Henrywood and Agarwal, Equation (12)

Average Error: 26.3 → 13.5

Time: 56.3s

Precision: binary64

Cost: 96976

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

Math

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

↓

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ t_1 := \sqrt{\frac{d}{\ell}}\\ t_2 := t_0 \cdot t_1\\ t_3 := \left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot -0.5\right)\right)\\ t_4 := \left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \mathbf{if}\;t_3 \leq -2 \cdot 10^{+265}:\\ \;\;\;\;t_2 \cdot \left(1 + {\left(\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2} \cdot -0.5\right)\\ \mathbf{elif}\;t_3 \leq -2 \cdot 10^{-170}:\\ \;\;\;\;t_0 \cdot \left(t_1 \cdot \mathsf{fma}\left({\left(0.5 \cdot \frac{D}{\frac{d}{M}}\right)}^{2}, \frac{h}{\ell} \cdot -0.5, 1\right)\right)\\ \mathbf{elif}\;t_3 \leq 4 \cdot 10^{-277}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;t_3 \leq 5 \cdot 10^{+212}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_4\\ \end{array} \]

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)))
        (t_1 (sqrt (/ d l)))
        (t_2 (* t_0 t_1))
        (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 (fabs (/ d (sqrt (* h l))))))
   (if (<= t_3 -2e+265)
     (*
      t_2
      (+ 1.0 (* (pow (* (* M (* 0.5 (/ D d))) (sqrt (/ h l))) 2.0) -0.5)))
     (if (<= t_3 -2e-170)
       (*
        t_0
        (* t_1 (fma (pow (* 0.5 (/ D (/ d M))) 2.0) (* (/ h l) -0.5) 1.0)))
       (if (<= t_3 4e-277) t_4 (if (<= t_3 5e+212) t_2 t_4))))))

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));
	double t_1 = sqrt((d / l));
	double t_2 = t_0 * t_1;
	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 = fabs((d / sqrt((h * l))));
	double tmp;
	if (t_3 <= -2e+265) {
		tmp = t_2 * (1.0 + (pow(((M * (0.5 * (D / d))) * sqrt((h / l))), 2.0) * -0.5));
	} else if (t_3 <= -2e-170) {
		tmp = t_0 * (t_1 * fma(pow((0.5 * (D / (d / M))), 2.0), ((h / l) * -0.5), 1.0));
	} else if (t_3 <= 4e-277) {
		tmp = t_4;
	} else if (t_3 <= 5e+212) {
		tmp = t_2;
	} else {
		tmp = t_4;
	}
	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 = sqrt(Float64(d / h))
	t_1 = sqrt(Float64(d / l))
	t_2 = Float64(t_0 * t_1)
	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 = abs(Float64(d / sqrt(Float64(h * l))))
	tmp = 0.0
	if (t_3 <= -2e+265)
		tmp = Float64(t_2 * Float64(1.0 + Float64((Float64(Float64(M * Float64(0.5 * Float64(D / d))) * sqrt(Float64(h / l))) ^ 2.0) * -0.5)));
	elseif (t_3 <= -2e-170)
		tmp = Float64(t_0 * Float64(t_1 * fma((Float64(0.5 * Float64(D / Float64(d / M))) ^ 2.0), Float64(Float64(h / l) * -0.5), 1.0)));
	elseif (t_3 <= 4e-277)
		tmp = t_4;
	elseif (t_3 <= 5e+212)
		tmp = t_2;
	else
		tmp = t_4;
	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[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 * t$95$1), $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[Abs[N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$3, -2e+265], N[(t$95$2 * N[(1.0 + N[(N[Power[N[(N[(M * N[(0.5 * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, -2e-170], N[(t$95$0 * N[(t$95$1 * N[(N[Power[N[(0.5 * N[(D / N[(d / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 4e-277], t$95$4, If[LessEqual[t$95$3, 5e+212], t$95$2, t$95$4]]]]]]]]]

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)))) < -2.00000000000000013e265

