrt_binary64 5.7

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

   19. Applied cbrt-prod_binary64 5.7

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

   20. Applied *-un-lft-identity_binary64 5.7

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

   21. Applied times-frac_binary64 5.7

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

   22. Applied sqrt-prod_binary64 5.4

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

   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. Simplified 64.0

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

   3. Applied add-cube-cbrt_binary64 64.0

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

   4. Applied *-un-lft-identity_binary64 64.0

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

   5. Applied times-frac_binary64 64.0

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

   6. Applied sqrt-prod_binary64 53.5

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

   7. Applied add-cube-cbrt_binary64 53.5

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

   8. Applied *-un-lft-identity_binary64 53.5

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

   9. Applied times-frac_binary64 53.5

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

   10. Applied sqrt-prod_binary64 51.2

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

   11. Applied sqrt-div_binary64 51.2

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

   12. Applied associate-*l/_binary64 51.2

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

   13. Applied sqrt-div_binary64 51.2

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

   14. Applied associate-*l/_binary64 51.2

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

   15. Applied frac-times_binary64 51.2

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

   16. Applied associate-*l/_binary64 47.6

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

   17. Simplified 47.6

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

   18. Taylor expanded in D around 0 39.4

       \[\leadsto \frac{\color{blue}{\left(1 - 0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)} \cdot \left(\sqrt{\frac{d}{\sqrt[3]{h}}} \cdot \sqrt{\frac{d}{\sqrt[3]{\ell}}}\right)}{\sqrt{\sqrt[3]{h} \cdot \sqrt[3]{h}} \cdot \sqrt{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}} \]

   19. Simplified 39.4

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot h\right)}{\left(d \cdot d\right) \cdot \ell}, -0.125, 1\right)} \cdot \left(\sqrt{\frac{d}{\sqrt[3]{h}}} \cdot \sqrt{\frac{d}{\sqrt[3]{\ell}}}\right)}{\sqrt{\sqrt[3]{h} \cdot \sqrt[3]{h}} \cdot \sqrt{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 11.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq \infty:\\ \;\;\;\;\frac{\mathsf{fma}\left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}, \frac{h}{\ell} \cdot -0.5, 1\right) \cdot \left(\sqrt{\frac{d}{\sqrt[3]{h}}} \cdot \left(\sqrt{\frac{1}{\sqrt[3]{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}} \cdot \sqrt{\frac{d}{\sqrt[3]{\sqrt[3]{\ell}}}}\right)\right)}{\sqrt{\sqrt[3]{h} \cdot \sqrt[3]{h}} \cdot \sqrt{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)}{\ell \cdot \left(d \cdot d\right)}, -0.125, 1\right) \cdot \left(\sqrt{\frac{d}{\sqrt[3]{h}}} \cdot \sqrt{\frac{d}{\sqrt[3]{\ell}}}\right)}{\sqrt{\sqrt[3]{h} \cdot \sqrt[3]{h}} \cdot \sqrt{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022068 
(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)))))