Henrywood and Agarwal, Equation (12)

Average Error: 26.9 → 18.5

Time: 1.0min

Precision: binary64

Cost: 40724

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

Math

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

↓

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := \sqrt{\frac{d}{h}}\\ t_2 := {\left(\frac{d}{h}\right)}^{0.5}\\ t_3 := \sqrt{-d}\\ \mathbf{if}\;d \leq -1 \cdot 10^{+81}:\\ \;\;\;\;\frac{t_3}{\sqrt{-h}} \cdot \left(t_0 \cdot \mathsf{fma}\left({\left(\frac{\frac{D}{d}}{\frac{2}{M}}\right)}^{2}, \frac{h}{\ell} \cdot -0.5, 1\right)\right)\\ \mathbf{elif}\;d \leq -1 \cdot 10^{-49}:\\ \;\;\;\;t_1 \cdot \left(t_0 \cdot \mathsf{fma}\left(-0.125, h \cdot \left(M \cdot \frac{D}{\frac{\ell}{M} \cdot \frac{d \cdot d}{D}}\right), 1\right)\right)\\ \mathbf{elif}\;d \leq -4 \cdot 10^{-244}:\\ \;\;\;\;\left(t_2 \cdot \frac{t_3}{\sqrt{-\ell}}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}\right)\right)\\ \mathbf{elif}\;d \leq 10^{-280}:\\ \;\;\;\;t_1 \cdot \left(t_0 \cdot \left(\left(M \cdot \frac{D \cdot \frac{h}{\frac{d}{D \cdot M}}}{\ell}\right) \cdot \frac{-0.125}{d}\right)\right)\\ \mathbf{elif}\;d \leq 10^{+49}:\\ \;\;\;\;\left(t_2 \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - {\left(\left(\left(\frac{D}{d} \cdot \frac{M}{2}\right) \cdot \sqrt{0.5}\right) \cdot \frac{\sqrt{h}}{\sqrt{\ell}}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

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 l)))
        (t_1 (sqrt (/ d h)))
        (t_2 (pow (/ d h) 0.5))
        (t_3 (sqrt (- d))))
   (if (<= d -1e+81)
     (*
      (/ t_3 (sqrt (- h)))
      (* t_0 (fma (pow (/ (/ D d) (/ 2.0 M)) 2.0) (* (/ h l) -0.5) 1.0)))
     (if (<= d -1e-49)
       (*
        t_1
        (* t_0 (fma -0.125 (* h (* M (/ D (* (/ l M) (/ (* d d) D))))) 1.0)))
       (if (<= d -4e-244)
         (*
          (* t_2 (/ t_3 (sqrt (- l))))
          (+ 1.0 (* (/ h l) (* -0.5 (pow (/ (* D M) (* d 2.0)) 2.0)))))
         (if (<= d 1e-280)
           (*
            t_1
            (* t_0 (* (* M (/ (* D (/ h (/ d (* D M)))) l)) (/ -0.125 d))))
           (if (<= d 1e+49)
             (*
              (* t_2 (pow (/ d l) 0.5))
              (-
               1.0
               (pow
                (* (* (* (/ D d) (/ M 2.0)) (sqrt 0.5)) (/ (sqrt h) (sqrt l)))
                2.0)))
             (/ d (* (sqrt h) (sqrt l))))))))))