   1. Initial program 57.2

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

   2. Simplified 54.8

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
      Proof
      (*.f64 (*.f64 (sqrt.f64 (/.f64 d h)) (sqrt.f64 (/.f64 d l))) (-.f64 1 (*.f64 1/2 (*.f64 (pow.f64 (*.f64 (/.f64 M 2) (/.f64 D d)) 2) (/.f64 h l))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (Rewrite<= unpow1/2_binary64 (pow.f64 (/.f64 d h) 1/2)) (sqrt.f64 (/.f64 d l))) (-.f64 1 (*.f64 1/2 (*.f64 (pow.f64 (*.f64 (/.f64 M 2) (/.f64 D d)) 2) (/.f64 h l))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (pow.f64 (/.f64 d h) (Rewrite<= metadata-eval (/.f64 1 2))) (sqrt.f64 (/.f64 d l))) (-.f64 1 (*.f64 1/2 (*.f64 (pow.f64 (*.f64 (/.f64 M 2) (/.f64 D d)) 2) (/.f64 h l))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (Rewrite<= unpow1/2_binary64 (pow.f64 (/.f64 d l) 1/2))) (-.f64 1 (*.f64 1/2 (*.f64 (pow.f64 (*.f64 (/.f64 M 2) (/.f64 D d)) 2) (/.f64 h l))))): 0 points increase in error, 1 points decrease in error
      (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (Rewrite<= metadata-eval (/.f64 1 2)))) (-.f64 1 (*.f64 1/2 (*.f64 (pow.f64 (*.f64 (/.f64 M 2) (/.f64 D d)) 2) (/.f64 h l))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (-.f64 1 (*.f64 (Rewrite<= metadata-eval (/.f64 1 2)) (*.f64 (pow.f64 (*.f64 (/.f64 M 2) (/.f64 D d)) 2) (/.f64 h l))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (-.f64 1 (*.f64 (/.f64 1 2) (*.f64 (pow.f64 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 M D) (*.f64 2 d))) 2) (/.f64 h l))))): 4 points increase in error, 12 points decrease in error
      (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (-.f64 1 (Rewrite<= associate-*l*_binary64 (*.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

   3. Applied egg-rr 41.1

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

   if -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)))) < -1.99999999999999997e-170

   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. Simplified 6.2

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(\frac{\frac{D}{d}}{\frac{2}{M}}\right)}^{2}, \frac{h}{\ell} \cdot -0.5, 1\right)\right)} \]
      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, 1 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))): 2 points increase in error, 4 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, 10 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))))): 6 points increase in error, 6 points decrease in error

   3. Taylor expanded in D around 0 1.7

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

   4. Simplified 6.0

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\color{blue}{\left(0.5 \cdot \frac{D}{\frac{d}{M}}\right)}}^{2}, \frac{h}{\ell} \cdot -0.5, 1\right)\right) \]
      Proof
      (*.f64 1/2 (/.f64 D (/.f64 d M))): 0 points increase in error, 0 points decrease in error
      (*.f64 1/2 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 D M) d))): 45 points increase in error, 45 points decrease in error

   if -1.99999999999999997e-170 < (*.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)))) < 3.99999999999999988e-277 or 4.99999999999999992e212 < (*.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 52.9

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

   2. Taylor expanded in d around inf 39.6

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

   3. Simplified 39.6

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}} \]
      Proof
      (*.f64 d (sqrt.f64 (/.f64 (/.f64 1 l) h))): 0 points increase in error, 0 points decrease in error
      (*.f64 d (sqrt.f64 (Rewrite<= associate-/r*_binary64 (/.f64 1 (*.f64 l h))))): 9 points increase in error, 13 points decrease in error
      (Rewrite=> *-commutative_binary64 (*.f64 (sqrt.f64 (/.f64 1 (*.f64 l h))) d)): 0 points increase in error, 0 points decrease in error

