Henrywood and Agarwal, Equation (12)

Average Error: 26.9 → 16.4

Time: 44.5s

Precision: binary64

Cost: 125268

\[ \begin{array}{c}[M, D] = \mathsf{sort}([M, D])\\ \end{array} \]

Math

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

↓

\[\begin{array}{l} t_0 := \frac{\left(M \cdot \frac{\sqrt{h}}{{\ell}^{1.5}}\right) \cdot \left(D \cdot 0.125\right)}{\frac{-d}{M \cdot D}}\\ t_1 := \left|\frac{d}{\sqrt{h \cdot \ell}}\right| \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot -0.5\right)\right)\\ t_2 := {\left(\frac{d}{h}\right)}^{0.5}\\ t_3 := 1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot -0.5\right)\\ t_4 := \left(t_2 \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot t_3\\ t_5 := {\left(\frac{d}{\ell}\right)}^{0.25}\\ \mathbf{if}\;t_4 \leq -\infty:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_4 \leq -5 \cdot 10^{-167}:\\ \;\;\;\;t_3 \cdot \left(t_2 \cdot \left(t_5 \cdot t_5\right)\right)\\ \mathbf{elif}\;t_4 \leq 0:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_4 \leq 2 \cdot 10^{+265}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.25 \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}, \frac{h}{\ell} \cdot -0.5, 1\right) \cdot \sqrt{\frac{d}{h}}}{\sqrt{\frac{\ell}{d}}}\\ \mathbf{elif}\;t_4 \leq \infty:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

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
         (/ (* (* M (/ (sqrt h) (pow l 1.5))) (* D 0.125)) (/ (- d) (* M D))))
        (t_1
         (*
          (fabs (/ d (sqrt (* h l))))
          (+ 1.0 (* (/ h l) (* (pow (* (/ M 2.0) (/ D d)) 2.0) -0.5)))))
        (t_2 (pow (/ d h) 0.5))
        (t_3 (+ 1.0 (* (/ h l) (* (pow (/ (* M D) (* d 2.0)) 2.0) -0.5))))
        (t_4 (* (* t_2 (pow (/ d l) 0.5)) t_3))
        (t_5 (pow (/ d l) 0.25)))
   (if (<= t_4 (- INFINITY))
     t_0
     (if (<= t_4 -5e-167)
       (* t_3 (* t_2 (* t_5 t_5)))
       (if (<= t_4 0.0)
         t_1
         (if (<= t_4 2e+265)
           (/
            (*
             (fma (* 0.25 (pow (* M (/ D d)) 2.0)) (* (/ h l) -0.5) 1.0)
             (sqrt (/ d h)))
            (sqrt (/ l d)))
           (if (<= t_4 INFINITY) t_1 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 = ((M * (sqrt(h) / pow(l, 1.5))) * (D * 0.125)) / (-d / (M * D));
	double t_1 = fabs((d / sqrt((h * l)))) * (1.0 + ((h / l) * (pow(((M / 2.0) * (D / d)), 2.0) * -0.5)));
	double t_2 = pow((d / h), 0.5);
	double t_3 = 1.0 + ((h / l) * (pow(((M * D) / (d * 2.0)), 2.0) * -0.5));
	double t_4 = (t_2 * pow((d / l), 0.5)) * t_3;