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 / l));
	double t_1 = sqrt((d / h));
	double t_2 = pow((d / h), 0.5);
	double t_3 = sqrt(-d);
	double tmp;
	if (d <= -1e+81) {
		tmp = (t_3 / sqrt(-h)) * (t_0 * fma(pow(((D / d) / (2.0 / M)), 2.0), ((h / l) * -0.5), 1.0));
	} else if (d <= -1e-49) {
		tmp = t_1 * (t_0 * fma(-0.125, (h * (M * (D / ((l / M) * ((d * d) / D))))), 1.0));
	} else if (d <= -4e-244) {
		tmp = (t_2 * (t_3 / sqrt(-l))) * (1.0 + ((h / l) * (-0.5 * pow(((D * M) / (d * 2.0)), 2.0))));
	} else if (d <= 1e-280) {
		tmp = t_1 * (t_0 * ((M * ((D * (h / (d / (D * M)))) / l)) * (-0.125 / d)));
	} else if (d <= 1e+49) {
		tmp = (t_2 * pow((d / l), 0.5)) * (1.0 - pow(((((D / d) * (M / 2.0)) * sqrt(0.5)) * (sqrt(h) / sqrt(l))), 2.0));
	} else {
		tmp = d / (sqrt(h) * sqrt(l));
	}
	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 / l))
	t_1 = sqrt(Float64(d / h))
	t_2 = Float64(d / h) ^ 0.5
	t_3 = sqrt(Float64(-d))
	tmp = 0.0
	if (d <= -1e+81)
		tmp = Float64(Float64(t_3 / sqrt(Float64(-h))) * Float64(t_0 * fma((Float64(Float64(D / d) / Float64(2.0 / M)) ^ 2.0), Float64(Float64(h / l) * -0.5), 1.0)));
	elseif (d <= -1e-49)
		tmp = Float64(t_1 * Float64(t_0 * fma(-0.125, Float64(h * Float64(M * Float64(D / Float64(Float64(l / M) * Float64(Float64(d * d) / D))))), 1.0)));
	elseif (d <= -4e-244)
		tmp = Float64(Float64(t_2 * Float64(t_3 / sqrt(Float64(-l)))) * Float64(1.0 + Float64(Float64(h / l) * Float64(-0.5 * (Float64(Float64(D * M) / Float64(d * 2.0)) ^ 2.0)))));
	elseif (d <= 1e-280)
		tmp = Float64(t_1 * Float64(t_0 * Float64(Float64(M * Float64(Float64(D * Float64(h / Float64(d / Float64(D * M)))) / l)) * Float64(-0.125 / d))));
	elseif (d <= 1e+49)
		tmp = Float64(Float64(t_2 * (Float64(d / l) ^ 0.5)) * Float64(1.0 - (Float64(Float64(Float64(Float64(D / d) * Float64(M / 2.0)) * sqrt(0.5)) * Float64(sqrt(h) / sqrt(l))) ^ 2.0)));
	else
		tmp = Float64(d / Float64(sqrt(h) * sqrt(l)));
	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 / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Power[N[(d / h), $MachinePrecision], 0.5], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[(-d)], $MachinePrecision]}, If[LessEqual[d, -1e+81], N[(N[(t$95$3 / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * N[(N[Power[N[(N[(D / d), $MachinePrecision] / N[(2.0 / M), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -1e-49], N[(t$95$1 * N[(t$95$0 * N[(-0.125 * N[(h * N[(M * N[(D / N[(N[(l / M), $MachinePrecision] * N[(N[(d * d), $MachinePrecision] / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -4e-244], N[(N[(t$95$2 * N[(t$95$3 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(-0.5 * N[Power[N[(N[(D * M), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1e-280], N[(t$95$1 * N[(t$95$0 * N[(N[(M * N[(N[(D * N[(h / N[(d / N[(D * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(-0.125 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1e+49], N[(N[(t$95$2 * N[Power[N[(d / l), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[Power[N[(N[(N[(N[(D / d), $MachinePrecision] * N[(M / 2.0), $MachinePrecision]), $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[h], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if d < -9.99999999999999921e80

   1. Initial program 26.8

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

   2. Simplified 26.3

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(\frac{\frac{D}{d}}{\frac{2}{M}}\right)}^{2}, \frac{h}{\ell} \cdot -0.5, 1\right)\right)} \]
      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))): 2 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 (/.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))): 2 points increase in error, 12 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))))): 3 points increase in error, 3 points decrease in error

   3. Applied egg-rr 12.0

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

   if -9.99999999999999921e80 < d < -9.99999999999999936e-50

   1. Initial program 14.9

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

   2. Simplified 15.4

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(\frac{\frac{D}{d}}{\frac{2}{M}}\right)}^{2}, \frac{h}{\ell} \cdot -0.5, 1\right)\right)} \]
      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))): 2 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 (/.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))): 2 points increase in error, 12 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))))): 3 points increase in error, 3 points decrease in error