   4. Applied egg-rr 51.4

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

   5. Simplified 22.6

      \[\leadsto \color{blue}{\left|\frac{d}{\sqrt{\ell \cdot h}}\right|} \]
      Proof
      (fabs.f64 (/.f64 d (sqrt.f64 (*.f64 l h)))): 0 points increase in error, 0 points decrease in error
      (fabs.f64 (/.f64 (Rewrite<= *-lft-identity_binary64 (*.f64 1 d)) (sqrt.f64 (*.f64 l h)))): 0 points increase in error, 0 points decrease in error
      (fabs.f64 (/.f64 (*.f64 1 d) (Rewrite<= unpow1/2_binary64 (pow.f64 (*.f64 l h) 1/2)))): 0 points increase in error, 0 points decrease in error
      (fabs.f64 (/.f64 (*.f64 1 d) (Rewrite<= exp-to-pow_binary64 (exp.f64 (*.f64 (log.f64 (*.f64 l h)) 1/2))))): 146 points increase in error, 28 points decrease in error
      (fabs.f64 (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 1 (exp.f64 (*.f64 (log.f64 (*.f64 l h)) 1/2))) d))): 25 points increase in error, 27 points decrease in error
      (fabs.f64 (*.f64 (Rewrite<= exp-neg_binary64 (exp.f64 (neg.f64 (*.f64 (log.f64 (*.f64 l h)) 1/2)))) d)): 15 points increase in error, 25 points decrease in error
      (fabs.f64 (*.f64 (exp.f64 (Rewrite<= distribute-lft-neg-out_binary64 (*.f64 (neg.f64 (log.f64 (*.f64 l h))) 1/2))) d)): 0 points increase in error, 0 points decrease in error
      (fabs.f64 (*.f64 (Rewrite=> exp-prod_binary64 (pow.f64 (exp.f64 (neg.f64 (log.f64 (*.f64 l h)))) 1/2)) d)): 11 points increase in error, 8 points decrease in error
      (fabs.f64 (*.f64 (pow.f64 (Rewrite=> exp-neg_binary64 (/.f64 1 (exp.f64 (log.f64 (*.f64 l h))))) 1/2) d)): 14 points increase in error, 14 points decrease in error
      (fabs.f64 (*.f64 (pow.f64 (/.f64 1 (Rewrite=> rem-exp-log_binary64 (*.f64 l h))) 1/2) d)): 28 points increase in error, 145 points decrease in error
      (fabs.f64 (*.f64 (pow.f64 (/.f64 1 (*.f64 l h)) (Rewrite<= metadata-eval (*.f64 2 1/4))) d)): 0 points increase in error, 0 points decrease in error
      (fabs.f64 (*.f64 (Rewrite<= pow-sqr_binary64 (*.f64 (pow.f64 (/.f64 1 (*.f64 l h)) 1/4) (pow.f64 (/.f64 1 (*.f64 l h)) 1/4))) d)): 49 points increase in error, 26 points decrease in error
      (fabs.f64 (*.f64 (Rewrite=> pow-sqr_binary64 (pow.f64 (/.f64 1 (*.f64 l h)) (*.f64 2 1/4))) d)): 26 points increase in error, 49 points decrease in error
      (fabs.f64 (*.f64 (pow.f64 (/.f64 1 (*.f64 l h)) (Rewrite=> metadata-eval 1/2)) d)): 0 points increase in error, 0 points decrease in error
      (fabs.f64 (*.f64 (Rewrite=> unpow1/2_binary64 (sqrt.f64 (/.f64 1 (*.f64 l h)))) d)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= rem-sqrt-square_binary64 (sqrt.f64 (*.f64 (*.f64 (sqrt.f64 (/.f64 1 (*.f64 l h))) d) (*.f64 (sqrt.f64 (/.f64 1 (*.f64 l h))) d)))): 49 points increase in error, 24 points decrease in error
      (sqrt.f64 (Rewrite<= unswap-sqr_binary64 (*.f64 (*.f64 (sqrt.f64 (/.f64 1 (*.f64 l h))) (sqrt.f64 (/.f64 1 (*.f64 l h)))) (*.f64 d d)))): 48 points increase in error, 9 points decrease in error
      (sqrt.f64 (*.f64 (Rewrite=> rem-square-sqrt_binary64 (/.f64 1 (*.f64 l h))) (*.f64 d d))): 3 points increase in error, 17 points decrease in error

   if 3.99999999999999988e-277 < (*.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)))) < 4.99999999999999992e212

   1. Initial program 0.8

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

   2. Simplified 0.8

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(\frac{h}{\ell}, {\left(\frac{\frac{D}{2} \cdot M}{d}\right)}^{2} \cdot -0.5, 1\right)\right)} \]
      Proof
      (*.f64 (sqrt.f64 (/.f64 d h)) (*.f64 (sqrt.f64 (/.f64 d l)) (fma.f64 (/.f64 h l) (*.f64 (pow.f64 (/.f64 (*.f64 (/.f64 D 2) M) d) 2) -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 (/.f64 h l) (*.f64 (pow.f64 (/.f64 (*.f64 (/.f64 D 2) M) d) 2) -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 (/.f64 h l) (*.f64 (pow.f64 (/.f64 (*.f64 (/.f64 D 2) M) d) 2) -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 (/.f64 h l) (*.f64 (pow.f64 (/.f64 (*.f64 (/.f64 D 2) M) d) 2) -1/2) 1))): 0 points increase in error, 1 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 (/.f64 h l) (*.f64 (pow.f64 (/.f64 (*.f64 (/.f64 D 2) M) d) 2) -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 (/.f64 h l) (*.f64 (pow.f64 (Rewrite<= associate-*r/_binary64 (*.f64 (/.f64 D 2) (/.f64 M d))) 2) -1/2) 1))): 8 points increase in error, 7 points decrease in error
      (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (*.f64 (pow.f64 (/.f64 d l) (/.f64 1 2)) (fma.f64 (/.f64 h l) (*.f64 (pow.f64 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 D M) (*.f64 2 d))) 2) -1/2) 1))): 7 points increase in error, 8 points decrease in error
      (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (*.f64 (pow.f64 (/.f64 d l) (/.f64 1 2)) (fma.f64 (/.f64 h l) (*.f64 (pow.f64 (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 M D)) (*.f64 2 d)) 2) -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 (/.f64 h l) (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (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 (/.f64 h l) (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (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 (/.f64 h l) (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.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 (/.f64 h l) (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 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)) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (/.f64 h l) (neg.f64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 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)) (+.f64 (Rewrite<= *-commutative_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))))): 6 points increase in error, 6 points decrease in error