	double t_5 = pow((d / l), 0.25);
	double tmp;
	if (t_4 <= -((double) INFINITY)) {
		tmp = t_0;
	} else if (t_4 <= -5e-167) {
		tmp = t_3 * (t_2 * (t_5 * t_5));
	} else if (t_4 <= 0.0) {
		tmp = t_1;
	} else if (t_4 <= 2e+265) {
		tmp = (fma((0.25 * pow((M * (D / d)), 2.0)), ((h / l) * -0.5), 1.0) * sqrt((d / h))) / sqrt((l / d));
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = t_1;
	} 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 = Float64(Float64(Float64(M * Float64(sqrt(h) / (l ^ 1.5))) * Float64(D * 0.125)) / Float64(Float64(-d) / Float64(M * D)))
	t_1 = Float64(abs(Float64(d / sqrt(Float64(h * l)))) * Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(Float64(M / 2.0) * Float64(D / d)) ^ 2.0) * -0.5))))
	t_2 = Float64(d / h) ^ 0.5
	t_3 = Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(Float64(M * D) / Float64(d * 2.0)) ^ 2.0) * -0.5)))
	t_4 = Float64(Float64(t_2 * (Float64(d / l) ^ 0.5)) * t_3)
	t_5 = Float64(d / l) ^ 0.25
	tmp = 0.0
	if (t_4 <= Float64(-Inf))
		tmp = t_0;
	elseif (t_4 <= -5e-167)
		tmp = Float64(t_3 * Float64(t_2 * Float64(t_5 * t_5)));
	elseif (t_4 <= 0.0)
		tmp = t_1;
	elseif (t_4 <= 2e+265)
		tmp = Float64(Float64(fma(Float64(0.25 * (Float64(M * Float64(D / d)) ^ 2.0)), Float64(Float64(h / l) * -0.5), 1.0) * sqrt(Float64(d / h))) / sqrt(Float64(l / d)));
	elseif (t_4 <= Inf)
		tmp = t_1;
	else
		tmp = t_0;
	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[(N[(M * N[(N[Sqrt[h], $MachinePrecision] / N[Power[l, 1.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(D * 0.125), $MachinePrecision]), $MachinePrecision] / N[((-d) / N[(M * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Abs[N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(M / 2.0), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[(d / h), $MachinePrecision], 0.5], $MachinePrecision]}, Block[{t$95$3 = 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]}, Block[{t$95$4 = N[(N[(t$95$2 * N[Power[N[(d / l), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision]}, Block[{t$95$5 = N[Power[N[(d / l), $MachinePrecision], 0.25], $MachinePrecision]}, If[LessEqual[t$95$4, (-Infinity)], t$95$0, If[LessEqual[t$95$4, -5e-167], N[(t$95$3 * N[(t$95$2 * N[(t$95$5 * t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 0.0], t$95$1, If[LessEqual[t$95$4, 2e+265], N[(N[(N[(N[(0.25 * N[Power[N[(M * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(l / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, Infinity], t$95$1, t$95$0]]]]]]]]]]]