   3. Taylor expanded in D around 0 26.0

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

   4. Simplified 12.8

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\mathsf{fma}\left(-0.125, h \cdot \left(\frac{D}{\frac{\ell}{M} \cdot \frac{d \cdot d}{D}} \cdot M\right), 1\right)}\right) \]
      Proof
      (fma.f64 -1/8 (*.f64 h (*.f64 (/.f64 D (*.f64 (/.f64 l M) (/.f64 (*.f64 d d) D))) M)) 1): 0 points increase in error, 0 points decrease in error
      (fma.f64 -1/8 (*.f64 h (*.f64 (/.f64 D (*.f64 (/.f64 l M) (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 d 2)) D))) M)) 1): 0 points increase in error, 0 points decrease in error
      (fma.f64 -1/8 (*.f64 h (*.f64 (Rewrite<= associate-/l/_binary64 (/.f64 (/.f64 D (/.f64 (pow.f64 d 2) D)) (/.f64 l M))) M)) 1): 10 points increase in error, 3 points decrease in error
      (fma.f64 -1/8 (*.f64 h (*.f64 (/.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 D D) (pow.f64 d 2))) (/.f64 l M)) M)) 1): 21 points increase in error, 4 points decrease in error
      (fma.f64 -1/8 (*.f64 h (*.f64 (/.f64 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 D 2)) (pow.f64 d 2)) (/.f64 l M)) M)) 1): 0 points increase in error, 0 points decrease in error
      (fma.f64 -1/8 (*.f64 h (*.f64 (Rewrite=> associate-/r/_binary64 (*.f64 (/.f64 (/.f64 (pow.f64 D 2) (pow.f64 d 2)) l) M)) M)) 1): 4 points increase in error, 6 points decrease in error
      (fma.f64 -1/8 (*.f64 h (*.f64 (*.f64 (Rewrite<= associate-/r*_binary64 (/.f64 (pow.f64 D 2) (*.f64 (pow.f64 d 2) l))) M) M)) 1): 10 points increase in error, 7 points decrease in error
      (fma.f64 -1/8 (*.f64 h (*.f64 (Rewrite<= associate-/r/_binary64 (/.f64 (pow.f64 D 2) (/.f64 (*.f64 (pow.f64 d 2) l) M))) M)) 1): 7 points increase in error, 5 points decrease in error
      (fma.f64 -1/8 (*.f64 h (Rewrite<= associate-/r/_binary64 (/.f64 (pow.f64 D 2) (/.f64 (/.f64 (*.f64 (pow.f64 d 2) l) M) M)))) 1): 4 points increase in error, 4 points decrease in error
      (fma.f64 -1/8 (*.f64 h (/.f64 (pow.f64 D 2) (Rewrite<= associate-/r*_binary64 (/.f64 (*.f64 (pow.f64 d 2) l) (*.f64 M M))))) 1): 24 points increase in error, 3 points decrease in error
      (fma.f64 -1/8 (*.f64 h (/.f64 (pow.f64 D 2) (/.f64 (*.f64 (pow.f64 d 2) l) (Rewrite<= unpow2_binary64 (pow.f64 M 2))))) 1): 0 points increase in error, 0 points decrease in error
      (fma.f64 -1/8 (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 (pow.f64 D 2) (/.f64 (*.f64 (pow.f64 d 2) l) (pow.f64 M 2))) h)) 1): 0 points increase in error, 0 points decrease in error
      (fma.f64 -1/8 (Rewrite<= associate-/r/_binary64 (/.f64 (pow.f64 D 2) (/.f64 (/.f64 (*.f64 (pow.f64 d 2) l) (pow.f64 M 2)) h))) 1): 4 points increase in error, 2 points decrease in error
      (fma.f64 -1/8 (/.f64 (pow.f64 D 2) (Rewrite<= associate-/r*_binary64 (/.f64 (*.f64 (pow.f64 d 2) l) (*.f64 (pow.f64 M 2) h)))) 1): 11 points increase in error, 4 points decrease in error
      (fma.f64 -1/8 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (pow.f64 D 2) (*.f64 (pow.f64 M 2) h)) (*.f64 (pow.f64 d 2) l))) 1): 9 points increase in error, 3 points decrease in error
      (Rewrite<= fma-def_binary64 (+.f64 (*.f64 -1/8 (/.f64 (*.f64 (pow.f64 D 2) (*.f64 (pow.f64 M 2) h)) (*.f64 (pow.f64 d 2) l))) 1)): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 -1/8 (*.f64 (pow.f64 D 2) (*.f64 (pow.f64 M 2) h))) (*.f64 (pow.f64 d 2) l))) 1): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (*.f64 -1/8 (*.f64 (pow.f64 D 2) (*.f64 (pow.f64 M 2) h))) (Rewrite<= *-commutative_binary64 (*.f64 l (pow.f64 d 2)))) 1): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite<= associate-*r/_binary64 (*.f64 -1/8 (/.f64 (*.f64 (pow.f64 D 2) (*.f64 (pow.f64 M 2) h)) (*.f64 l (pow.f64 d 2))))) 1): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 -1/8 (/.f64 (*.f64 (pow.f64 D 2) (Rewrite<= *-commutative_binary64 (*.f64 h (pow.f64 M 2)))) (*.f64 l (pow.f64 d 2)))) 1): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 -1/8 (/.f64 (*.f64 (pow.f64 D 2) (*.f64 h (pow.f64 M 2))) (Rewrite=> *-commutative_binary64 (*.f64 (pow.f64 d 2) l)))) 1): 0 points increase in error, 0 points decrease in error
      (Rewrite<= +-commutative_binary64 (+.f64 1 (*.f64 -1/8 (/.f64 (*.f64 (pow.f64 D 2) (*.f64 h (pow.f64 M 2))) (*.f64 (pow.f64 d 2) l))))): 0 points increase in error, 0 points decrease in error

   if -9.99999999999999936e-50 < d < -3.9999999999999997e-244

   1. Initial program 31.2

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

   2. Applied egg-rr 25.4

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

   if -3.9999999999999997e-244 < d < 9.9999999999999996e-281

   1. Initial program 44.8

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

   2. Simplified 46.8

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(\frac{\frac{D}{d}}{\frac{2}{M}}\right)}^{2}, \frac{h}{\ell} \cdot -0.5, 1\right)\right)} \]
      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))): 2 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 (/.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))): 2 points increase in error, 12 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))))): 3 points increase in error, 3 points decrease in error

