Henrywood and Agarwal, Equation (12)

Average Accuracy: 36.0% → 48.4%

Time: 35.5s

Precision: binary64

Cost: 96976

Math

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

↓

\[\begin{array}{l} t_0 := \left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ t_1 := \left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;t_0 \cdot \left(-0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \left(M \cdot \frac{\frac{M}{\frac{d}{h}}}{d}\right)\right)\right)\\ \mathbf{elif}\;t_1 \leq -1 \cdot 10^{-46}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;\left(1 + \frac{h}{\ell} \cdot \left(-0.125 \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}\right)\right) \cdot t_0\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+184}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

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 (fabs (/ d (sqrt (* h l)))))
        (t_1
         (*
          (* (pow (/ d h) 0.5) (pow (/ d l) 0.5))
          (- 1.0 (* (* 0.5 (pow (/ (* M D) (* d 2.0)) 2.0)) (/ h l))))))
   (if (<= t_1 (- INFINITY))
     (* t_0 (* -0.125 (* (/ (* D D) l) (* M (/ (/ M (/ d h)) d)))))
     (if (<= t_1 -1e-46)
       t_1
       (if (<= t_1 0.0)
         (* (+ 1.0 (* (/ h l) (* -0.125 (pow (* D (/ M d)) 2.0)))) t_0)
         (if (<= t_1 5e+184) (* (sqrt (/ d h)) (sqrt (/ d l))) t_0))))))

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 = fabs((d / sqrt((h * l))));
	double t_1 = (pow((d / h), 0.5) * pow((d / l), 0.5)) * (1.0 - ((0.5 * pow(((M * D) / (d * 2.0)), 2.0)) * (h / l)));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_0 * (-0.125 * (((D * D) / l) * (M * ((M / (d / h)) / d))));
	} else if (t_1 <= -1e-46) {
		tmp = t_1;
	} else if (t_1 <= 0.0) {
		tmp = (1.0 + ((h / l) * (-0.125 * pow((D * (M / d)), 2.0)))) * t_0;
	} else if (t_1 <= 5e+184) {
		tmp = sqrt((d / h)) * sqrt((d / l));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double d, double h, double l, double M, double D) {
	return (Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * Math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
}

↓

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.abs((d / Math.sqrt((h * l))));
	double t_1 = (Math.pow((d / h), 0.5) * Math.pow((d / l), 0.5)) * (1.0 - ((0.5 * Math.pow(((M * D) / (d * 2.0)), 2.0)) * (h / l)));
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = t_0 * (-0.125 * (((D * D) / l) * (M * ((M / (d / h)) / d))));
	} else if (t_1 <= -1e-46) {
		tmp = t_1;
	} else if (t_1 <= 0.0) {
		tmp = (1.0 + ((h / l) * (-0.125 * Math.pow((D * (M / d)), 2.0)))) * t_0;
	} else if (t_1 <= 5e+184) {
		tmp = Math.sqrt((d / h)) * Math.sqrt((d / l));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	return (math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))

↓

def code(d, h, l, M, D):
	t_0 = math.fabs((d / math.sqrt((h * l))))
	t_1 = (math.pow((d / h), 0.5) * math.pow((d / l), 0.5)) * (1.0 - ((0.5 * math.pow(((M * D) / (d * 2.0)), 2.0)) * (h / l)))
	tmp = 0
	if t_1 <= -math.inf:
		tmp = t_0 * (-0.125 * (((D * D) / l) * (M * ((M / (d / h)) / d))))
	elif t_1 <= -1e-46:
		tmp = t_1
	elif t_1 <= 0.0:
		tmp = (1.0 + ((h / l) * (-0.125 * math.pow((D * (M / d)), 2.0)))) * t_0
	elif t_1 <= 5e+184:
		tmp = math.sqrt((d / h)) * math.sqrt((d / l))
	else:
		tmp = t_0
	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 = abs(Float64(d / sqrt(Float64(h * l))))
	t_1 = Float64(Float64((Float64(d / h) ^ 0.5) * (Float64(d / l) ^ 0.5)) * Float64(1.0 - Float64(Float64(0.5 * (Float64(Float64(M * D) / Float64(d * 2.0)) ^ 2.0)) * Float64(h / l))))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(t_0 * Float64(-0.125 * Float64(Float64(Float64(D * D) / l) * Float64(M * Float64(Float64(M / Float64(d / h)) / d)))));
	elseif (t_1 <= -1e-46)
		tmp = t_1;
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(1.0 + Float64(Float64(h / l) * Float64(-0.125 * (Float64(D * Float64(M / d)) ^ 2.0)))) * t_0);
	elseif (t_1 <= 5e+184)
		tmp = Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp = code(d, h, l, M, D)
	tmp = (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((M * D) / (2.0 * d)) ^ 2.0)) * (h / l)));
end