   3. Taylor expanded in h around 0 1.2

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{1}\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 13.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot -0.5\right)\right) \leq -2 \cdot 10^{+265}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 + {\left(\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2} \cdot -0.5\right)\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot -0.5\right)\right) \leq -2 \cdot 10^{-170}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \frac{D}{\frac{d}{M}}\right)}^{2}, \frac{h}{\ell} \cdot -0.5, 1\right)\right)\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot -0.5\right)\right) \leq 4 \cdot 10^{-277}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot -0.5\right)\right) \leq 5 \cdot 10^{+212}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \end{array} \]

Alternatives

Alternative 1
Error	13.5
Cost	96976

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ t_1 := \left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ t_2 := \left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot -0.5\right)\right)\\ \mathbf{if}\;t_2 \leq -\infty:\\ \;\;\;\;t_0 \cdot \left(1 + 0.5 \cdot \left(\left(M \cdot \frac{1}{\frac{\ell}{h \cdot M}}\right) \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot -0.25\right)\right)\right)\\ \mathbf{elif}\;t_2 \leq -2 \cdot 10^{-170}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_2 \leq 4 \cdot 10^{-277}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_2 \leq 5 \cdot 10^{+212}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	13.6
Cost	96976

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ t_1 := \sqrt{\frac{d}{\ell}}\\ t_2 := t_0 \cdot t_1\\ t_3 := \left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot -0.5\right)\right)\\ t_4 := \left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \mathbf{if}\;t_3 \leq -\infty:\\ \;\;\;\;t_2 \cdot \left(1 + 0.5 \cdot \left(\left(M \cdot \frac{1}{\frac{\ell}{h \cdot M}}\right) \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot -0.25\right)\right)\right)\\ \mathbf{elif}\;t_3 \leq -2 \cdot 10^{-170}:\\ \;\;\;\;t_0 \cdot \left(t_1 \cdot \mathsf{fma}\left(\frac{h}{\ell}, -0.5 \cdot {\left(\frac{M \cdot \frac{D}{2}}{d}\right)}^{2}, 1\right)\right)\\ \mathbf{elif}\;t_3 \leq 4 \cdot 10^{-277}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;t_3 \leq 5 \cdot 10^{+212}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_4\\ \end{array} \]

Alternative 3
Error	14.2
Cost	96976

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ t_1 := \sqrt{\frac{d}{\ell}}\\ t_2 := t_0 \cdot t_1\\ t_3 := \left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot -0.5\right)\right)\\ t_4 := \left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \mathbf{if}\;t_3 \leq -2 \cdot 10^{+265}:\\ \;\;\;\;t_2 \cdot \left(1 + 0.5 \cdot \left(\left(M \cdot \frac{1}{\frac{\ell}{h \cdot M}}\right) \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot -0.25\right)\right)\right)\\ \mathbf{elif}\;t_3 \leq -2 \cdot 10^{-170}:\\ \;\;\;\;t_0 \cdot \left(t_1 \cdot \mathsf{fma}\left({\left(0.5 \cdot \frac{D}{\frac{d}{M}}\right)}^{2}, \frac{h}{\ell} \cdot -0.5, 1\right)\right)\\ \mathbf{elif}\;t_3 \leq 4 \cdot 10^{-277}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;t_3 \leq 5 \cdot 10^{+212}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_4\\ \end{array} \]

Alternative 4
Error	19.3
Cost	21448

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;h \leq -2.5 \cdot 10^{-121}:\\ \;\;\;\;t_0 \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + -0.5 \cdot \left(h \cdot \frac{{\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2}}{\ell}\right)\right)\right)\\ \mathbf{elif}\;h \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\left(t_0 \cdot \frac{\sqrt{-d}}{\sqrt{-h}}\right) \cdot \left(1 + 0.5 \cdot \left(\frac{M \cdot M}{\frac{\ell}{h}} \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot -0.25\right)\right)\right)\\ \mathbf{elif}\;h \leq 1.25 \cdot 10^{+184}:\\ \;\;\;\;\left(1 + -0.5 \cdot \frac{{\left(M \cdot \frac{0.5}{\frac{d}{D}}\right)}^{2}}{\frac{\ell}{h}}\right) \cdot \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{1}{\frac{\sqrt{h}}{{\ell}^{-0.5}}}\\ \end{array} \]

Alternative 5
Error	24.1
Cost	21400

\[\begin{array}{l} t_0 := \left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ t_1 := \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 + 0.5 \cdot \left(\left(M \cdot \frac{1}{\frac{\ell}{h \cdot M}}\right) \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot -0.25\right)\right)\right)\\ \mathbf{if}\;d \leq -3.3 \cdot 10^{+262}:\\ \;\;\;\;\sqrt{\frac{d}{h \cdot \frac{\ell}{d}}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{M \cdot \frac{D}{d}}{2}\right)}^{2}\right)\right)\\ \mathbf{elif}\;d \leq -1.05 \cdot 10^{+51}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq -3.8 \cdot 10^{-166}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq 5.8 \cdot 10^{-226}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 6.1 \cdot 10^{-170}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq 9.5 \cdot 10^{-68}:\\ \;\;\;\;\mathsf{fma}\left(h \cdot \frac{{\left(M \cdot \frac{0.5}{\frac{d}{D}}\right)}^{2}}{\ell}, -0.5, 1\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{elif}\;d \leq 1.6 \cdot 10^{-34}:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 + \frac{D \cdot D}{\ell} \cdot \left(\left(\frac{h}{d} \cdot \frac{M \cdot M}{d}\right) \cdot -0.125\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{1}{\frac{\sqrt{h}}{{\ell}^{-0.5}}}\\ \end{array} \]