   3. Taylor expanded in D around inf 64.0

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

   4. Simplified 52.5

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(\frac{\left(D \cdot D\right) \cdot \left(h \cdot M\right)}{d \cdot \frac{\ell}{M}} \cdot \frac{-0.125}{d}\right)}\right) \]
      Proof
      (*.f64 (/.f64 (*.f64 (*.f64 D D) (*.f64 h M)) (*.f64 d (/.f64 l M))) (/.f64 -1/8 d)): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 D 2)) (*.f64 h M)) (*.f64 d (/.f64 l M))) (/.f64 -1/8 d)): 1 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (pow.f64 D 2) h) M)) (*.f64 d (/.f64 l M))) (/.f64 -1/8 d)): 12 points increase in error, 20 points decrease in error
      (*.f64 (Rewrite<= associate-/l/_binary64 (/.f64 (/.f64 (*.f64 (*.f64 (pow.f64 D 2) h) M) (/.f64 l M)) d)) (/.f64 -1/8 d)): 11 points increase in error, 28 points decrease in error
      (*.f64 (/.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (*.f64 (*.f64 (pow.f64 D 2) h) M) M) l)) d) (/.f64 -1/8 d)): 14 points increase in error, 7 points decrease in error
      (*.f64 (/.f64 (/.f64 (Rewrite<= associate-*r*_binary64 (*.f64 (*.f64 (pow.f64 D 2) h) (*.f64 M M))) l) d) (/.f64 -1/8 d)): 28 points increase in error, 7 points decrease in error
      (*.f64 (/.f64 (/.f64 (*.f64 (*.f64 (pow.f64 D 2) h) (Rewrite<= unpow2_binary64 (pow.f64 M 2))) l) d) (/.f64 -1/8 d)): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (/.f64 (Rewrite<= associate-*r*_binary64 (*.f64 (pow.f64 D 2) (*.f64 h (pow.f64 M 2)))) l) d) (/.f64 -1/8 d)): 14 points increase in error, 12 points decrease in error
      (*.f64 (/.f64 (/.f64 (*.f64 (pow.f64 D 2) (Rewrite=> *-commutative_binary64 (*.f64 (pow.f64 M 2) h))) l) d) (/.f64 -1/8 d)): 0 points increase in error, 0 points decrease in error
      (Rewrite=> *-commutative_binary64 (*.f64 (/.f64 -1/8 d) (/.f64 (/.f64 (*.f64 (pow.f64 D 2) (*.f64 (pow.f64 M 2) h)) l) d))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 -1/8 d) (Rewrite=> associate-/l/_binary64 (/.f64 (*.f64 (pow.f64 D 2) (*.f64 (pow.f64 M 2) h)) (*.f64 d l)))): 26 points increase in error, 6 points decrease in error
      (Rewrite<= times-frac_binary64 (/.f64 (*.f64 -1/8 (*.f64 (pow.f64 D 2) (*.f64 (pow.f64 M 2) h))) (*.f64 d (*.f64 d l)))): 33 points increase in error, 7 points decrease in error
      (/.f64 (*.f64 -1/8 (*.f64 (pow.f64 D 2) (Rewrite<= *-commutative_binary64 (*.f64 h (pow.f64 M 2))))) (*.f64 d (*.f64 d l))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 -1/8 (*.f64 (pow.f64 D 2) (*.f64 h (pow.f64 M 2)))) (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 d d) l))): 10 points increase in error, 7 points decrease in error
      (/.f64 (*.f64 -1/8 (*.f64 (pow.f64 D 2) (*.f64 h (pow.f64 M 2)))) (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 d 2)) l)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*r/_binary64 (*.f64 -1/8 (/.f64 (*.f64 (pow.f64 D 2) (*.f64 h (pow.f64 M 2))) (*.f64 (pow.f64 d 2) l)))): 1 points increase in error, 0 points decrease in error

   5. Applied egg-rr 52.7

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

   6. Taylor expanded in h around 0 52.5

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

   7. Simplified 49.1

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\left(\frac{\color{blue}{D \cdot \frac{h}{\frac{d}{D \cdot M}}}}{\ell} \cdot M\right) \cdot \frac{-0.125}{d}\right)\right) \]
      Proof
      (*.f64 D (/.f64 h (/.f64 d (*.f64 D M)))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 D h) (/.f64 d (*.f64 D M)))): 35 points increase in error, 29 points decrease in error
      (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 D (/.f64 d (*.f64 D M))) h)): 28 points increase in error, 38 points decrease in error
      (*.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 D (*.f64 D M)) d)) h): 33 points increase in error, 31 points decrease in error
      (*.f64 (/.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 D D) M)) d) h): 28 points increase in error, 15 points decrease in error
      (*.f64 (/.f64 (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 D 2)) M) d) h): 0 points increase in error, 0 points decrease in error
      (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 (*.f64 (pow.f64 D 2) M) h) d)): 30 points increase in error, 20 points decrease in error
      (/.f64 (Rewrite<= associate-*r*_binary64 (*.f64 (pow.f64 D 2) (*.f64 M h))) d): 33 points increase in error, 14 points decrease in error