↓

function tmp_2 = code(d, h, l, M, D)
	t_0 = abs((d / sqrt((h * l))));
	t_1 = (((d / h) ^ 0.5) * ((d / l) ^ 0.5)) * (1.0 - ((0.5 * (((M * D) / (d * 2.0)) ^ 2.0)) * (h / l)));
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = t_0 * (-0.125 * (((D * D) / l) * (M * ((M / (d / h)) / d))));
	elseif (t_1 <= -1e-46)
		tmp = t_1;
	elseif (t_1 <= 0.0)
		tmp = (1.0 + ((h / l) * (-0.125 * ((D * (M / d)) ^ 2.0)))) * t_0;
	elseif (t_1 <= 5e+184)
		tmp = sqrt((d / h)) * sqrt((d / l));
	else
		tmp = t_0;
	end
	tmp_2 = 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[Abs[N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = 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[(0.5 * N[Power[N[(N[(M * D), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(t$95$0 * N[(-0.125 * N[(N[(N[(D * D), $MachinePrecision] / l), $MachinePrecision] * N[(M * N[(N[(M / N[(d / h), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -1e-46], t$95$1, If[LessEqual[t$95$1, 0.0], N[(N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(-0.125 * N[Power[N[(D * N[(M / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[t$95$1, 5e+184], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]]]]

Derivation

1. Split input into 5 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 0.0%

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

   2. Applied egg-rr 0.0%

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

      [Start] 0.0	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      sub-neg [=>] 0.0	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\left(1 + \left(-\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]
      distribute-lft-in [=>] 0.0	\[ \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot 1 + \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(-\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)} \]
      *-commutative [<=] 0.0	\[ \color{blue}{1 \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} + \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(-\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      *-un-lft-identity [<=] 0.0	\[ \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} + \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(-\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 0.0	\[ {\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} + \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(-\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 0.0	\[ \color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} + \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(-\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 0.0	\[ \sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}} + \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(-\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 0.0	\[ \sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} + \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(-\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      sqrt-unprod [=>] 0.0	\[ \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} + \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(-\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 0.4%

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

      [Start] 0.0	\[ \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} + \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(\frac{h}{\ell} \cdot \left({\left(0.5 \cdot \frac{M \cdot D}{d}\right)}^{2} \cdot -0.5\right)\right) \]
      *-rgt-identity [<=] 0.0	\[ \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot 1} + \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(\frac{h}{\ell} \cdot \left({\left(0.5 \cdot \frac{M \cdot D}{d}\right)}^{2} \cdot -0.5\right)\right) \]
      distribute-lft-in [<=] 0.0	\[ \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(0.5 \cdot \frac{M \cdot D}{d}\right)}^{2} \cdot -0.5\right)\right)} \]
      +-commutative [=>] 0.0	\[ \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \color{blue}{\left(\frac{h}{\ell} \cdot \left({\left(0.5 \cdot \frac{M \cdot D}{d}\right)}^{2} \cdot -0.5\right) + 1\right)} \]
      fma-def [=>] 0.0	\[ \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \color{blue}{\mathsf{fma}\left(\frac{h}{\ell}, {\left(0.5 \cdot \frac{M \cdot D}{d}\right)}^{2} \cdot -0.5, 1\right)} \]
      *-commutative [=>] 0.0	\[ \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \mathsf{fma}\left(\frac{h}{\ell}, {\left(0.5 \cdot \frac{\color{blue}{D \cdot M}}{d}\right)}^{2} \cdot -0.5, 1\right) \]
      associate-/l* [=>] 0.4	\[ \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \mathsf{fma}\left(\frac{h}{\ell}, {\left(0.5 \cdot \color{blue}{\frac{D}{\frac{d}{M}}}\right)}^{2} \cdot -0.5, 1\right) \]

   4. Applied egg-rr 0.8%

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

      [Start] 0.4	\[ \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \mathsf{fma}\left(\frac{h}{\ell}, {\left(0.5 \cdot \frac{D}{\frac{d}{M}}\right)}^{2} \cdot -0.5, 1\right) \]
      add-sqr-sqrt [=>] 0.4	\[ \sqrt{\color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}}} \cdot \mathsf{fma}\left(\frac{h}{\ell}, {\left(0.5 \cdot \frac{D}{\frac{d}{M}}\right)}^{2} \cdot -0.5, 1\right) \]
      rem-sqrt-square [=>] 0.4	\[ \color{blue}{\left|\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right|} \cdot \mathsf{fma}\left(\frac{h}{\ell}, {\left(0.5 \cdot \frac{D}{\frac{d}{M}}\right)}^{2} \cdot -0.5, 1\right) \]
      frac-times [=>] 0.0	\[ \left|\sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}}\right| \cdot \mathsf{fma}\left(\frac{h}{\ell}, {\left(0.5 \cdot \frac{D}{\frac{d}{M}}\right)}^{2} \cdot -0.5, 1\right) \]
      sqrt-div [=>] 0.1	\[ \left|\color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}}\right| \cdot \mathsf{fma}\left(\frac{h}{\ell}, {\left(0.5 \cdot \frac{D}{\frac{d}{M}}\right)}^{2} \cdot -0.5, 1\right) \]
      sqrt-unprod [<=] 0.4	\[ \left|\frac{\color{blue}{\sqrt{d} \cdot \sqrt{d}}}{\sqrt{h \cdot \ell}}\right| \cdot \mathsf{fma}\left(\frac{h}{\ell}, {\left(0.5 \cdot \frac{D}{\frac{d}{M}}\right)}^{2} \cdot -0.5, 1\right) \]
      add-sqr-sqrt [<=] 0.8	\[ \left|\frac{\color{blue}{d}}{\sqrt{h \cdot \ell}}\right| \cdot \mathsf{fma}\left(\frac{h}{\ell}, {\left(0.5 \cdot \frac{D}{\frac{d}{M}}\right)}^{2} \cdot -0.5, 1\right) \]

