Average Error: 26.7 → 16.6
Time: 17.2s
Precision: 64
\[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;d \le -9.6145456522853459 \cdot 10^{128}:\\ \;\;\;\;\left(e^{0.5 \cdot \left(\log \left(\frac{-1}{h}\right) - \log \left(\frac{-1}{d}\right)\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)}^{\left(\frac{2}{2}\right)}\right) \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{h}{\ell}\right)\right)\\ \mathbf{elif}\;d \le -1.86853407285333 \cdot 10^{-310}:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot e^{0.5 \cdot \left(\log \left(\frac{-1}{\ell}\right) - \log \left(\frac{-1}{d}\right)\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(-h\right)}{-\ell}\right)\\ \mathbf{elif}\;d \le 3.70657678297185702 \cdot 10^{82}:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {d}^{\left(\frac{1}{2}\right)}\right) \cdot \left({\left(\frac{1}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(-h\right)}{-\ell}\right)\right)\\ \mathbf{elif}\;d \le 1.7042909891776182 \cdot 10^{187}:\\ \;\;\;\;{d}^{\left(\frac{1}{2}\right)} \cdot \left(\left({\left(\frac{1}{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)}^{\left(\frac{2}{2}\right)}\right) \cdot \frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(-h\right)}{-\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left({\left({d}^{1}\right)}^{1} \cdot {\left(\frac{1}{{h}^{1} \cdot {\ell}^{1}}\right)}^{0.5}\right)\\ \end{array}\]
\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}\;d \le -9.6145456522853459 \cdot 10^{128}:\\
\;\;\;\;\left(e^{0.5 \cdot \left(\log \left(\frac{-1}{h}\right) - \log \left(\frac{-1}{d}\right)\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)}^{\left(\frac{2}{2}\right)}\right) \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{h}{\ell}\right)\right)\\

\mathbf{elif}\;d \le -1.86853407285333 \cdot 10^{-310}:\\
\;\;\;\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot e^{0.5 \cdot \left(\log \left(\frac{-1}{\ell}\right) - \log \left(\frac{-1}{d}\right)\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(-h\right)}{-\ell}\right)\\

\mathbf{elif}\;d \le 3.70657678297185702 \cdot 10^{82}:\\
\;\;\;\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {d}^{\left(\frac{1}{2}\right)}\right) \cdot \left({\left(\frac{1}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(-h\right)}{-\ell}\right)\right)\\

\mathbf{elif}\;d \le 1.7042909891776182 \cdot 10^{187}:\\
\;\;\;\;{d}^{\left(\frac{1}{2}\right)} \cdot \left(\left({\left(\frac{1}{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)}^{\left(\frac{2}{2}\right)}\right) \cdot \frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(-h\right)}{-\ell}\right)\right)\\

\mathbf{else}:\\
\;\;\;\;1 \cdot \left({\left({d}^{1}\right)}^{1} \cdot {\left(\frac{1}{{h}^{1} \cdot {\ell}^{1}}\right)}^{0.5}\right)\\

\end{array}
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 VAR;
	if ((d <= -9.614545652285346e+128)) {
		VAR = ((exp((0.5 * (log((-1.0 / h)) - log((-1.0 / d))))) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), (2.0 / 2.0))) * (pow(((M * D) / (2.0 * d)), (2.0 / 2.0)) * (h / l)))));
	} else {
		double VAR_1;
		if ((d <= -1.86853407285333e-310)) {
			VAR_1 = ((pow((d / h), (1.0 / 2.0)) * exp((0.5 * (log((-1.0 / l)) - log((-1.0 / d)))))) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), (2.0 / 2.0))) * ((pow(((M * D) / (2.0 * d)), (2.0 / 2.0)) * -h) / -l))));
		} else {
			double VAR_2;
			if ((d <= 3.706576782971857e+82)) {
				VAR_2 = ((pow((d / h), (1.0 / 2.0)) * pow(d, (1.0 / 2.0))) * (pow((1.0 / l), (1.0 / 2.0)) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), (2.0 / 2.0))) * ((pow(((M * D) / (2.0 * d)), (2.0 / 2.0)) * -h) / -l)))));
			} else {
				double VAR_3;
				if ((d <= 1.7042909891776182e+187)) {
					VAR_3 = (pow(d, (1.0 / 2.0)) * ((pow((1.0 / 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 / 2.0))) * ((pow(((M * D) / (2.0 * d)), (2.0 / 2.0)) * -h) / -l)))));
				} else {
					VAR_3 = (1.0 * (pow(pow(d, 1.0), 1.0) * pow((1.0 / (pow(h, 1.0) * pow(l, 1.0))), 0.5)));
				}
				VAR_2 = VAR_3;
			}
			VAR_1 = VAR_2;
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus d