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

   1. Initial program 64.0

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

   2. Applied egg-rr 64.0

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

   3. Taylor expanded in d around 0 62.0

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

   4. Simplified 61.0

      \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(M \cdot \frac{-0.125 \cdot D}{\frac{d}{D \cdot M}}\right)} \]
      Proof
      (*.f64 (sqrt.f64 (/.f64 h (pow.f64 l 3))) (*.f64 M (/.f64 (*.f64 -1/8 D) (/.f64 d (*.f64 D M))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 (/.f64 h (pow.f64 l 3))) (*.f64 M (/.f64 (*.f64 -1/8 D) (/.f64 d (Rewrite<= *-commutative_binary64 (*.f64 M D)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 (/.f64 h (pow.f64 l 3))) (*.f64 M (/.f64 (*.f64 -1/8 D) (Rewrite=> associate-/r*_binary64 (/.f64 (/.f64 d M) D))))): 13 points increase in error, 3 points decrease in error
      (*.f64 (sqrt.f64 (/.f64 h (pow.f64 l 3))) (*.f64 M (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (*.f64 -1/8 D) D) (/.f64 d M))))): 17 points increase in error, 6 points decrease in error
      (*.f64 (sqrt.f64 (/.f64 h (pow.f64 l 3))) (*.f64 M (/.f64 (Rewrite<= associate-*r*_binary64 (*.f64 -1/8 (*.f64 D D))) (/.f64 d M)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 (/.f64 h (pow.f64 l 3))) (*.f64 M (/.f64 (*.f64 -1/8 (Rewrite<= unpow2_binary64 (pow.f64 D 2))) (/.f64 d M)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 (/.f64 h (pow.f64 l 3))) (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 (*.f64 -1/8 (pow.f64 D 2)) (/.f64 d M)) M))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 (/.f64 h (pow.f64 l 3))) (Rewrite<= associate-/r/_binary64 (/.f64 (*.f64 -1/8 (pow.f64 D 2)) (/.f64 (/.f64 d M) M)))): 20 points increase in error, 3 points decrease in error
      (*.f64 (sqrt.f64 (/.f64 h (pow.f64 l 3))) (/.f64 (*.f64 -1/8 (pow.f64 D 2)) (Rewrite<= associate-/r*_binary64 (/.f64 d (*.f64 M M))))): 14 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 (/.f64 h (pow.f64 l 3))) (/.f64 (*.f64 -1/8 (pow.f64 D 2)) (/.f64 d (Rewrite<= unpow2_binary64 (pow.f64 M 2))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 (/.f64 h (pow.f64 l 3))) (Rewrite<= associate-*r/_binary64 (*.f64 -1/8 (/.f64 (pow.f64 D 2) (/.f64 d (pow.f64 M 2)))))): 0 points increase in error, 1 points decrease in error
      (*.f64 (sqrt.f64 (/.f64 h (pow.f64 l 3))) (*.f64 -1/8 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (pow.f64 D 2) (pow.f64 M 2)) d)))): 5 points increase in error, 18 points decrease in error
      (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 -1/8 (/.f64 (*.f64 (pow.f64 D 2) (pow.f64 M 2)) d)) (sqrt.f64 (/.f64 h (pow.f64 l 3))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*r*_binary64 (*.f64 -1/8 (*.f64 (/.f64 (*.f64 (pow.f64 D 2) (pow.f64 M 2)) d) (sqrt.f64 (/.f64 h (pow.f64 l 3)))))): 0 points increase in error, 0 points decrease in error

   5. Applied egg-rr 54.9

      \[\leadsto \color{blue}{\frac{\left(\frac{\sqrt{h}}{{\ell}^{1.5}} \cdot M\right) \cdot \left(D \cdot 0.125\right)}{\frac{-d}{M \cdot D}}} \]

   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)))) < -5.0000000000000002e-167

   1. Initial program 1.2

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

   2. Applied egg-rr 1.2

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{0.25} \cdot {\left(\frac{d}{\ell}\right)}^{0.25}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   if -5.0000000000000002e-167 < (*.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 2.00000000000000013e265 < (*.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 52.4

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

   2. Applied egg-rr 52.4

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

   3. Applied egg-rr 55.8

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

   4. Applied egg-rr 10.7

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

   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)))) < 2.00000000000000013e265