   5. Taylor expanded in h around inf 1.3%

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

   6. Simplified 1.8%

      \[\leadsto \left|\frac{d}{\sqrt{h \cdot \ell}}\right| \cdot \color{blue}{\left(-0.125 \cdot \left(\frac{D}{\frac{\ell}{D}} \cdot \frac{M \cdot M}{\frac{d \cdot d}{h}}\right)\right)} \]
      Proof

      [Start] 1.3	\[ \left|\frac{d}{\sqrt{h \cdot \ell}}\right| \cdot \left(-0.125 \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{\ell \cdot {d}^{2}}\right) \]
      associate-*r/ [=>] 1.3	\[ \left|\frac{d}{\sqrt{h \cdot \ell}}\right| \cdot \color{blue}{\frac{-0.125 \cdot \left({D}^{2} \cdot \left(h \cdot {M}^{2}\right)\right)}{\ell \cdot {d}^{2}}} \]
      *-commutative [<=] 1.3	\[ \left|\frac{d}{\sqrt{h \cdot \ell}}\right| \cdot \frac{-0.125 \cdot \left({D}^{2} \cdot \color{blue}{\left({M}^{2} \cdot h\right)}\right)}{\ell \cdot {d}^{2}} \]
      associate-*r/ [<=] 1.3	\[ \left|\frac{d}{\sqrt{h \cdot \ell}}\right| \cdot \color{blue}{\left(-0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}\right)} \]
      times-frac [=>] 1.3	\[ \left|\frac{d}{\sqrt{h \cdot \ell}}\right| \cdot \left(-0.125 \cdot \color{blue}{\left(\frac{{D}^{2}}{\ell} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right)}\right) \]
      unpow2 [=>] 1.3	\[ \left|\frac{d}{\sqrt{h \cdot \ell}}\right| \cdot \left(-0.125 \cdot \left(\frac{\color{blue}{D \cdot D}}{\ell} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right)\right) \]
      associate-/l* [=>] 1.5	\[ \left|\frac{d}{\sqrt{h \cdot \ell}}\right| \cdot \left(-0.125 \cdot \left(\color{blue}{\frac{D}{\frac{\ell}{D}}} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right)\right) \]
      associate-/l* [=>] 1.8	\[ \left|\frac{d}{\sqrt{h \cdot \ell}}\right| \cdot \left(-0.125 \cdot \left(\frac{D}{\frac{\ell}{D}} \cdot \color{blue}{\frac{{M}^{2}}{\frac{{d}^{2}}{h}}}\right)\right) \]
      unpow2 [=>] 1.8	\[ \left|\frac{d}{\sqrt{h \cdot \ell}}\right| \cdot \left(-0.125 \cdot \left(\frac{D}{\frac{\ell}{D}} \cdot \frac{\color{blue}{M \cdot M}}{\frac{{d}^{2}}{h}}\right)\right) \]
      unpow2 [=>] 1.8	\[ \left|\frac{d}{\sqrt{h \cdot \ell}}\right| \cdot \left(-0.125 \cdot \left(\frac{D}{\frac{\ell}{D}} \cdot \frac{M \cdot M}{\frac{\color{blue}{d \cdot d}}{h}}\right)\right) \]

   7. Taylor expanded in D around 0 1.3%

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

   8. Simplified 2.7%

      \[\leadsto \left|\frac{d}{\sqrt{h \cdot \ell}}\right| \cdot \left(-0.125 \cdot \color{blue}{\left(\frac{D \cdot D}{\ell} \cdot \left(M \cdot \frac{\frac{M}{\frac{d}{h}}}{d}\right)\right)}\right) \]
      Proof