   if 9.9999999999999996e-281 < d < 9.99999999999999946e48

   1. Initial program 26.1

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

   2. Applied egg-rr 25.2

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

   3. Applied egg-rr 22.5

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

   if 9.99999999999999946e48 < d

   1. Initial program 26.4

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

   2. Taylor expanded in d around inf 17.6

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

   3. Applied egg-rr 17.4

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

   4. Applied egg-rr 29.3

      \[\leadsto \frac{d}{\color{blue}{{\left({\left(\ell \cdot h\right)}^{1.5}\right)}^{0.3333333333333333}}} \]

   5. Applied egg-rr 8.5

      \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell} \cdot \sqrt{h}}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 18.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1 \cdot 10^{+81}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(\frac{\frac{D}{d}}{\frac{2}{M}}\right)}^{2}, \frac{h}{\ell} \cdot -0.5, 1\right)\right)\\ \mathbf{elif}\;d \leq -1 \cdot 10^{-49}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(-0.125, h \cdot \left(M \cdot \frac{D}{\frac{\ell}{M} \cdot \frac{d \cdot d}{D}}\right), 1\right)\right)\\ \mathbf{elif}\;d \leq -4 \cdot 10^{-244}:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}\right)\right)\\ \mathbf{elif}\;d \leq 10^{-280}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\left(M \cdot \frac{D \cdot \frac{h}{\frac{d}{D \cdot M}}}{\ell}\right) \cdot \frac{-0.125}{d}\right)\right)\\ \mathbf{elif}\;d \leq 10^{+49}:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - {\left(\left(\left(\frac{D}{d} \cdot \frac{M}{2}\right) \cdot \sqrt{0.5}\right) \cdot \frac{\sqrt{h}}{\sqrt{\ell}}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternatives

Alternative 1
Error	21.3
Cost	62600

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ t_1 := 1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}\right)\\ t_2 := \sqrt{\frac{d}{\ell}}\\ t_3 := \left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot t_1\\ \mathbf{if}\;t_3 \leq -5 \cdot 10^{+252}:\\ \;\;\;\;t_0 \cdot \left(t_2 \cdot \left(\left(M \cdot \frac{D \cdot \frac{h}{\frac{d}{D \cdot M}}}{\ell}\right) \cdot \frac{-0.125}{d}\right)\right)\\ \mathbf{elif}\;t_3 \leq 5 \cdot 10^{+226}:\\ \;\;\;\;t_1 \cdot \left(t_2 \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}}\\ \end{array} \]

Alternative 2
Error	18.6
Cost	40660

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := \sqrt{\frac{d}{h}}\\ t_2 := {\left(\frac{d}{h}\right)}^{0.5}\\ t_3 := \sqrt{-d}\\ \mathbf{if}\;d \leq -1 \cdot 10^{+81}:\\ \;\;\;\;\frac{t_3}{\sqrt{-h}} \cdot \left(t_0 \cdot \mathsf{fma}\left({\left(\frac{\frac{D}{d}}{\frac{2}{M}}\right)}^{2}, \frac{h}{\ell} \cdot -0.5, 1\right)\right)\\ \mathbf{elif}\;d \leq -1 \cdot 10^{-49}:\\ \;\;\;\;t_1 \cdot \left(t_0 \cdot \mathsf{fma}\left(-0.125, h \cdot \left(M \cdot \frac{D}{\frac{\ell}{M} \cdot \frac{d \cdot d}{D}}\right), 1\right)\right)\\ \mathbf{elif}\;d \leq -4 \cdot 10^{-244}:\\ \;\;\;\;\left(t_2 \cdot \frac{t_3}{\sqrt{-\ell}}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}\right)\right)\\ \mathbf{elif}\;d \leq 10^{-240}:\\ \;\;\;\;t_1 \cdot \left(t_0 \cdot \left(\left(M \cdot \frac{D \cdot \frac{h}{\frac{d}{D \cdot M}}}{\ell}\right) \cdot \frac{-0.125}{d}\right)\right)\\ \mathbf{elif}\;d \leq 1000000000000:\\ \;\;\;\;\left(t_2 \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \left(1 - {\left(\left(\left(\frac{D}{d} \cdot \frac{M}{2}\right) \cdot \sqrt{0.5}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 3
Error	17.9
Cost	40592