Bits error versus h

Bits error versus l

Bits error versus M

Bits error versus D

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 5 regimes

  2. if d < -9.614545652285346e+128

    1. Initial program 27.6

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

    3. Applied sqr-pow27.6

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

    4. Applied associate-*r*27.6

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

    5. Applied associate-*l*27.6

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

    6. Taylor expanded around -inf 17.7

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

    if -9.614545652285346e+128 < d < -1.86853407285333e-310

    1. Initial program 25.4

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

    3. Applied sqr-pow25.4

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

    4. Applied associate-*r*25.4

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

    5. Applied associate-*l*23.1

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

    7. Applied frac-2neg23.1

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

    8. Applied associate-*r/21.0

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

    9. Taylor expanded around -inf 18.7

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

    if -1.86853407285333e-310 < d < 3.706576782971857e+82

    1. Initial program 27.7

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

    3. Applied sqr-pow27.7

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

    4. Applied associate-*r*27.7

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

    5. Applied associate-*l*25.0

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

    7. Applied frac-2neg25.0

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

    8. Applied associate-*r/23.0

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

    10. Applied div-inv23.0

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

    11. Applied unpow-prod-down16.5

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

    12. Applied associate-*r*16.5

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

    13. Applied associate-*l*16.7

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

    if 3.706576782971857e+82 < d < 1.7042909891776182e+187

    1. Initial program 23.2

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

    3. Applied sqr-pow23.2

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

    4. Applied associate-*r*23.2

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

    5. Applied associate-*l*22.7

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

    7. Applied frac-2neg22.7

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

    8. Applied associate-*r/20.6

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

    10. Applied div-inv20.6

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

    11. Applied unpow-prod-down9.6

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

    12. Applied associate-*l*9.5

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

    13. Applied associate-*l*9.5

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

    if 1.7042909891776182e+187 < d

    1. Initial program 30.8

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

    2. Taylor expanded around 0 14.1

      \[\leadsto \color{blue}{1 \cdot \left({\left({d}^{1}\right)}^{1} \cdot {\left(\frac{1}{{h}^{1} \cdot {\ell}^{1}}\right)}^{0.5}\right)}\]
  3. Recombined 5 regimes into one program.

  4. Final simplification16.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;d \le -9.6145456522853459 \cdot 10^{128}:\\ \;\;\;\;\left(e^{0.5 \cdot \left(\log \left(\frac{-1}{h}\right) - \log \left(\frac{-1}{d}\right)\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)}^{\left(\frac{2}{2}\right)}\right) \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{h}{\ell}\right)\right)\\ \mathbf{elif}\;d \le -1.86853407285333 \cdot 10^{-310}:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot e^{0.5 \cdot \left(\log \left(\frac{-1}{\ell}\right) - \log \left(\frac{-1}{d}\right)\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(-h\right)}{-\ell}\right)\\ \mathbf{elif}\;d \le 3.70657678297185702 \cdot 10^{82}:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {d}^{\left(\frac{1}{2}\right)}\right) \cdot \left({\left(\frac{1}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(-h\right)}{-\ell}\right)\right)\\ \mathbf{elif}\;d \le 1.7042909891776182 \cdot 10^{187}:\\ \;\;\;\;{d}^{\left(\frac{1}{2}\right)} \cdot \left(\left({\left(\frac{1}{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)}^{\left(\frac{2}{2}\right)}\right) \cdot \frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(-h\right)}{-\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left({\left({d}^{1}\right)}^{1} \cdot {\left(\frac{1}{{h}^{1} \cdot {\ell}^{1}}\right)}^{0.5}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020078 +o rules:numerics
(FPCore (d h l M D)
  :name "Henrywood and Agarwal, Equation (12)"
  :precision binary64
  (* (* (pow (/ d h) (/ 1 2)) (pow (/ d l) (/ 1 2))) (- 1 (* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) (/ h l)))))