      [Start] 1.3	\[ \left|\frac{d}{\sqrt{h \cdot \ell}}\right| \cdot \left(-0.125 \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{{d}^{2} \cdot \ell}\right) \]
      unpow2 [=>] 1.3	\[ \left|\frac{d}{\sqrt{h \cdot \ell}}\right| \cdot \left(-0.125 \cdot \frac{{D}^{2} \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)}{{d}^{2} \cdot \ell}\right) \]
      *-commutative [<=] 1.3	\[ \left|\frac{d}{\sqrt{h \cdot \ell}}\right| \cdot \left(-0.125 \cdot \frac{{D}^{2} \cdot \color{blue}{\left(\left(M \cdot M\right) \cdot h\right)}}{{d}^{2} \cdot \ell}\right) \]
      unpow2 [=>] 1.3	\[ \left|\frac{d}{\sqrt{h \cdot \ell}}\right| \cdot \left(-0.125 \cdot \frac{{D}^{2} \cdot \left(\left(M \cdot M\right) \cdot h\right)}{\color{blue}{\left(d \cdot d\right)} \cdot \ell}\right) \]
      *-commutative [<=] 1.3	\[ \left|\frac{d}{\sqrt{h \cdot \ell}}\right| \cdot \left(-0.125 \cdot \frac{{D}^{2} \cdot \left(\left(M \cdot M\right) \cdot h\right)}{\color{blue}{\ell \cdot \left(d \cdot d\right)}}\right) \]
      times-frac [=>] 1.3	\[ \left|\frac{d}{\sqrt{h \cdot \ell}}\right| \cdot \left(-0.125 \cdot \color{blue}{\left(\frac{{D}^{2}}{\ell} \cdot \frac{\left(M \cdot M\right) \cdot h}{d \cdot d}\right)}\right) \]
      unpow2 [=>] 1.3	\[ \left|\frac{d}{\sqrt{h \cdot \ell}}\right| \cdot \left(-0.125 \cdot \left(\frac{\color{blue}{D \cdot D}}{\ell} \cdot \frac{\left(M \cdot M\right) \cdot h}{d \cdot d}\right)\right) \]
      associate-/l* [=>] 1.5	\[ \left|\frac{d}{\sqrt{h \cdot \ell}}\right| \cdot \left(-0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \color{blue}{\frac{M \cdot M}{\frac{d \cdot d}{h}}}\right)\right) \]
      associate-*l/ [<=] 1.9	\[ \left|\frac{d}{\sqrt{h \cdot \ell}}\right| \cdot \left(-0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \frac{M \cdot M}{\color{blue}{\frac{d}{h} \cdot d}}\right)\right) \]
      associate-*r/ [<=] 2.5	\[ \left|\frac{d}{\sqrt{h \cdot \ell}}\right| \cdot \left(-0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \color{blue}{\left(M \cdot \frac{M}{\frac{d}{h} \cdot d}\right)}\right)\right) \]
      associate-/r* [=>] 2.7	\[ \left|\frac{d}{\sqrt{h \cdot \ell}}\right| \cdot \left(-0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \left(M \cdot \color{blue}{\frac{\frac{M}{\frac{d}{h}}}{d}}\right)\right)\right) \]

   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)))) < -1.00000000000000002e-46

   1. Initial program 98.1%

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

   if -1.00000000000000002e-46 < (*.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

   1. Initial program 45.4%

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

   2. Applied egg-rr 35.3%

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

      [Start] 45.4	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      sub-neg [=>] 45.4	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\left(1 + \left(-\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]
      distribute-lft-in [=>] 45.4	\[ \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot 1 + \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(-\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)} \]
      *-commutative [<=] 45.4	\[ \color{blue}{1 \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} + \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(-\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      *-un-lft-identity [<=] 45.4	\[ \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} + \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(-\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 45.4	\[ {\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} + \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(-\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 45.4	\[ \color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} + \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(-\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 45.4	\[ \sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}} + \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(-\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 45.4	\[ \sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} + \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(-\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      sqrt-unprod [=>] 45.0	\[ \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} + \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(-\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 33.4%

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

      [Start] 35.3	\[ \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} + \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(\frac{h}{\ell} \cdot \left({\left(0.5 \cdot \frac{M \cdot D}{d}\right)}^{2} \cdot -0.5\right)\right) \]
      *-rgt-identity [<=] 35.3	\[ \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot 1} + \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(\frac{h}{\ell} \cdot \left({\left(0.5 \cdot \frac{M \cdot D}{d}\right)}^{2} \cdot -0.5\right)\right) \]
      distribute-lft-in [<=] 35.3	\[ \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(0.5 \cdot \frac{M \cdot D}{d}\right)}^{2} \cdot -0.5\right)\right)} \]
      +-commutative [=>] 35.3	\[ \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \color{blue}{\left(\frac{h}{\ell} \cdot \left({\left(0.5 \cdot \frac{M \cdot D}{d}\right)}^{2} \cdot -0.5\right) + 1\right)} \]
      fma-def [=>] 35.3	\[ \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \color{blue}{\mathsf{fma}\left(\frac{h}{\ell}, {\left(0.5 \cdot \frac{M \cdot D}{d}\right)}^{2} \cdot -0.5, 1\right)} \]
      *-commutative [=>] 35.3	\[ \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \mathsf{fma}\left(\frac{h}{\ell}, {\left(0.5 \cdot \frac{\color{blue}{D \cdot M}}{d}\right)}^{2} \cdot -0.5, 1\right) \]
      associate-/l* [=>] 33.4	\[ \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \mathsf{fma}\left(\frac{h}{\ell}, {\left(0.5 \cdot \color{blue}{\frac{D}{\frac{d}{M}}}\right)}^{2} \cdot -0.5, 1\right) \]