\[\begin{array}{l} t_0 := {\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\\ t_1 := \sqrt{-h}\\ t_2 := \left(\frac{D}{d} \cdot \frac{M}{2}\right) \cdot \sqrt{0.5}\\ \mathbf{if}\;d \leq -1 \cdot 10^{+81}:\\ \;\;\;\;\frac{\sqrt{-d}}{t_1} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(\frac{\frac{D}{d}}{\frac{2}{M}}\right)}^{2}, \frac{h}{\ell} \cdot -0.5, 1\right)\right)\\ \mathbf{elif}\;d \leq -9.5 \cdot 10^{-141}:\\ \;\;\;\;t_0 \cdot \left(1 - {\left(\frac{t_1}{\sqrt{-\ell}} \cdot t_2\right)}^{2}\right)\\ \mathbf{elif}\;d \leq 1.1 \cdot 10^{-297}:\\ \;\;\;\;\mathsf{fma}\left(D \cdot \frac{M \cdot \left(D \cdot M\right)}{d}, \sqrt{\frac{h}{{\ell}^{3}}} \cdot 0.125, \left(-d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\right)\\ \mathbf{elif}\;d \leq 10^{+49}:\\ \;\;\;\;t_0 \cdot \left(1 - {\left(t_2 \cdot \frac{\sqrt{h}}{\sqrt{\ell}}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 4
Error	18.2
Cost	33668

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := {\left(\frac{d}{h}\right)}^{0.5}\\ t_2 := 1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}\right)\\ t_3 := \sqrt{-d}\\ t_4 := t_1 \cdot \frac{t_3}{\sqrt{-\ell}}\\ \mathbf{if}\;d \leq -1 \cdot 10^{+81}:\\ \;\;\;\;\frac{t_3}{\sqrt{-h}} \cdot \left(t_0 \cdot \mathsf{fma}\left({\left(\frac{\frac{D}{d}}{\frac{2}{M}}\right)}^{2}, \frac{h}{\ell} \cdot -0.5, 1\right)\right)\\ \mathbf{elif}\;d \leq -1 \cdot 10^{-49}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(t_0 \cdot \mathsf{fma}\left(-0.125, h \cdot \left(M \cdot \frac{D}{\frac{\ell}{M} \cdot \frac{d \cdot d}{D}}\right), 1\right)\right)\\ \mathbf{elif}\;d \leq -1 \cdot 10^{-245}:\\ \;\;\;\;t_4 \cdot t_2\\ \mathbf{elif}\;d \leq 0:\\ \;\;\;\;t_4 \cdot \left(1 - 0.125 \cdot \left(M \cdot \left(M \cdot \frac{\frac{D}{\frac{d}{D}}}{\frac{\ell}{\frac{h}{d}}}\right)\right)\right)\\ \mathbf{elif}\;d \leq 1000000000000:\\ \;\;\;\;t_2 \cdot \left(t_1 \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 5
Error	20.8
Cost	27992

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{h \cdot \frac{\ell}{d}}}\\ t_1 := 1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}\right)\\ \mathbf{if}\;\ell \leq -1 \cdot 10^{-123}:\\ \;\;\;\;t_1 \cdot \left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\right)\\ \mathbf{elif}\;\ell \leq 10^{-291}:\\ \;\;\;\;t_0 \cdot \left(1 + h \cdot \left(\left(\frac{D}{d} \cdot \left(\frac{D}{d} \cdot -0.125\right)\right) \cdot \left(M \cdot \frac{M}{\ell}\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 10^{-120}:\\ \;\;\;\;t_1 \cdot \left({\left(\frac{d}{h}\right)}^{0.5} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right)\\ \mathbf{elif}\;\ell \leq 3.6 \cdot 10^{-115}:\\ \;\;\;\;t_0 \cdot \left(1 + -0.125 \cdot \left(\frac{D}{d \cdot \frac{d}{D}} \cdot \left(h \cdot \frac{M}{\frac{\ell}{M}}\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 1.35 \cdot 10^{-19}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \mathbf{elif}\;\ell \leq 10^{+199}:\\ \;\;\;\;\left({\left(\frac{d}{\ell}\right)}^{0.5} \cdot \frac{\sqrt{d}}{\sqrt{h}}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{D}{\frac{\frac{d}{0.5}}{M}}\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}\\ \end{array} \]

Alternative 6
Error	20.7
Cost	27992

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{h \cdot \frac{\ell}{d}}}\\ t_1 := 1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}\right)\\ \mathbf{if}\;\ell \leq -1 \cdot 10^{-123}:\\ \;\;\;\;t_1 \cdot \left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\right)\\ \mathbf{elif}\;\ell \leq 10^{-291}:\\ \;\;\;\;t_0 \cdot \left(1 + h \cdot \left(\left(\frac{D}{d} \cdot \left(\frac{D}{d} \cdot -0.125\right)\right) \cdot \left(M \cdot \frac{M}{\ell}\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 10^{-120}:\\ \;\;\;\;t_1 \cdot \left({\left(\frac{d}{h}\right)}^{0.5} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right)\\ \mathbf{elif}\;\ell \leq 3.6 \cdot 10^{-115}:\\ \;\;\;\;t_0 \cdot \left(1 + -0.125 \cdot \left(\frac{D}{d \cdot \frac{d}{D}} \cdot \left(h \cdot \frac{M}{\frac{\ell}{M}}\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 1.35 \cdot 10^{-19}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \mathbf{elif}\;\ell \leq 10^{+199}:\\ \;\;\;\;t_1 \cdot \left({\left(\frac{d}{\ell}\right)}^{0.5} \cdot \frac{\sqrt{d}}{\sqrt{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}\\ \end{array} \]

