Henrywood and Agarwal, Equation (12)

Average Error: 41.72% → 24.42%

Time: 40.0s

Precision: binary64

Cost: 110608

Math

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

↓

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ t_1 := \sqrt{h \cdot \ell}\\ t_2 := M \cdot \left(0.5 \cdot \frac{D}{d}\right)\\ t_3 := \left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot -0.5\right)\right)\\ t_4 := 2 \cdot \frac{\ell}{h}\\ t_5 := 0.5 \cdot \frac{D}{\frac{d}{M}}\\ t_6 := \left|d \cdot \frac{\mathsf{fma}\left({t_2}^{2}, \frac{h}{\ell} \cdot -0.5, 1\right)}{t_1}\right|\\ \mathbf{if}\;t_3 \leq -5 \cdot 10^{-227}:\\ \;\;\;\;t_0 \cdot \left(1 - {\left(\frac{t_2}{\sqrt{t_4}}\right)}^{2}\right)\\ \mathbf{elif}\;t_3 \leq 0:\\ \;\;\;\;t_6\\ \mathbf{elif}\;t_3 \leq 2 \cdot 10^{+270}:\\ \;\;\;\;t_0 \cdot \left(1 - t_5 \cdot \frac{t_5}{t_4}\right)\\ \mathbf{elif}\;t_3 \leq \infty:\\ \;\;\;\;t_6\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{t_1}\\ \end{array} \]

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 (* (sqrt (/ d h)) (sqrt (/ d l))))
        (t_1 (sqrt (* h l)))
        (t_2 (* M (* 0.5 (/ D d))))
        (t_3
         (*
          (* (pow (/ d h) 0.5) (pow (/ d l) 0.5))
          (+ 1.0 (* (/ h l) (* (pow (/ (* M D) (* d 2.0)) 2.0) -0.5)))))
        (t_4 (* 2.0 (/ l h)))
        (t_5 (* 0.5 (/ D (/ d M))))
        (t_6 (fabs (* d (/ (fma (pow t_2 2.0) (* (/ h l) -0.5) 1.0) t_1)))))
   (if (<= t_3 -5e-227)
     (* t_0 (- 1.0 (pow (/ t_2 (sqrt t_4)) 2.0)))
     (if (<= t_3 0.0)
       t_6
       (if (<= t_3 2e+270)
         (* t_0 (- 1.0 (* t_5 (/ t_5 t_4))))
         (if (<= t_3 INFINITY) t_6 (/ d t_1)))))))

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 = sqrt((d / h)) * sqrt((d / l));
	double t_1 = sqrt((h * l));
	double t_2 = M * (0.5 * (D / d));
	double t_3 = (pow((d / h), 0.5) * pow((d / l), 0.5)) * (1.0 + ((h / l) * (pow(((M * D) / (d * 2.0)), 2.0) * -0.5)));
	double t_4 = 2.0 * (l / h);
	double t_5 = 0.5 * (D / (d / M));
	double t_6 = fabs((d * (fma(pow(t_2, 2.0), ((h / l) * -0.5), 1.0) / t_1)));
	double tmp;
	if (t_3 <= -5e-227) {
		tmp = t_0 * (1.0 - pow((t_2 / sqrt(t_4)), 2.0));
	} else if (t_3 <= 0.0) {
		tmp = t_6;
	} else if (t_3 <= 2e+270) {
		tmp = t_0 * (1.0 - (t_5 * (t_5 / t_4)));
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = t_6;
	} else {
		tmp = d / t_1;
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	return Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
end