   4. Applied egg-rr 71.1%

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

      [Start] 33.4	\[ \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \mathsf{fma}\left(\frac{h}{\ell}, {\left(0.5 \cdot \frac{D}{\frac{d}{M}}\right)}^{2} \cdot -0.5, 1\right) \]
      add-sqr-sqrt [=>] 33.4	\[ \sqrt{\color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}}} \cdot \mathsf{fma}\left(\frac{h}{\ell}, {\left(0.5 \cdot \frac{D}{\frac{d}{M}}\right)}^{2} \cdot -0.5, 1\right) \]
      rem-sqrt-square [=>] 33.4	\[ \color{blue}{\left|\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right|} \cdot \mathsf{fma}\left(\frac{h}{\ell}, {\left(0.5 \cdot \frac{D}{\frac{d}{M}}\right)}^{2} \cdot -0.5, 1\right) \]
      frac-times [=>] 34.1	\[ \left|\sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}}\right| \cdot \mathsf{fma}\left(\frac{h}{\ell}, {\left(0.5 \cdot \frac{D}{\frac{d}{M}}\right)}^{2} \cdot -0.5, 1\right) \]
      sqrt-div [=>] 46.3	\[ \left|\color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}}\right| \cdot \mathsf{fma}\left(\frac{h}{\ell}, {\left(0.5 \cdot \frac{D}{\frac{d}{M}}\right)}^{2} \cdot -0.5, 1\right) \]
      sqrt-unprod [<=] 36.7	\[ \left|\frac{\color{blue}{\sqrt{d} \cdot \sqrt{d}}}{\sqrt{h \cdot \ell}}\right| \cdot \mathsf{fma}\left(\frac{h}{\ell}, {\left(0.5 \cdot \frac{D}{\frac{d}{M}}\right)}^{2} \cdot -0.5, 1\right) \]
      add-sqr-sqrt [<=] 71.1	\[ \left|\frac{\color{blue}{d}}{\sqrt{h \cdot \ell}}\right| \cdot \mathsf{fma}\left(\frac{h}{\ell}, {\left(0.5 \cdot \frac{D}{\frac{d}{M}}\right)}^{2} \cdot -0.5, 1\right) \]

   5. Applied egg-rr 70.4%

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

      [Start] 71.1	\[ \left|\frac{d}{\sqrt{h \cdot \ell}}\right| \cdot \mathsf{fma}\left(\frac{h}{\ell}, {\left(0.5 \cdot \frac{D}{\frac{d}{M}}\right)}^{2} \cdot -0.5, 1\right) \]
      fma-udef [=>] 71.1	\[ \left|\frac{d}{\sqrt{h \cdot \ell}}\right| \cdot \color{blue}{\left(\frac{h}{\ell} \cdot \left({\left(0.5 \cdot \frac{D}{\frac{d}{M}}\right)}^{2} \cdot -0.5\right) + 1\right)} \]
      *-commutative [=>] 71.1	\[ \left|\frac{d}{\sqrt{h \cdot \ell}}\right| \cdot \left(\frac{h}{\ell} \cdot \color{blue}{\left(-0.5 \cdot {\left(0.5 \cdot \frac{D}{\frac{d}{M}}\right)}^{2}\right)} + 1\right) \]
      unpow2 [=>] 71.1	\[ \left|\frac{d}{\sqrt{h \cdot \ell}}\right| \cdot \left(\frac{h}{\ell} \cdot \left(-0.5 \cdot \color{blue}{\left(\left(0.5 \cdot \frac{D}{\frac{d}{M}}\right) \cdot \left(0.5 \cdot \frac{D}{\frac{d}{M}}\right)\right)}\right) + 1\right) \]
      swap-sqr [=>] 71.1	\[ \left|\frac{d}{\sqrt{h \cdot \ell}}\right| \cdot \left(\frac{h}{\ell} \cdot \left(-0.5 \cdot \color{blue}{\left(\left(0.5 \cdot 0.5\right) \cdot \left(\frac{D}{\frac{d}{M}} \cdot \frac{D}{\frac{d}{M}}\right)\right)}\right) + 1\right) \]
      metadata-eval [=>] 71.1	\[ \left|\frac{d}{\sqrt{h \cdot \ell}}\right| \cdot \left(\frac{h}{\ell} \cdot \left(-0.5 \cdot \left(\color{blue}{0.25} \cdot \left(\frac{D}{\frac{d}{M}} \cdot \frac{D}{\frac{d}{M}}\right)\right)\right) + 1\right) \]
      metadata-eval [<=] 71.1	\[ \left|\frac{d}{\sqrt{h \cdot \ell}}\right| \cdot \left(\frac{h}{\ell} \cdot \left(-0.5 \cdot \left(\color{blue}{\left(-0.5 \cdot -0.5\right)} \cdot \left(\frac{D}{\frac{d}{M}} \cdot \frac{D}{\frac{d}{M}}\right)\right)\right) + 1\right) \]
      associate-*r* [=>] 71.1	\[ \left|\frac{d}{\sqrt{h \cdot \ell}}\right| \cdot \left(\frac{h}{\ell} \cdot \color{blue}{\left(\left(-0.5 \cdot \left(-0.5 \cdot -0.5\right)\right) \cdot \left(\frac{D}{\frac{d}{M}} \cdot \frac{D}{\frac{d}{M}}\right)\right)} + 1\right) \]
      metadata-eval [=>] 71.1	\[ \left|\frac{d}{\sqrt{h \cdot \ell}}\right| \cdot \left(\frac{h}{\ell} \cdot \left(\left(-0.5 \cdot \color{blue}{0.25}\right) \cdot \left(\frac{D}{\frac{d}{M}} \cdot \frac{D}{\frac{d}{M}}\right)\right) + 1\right) \]
      metadata-eval [=>] 71.1	\[ \left|\frac{d}{\sqrt{h \cdot \ell}}\right| \cdot \left(\frac{h}{\ell} \cdot \left(\color{blue}{-0.125} \cdot \left(\frac{D}{\frac{d}{M}} \cdot \frac{D}{\frac{d}{M}}\right)\right) + 1\right) \]
      pow2 [=>] 71.1	\[ \left|\frac{d}{\sqrt{h \cdot \ell}}\right| \cdot \left(\frac{h}{\ell} \cdot \left(-0.125 \cdot \color{blue}{{\left(\frac{D}{\frac{d}{M}}\right)}^{2}}\right) + 1\right) \]
      div-inv [=>] 70.4	\[ \left|\frac{d}{\sqrt{h \cdot \ell}}\right| \cdot \left(\frac{h}{\ell} \cdot \left(-0.125 \cdot {\color{blue}{\left(D \cdot \frac{1}{\frac{d}{M}}\right)}}^{2}\right) + 1\right) \]
      clear-num [<=] 70.4	\[ \left|\frac{d}{\sqrt{h \cdot \ell}}\right| \cdot \left(\frac{h}{\ell} \cdot \left(-0.125 \cdot {\left(D \cdot \color{blue}{\frac{M}{d}}\right)}^{2}\right) + 1\right) \]