   1. Initial program 1.1

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

   2. Simplified 1.3

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(\frac{\frac{D}{d}}{\frac{2}{M}}\right)}^{2}, \frac{h}{\ell} \cdot -0.5, 1\right)\right)} \]
      Proof
      (*.f64 (sqrt.f64 (/.f64 d h)) (*.f64 (sqrt.f64 (/.f64 d l)) (fma.f64 (pow.f64 (/.f64 (/.f64 D d) (/.f64 2 M)) 2) (*.f64 (/.f64 h l) -1/2) 1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= unpow1/2_binary64 (pow.f64 (/.f64 d h) 1/2)) (*.f64 (sqrt.f64 (/.f64 d l)) (fma.f64 (pow.f64 (/.f64 (/.f64 D d) (/.f64 2 M)) 2) (*.f64 (/.f64 h l) -1/2) 1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (pow.f64 (/.f64 d h) (Rewrite<= metadata-eval (/.f64 1 2))) (*.f64 (sqrt.f64 (/.f64 d l)) (fma.f64 (pow.f64 (/.f64 (/.f64 D d) (/.f64 2 M)) 2) (*.f64 (/.f64 h l) -1/2) 1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (*.f64 (Rewrite<= unpow1/2_binary64 (pow.f64 (/.f64 d l) 1/2)) (fma.f64 (pow.f64 (/.f64 (/.f64 D d) (/.f64 2 M)) 2) (*.f64 (/.f64 h l) -1/2) 1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (*.f64 (pow.f64 (/.f64 d l) (Rewrite<= metadata-eval (/.f64 1 2))) (fma.f64 (pow.f64 (/.f64 (/.f64 D d) (/.f64 2 M)) 2) (*.f64 (/.f64 h l) -1/2) 1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (*.f64 (pow.f64 (/.f64 d l) (/.f64 1 2)) (fma.f64 (pow.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (/.f64 D d) M) 2)) 2) (*.f64 (/.f64 h l) -1/2) 1))): 0 points increase in error, 2 points decrease in error
      (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (*.f64 (pow.f64 (/.f64 d l) (/.f64 1 2)) (fma.f64 (pow.f64 (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 M (/.f64 D d))) 2) 2) (*.f64 (/.f64 h l) -1/2) 1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (*.f64 (pow.f64 (/.f64 d l) (/.f64 1 2)) (fma.f64 (pow.f64 (/.f64 (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 M D) d)) 2) 2) (*.f64 (/.f64 h l) -1/2) 1))): 6 points increase in error, 3 points decrease in error
      (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (*.f64 (pow.f64 (/.f64 d l) (/.f64 1 2)) (fma.f64 (pow.f64 (Rewrite=> associate-/l/_binary64 (/.f64 (*.f64 M D) (*.f64 2 d))) 2) (*.f64 (/.f64 h l) -1/2) 1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (*.f64 (pow.f64 (/.f64 d l) (/.f64 1 2)) (fma.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (*.f64 (/.f64 h l) (Rewrite<= metadata-eval (neg.f64 1/2))) 1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (*.f64 (pow.f64 (/.f64 d l) (/.f64 1 2)) (fma.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (*.f64 (/.f64 h l) (neg.f64 (Rewrite<= metadata-eval (/.f64 1 2)))) 1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (*.f64 (pow.f64 (/.f64 d l) (/.f64 1 2)) (fma.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (/.f64 h l) (/.f64 1 2)))) 1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (*.f64 (pow.f64 (/.f64 d l) (/.f64 1 2)) (fma.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 1 2) (/.f64 h l)))) 1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (*.f64 (pow.f64 (/.f64 d l) (/.f64 1 2)) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (neg.f64 (*.f64 (/.f64 1 2) (/.f64 h l)))) 1)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (*.f64 (pow.f64 (/.f64 d l) (/.f64 1 2)) (+.f64 (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (*.f64 (/.f64 1 2) (/.f64 h l))))) 1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (*.f64 (pow.f64 (/.f64 d l) (/.f64 1 2)) (+.f64 (neg.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 1 2)) (/.f64 h l)))) 1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (*.f64 (pow.f64 (/.f64 d l) (/.f64 1 2)) (+.f64 (neg.f64 (*.f64 (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2))) (/.f64 h l))) 1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (*.f64 (pow.f64 (/.f64 d l) (/.f64 1 2)) (+.f64 (Rewrite<= distribute-lft-neg-out_binary64 (*.f64 (neg.f64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2))) (/.f64 h l))) 1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (*.f64 (pow.f64 (/.f64 d l) (/.f64 1 2)) (Rewrite<= +-commutative_binary64 (+.f64 1 (*.f64 (neg.f64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2))) (/.f64 h l)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (*.f64 (pow.f64 (/.f64 d l) (/.f64 1 2)) (Rewrite<= cancel-sign-sub-inv_binary64 (-.f64 1 (*.f64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)) (/.f64 h l)))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*l*_binary64 (*.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 points increase in error, 2 points decrease in error