↓

function code(d, h, l, M, D)
	t_0 = Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l)))
	t_1 = sqrt(Float64(h * l))
	t_2 = Float64(M * Float64(0.5 * Float64(D / d)))
	t_3 = Float64(Float64((Float64(d / h) ^ 0.5) * (Float64(d / l) ^ 0.5)) * Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(Float64(M * D) / Float64(d * 2.0)) ^ 2.0) * -0.5))))
	t_4 = Float64(2.0 * Float64(l / h))
	t_5 = Float64(0.5 * Float64(D / Float64(d / M)))
	t_6 = abs(Float64(d * Float64(fma((t_2 ^ 2.0), Float64(Float64(h / l) * -0.5), 1.0) / t_1)))
	tmp = 0.0
	if (t_3 <= -5e-227)
		tmp = Float64(t_0 * Float64(1.0 - (Float64(t_2 / sqrt(t_4)) ^ 2.0)));
	elseif (t_3 <= 0.0)
		tmp = t_6;
	elseif (t_3 <= 2e+270)
		tmp = Float64(t_0 * Float64(1.0 - Float64(t_5 * Float64(t_5 / t_4))));
	elseif (t_3 <= Inf)
		tmp = t_6;
	else
		tmp = Float64(d / t_1);
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(M * N[(0.5 * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[Power[N[(d / h), $MachinePrecision], 0.5], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(2.0 * N[(l / h), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(0.5 * N[(D / N[(d / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[Abs[N[(d * N[(N[(N[Power[t$95$2, 2.0], $MachinePrecision] * N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision] + 1.0), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$3, -5e-227], N[(t$95$0 * N[(1.0 - N[Power[N[(t$95$2 / N[Sqrt[t$95$4], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.0], t$95$6, If[LessEqual[t$95$3, 2e+270], N[(t$95$0 * N[(1.0 - N[(t$95$5 * N[(t$95$5 / t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, Infinity], t$95$6, N[(d / t$95$1), $MachinePrecision]]]]]]]]]]]]

Derivation

1. Split input into 4 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)))) < -4.99999999999999961e-227

   1. Initial program 45.22

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

   2. Applied egg-rr 45.19

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

   3. Applied egg-rr 33.41

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

   4. Simplified 36.95

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

      [Start] 33.41	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{0.5}{\sqrt{2 \cdot \frac{\ell}{h}} \cdot \frac{d}{M \cdot D}} \cdot \frac{0.5}{\sqrt{2 \cdot \frac{\ell}{h}} \cdot \frac{d}{M \cdot D}}\right) \]
      unpow2 [<=] 33.41	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{{\left(\frac{0.5}{\sqrt{2 \cdot \frac{\ell}{h}} \cdot \frac{d}{M \cdot D}}\right)}^{2}}\right) \]
      *-commutative [=>] 33.41	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - {\left(\frac{0.5}{\color{blue}{\frac{d}{M \cdot D} \cdot \sqrt{2 \cdot \frac{\ell}{h}}}}\right)}^{2}\right) \]
      associate-/r* [=>] 33.39	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - {\color{blue}{\left(\frac{\frac{0.5}{\frac{d}{M \cdot D}}}{\sqrt{2 \cdot \frac{\ell}{h}}}\right)}}^{2}\right) \]
      associate-/r* [=>] 35.63	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - {\left(\frac{\frac{0.5}{\color{blue}{\frac{\frac{d}{M}}{D}}}}{\sqrt{2 \cdot \frac{\ell}{h}}}\right)}^{2}\right) \]
      associate-/l* [<=] 35.55	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - {\left(\frac{\color{blue}{\frac{0.5 \cdot D}{\frac{d}{M}}}}{\sqrt{2 \cdot \frac{\ell}{h}}}\right)}^{2}\right) \]
      associate-*r/ [<=] 35.55	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - {\left(\frac{\color{blue}{0.5 \cdot \frac{D}{\frac{d}{M}}}}{\sqrt{2 \cdot \frac{\ell}{h}}}\right)}^{2}\right) \]
      *-commutative [=>] 35.55	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - {\left(\frac{\color{blue}{\frac{D}{\frac{d}{M}} \cdot 0.5}}{\sqrt{2 \cdot \frac{\ell}{h}}}\right)}^{2}\right) \]
      associate-/r/ [=>] 36.95	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - {\left(\frac{\color{blue}{\left(\frac{D}{d} \cdot M\right)} \cdot 0.5}{\sqrt{2 \cdot \frac{\ell}{h}}}\right)}^{2}\right) \]
      *-commutative [=>] 36.95	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - {\left(\frac{\color{blue}{\left(M \cdot \frac{D}{d}\right)} \cdot 0.5}{\sqrt{2 \cdot \frac{\ell}{h}}}\right)}^{2}\right) \]
      associate-*l* [=>] 36.95	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - {\left(\frac{\color{blue}{M \cdot \left(\frac{D}{d} \cdot 0.5\right)}}{\sqrt{2 \cdot \frac{\ell}{h}}}\right)}^{2}\right) \]