   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)))) < 4.9999999999999999e184

   1. Initial program 98.6%

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

   2. Simplified 98.1%

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

      [Start] 98.6	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      associate-*l* [=>] 98.6	\[ \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]
      metadata-eval [=>] 98.6	\[ {\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]
      unpow1/2 [=>] 98.6	\[ \color{blue}{\sqrt{\frac{d}{h}}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]
      metadata-eval [=>] 98.6	\[ \sqrt{\frac{d}{h}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]
      unpow1/2 [=>] 98.6	\[ \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]
      sub-neg [=>] 98.6	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(1 + \left(-\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]
      +-commutative [=>] 98.6	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(\left(-\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) + 1\right)}\right) \]
      *-commutative [=>] 98.6	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\left(-\color{blue}{\frac{h}{\ell} \cdot \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)}\right) + 1\right)\right) \]
      distribute-rgt-neg-in [=>] 98.6	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\color{blue}{\frac{h}{\ell} \cdot \left(-\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} + 1\right)\right) \]
      fma-def [=>] 98.6	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\mathsf{fma}\left(\frac{h}{\ell}, -\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}, 1\right)}\right) \]

   3. Taylor expanded in h around 0 98.1%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{1}\right) \]

   if 4.9999999999999999e184 < (*.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 9.3%

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

   2. Applied egg-rr 0.0%

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

      [Start] 9.3	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      sub-neg [=>] 9.3	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\left(1 + \left(-\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]
      distribute-lft-in [=>] 9.3	\[ \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot 1 + \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(-\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)} \]
      *-commutative [<=] 9.3	\[ \color{blue}{1 \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} + \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(-\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      *-un-lft-identity [<=] 9.3	\[ \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} + \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(-\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 9.3	\[ {\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} + \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(-\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 9.3	\[ \color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} + \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(-\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 9.3	\[ \sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}} + \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(-\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 9.3	\[ \sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} + \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(-\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      sqrt-unprod [=>] 0.0	\[ \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} + \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(-\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 0.0%

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

      [Start] 0.0	\[ \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} + \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(\frac{h}{\ell} \cdot \left({\left(0.5 \cdot \frac{M \cdot D}{d}\right)}^{2} \cdot -0.5\right)\right) \]
      *-rgt-identity [<=] 0.0	\[ \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot 1} + \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(\frac{h}{\ell} \cdot \left({\left(0.5 \cdot \frac{M \cdot D}{d}\right)}^{2} \cdot -0.5\right)\right) \]
      distribute-lft-in [<=] 0.0	\[ \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(0.5 \cdot \frac{M \cdot D}{d}\right)}^{2} \cdot -0.5\right)\right)} \]
      +-commutative [=>] 0.0	\[ \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \color{blue}{\left(\frac{h}{\ell} \cdot \left({\left(0.5 \cdot \frac{M \cdot D}{d}\right)}^{2} \cdot -0.5\right) + 1\right)} \]
      fma-def [=>] 0.0	\[ \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \color{blue}{\mathsf{fma}\left(\frac{h}{\ell}, {\left(0.5 \cdot \frac{M \cdot D}{d}\right)}^{2} \cdot -0.5, 1\right)} \]
      *-commutative [=>] 0.0	\[ \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \mathsf{fma}\left(\frac{h}{\ell}, {\left(0.5 \cdot \frac{\color{blue}{D \cdot M}}{d}\right)}^{2} \cdot -0.5, 1\right) \]
      associate-/l* [=>] 0.0	\[ \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \mathsf{fma}\left(\frac{h}{\ell}, {\left(0.5 \cdot \color{blue}{\frac{D}{\frac{d}{M}}}\right)}^{2} \cdot -0.5, 1\right) \]