Alternative 7
Error	20.7
Cost	27992

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{h \cdot \frac{\ell}{d}}}\\ t_1 := 1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}\right)\\ t_2 := \sqrt{\frac{1}{\ell}}\\ \mathbf{if}\;\ell \leq -1 \cdot 10^{-123}:\\ \;\;\;\;t_1 \cdot \left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\right)\\ \mathbf{elif}\;\ell \leq 10^{-291}:\\ \;\;\;\;t_0 \cdot \left(1 + h \cdot \left(\left(\frac{D}{d} \cdot \left(\frac{D}{d} \cdot -0.125\right)\right) \cdot \left(M \cdot \frac{M}{\ell}\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 10^{-120}:\\ \;\;\;\;t_1 \cdot \left({\left(\frac{d}{h}\right)}^{0.5} \cdot \left(\sqrt{d} \cdot t_2\right)\right)\\ \mathbf{elif}\;\ell \leq 3.6 \cdot 10^{-115}:\\ \;\;\;\;t_0 \cdot \left(1 + -0.125 \cdot \left(\frac{D}{d \cdot \frac{d}{D}} \cdot \left(h \cdot \frac{M}{\frac{\ell}{M}}\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 1.35 \cdot 10^{-19}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \mathbf{elif}\;\ell \leq 10^{+199}:\\ \;\;\;\;t_1 \cdot \left({\left(\frac{d}{\ell}\right)}^{0.5} \cdot \frac{\sqrt{d}}{\sqrt{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{t_2}{\sqrt{h}}\\ \end{array} \]

Alternative 8
Error	24.9
Cost	21396

\[\begin{array}{l} t_0 := \frac{\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(0.5 \cdot \frac{D \cdot M}{d}\right)}^{2}\right)\right)}{\sqrt{\frac{\ell}{d}}}\\ \mathbf{if}\;\ell \leq -2.4 \cdot 10^{+197}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq -1.46 \cdot 10^{+52}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{elif}\;\ell \leq -1 \cdot 10^{-123}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq 10^{-291}:\\ \;\;\;\;\sqrt{\frac{d}{h \cdot \frac{\ell}{d}}} \cdot \left(1 + h \cdot \left(\left(\frac{D}{d} \cdot \left(\frac{D}{d} \cdot -0.125\right)\right) \cdot \left(M \cdot \frac{M}{\ell}\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 3.6 \cdot 10^{-115}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 9
Error	24.6
Cost	20932

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ t_1 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;d \leq -9.5 \cdot 10^{-141}:\\ \;\;\;\;t_0 \cdot \left(t_1 \cdot \mathsf{fma}\left(-0.125, h \cdot \left(M \cdot \frac{D}{\frac{\ell}{M} \cdot \frac{d \cdot d}{D}}\right), 1\right)\right)\\ \mathbf{elif}\;d \leq -5.6 \cdot 10^{-227}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{elif}\;d \leq 7.5 \cdot 10^{-250}:\\ \;\;\;\;t_0 \cdot \left(t_1 \cdot \left(\left(M \cdot \frac{D \cdot \frac{h}{\frac{d}{D \cdot M}}}{\ell}\right) \cdot \frac{-0.125}{d}\right)\right)\\ \mathbf{elif}\;d \leq 1.14 \cdot 10^{-126}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right)\\ \mathbf{elif}\;d \leq 1.2 \cdot 10^{-104}:\\ \;\;\;\;t_0 \cdot \left(t_1 \cdot \left(\frac{-0.125}{d} \cdot \left(\frac{D}{\frac{\ell}{M}} \cdot \frac{D \cdot \left(h \cdot M\right)}{d}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 10
Error	24.5
Cost	15192

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := \sqrt{\frac{d}{h}}\\ t_2 := \frac{d}{D \cdot M}\\ t_3 := \left(-d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{if}\;d \leq -3.2 \cdot 10^{+79}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;d \leq -1.85 \cdot 10^{-100}:\\ \;\;\;\;t_0 \cdot t_1\\ \mathbf{elif}\;d \leq -5.6 \cdot 10^{-227}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;d \leq 3.5 \cdot 10^{-283}:\\ \;\;\;\;t_1 \cdot \left(t_0 \cdot \left(\left(M \cdot \frac{D \cdot \frac{h}{t_2}}{\ell}\right) \cdot \frac{-0.125}{d}\right)\right)\\ \mathbf{elif}\;d \leq 1.14 \cdot 10^{-126}:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{h} \cdot \frac{1}{\ell}}\\ \mathbf{elif}\;d \leq 2.1 \cdot 10^{-109}:\\ \;\;\;\;t_1 \cdot \left(t_0 \cdot \left(\frac{-0.125}{d} \cdot \left(\frac{D}{d} \cdot \frac{D \cdot \left(h \cdot M\right)}{\frac{\ell}{M}}\right)\right)\right)\\ \mathbf{elif}\;d \leq 2 \cdot 10^{-81}:\\ \;\;\;\;\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(M \cdot \frac{D \cdot -0.125}{t_2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 11
Error	23.8
Cost	15192