   5. Applied egg-rr 70.85

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{d}{\ell}}\right)} - 1\right)}\right) \cdot \left(1 - {\left(\frac{M \cdot \left(\frac{D}{d} \cdot 0.5\right)}{\sqrt{2 \cdot \frac{\ell}{h}}}\right)}^{2}\right) \]

   6. Simplified 36.95

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

      [Start] 70.85	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(e^{\mathsf{log1p}\left(\sqrt{\frac{d}{\ell}}\right)} - 1\right)\right) \cdot \left(1 - {\left(\frac{M \cdot \left(\frac{D}{d} \cdot 0.5\right)}{\sqrt{2 \cdot \frac{\ell}{h}}}\right)}^{2}\right) \]
      expm1-def [=>] 38.29	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{\ell}}\right)\right)}\right) \cdot \left(1 - {\left(\frac{M \cdot \left(\frac{D}{d} \cdot 0.5\right)}{\sqrt{2 \cdot \frac{\ell}{h}}}\right)}^{2}\right) \]
      expm1-log1p [=>] 36.95	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - {\left(\frac{M \cdot \left(\frac{D}{d} \cdot 0.5\right)}{\sqrt{2 \cdot \frac{\ell}{h}}}\right)}^{2}\right) \]

   7. Applied egg-rr 74.22

      \[\leadsto \left(\color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h}}\right)} - 1\right)} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M \cdot \left(\frac{D}{d} \cdot 0.5\right)}{\sqrt{2 \cdot \frac{\ell}{h}}}\right)}^{2}\right) \]

   8. Simplified 36.95

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

      [Start] 74.22	\[ \left(\left(e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h}}\right)} - 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M \cdot \left(\frac{D}{d} \cdot 0.5\right)}{\sqrt{2 \cdot \frac{\ell}{h}}}\right)}^{2}\right) \]
      expm1-def [=>] 38.11	\[ \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h}}\right)\right)} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M \cdot \left(\frac{D}{d} \cdot 0.5\right)}{\sqrt{2 \cdot \frac{\ell}{h}}}\right)}^{2}\right) \]
      expm1-log1p [=>] 36.95	\[ \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M \cdot \left(\frac{D}{d} \cdot 0.5\right)}{\sqrt{2 \cdot \frac{\ell}{h}}}\right)}^{2}\right) \]

   if -4.99999999999999961e-227 < (*.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)))) < 0.0 or 2.0000000000000001e270 < (*.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 80.26

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

   2. Simplified 81.04

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

      [Start] 80.26	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 80.26	\[ \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 80.26	\[ \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 80.26	\[ \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 80.26	\[ \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      associate-*l* [=>] 80.26	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{1}{2} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)}\right) \]
      metadata-eval [=>] 80.26	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{0.5} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
      times-frac [=>] 81.04	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   3. Applied egg-rr 55.7

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

   4. Simplified 55.7

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

      [Start] 55.7	\[ \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} + \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(\left(-0.5 \cdot \frac{h}{\ell}\right) \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right) \]
      *-lft-identity [<=] 55.7	\[ \color{blue}{1 \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}}} + \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(\left(-0.5 \cdot \frac{h}{\ell}\right) \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right) \]
      *-commutative [<=] 55.7	\[ 1 \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} + \color{blue}{\left(\left(-0.5 \cdot \frac{h}{\ell}\right) \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right) \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}}} \]
      distribute-rgt-in [<=] 55.7	\[ \color{blue}{\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 + \left(-0.5 \cdot \frac{h}{\ell}\right) \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right)} \]
      *-commutative [=>] 55.7	\[ \frac{d}{\color{blue}{\sqrt{\ell} \cdot \sqrt{h}}} \cdot \left(1 + \left(-0.5 \cdot \frac{h}{\ell}\right) \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right) \]
      *-commutative [=>] 55.7	\[ \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 + \color{blue}{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(-0.5 \cdot \frac{h}{\ell}\right)}\right) \]
      *-commutative [=>] 55.7	\[ \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 + {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \color{blue}{\left(\frac{h}{\ell} \cdot -0.5\right)}\right) \]