Alternative 6
Error	24.1
Cost	21400

\[\begin{array}{l} \mathbf{if}\;d \leq -1.58 \cdot 10^{+262}:\\ \;\;\;\;\sqrt{\frac{d}{h \cdot \frac{\ell}{d}}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{M \cdot \frac{D}{d}}{2}\right)}^{2}\right)\right)\\ \mathbf{elif}\;d \leq -5 \cdot 10^{+52}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \mathbf{elif}\;d \leq -4.9 \cdot 10^{-278}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 + 0.5 \cdot \left(\left(M \cdot \frac{1}{\frac{\ell}{h \cdot M}}\right) \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot -0.25\right)\right)\right)\\ \mathbf{elif}\;d \leq 1.2 \cdot 10^{-240}:\\ \;\;\;\;\frac{D}{\frac{\frac{d}{M}}{M \cdot D}} \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)\\ \mathbf{elif}\;d \leq 6.6 \cdot 10^{-103}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 + -0.125 \cdot \left({\left(\frac{D}{d}\right)}^{2} \cdot \left(M \cdot \left(h \cdot \frac{M}{\ell}\right)\right)\right)\right)\\ \mathbf{elif}\;d \leq 1.05 \cdot 10^{-67}:\\ \;\;\;\;\mathsf{fma}\left(h \cdot \frac{{\left(M \cdot \frac{0.5}{\frac{d}{D}}\right)}^{2}}{\ell}, -0.5, 1\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{elif}\;d \leq 6.8 \cdot 10^{-36}:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 + \frac{D \cdot D}{\ell} \cdot \left(\left(\frac{h}{d} \cdot \frac{M \cdot M}{d}\right) \cdot -0.125\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{1}{\frac{\sqrt{h}}{{\ell}^{-0.5}}}\\ \end{array} \]

Alternative 7
Error	21.3
Cost	21400

\[\begin{array}{l} t_0 := \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ t_1 := {\left(M \cdot \frac{0.5}{\frac{d}{D}}\right)}^{2}\\ \mathbf{if}\;h \leq -3.3 \cdot 10^{+92}:\\ \;\;\;\;\mathsf{fma}\left(h \cdot \frac{t_1}{\ell}, -0.5, 1\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{elif}\;h \leq -2.7 \cdot 10^{-260}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \mathbf{elif}\;h \leq 1.55 \cdot 10^{-280}:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 + \frac{D \cdot D}{\ell} \cdot \left(\left(\frac{h}{d} \cdot \frac{M \cdot M}{d}\right) \cdot -0.125\right)\right)\\ \mathbf{elif}\;h \leq 5 \cdot 10^{-259}:\\ \;\;\;\;t_0 \cdot \left(1 + \left(\frac{h}{\ell} \cdot -0.5\right) \cdot {\left(M \cdot \frac{0.5 \cdot D}{d}\right)}^{2}\right)\\ \mathbf{elif}\;h \leq 2 \cdot 10^{-213}:\\ \;\;\;\;t_0 \cdot \left(1 + -0.125 \cdot \left({\left(\frac{D}{d}\right)}^{2} \cdot \left(M \cdot \left(h \cdot \frac{M}{\ell}\right)\right)\right)\right)\\ \mathbf{elif}\;h \leq 1.05 \cdot 10^{+180}:\\ \;\;\;\;\left(1 + -0.5 \cdot \frac{t_1}{\frac{\ell}{h}}\right) \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{1}{\frac{\sqrt{h}}{{\ell}^{-0.5}}}\\ \end{array} \]

Alternative 8
Error	19.2
Cost	21396

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := \sqrt{\frac{d}{h}}\\ t_2 := t_0 \cdot \left(t_1 \cdot \left(1 + -0.5 \cdot \left(h \cdot \frac{{\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2}}{\ell}\right)\right)\right)\\ \mathbf{if}\;\ell \leq -1.4 \cdot 10^{+102}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-76}:\\ \;\;\;\;\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{d} \cdot 0.125\right) - d \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(t_1 \cdot t_0\right) \cdot \left(1 + -0.5 \cdot \frac{h \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell}\right)\\ \mathbf{elif}\;\ell \leq 2.02 \cdot 10^{-224}:\\ \;\;\;\;d \cdot \frac{1}{\frac{\sqrt{h}}{{\ell}^{-0.5}}}\\ \mathbf{elif}\;\ell \leq 6 \cdot 10^{-198}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\left(1 + -0.5 \cdot \frac{{\left(M \cdot \frac{0.5}{\frac{d}{D}}\right)}^{2}}{\frac{\ell}{h}}\right) \cdot \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