   4. Applied egg-rr 26.2%

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

      [Start] 0.0	\[ \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \mathsf{fma}\left(\frac{h}{\ell}, {\left(0.5 \cdot \frac{D}{\frac{d}{M}}\right)}^{2} \cdot -0.5, 1\right) \]
      add-sqr-sqrt [=>] 0.0	\[ \sqrt{\color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}}} \cdot \mathsf{fma}\left(\frac{h}{\ell}, {\left(0.5 \cdot \frac{D}{\frac{d}{M}}\right)}^{2} \cdot -0.5, 1\right) \]
      rem-sqrt-square [=>] 0.0	\[ \color{blue}{\left|\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right|} \cdot \mathsf{fma}\left(\frac{h}{\ell}, {\left(0.5 \cdot \frac{D}{\frac{d}{M}}\right)}^{2} \cdot -0.5, 1\right) \]
      frac-times [=>] 2.1	\[ \left|\sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}}\right| \cdot \mathsf{fma}\left(\frac{h}{\ell}, {\left(0.5 \cdot \frac{D}{\frac{d}{M}}\right)}^{2} \cdot -0.5, 1\right) \]
      sqrt-div [=>] 7.0	\[ \left|\color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}}\right| \cdot \mathsf{fma}\left(\frac{h}{\ell}, {\left(0.5 \cdot \frac{D}{\frac{d}{M}}\right)}^{2} \cdot -0.5, 1\right) \]
      sqrt-unprod [<=] 13.0	\[ \left|\frac{\color{blue}{\sqrt{d} \cdot \sqrt{d}}}{\sqrt{h \cdot \ell}}\right| \cdot \mathsf{fma}\left(\frac{h}{\ell}, {\left(0.5 \cdot \frac{D}{\frac{d}{M}}\right)}^{2} \cdot -0.5, 1\right) \]
      add-sqr-sqrt [<=] 26.2	\[ \left|\frac{\color{blue}{d}}{\sqrt{h \cdot \ell}}\right| \cdot \mathsf{fma}\left(\frac{h}{\ell}, {\left(0.5 \cdot \frac{D}{\frac{d}{M}}\right)}^{2} \cdot -0.5, 1\right) \]

   5. Taylor expanded in h around 0 38.0%

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

4. Final simplification 48.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -\infty:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right| \cdot \left(-0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \left(M \cdot \frac{\frac{M}{\frac{d}{h}}}{d}\right)\right)\right)\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -1 \cdot 10^{-46}:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 0:\\ \;\;\;\;\left(1 + \frac{h}{\ell} \cdot \left(-0.125 \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}\right)\right) \cdot \left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 5 \cdot 10^{+184}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	48.4%
Cost	76044

\[\begin{array}{l} t_0 := \left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ t_1 := \left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_2 := \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;t_1 \leq -2 \cdot 10^{+32}:\\ \;\;\;\;t_2 \cdot \left(1 - {\left(\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right) \cdot \sqrt{\frac{h \cdot 0.5}{\ell}}\right)}^{2}\right)\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;\left(1 + \frac{h}{\ell} \cdot \left(-0.125 \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}\right)\right) \cdot t_0\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+184}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Accuracy	40.7%
Cost	21264

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{\frac{M}{2}}{d}\right)}^{2}\right)\right)\right)\\ \mathbf{if}\;h \leq -1.85 \cdot 10^{+204}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;h \leq 3 \cdot 10^{-309}:\\ \;\;\;\;\left(1 + \frac{h}{\ell} \cdot \left(-0.125 \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}\right)\right) \cdot \left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \mathbf{elif}\;h \leq 8 \cdot 10^{-133}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \mathbf{elif}\;h \leq 2 \cdot 10^{+69}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \mathsf{fma}\left(\frac{h}{\ell}, {\left(0.5 \cdot \frac{D}{\frac{d}{M}}\right)}^{2} \cdot -0.5, 1\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Accuracy	40.7%
Cost	21264

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ t_1 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;h \leq -3 \cdot 10^{+204}:\\ \;\;\;\;t_0 \cdot \left(t_1 \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{\frac{M}{2}}{d}\right)}^{2}\right)\right)\right)\\ \mathbf{elif}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(1 + \frac{h}{\ell} \cdot \left(-0.125 \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}\right)\right) \cdot \left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \mathbf{elif}\;h \leq 10^{-132}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \mathbf{elif}\;h \leq 1.02 \cdot 10^{+68}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \mathsf{fma}\left(\frac{h}{\ell}, {\left(0.5 \cdot \frac{D}{\frac{d}{M}}\right)}^{2} \cdot -0.5, 1\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(t_1 \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right)\right)\\ \end{array} \]