   5. Applied egg-rr 71.57

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

   6. Simplified 18.66

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

      [Start] 71.57	\[ \sqrt{{\left(\frac{d}{\frac{\sqrt{h \cdot \ell}}{\mathsf{fma}\left({\left(0.5 \cdot \left(\frac{D}{d} \cdot M\right)\right)}^{2}, \frac{h}{\ell} \cdot -0.5, 1\right)}}\right)}^{2}} \]
      unpow2 [=>] 71.57	\[ \sqrt{\color{blue}{\frac{d}{\frac{\sqrt{h \cdot \ell}}{\mathsf{fma}\left({\left(0.5 \cdot \left(\frac{D}{d} \cdot M\right)\right)}^{2}, \frac{h}{\ell} \cdot -0.5, 1\right)}} \cdot \frac{d}{\frac{\sqrt{h \cdot \ell}}{\mathsf{fma}\left({\left(0.5 \cdot \left(\frac{D}{d} \cdot M\right)\right)}^{2}, \frac{h}{\ell} \cdot -0.5, 1\right)}}}} \]
      rem-sqrt-square [=>] 18.56	\[ \color{blue}{\left|\frac{d}{\frac{\sqrt{h \cdot \ell}}{\mathsf{fma}\left({\left(0.5 \cdot \left(\frac{D}{d} \cdot M\right)\right)}^{2}, \frac{h}{\ell} \cdot -0.5, 1\right)}}\right|} \]
      associate-/l* [<=] 18.56	\[ \left|\color{blue}{\frac{d \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(\frac{D}{d} \cdot M\right)\right)}^{2}, \frac{h}{\ell} \cdot -0.5, 1\right)}{\sqrt{h \cdot \ell}}}\right| \]
      *-commutative [=>] 18.56	\[ \left|\frac{\color{blue}{\mathsf{fma}\left({\left(0.5 \cdot \left(\frac{D}{d} \cdot M\right)\right)}^{2}, \frac{h}{\ell} \cdot -0.5, 1\right) \cdot d}}{\sqrt{h \cdot \ell}}\right| \]
      associate-*l/ [<=] 18.66	\[ \left|\color{blue}{\frac{\mathsf{fma}\left({\left(0.5 \cdot \left(\frac{D}{d} \cdot M\right)\right)}^{2}, \frac{h}{\ell} \cdot -0.5, 1\right)}{\sqrt{h \cdot \ell}} \cdot d}\right| \]
      *-commutative [=>] 18.66	\[ \left|\color{blue}{d \cdot \frac{\mathsf{fma}\left({\left(0.5 \cdot \left(\frac{D}{d} \cdot M\right)\right)}^{2}, \frac{h}{\ell} \cdot -0.5, 1\right)}{\sqrt{h \cdot \ell}}}\right| \]
      associate-*r* [=>] 18.66	\[ \left|d \cdot \frac{\mathsf{fma}\left({\color{blue}{\left(\left(0.5 \cdot \frac{D}{d}\right) \cdot M\right)}}^{2}, \frac{h}{\ell} \cdot -0.5, 1\right)}{\sqrt{h \cdot \ell}}\right| \]
      *-commutative [<=] 18.66	\[ \left|d \cdot \frac{\mathsf{fma}\left({\color{blue}{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}}^{2}, \frac{h}{\ell} \cdot -0.5, 1\right)}{\sqrt{h \cdot \ell}}\right| \]

   if 0.0 < (*.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)))) < 2.0000000000000001e270

   1. Initial program 1.35

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

   2. Applied egg-rr 1.36

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

   3. Applied egg-rr 51.09

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{d}{\ell}}\right)} - 1\right)}\right) \cdot \left(1 - \frac{{\left(0.5 \cdot \frac{M \cdot D}{d}\right)}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}\right) \]