Alternative 9
Error	22.8
Cost	21268

\[\begin{array}{l} t_0 := \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 + \left(\frac{h}{\ell} \cdot -0.5\right) \cdot {\left(M \cdot \frac{0.5 \cdot D}{d}\right)}^{2}\right)\\ \mathbf{if}\;d \leq -2.36 \cdot 10^{+260}:\\ \;\;\;\;\sqrt{\frac{d}{h \cdot \frac{\ell}{d}}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{M \cdot \frac{D}{d}}{2}\right)}^{2}\right)\right)\\ \mathbf{elif}\;d \leq -3.9 \cdot 10^{+52}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \mathbf{elif}\;d \leq -6.8 \cdot 10^{-292}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 + 0.5 \cdot \left(\left(M \cdot \frac{1}{\frac{\ell}{h \cdot M}}\right) \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot -0.25\right)\right)\right)\\ \mathbf{elif}\;d \leq 1.06 \cdot 10^{-33}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 1.1 \cdot 10^{+282}:\\ \;\;\;\;d \cdot \frac{1}{\frac{\sqrt{h}}{{\ell}^{-0.5}}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 10
Error	20.4
Cost	21136

\[\begin{array}{l} \mathbf{if}\;h \leq -8.6 \cdot 10^{-186}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \left(\frac{h}{\ell} \cdot -0.5\right) \cdot {\left(\frac{D}{d} \cdot \left(0.5 \cdot M\right)\right)}^{2}\right)\right)\\ \mathbf{elif}\;h \leq -2.7 \cdot 10^{-260}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{elif}\;h \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 + \frac{D \cdot D}{\ell} \cdot \left(\left(\frac{h}{d} \cdot \frac{M \cdot M}{d}\right) \cdot -0.125\right)\right)\\ \mathbf{elif}\;h \leq 5.6 \cdot 10^{+181}:\\ \;\;\;\;\left(1 + -0.5 \cdot \frac{{\left(M \cdot \frac{0.5}{\frac{d}{D}}\right)}^{2}}{\frac{\ell}{h}}\right) \cdot \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{1}{\frac{\sqrt{h}}{{\ell}^{-0.5}}}\\ \end{array} \]

Alternative 11
Error	19.6
Cost	21136

\[\begin{array}{l} \mathbf{if}\;h \leq -1.12 \cdot 10^{-183}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + -0.5 \cdot \left(h \cdot \frac{{\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2}}{\ell}\right)\right)\right)\\ \mathbf{elif}\;h \leq -2.7 \cdot 10^{-260}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{elif}\;h \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 + \frac{D \cdot D}{\ell} \cdot \left(\left(\frac{h}{d} \cdot \frac{M \cdot M}{d}\right) \cdot -0.125\right)\right)\\ \mathbf{elif}\;h \leq 3 \cdot 10^{+184}:\\ \;\;\;\;\left(1 + -0.5 \cdot \frac{{\left(M \cdot \frac{0.5}{\frac{d}{D}}\right)}^{2}}{\frac{\ell}{h}}\right) \cdot \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{1}{\frac{\sqrt{h}}{{\ell}^{-0.5}}}\\ \end{array} \]

Alternative 12
Error	24.7
Cost	15580

\[\begin{array}{l} t_0 := \left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ t_1 := \sqrt{\frac{d \cdot \frac{d}{\ell}}{h}} \cdot \left(1 + -0.5 \cdot \left(h \cdot \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell}\right)\right)\\ t_2 := \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + 0.5 \cdot \left(\left(\frac{D \cdot D}{\ell} \cdot \frac{h \cdot \left(M \cdot M\right)}{d \cdot d}\right) \cdot -0.25\right)\right)\right)\\ \mathbf{if}\;d \leq -8.6 \cdot 10^{+262}:\\ \;\;\;\;\sqrt{\frac{d}{h \cdot \frac{\ell}{d}}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{M \cdot \frac{D}{d}}{2}\right)}^{2}\right)\right)\\ \mathbf{elif}\;d \leq -1.05 \cdot 10^{+99}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq -1.04 \cdot 10^{+24}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq -3.5 \cdot 10^{-129}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;d \leq 4.9 \cdot 10^{-162}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 8.5 \cdot 10^{-68}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq 1.7 \cdot 10^{-34}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{1}{\frac{\sqrt{h}}{{\ell}^{-0.5}}}\\ \end{array} \]