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := \sqrt{\frac{d}{h}}\\ t_2 := \left(-d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{if}\;d \leq -3.2 \cdot 10^{+79}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;d \leq -1.85 \cdot 10^{-100}:\\ \;\;\;\;t_0 \cdot t_1\\ \mathbf{elif}\;d \leq -5.6 \cdot 10^{-227}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;d \leq 3.5 \cdot 10^{-283}:\\ \;\;\;\;t_1 \cdot \left(t_0 \cdot \left(\left(M \cdot \frac{D \cdot \frac{h}{\frac{d}{D \cdot M}}}{\ell}\right) \cdot \frac{-0.125}{d}\right)\right)\\ \mathbf{elif}\;d \leq 1.14 \cdot 10^{-126}:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{h} \cdot \frac{1}{\ell}}\\ \mathbf{elif}\;d \leq 1.2 \cdot 10^{-104}:\\ \;\;\;\;t_1 \cdot \left(t_0 \cdot \left(\frac{-0.125}{d} \cdot \left(\frac{D}{\frac{\ell}{M}} \cdot \frac{D \cdot \left(h \cdot M\right)}{d}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 12
Error	24.1
Cost	15192

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := \left(-d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ t_2 := \sqrt{\frac{d}{h}}\\ \mathbf{if}\;d \leq -3.2 \cdot 10^{+79}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq -9.2 \cdot 10^{-69}:\\ \;\;\;\;\left(1 - 0.125 \cdot \left(M \cdot \left(M \cdot \frac{\frac{D}{\frac{d}{D}}}{\frac{\ell}{\frac{h}{d}}}\right)\right)\right) \cdot \left(t_0 \cdot {\left(\frac{d}{h}\right)}^{0.5}\right)\\ \mathbf{elif}\;d \leq -5.6 \cdot 10^{-227}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq 3.5 \cdot 10^{-283}:\\ \;\;\;\;t_2 \cdot \left(t_0 \cdot \left(\left(M \cdot \frac{D \cdot \frac{h}{\frac{d}{D \cdot M}}}{\ell}\right) \cdot \frac{-0.125}{d}\right)\right)\\ \mathbf{elif}\;d \leq 1.14 \cdot 10^{-126}:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{h} \cdot \frac{1}{\ell}}\\ \mathbf{elif}\;d \leq 1.2 \cdot 10^{-104}:\\ \;\;\;\;t_2 \cdot \left(t_0 \cdot \left(\frac{-0.125}{d} \cdot \left(\frac{D}{\frac{\ell}{M}} \cdot \frac{D \cdot \left(h \cdot M\right)}{d}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 13
Error	23.1
Cost	14928

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := \sqrt{\frac{d}{h}}\\ t_2 := \left(-d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{if}\;d \leq -3.2 \cdot 10^{+79}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;d \leq -1.85 \cdot 10^{-100}:\\ \;\;\;\;t_0 \cdot t_1\\ \mathbf{elif}\;d \leq -5.6 \cdot 10^{-227}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;d \leq 3.5 \cdot 10^{-283}:\\ \;\;\;\;t_1 \cdot \left(t_0 \cdot \left(\left(M \cdot \frac{D \cdot \frac{h}{\frac{d}{D \cdot M}}}{\ell}\right) \cdot \frac{-0.125}{d}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 14
Error	23.4
Cost	13516

\[\begin{array}{l} t_0 := \left(-d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{if}\;d \leq -3.2 \cdot 10^{+79}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq -1.85 \cdot 10^{-100}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;d \leq 1.4 \cdot 10^{-232}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 15
Error	23.8
Cost	13252

\[\begin{array}{l} \mathbf{if}\;\ell \leq 4.2 \cdot 10^{-292}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 16
Error	27.3
Cost	7044

\[\begin{array}{l} \mathbf{if}\;\ell \leq 4.2 \cdot 10^{-292}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}}\\ \end{array} \]

Alternative 17
Error	36.9
Cost	6980

\[\begin{array}{l} \mathbf{if}\;\ell \leq -1.55 \cdot 10^{-271}:\\ \;\;\;\;\sqrt{\frac{d \cdot d}{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}}\\ \end{array} \]

Alternative 18
Error	43.8
Cost	6720

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

Reproduce

herbie shell --seed 2022298 
(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)))))