   4. Simplified 1.36

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

      [Start] 51.09	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(e^{\mathsf{log1p}\left(\sqrt{\frac{d}{\ell}}\right)} - 1\right)\right) \cdot \left(1 - \frac{{\left(0.5 \cdot \frac{M \cdot D}{d}\right)}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}\right) \]
      expm1-def [=>] 4.36	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{\ell}}\right)\right)}\right) \cdot \left(1 - \frac{{\left(0.5 \cdot \frac{M \cdot D}{d}\right)}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}\right) \]
      expm1-log1p [=>] 1.36	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{{\left(0.5 \cdot \frac{M \cdot D}{d}\right)}^{2} \cdot \left(0.5 \cdot h\right)}{\ell}\right) \]

   5. Applied egg-rr 1.53

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

   6. Applied egg-rr 34.94

      \[\leadsto \left(\color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h}}\right)} - 1\right)} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.5 \cdot \frac{D}{\frac{d}{M}}}{2 \cdot \frac{\ell}{h}} \cdot \left(0.5 \cdot \frac{D}{\frac{d}{M}}\right)\right) \]

   7. Simplified 1.53

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

      [Start] 34.94	\[ \left(\left(e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h}}\right)} - 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.5 \cdot \frac{D}{\frac{d}{M}}}{2 \cdot \frac{\ell}{h}} \cdot \left(0.5 \cdot \frac{D}{\frac{d}{M}}\right)\right) \]
      expm1-def [=>] 5.93	\[ \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h}}\right)\right)} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.5 \cdot \frac{D}{\frac{d}{M}}}{2 \cdot \frac{\ell}{h}} \cdot \left(0.5 \cdot \frac{D}{\frac{d}{M}}\right)\right) \]
      expm1-log1p [=>] 1.53	\[ \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.5 \cdot \frac{D}{\frac{d}{M}}}{2 \cdot \frac{\ell}{h}} \cdot \left(0.5 \cdot \frac{D}{\frac{d}{M}}\right)\right) \]

   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 100

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

   2. Simplified 99.93

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

      [Start] 100	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 100	\[ \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 100	\[ \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 100	\[ \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 100	\[ \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      associate-*l* [=>] 100	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{1}{2} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)}\right) \]
      metadata-eval [=>] 100	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{0.5} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
      times-frac [=>] 99.93	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   3. Taylor expanded in d around inf 80.42

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]

   4. Simplified 80.43

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}} \]
      Proof

      [Start] 80.42	\[ \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]
      *-commutative [=>] 80.42	\[ \color{blue}{d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]
      associate-/r* [=>] 80.43	\[ d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \]

   5. Applied egg-rr 86.07

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{d}{\sqrt{\ell \cdot h}}\right)} - 1} \]

   6. Simplified 80.41

      \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
      Proof

      [Start] 86.07	\[ e^{\mathsf{log1p}\left(\frac{d}{\sqrt{\ell \cdot h}}\right)} - 1 \]
      expm1-def [=>] 81.6	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{d}{\sqrt{\ell \cdot h}}\right)\right)} \]
      expm1-log1p [=>] 80.41	\[ \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]

3. Recombined 4 regimes into one program.

4. Final simplification 24.42

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot -0.5\right)\right) \leq -5 \cdot 10^{-227}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M \cdot \left(0.5 \cdot \frac{D}{d}\right)}{\sqrt{2 \cdot \frac{\ell}{h}}}\right)}^{2}\right)\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot -0.5\right)\right) \leq 0:\\ \;\;\;\;\left|d \cdot \frac{\mathsf{fma}\left({\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}, \frac{h}{\ell} \cdot -0.5, 1\right)}{\sqrt{h \cdot \ell}}\right|\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot -0.5\right)\right) \leq 2 \cdot 10^{+270}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \left(0.5 \cdot \frac{D}{\frac{d}{M}}\right) \cdot \frac{0.5 \cdot \frac{D}{\frac{d}{M}}}{2 \cdot \frac{\ell}{h}}\right)\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot -0.5\right)\right) \leq \infty:\\ \;\;\;\;\left|d \cdot \frac{\mathsf{fma}\left({\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}, \frac{h}{\ell} \cdot -0.5, 1\right)}{\sqrt{h \cdot \ell}}\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}}\\ \end{array} \]