Alternative 13
Error	24.6
Cost	15580

\[\begin{array}{l} t_0 := \left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ t_1 := \sqrt{\frac{d \cdot \frac{d}{\ell}}{h}} \cdot \left(1 + -0.5 \cdot \left(h \cdot \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell}\right)\right)\\ t_2 := \sqrt{\frac{d}{\ell}}\\ t_3 := \sqrt{\frac{d}{h}}\\ t_4 := 1 + 0.5 \cdot \left(\left(\frac{D \cdot D}{\ell} \cdot \frac{h \cdot \left(M \cdot M\right)}{d \cdot d}\right) \cdot -0.25\right)\\ \mathbf{if}\;d \leq -8.6 \cdot 10^{+262}:\\ \;\;\;\;\sqrt{\frac{d}{h \cdot \frac{\ell}{d}}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{M \cdot \frac{D}{d}}{2}\right)}^{2}\right)\right)\\ \mathbf{elif}\;d \leq -5.8 \cdot 10^{+97}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq -3.2 \cdot 10^{+23}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq -6.2 \cdot 10^{-133}:\\ \;\;\;\;t_2 \cdot \left(t_3 \cdot t_4\right)\\ \mathbf{elif}\;d \leq 4.9 \cdot 10^{-162}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 1.15 \cdot 10^{-68}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq 4.4 \cdot 10^{-36}:\\ \;\;\;\;\left(t_3 \cdot t_2\right) \cdot t_4\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{1}{\frac{\sqrt{h}}{{\ell}^{-0.5}}}\\ \end{array} \]

Alternative 14
Error	24.6
Cost	15580

\[\begin{array}{l} t_0 := \left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ t_1 := \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ t_2 := t_1 \cdot \left(1 + 0.5 \cdot \left(\left(M \cdot \left(M \cdot \frac{h}{\ell}\right)\right) \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot -0.25\right)\right)\right)\\ \mathbf{if}\;d \leq -8.6 \cdot 10^{+262}:\\ \;\;\;\;\sqrt{\frac{d}{h \cdot \frac{\ell}{d}}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{M \cdot \frac{D}{d}}{2}\right)}^{2}\right)\right)\\ \mathbf{elif}\;d \leq -4.8 \cdot 10^{+52}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq -1.42 \cdot 10^{-145}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;d \leq 1.75 \cdot 10^{-227}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 2.15 \cdot 10^{-162}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;d \leq 3.5 \cdot 10^{-68}:\\ \;\;\;\;\sqrt{\frac{d \cdot \frac{d}{\ell}}{h}} \cdot \left(1 + -0.5 \cdot \left(h \cdot \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell}\right)\right)\\ \mathbf{elif}\;d \leq 1.05 \cdot 10^{-35}:\\ \;\;\;\;t_1 \cdot \left(1 + 0.5 \cdot \left(\left(\frac{D \cdot D}{\ell} \cdot \frac{h \cdot \left(M \cdot M\right)}{d \cdot d}\right) \cdot -0.25\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{1}{\frac{\sqrt{h}}{{\ell}^{-0.5}}}\\ \end{array} \]

Alternative 15
Error	24.1
Cost	15444

\[\begin{array}{l} t_0 := \left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ t_1 := \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 + 0.5 \cdot \left(\left(M \cdot \frac{1}{\frac{\ell}{h \cdot M}}\right) \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot -0.25\right)\right)\right)\\ \mathbf{if}\;d \leq -3.4 \cdot 10^{+257}:\\ \;\;\;\;\sqrt{\frac{d}{h \cdot \frac{\ell}{d}}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{M \cdot \frac{D}{d}}{2}\right)}^{2}\right)\right)\\ \mathbf{elif}\;d \leq -4.5 \cdot 10^{+52}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq -3 \cdot 10^{-165}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq 7.2 \cdot 10^{-228}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 4.4 \cdot 10^{-36}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{1}{\frac{\sqrt{h}}{{\ell}^{-0.5}}}\\ \end{array} \]

Alternative 16
Error	24.4
Cost	15128

\[\begin{array}{l} t_0 := \left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ t_1 := \sqrt{\frac{d}{h \cdot \frac{\ell}{d}}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{M \cdot \frac{D}{d}}{2}\right)}^{2}\right)\right)\\ \mathbf{if}\;d \leq -8.6 \cdot 10^{+262}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq -4.1 \cdot 10^{+89}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq -2.8 \cdot 10^{-165}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;d \leq 5.3 \cdot 10^{-229}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 1.36 \cdot 10^{-207}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}}\\ \mathbf{elif}\;d \leq 5 \cdot 10^{-68}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{\ell}^{-0.5}}{\sqrt{h}}\\ \end{array} \]

Alternative 17
Error	24.8
Cost	14996

\[\begin{array}{l} t_0 := \left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ t_1 := {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\\ \mathbf{if}\;d \leq -1.95 \cdot 10^{+262}:\\ \;\;\;\;\sqrt{\frac{d}{h \cdot \frac{\ell}{d}}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{M \cdot \frac{D}{d}}{2}\right)}^{2}\right)\right)\\ \mathbf{elif}\;d \leq -1.6 \cdot 10^{+75}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq -3.4 \cdot 10^{-145}:\\ \;\;\;\;\left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot t_1\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{elif}\;d \leq 2.35 \cdot 10^{-163}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 3.8 \cdot 10^{-68}:\\ \;\;\;\;\sqrt{\frac{d \cdot \frac{d}{\ell}}{h}} \cdot \left(1 + -0.5 \cdot \left(h \cdot \frac{t_1}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{\ell}^{-0.5}}{\sqrt{h}}\\ \end{array} \]