Alternative 4
Accuracy	42.3%
Cost	21264

\[\begin{array}{l} t_0 := \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - h \cdot \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{0.5}}\right)\\ \mathbf{if}\;h \leq -2.9 \cdot 10^{+187}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(1 + \frac{h}{\ell} \cdot \left(-0.125 \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}\right)\right) \cdot \left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \mathbf{elif}\;h \leq 5 \cdot 10^{-134}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \mathbf{elif}\;h \leq 1.85 \cdot 10^{+70}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \mathsf{fma}\left(\frac{h}{\ell}, {\left(0.5 \cdot \frac{D}{\frac{d}{M}}\right)}^{2} \cdot -0.5, 1\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Accuracy	41.2%
Cost	21008

\[\begin{array}{l} t_0 := \left(1 + \frac{h}{\ell} \cdot \left(-0.125 \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}\right)\right) \cdot \left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \mathbf{if}\;h \leq -4.7 \cdot 10^{+145}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \frac{\sqrt{-d}}{\sqrt{-h}}\\ \mathbf{elif}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;h \leq 2.4 \cdot 10^{-128}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \mathbf{elif}\;h \leq 3.1 \cdot 10^{+208}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{d \cdot {h}^{-0.5}}{\sqrt{\ell}}\\ \end{array} \]

Alternative 6
Accuracy	40.9%
Cost	20036

\[\begin{array}{l} t_0 := \frac{d}{\sqrt{h \cdot \ell}}\\ \mathbf{if}\;d \leq -1.65 \cdot 10^{-68}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\\ \mathbf{elif}\;d \leq 10^{-183}:\\ \;\;\;\;\left|t_0\right|\\ \mathbf{elif}\;d \leq 230:\\ \;\;\;\;t_0 \cdot \left(1 + {\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \frac{h \cdot -0.125}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

Alternative 7
Accuracy	40.7%
Cost	19908

\[\begin{array}{l} t_0 := \frac{d}{\sqrt{h \cdot \ell}}\\ \mathbf{if}\;d \leq -1.16 \cdot 10^{-68}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \frac{\sqrt{-d}}{\sqrt{-h}}\\ \mathbf{elif}\;d \leq 1.1 \cdot 10^{-183}:\\ \;\;\;\;\left|t_0\right|\\ \mathbf{elif}\;d \leq 235:\\ \;\;\;\;t_0 \cdot \left(1 + {\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \frac{h \cdot -0.125}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

Alternative 8
Accuracy	39.6%
Cost	14344

\[\begin{array}{l} t_0 := \frac{d}{\sqrt{h \cdot \ell}}\\ \mathbf{if}\;d \leq 7.2 \cdot 10^{-185}:\\ \;\;\;\;\left|t_0\right|\\ \mathbf{elif}\;d \leq 230:\\ \;\;\;\;t_0 \cdot \left(1 + {\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \frac{h \cdot -0.125}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

Alternative 9
Accuracy	39.6%
Cost	13316

\[\begin{array}{l} \mathbf{if}\;\ell \leq 1.8 \cdot 10^{-261}:\\ \;\;\;\;\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-d\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\frac{\sqrt{\ell}}{{h}^{-0.5}}}\\ \end{array} \]

Alternative 10
Accuracy	39.6%
Cost	13252

\[\begin{array}{l} \mathbf{if}\;\ell \leq 1.8 \cdot 10^{-261}:\\ \;\;\;\;\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-d\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

Alternative 11
Accuracy	35.5%
Cost	7044

\[\begin{array}{l} \mathbf{if}\;\ell \leq 1.75 \cdot 10^{-210}:\\ \;\;\;\;\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-d\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \end{array} \]

Alternative 12
Accuracy	29.2%
Cost	6980

\[\begin{array}{l} \mathbf{if}\;\ell \leq -4.2 \cdot 10^{-238}:\\ \;\;\;\;\sqrt{d \cdot \frac{d}{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot {\left(h \cdot \ell\right)}^{-0.5}\\ \end{array} \]

Alternative 13
Accuracy	29.3%
Cost	6980

\[\begin{array}{l} \mathbf{if}\;d \leq 5.5 \cdot 10^{+54}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot {\left(h \cdot \ell\right)}^{-0.5}\\ \end{array} \]

Alternative 14
Accuracy	29.3%
Cost	6980

\[\begin{array}{l} \mathbf{if}\;d \leq 9 \cdot 10^{+54}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \end{array} \]

Alternative 15
Accuracy	20.1%
Cost	6784

\[d \cdot {\left(h \cdot \ell\right)}^{-0.5} \]

Alternative 16
Accuracy	20.1%
Cost	6720

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

Reproduce

herbie shell --seed 2023153 
(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)))))