Alternatives

Alternative 1
Error	24.5%
Cost	110608

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ t_1 := \sqrt{h \cdot \ell}\\ t_2 := 0.5 \cdot \frac{D}{\frac{d}{M}}\\ t_3 := \left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot -0.5\right)\right)\\ t_4 := M \cdot \left(0.5 \cdot \frac{D}{d}\right)\\ t_5 := \left|d \cdot \frac{\mathsf{fma}\left({t_4}^{2}, \frac{h}{\ell} \cdot -0.5, 1\right)}{t_1}\right|\\ \mathbf{if}\;t_3 \leq -5 \cdot 10^{-227}:\\ \;\;\;\;t_0 \cdot \left(1 - 0.5 \cdot {\left(t_4 \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right)\\ \mathbf{elif}\;t_3 \leq 0:\\ \;\;\;\;t_5\\ \mathbf{elif}\;t_3 \leq 2 \cdot 10^{+270}:\\ \;\;\;\;t_0 \cdot \left(1 - t_2 \cdot \frac{t_2}{2 \cdot \frac{\ell}{h}}\right)\\ \mathbf{elif}\;t_3 \leq \infty:\\ \;\;\;\;t_5\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{t_1}\\ \end{array} \]

Alternative 2
Error	26.62%
Cost	27540

\[\begin{array}{l} t_0 := {\left(\frac{d}{h}\right)}^{0.5}\\ t_1 := \left(1 + \left(\frac{h}{\ell} \cdot -0.5\right) \cdot \left(\frac{D}{d} \cdot \left(\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right) \cdot \left(0.5 \cdot M\right)\right)\right)\right) \cdot \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ t_2 := \frac{D}{\frac{d}{M}}\\ t_3 := \sqrt{\frac{d}{\ell}}\\ t_4 := 0.5 \cdot t_2\\ t_5 := 1 - t_4 \cdot \frac{t_4}{2 \cdot \frac{\ell}{h}}\\ t_6 := \sqrt{-d}\\ \mathbf{if}\;d \leq -1.78:\\ \;\;\;\;t_5 \cdot \left(t_3 \cdot \frac{t_6}{\sqrt{-h}}\right)\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;t_5 \cdot \left(t_0 \cdot \frac{t_6}{\sqrt{-\ell}}\right)\\ \mathbf{elif}\;d \leq 7.8 \cdot 10^{-110}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq 8.5 \cdot 10^{+23}:\\ \;\;\;\;\left(t_0 \cdot t_3\right) \cdot \left(1 + \left(h \cdot \frac{\frac{D \cdot \frac{M}{d}}{4}}{\ell}\right) \cdot \left(-0.5 \cdot t_2\right)\right)\\ \mathbf{elif}\;d \leq 4 \cdot 10^{+144}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}} \cdot \mathsf{fma}\left({\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}, \frac{-0.5}{\frac{\ell}{h}}, 1\right)\\ \mathbf{elif}\;d \leq 6.5 \cdot 10^{+273}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	26.69%
Cost	21640

\[\begin{array}{l} t_0 := {\left(\frac{d}{h}\right)}^{0.5}\\ t_1 := \frac{D}{\frac{d}{M}}\\ t_2 := \sqrt{\frac{d}{\ell}}\\ t_3 := \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ t_4 := \left(1 + \left(\frac{h}{\ell} \cdot -0.5\right) \cdot \left(\frac{D}{d} \cdot \left(\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right) \cdot \left(0.5 \cdot M\right)\right)\right)\right) \cdot t_3\\ t_5 := 0.5 \cdot t_1\\ t_6 := 1 - t_5 \cdot \frac{t_5}{2 \cdot \frac{\ell}{h}}\\ t_7 := \sqrt{-d}\\ \mathbf{if}\;d \leq -1.78:\\ \;\;\;\;t_6 \cdot \left(t_2 \cdot \frac{t_7}{\sqrt{-h}}\right)\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;t_6 \cdot \left(t_0 \cdot \frac{t_7}{\sqrt{-\ell}}\right)\\ \mathbf{elif}\;d \leq 10^{-109}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;d \leq 3000000000:\\ \;\;\;\;\left(t_0 \cdot t_2\right) \cdot \left(1 + \left(h \cdot \frac{\frac{D \cdot \frac{M}{d}}{4}}{\ell}\right) \cdot \left(-0.5 \cdot t_1\right)\right)\\ \mathbf{elif}\;d \leq 2.5 \cdot 10^{+139}:\\ \;\;\;\;\left(1 + \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{h \cdot -0.5}}\right) \cdot t_3\\ \mathbf{elif}\;d \leq 2.8 \cdot 10^{+269}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{else}:\\ \;\;\;\;t_4\\ \end{array} \]