Alternative 18
Error	23.7
Cost	14732

\[\begin{array}{l} t_0 := \left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \mathbf{if}\;d \leq -5 \cdot 10^{+262}:\\ \;\;\;\;\sqrt{\frac{d}{h \cdot \frac{\ell}{d}}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{M \cdot \frac{D}{d}}{2}\right)}^{2}\right)\right)\\ \mathbf{elif}\;d \leq -2.95 \cdot 10^{+97}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq -4.8 \cdot 10^{-165}:\\ \;\;\;\;\sqrt{\frac{d \cdot \frac{d}{\ell}}{h}} \cdot \left(1 + -0.5 \cdot \left(h \cdot \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell}\right)\right)\\ \mathbf{elif}\;d \leq 1.25 \cdot 10^{-229}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{1}{\frac{\sqrt{h}}{{\ell}^{-0.5}}}\\ \end{array} \]

Alternative 19
Error	24.0
Cost	13708

\[\begin{array}{l} \mathbf{if}\;d \leq -2.8 \cdot 10^{+89}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{elif}\;d \leq -4.8 \cdot 10^{-165}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{elif}\;d \leq 1.4 \cdot 10^{-230}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{1}{\frac{\sqrt{h}}{{\ell}^{-0.5}}}\\ \end{array} \]

Alternative 20
Error	24.0
Cost	13580

\[\begin{array}{l} \mathbf{if}\;d \leq -2.8 \cdot 10^{+89}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{elif}\;d \leq -4.5 \cdot 10^{-165}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{elif}\;d \leq 3.1 \cdot 10^{-229}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{\ell}^{-0.5}}{\sqrt{h}}\\ \end{array} \]

Alternative 21
Error	23.0
Cost	13576

\[\begin{array}{l} \mathbf{if}\;d \leq -3.1 \cdot 10^{+90}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{elif}\;d \leq 2.8 \cdot 10^{-294}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{1}{\frac{\sqrt{h}}{{\ell}^{-0.5}}}\\ \end{array} \]

Alternative 22
Error	24.0
Cost	13516

\[\begin{array}{l} \mathbf{if}\;d \leq -2.9 \cdot 10^{+90}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{elif}\;d \leq -4 \cdot 10^{-165}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{elif}\;d \leq 2.9 \cdot 10^{-228}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

Alternative 23
Error	27.9
Cost	13384

\[\begin{array}{l} \mathbf{if}\;d \leq -7.4 \cdot 10^{+89}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{elif}\;d \leq -1 \cdot 10^{-169}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \end{array} \]

Alternative 24
Error	27.9
Cost	7308

\[\begin{array}{l} t_0 := d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{if}\;d \leq -2.8 \cdot 10^{+89}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq -1.3 \cdot 10^{-165}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{elif}\;d \leq 2.45 \cdot 10^{-294}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \end{array} \]

Alternative 25
Error	34.4
Cost	6980

\[\begin{array}{l} \mathbf{if}\;d \leq -8.6 \cdot 10^{-190}:\\ \;\;\;\;\sqrt{d \cdot \frac{d}{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot {\left(h \cdot \ell\right)}^{-0.5}\\ \end{array} \]

Alternative 26
Error	32.8
Cost	6980

\[\begin{array}{l} \mathbf{if}\;h \leq 1.55 \cdot 10^{-280}:\\ \;\;\;\;\sqrt{d \cdot \frac{\frac{d}{\ell}}{h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot {\left(h \cdot \ell\right)}^{-0.5}\\ \end{array} \]

Alternative 27
Error	33.0
Cost	6980

\[\begin{array}{l} \mathbf{if}\;h \leq 1.55 \cdot 10^{-280}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot {\left(h \cdot \ell\right)}^{-0.5}\\ \end{array} \]

Alternative 28
Error	32.9
Cost	6980

\[\begin{array}{l} \mathbf{if}\;h \leq 5 \cdot 10^{-277}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \end{array} \]

Alternative 29
Error	33.0
Cost	6980

\[\begin{array}{l} \mathbf{if}\;h \leq 1.15 \cdot 10^{-279}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \end{array} \]

Alternative 30
Error	32.9
Cost	6980

\[\begin{array}{l} \mathbf{if}\;h \leq 4.5 \cdot 10^{-280}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \end{array} \]

Alternative 31
Error	43.8
Cost	6784

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

Alternative 32
Error	43.8
Cost	6720

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

Reproduce

herbie shell --seed 2022328 
(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)))))