   3. Applied egg-rr 31.9

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

   4. Applied egg-rr 2.1

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({\left(\frac{D}{d} \cdot M\right)}^{2} \cdot 0.25, \frac{h}{\ell} \cdot -0.5, 1\right) \cdot \sqrt{\frac{d}{h}}}{\sqrt{\frac{\ell}{d}}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 16.4

   \[\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 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot -0.5\right)\right) \leq -\infty:\\ \;\;\;\;\frac{\left(M \cdot \frac{\sqrt{h}}{{\ell}^{1.5}}\right) \cdot \left(D \cdot 0.125\right)}{\frac{-d}{M \cdot D}}\\ \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 -5 \cdot 10^{-167}:\\ \;\;\;\;\left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot -0.5\right)\right) \cdot \left({\left(\frac{d}{h}\right)}^{0.5} \cdot \left({\left(\frac{d}{\ell}\right)}^{0.25} \cdot {\left(\frac{d}{\ell}\right)}^{0.25}\right)\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|\frac{d}{\sqrt{h \cdot \ell}}\right| \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot -0.5\right)\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^{+265}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.25 \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}, \frac{h}{\ell} \cdot -0.5, 1\right) \cdot \sqrt{\frac{d}{h}}}{\sqrt{\frac{\ell}{d}}}\\ \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|\frac{d}{\sqrt{h \cdot \ell}}\right| \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(M \cdot \frac{\sqrt{h}}{{\ell}^{1.5}}\right) \cdot \left(D \cdot 0.125\right)}{\frac{-d}{M \cdot D}}\\ \end{array} \]

Alternatives

Alternative 1
Error	16.7
Cost	125268

\[\begin{array}{l} t_0 := \frac{\left(M \cdot \frac{\sqrt{h}}{{\ell}^{1.5}}\right) \cdot \left(D \cdot 0.125\right)}{\frac{-d}{M \cdot D}}\\ t_1 := {\left(\frac{d}{h}\right)}^{0.5}\\ t_2 := \left(t_1 \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot -0.5\right)\right)\\ t_3 := \left|\frac{d}{\sqrt{h \cdot \ell}}\right| \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot -0.5\right)\right)\\ \mathbf{if}\;t_2 \leq -\infty:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_2 \leq -5 \cdot 10^{-167}:\\ \;\;\;\;\left(t_1 \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 + \frac{-0.5 \cdot {\left(\frac{D}{\frac{d \cdot 2}{M}}\right)}^{2}}{\frac{\ell}{h}}\right)\\ \mathbf{elif}\;t_2 \leq 0:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t_2 \leq 2 \cdot 10^{+265}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.25 \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}, \frac{h}{\ell} \cdot -0.5, 1\right) \cdot \sqrt{\frac{d}{h}}}{\sqrt{\frac{\ell}{d}}}\\ \mathbf{elif}\;t_2 \leq \infty:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	21.1
Cost	118996

\[\begin{array}{l} t_0 := {\left(\frac{d}{h}\right)}^{0.5}\\ t_1 := \left(t_0 \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot -0.5\right)\right)\\ t_2 := \frac{\left(M \cdot \frac{\sqrt{h}}{{\ell}^{1.5}}\right) \cdot \left(D \cdot 0.125\right)}{\frac{-d}{M \cdot D}}\\ t_3 := \sqrt{\frac{d}{\ell}}\\ t_4 := \frac{h}{\ell} \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot -0.5\right)\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq -5 \cdot 10^{-167}:\\ \;\;\;\;\left(t_0 \cdot t_3\right) \cdot \left(1 + \frac{-0.5 \cdot {\left(\frac{D}{\frac{d \cdot 2}{M}}\right)}^{2}}{\frac{\ell}{h}}\right)\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 + t_4\right)\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+285}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(t_3 \cdot \left(1 + \left(\frac{h}{\ell} \cdot -0.5\right) \cdot {\left(\frac{D}{d} \cdot \left(0.5 \cdot M\right)\right)}^{2}\right)\right)\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right) \cdot \left(-1 - t_4\right)\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 3
Error	21.4
Cost	118996