Alternative 4
Error	28.48%
Cost	21444

\[\begin{array}{l} t_0 := \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ t_1 := \left(1 + \left(\frac{h}{\ell} \cdot -0.5\right) \cdot \left(\frac{D}{d} \cdot \left(\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right) \cdot \left(0.5 \cdot M\right)\right)\right)\right) \cdot t_0\\ t_2 := \sqrt{\frac{d}{\ell}}\\ t_3 := \frac{D}{\frac{d}{M}}\\ t_4 := 0.5 \cdot t_3\\ \mathbf{if}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(1 - t_4 \cdot \frac{t_4}{2 \cdot \frac{\ell}{h}}\right) \cdot \left(t_2 \cdot \frac{\sqrt{-d}}{\sqrt{-h}}\right)\\ \mathbf{elif}\;d \leq 9.5 \cdot 10^{-110}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq 2800000000:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot t_2\right) \cdot \left(1 + \left(h \cdot \frac{\frac{D \cdot \frac{M}{d}}{4}}{\ell}\right) \cdot \left(-0.5 \cdot t_3\right)\right)\\ \mathbf{elif}\;d \leq 4.05 \cdot 10^{+145}:\\ \;\;\;\;\left(1 + \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{h \cdot -0.5}}\right) \cdot t_0\\ \mathbf{elif}\;d \leq 2 \cdot 10^{+270}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	33.82%
Cost	21136

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ t_2 := \left(1 + \left(\frac{h}{\ell} \cdot -0.5\right) \cdot \left(\frac{D}{d} \cdot \left(\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right) \cdot \left(0.5 \cdot M\right)\right)\right)\right) \cdot t_1\\ t_3 := \frac{D}{\frac{d}{M}}\\ t_4 := 0.5 \cdot t_3\\ \mathbf{if}\;d \leq 2 \cdot 10^{-274}:\\ \;\;\;\;\left(1 - t_4 \cdot \frac{t_4}{2 \cdot \frac{\ell}{h}}\right) \cdot \left(t_0 \cdot \frac{1}{\sqrt{\frac{h}{d}}}\right)\\ \mathbf{elif}\;d \leq 9.8 \cdot 10^{-110}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;d \leq 3000000000:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot t_0\right) \cdot \left(1 + \left(h \cdot \frac{\frac{D \cdot \frac{M}{d}}{4}}{\ell}\right) \cdot \left(-0.5 \cdot t_3\right)\right)\\ \mathbf{elif}\;d \leq 2.5 \cdot 10^{+141}:\\ \;\;\;\;\left(1 + \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{h \cdot -0.5}}\right) \cdot t_1\\ \mathbf{elif}\;d \leq 4 \cdot 10^{+269}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 6
Error	30.81%
Cost	20872

\[\begin{array}{l} \mathbf{if}\;h \leq -1.85 \cdot 10^{-179}:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 + \left(h \cdot \frac{\frac{D \cdot \frac{M}{d}}{4}}{\ell}\right) \cdot \left(-0.5 \cdot \frac{D}{\frac{d}{M}}\right)\right)\\ \mathbf{elif}\;h \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{\frac{-\ell}{d}}}}{\sqrt{-h}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right) \cdot \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

Alternative 7
Error	31.2%
Cost	15116

\[\begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{0.5} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 + \left(h \cdot \frac{\frac{D \cdot \frac{M}{d}}{4}}{\ell}\right) \cdot \left(-0.5 \cdot \frac{D}{\frac{d}{M}}\right)\right)\\ t_1 := \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \mathbf{if}\;d \leq 1.3 \cdot 10^{-308}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 7.2 \cdot 10^{-110}:\\ \;\;\;\;\left(1 + \left(\frac{h}{\ell} \cdot -0.5\right) \cdot \left(\frac{D}{d} \cdot \left(\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right) \cdot \left(0.5 \cdot M\right)\right)\right)\right) \cdot t_1\\ \mathbf{elif}\;d \leq 4.2 \cdot 10^{+75}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Error	32.33%
Cost	15116

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ t_2 := \frac{D}{\frac{d}{M}}\\ t_3 := 0.5 \cdot t_2\\ \mathbf{if}\;d \leq 2.2 \cdot 10^{-278}:\\ \;\;\;\;\left(1 - t_3 \cdot \frac{t_3}{2 \cdot \frac{\ell}{h}}\right) \cdot \left(t_0 \cdot \frac{1}{\sqrt{\frac{h}{d}}}\right)\\ \mathbf{elif}\;d \leq 9.2 \cdot 10^{-110}:\\ \;\;\;\;\left(1 + \left(\frac{h}{\ell} \cdot -0.5\right) \cdot \left(\frac{D}{d} \cdot \left(\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right) \cdot \left(0.5 \cdot M\right)\right)\right)\right) \cdot t_1\\ \mathbf{elif}\;d \leq 6.6 \cdot 10^{+76}:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot t_0\right) \cdot \left(1 + \left(h \cdot \frac{\frac{D \cdot \frac{M}{d}}{4}}{\ell}\right) \cdot \left(-0.5 \cdot t_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
Error	33.68%
Cost	15053

\[\begin{array}{l} t_0 := \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ t_1 := 0.5 \cdot \frac{D}{\frac{d}{M}}\\ \mathbf{if}\;d \leq 2.3 \cdot 10^{-290}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - t_1 \cdot \frac{t_1}{2 \cdot \frac{\ell}{h}}\right)\\ \mathbf{elif}\;d \leq 5 \cdot 10^{+76} \lor \neg \left(d \leq 2.1 \cdot 10^{+261}\right):\\ \;\;\;\;\left(1 + \left(\frac{h}{\ell} \cdot -0.5\right) \cdot \left(\frac{D}{d} \cdot \left(\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right) \cdot \left(0.5 \cdot M\right)\right)\right)\right) \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 10
Error	37.19%
Cost	15052

\[\begin{array}{l} \mathbf{if}\;h \leq -1.55 \cdot 10^{+92}:\\ \;\;\;\;\left(1 - h \cdot \frac{{\left(M \cdot \frac{0.5 \cdot D}{d}\right)}^{2}}{2 \cdot \ell}\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{elif}\;h \leq -7.5 \cdot 10^{-177}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;h \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{\frac{-\ell}{d}}}}{\sqrt{-h}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \left(\frac{h}{\ell} \cdot -0.5\right) \cdot \left(\frac{D}{d} \cdot \left(\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right) \cdot \left(0.5 \cdot M\right)\right)\right)\right) \cdot \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

Alternative 11
Error	38.44%
Cost	14468

\[\begin{array}{l} \mathbf{if}\;h \leq -2 \cdot 10^{+88}:\\ \;\;\;\;\left(1 - h \cdot \frac{{\left(M \cdot \frac{0.5 \cdot D}{d}\right)}^{2}}{2 \cdot \ell}\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{elif}\;h \leq -4.2 \cdot 10^{-170}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;h \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{\frac{-\ell}{d}}}}{\sqrt{-h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

Alternative 12
Error	38.31%
Cost	13640

\[\begin{array}{l} \mathbf{if}\;h \leq -7.4 \cdot 10^{-174}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;h \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{\frac{-\ell}{d}}}}{\sqrt{-h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

Alternative 13
Error	40.09%
Cost	13380

\[\begin{array}{l} \mathbf{if}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

Alternative 14
Error	46.07%
Cost	13252

\[\begin{array}{l} \mathbf{if}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

Alternative 15
Error	52.33%
Cost	6980

\[\begin{array}{l} \mathbf{if}\;h \leq -1.9 \cdot 10^{-308}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}}\\ \end{array} \]

Alternative 16
Error	52.2%
Cost	6980

\[\begin{array}{l} \mathbf{if}\;h \leq -1.9 \cdot 10^{-308}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \end{array} \]

Alternative 17
Error	69.16%
Cost	6720

\[\frac{d}{\sqrt{h \cdot \ell}} \]

Reproduce

herbie shell --seed 2023121 
(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)))))