\[\begin{array}{l} t_0 := {\left(\frac{d}{h}\right)}^{0.5}\\ t_1 := \left(t_0 \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot -0.5\right)\right)\\ t_2 := \frac{\left(M \cdot \frac{\sqrt{h}}{{\ell}^{1.5}}\right) \cdot \left(D \cdot 0.125\right)}{\frac{-d}{M \cdot D}}\\ t_3 := \frac{h}{\ell} \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot -0.5\right)\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq -5 \cdot 10^{-167}:\\ \;\;\;\;\left(t_0 \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 + \frac{-0.5 \cdot {\left(\frac{D}{\frac{d \cdot 2}{M}}\right)}^{2}}{\frac{\ell}{h}}\right)\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 + t_3\right)\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+285}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.25 \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}, \frac{h}{\ell} \cdot -0.5, 1\right) \cdot \sqrt{\frac{d}{h}}}{\sqrt{\frac{\ell}{d}}}\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right) \cdot \left(-1 - t_3\right)\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 4
Error	21.1
Cost	21128

\[\begin{array}{l} t_0 := 1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(0.5 \cdot \frac{M \cdot D}{d}\right)}^{2}\right)\\ \mathbf{if}\;d \leq -1.18 \cdot 10^{+232}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right) \cdot \left(-1 - \frac{h}{\ell} \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot -0.5\right)\right)\\ \mathbf{elif}\;d \leq -4 \cdot 10^{-303}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{\frac{\ell}{d}}} \cdot t_0}{\sqrt{\frac{h}{d}}}\\ \mathbf{elif}\;d \leq 10^{-235}:\\ \;\;\;\;\frac{\sqrt{h}}{{\ell}^{1.5}} \cdot \left(M \cdot \frac{D \cdot -0.125}{\frac{d}{M \cdot D}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0 \cdot \frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \]

Alternative 5
Error	21.2
Cost	21128

\[\begin{array}{l} t_0 := 1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(0.5 \cdot \frac{M \cdot D}{d}\right)}^{2}\right)\\ \mathbf{if}\;d \leq -1.18 \cdot 10^{+232}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(-1 - \frac{h}{\ell} \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot -0.5\right)\right)\\ \mathbf{elif}\;d \leq -4 \cdot 10^{-303}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{\frac{\ell}{d}}} \cdot t_0}{\sqrt{\frac{h}{d}}}\\ \mathbf{elif}\;d \leq 10^{-235}:\\ \;\;\;\;\frac{\sqrt{h}}{{\ell}^{1.5}} \cdot \left(M \cdot \frac{D \cdot -0.125}{\frac{d}{M \cdot D}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0 \cdot \frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \]

Alternative 6
Error	22.6
Cost	21004

\[\begin{array}{l} t_0 := \frac{\sqrt{\frac{d}{\ell}} \cdot \left(1 + \frac{h \cdot \left(-0.125 \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}\right)}{\ell}\right)}{\sqrt{\frac{h}{d}}}\\ \mathbf{if}\;d \leq -4 \cdot 10^{-303}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 3 \cdot 10^{-160}:\\ \;\;\;\;\frac{\left(M \cdot \frac{\sqrt{h}}{{\ell}^{1.5}}\right) \cdot \left(D \cdot 0.125\right)}{\frac{-d}{M \cdot D}}\\ \mathbf{elif}\;d \leq 10^{-20}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 7
Error	21.5
Cost	20872

\[\begin{array}{l} \mathbf{if}\;d \leq -4 \cdot 10^{-303}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{\ell}} \cdot \left(1 + \frac{h \cdot \left(-0.125 \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}\right)}{\ell}\right)}{\sqrt{\frac{h}{d}}}\\ \mathbf{elif}\;d \leq 10^{-235}:\\ \;\;\;\;\frac{\sqrt{h}}{{\ell}^{1.5}} \cdot \left(M \cdot \frac{D \cdot -0.125}{\frac{d}{M \cdot D}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(0.5 \cdot \frac{M \cdot D}{d}\right)}^{2}\right)\right) \cdot \frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \]

Alternative 8
Error	21.8
Cost	20872

\[\begin{array}{l} t_0 := 1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(0.5 \cdot \frac{M \cdot D}{d}\right)}^{2}\right)\\ \mathbf{if}\;d \leq -4 \cdot 10^{-303}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{\ell}} \cdot t_0}{\sqrt{\frac{h}{d}}}\\ \mathbf{elif}\;d \leq 10^{-235}:\\ \;\;\;\;\frac{\sqrt{h}}{{\ell}^{1.5}} \cdot \left(M \cdot \frac{D \cdot -0.125}{\frac{d}{M \cdot D}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0 \cdot \frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \]

Alternative 9
Error	25.4
Cost	14152

\[\begin{array}{l} \mathbf{if}\;d \leq 3.6 \cdot 10^{-298}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;d \leq 1.3 \cdot 10^{-177}:\\ \;\;\;\;\frac{\left(M \cdot \frac{\sqrt{h}}{{\ell}^{1.5}}\right) \cdot \left(D \cdot 0.125\right)}{\frac{-d}{M \cdot D}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 10
Error	25.6
Cost	14088

\[\begin{array}{l} \mathbf{if}\;d \leq 4.5 \cdot 10^{-298}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;d \leq 7.8 \cdot 10^{-205}:\\ \;\;\;\;\frac{\sqrt{h} \cdot \left(M \cdot \left(D \cdot -0.125\right)\right)}{{\ell}^{1.5} \cdot \frac{\frac{d}{M}}{D}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 11
Error	25.5
Cost	14088

\[\begin{array}{l} \mathbf{if}\;d \leq 3.6 \cdot 10^{-298}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;d \leq 7.8 \cdot 10^{-205}:\\ \;\;\;\;\frac{\sqrt{h}}{{\ell}^{1.5}} \cdot \left(M \cdot \frac{D \cdot -0.125}{\frac{d}{M \cdot D}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 12
Error	32.7
Cost	13384

\[\begin{array}{l} \mathbf{if}\;d \leq -1.1 \cdot 10^{-307}:\\ \;\;\;\;\sqrt{\frac{d \cdot d}{h \cdot \ell}}\\ \mathbf{elif}\;d \leq 2.8 \cdot 10^{-236}:\\ \;\;\;\;\left(M \cdot \frac{D \cdot -0.125}{\frac{d}{M \cdot D}}\right) \cdot \sqrt{\frac{h}{\ell} \cdot \frac{1}{\ell \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 13
Error	25.7
Cost	13384

\[\begin{array}{l} \mathbf{if}\;d \leq -1.7 \cdot 10^{-305}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;d \leq 2.8 \cdot 10^{-236}:\\ \;\;\;\;\left(M \cdot \frac{D \cdot -0.125}{\frac{d}{M \cdot D}}\right) \cdot \sqrt{\frac{h}{\ell} \cdot \frac{1}{\ell \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 14
Error	36.9
Cost	8008

\[\begin{array}{l} \mathbf{if}\;d \leq -1.1 \cdot 10^{-307}:\\ \;\;\;\;\sqrt{\frac{d \cdot d}{h \cdot \ell}}\\ \mathbf{elif}\;d \leq 2.8 \cdot 10^{-236}:\\ \;\;\;\;\left(M \cdot \frac{D \cdot -0.125}{\frac{d}{M \cdot D}}\right) \cdot \sqrt{\frac{1}{\ell} \cdot \frac{h}{\ell \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}}\\ \end{array} \]

Alternative 15
Error	36.9
Cost	8008

\[\begin{array}{l} \mathbf{if}\;d \leq -1.1 \cdot 10^{-307}:\\ \;\;\;\;\sqrt{\frac{d \cdot d}{h \cdot \ell}}\\ \mathbf{elif}\;d \leq 2.8 \cdot 10^{-236}:\\ \;\;\;\;\left(M \cdot \frac{D \cdot -0.125}{\frac{d}{M \cdot D}}\right) \cdot \sqrt{\frac{h}{\ell} \cdot \frac{1}{\ell \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}}\\ \end{array} \]

Alternative 16
Error	37.3
Cost	6980

\[\begin{array}{l} \mathbf{if}\;\ell \leq -6.8 \cdot 10^{-195}:\\ \;\;\;\;\sqrt{\frac{d \cdot d}{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}}\\ \end{array} \]

Alternative 17
Error	44.2
Cost	6720

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

Reproduce

herbie shell --seed 2022310 
(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)))))