Henrywood and Agarwal, Equation (12)

Average Error: 26.8 → 18.3

Time: 1.1min

Precision: binary64

Cost: 28252

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

Math

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

↓

\[\begin{array}{l} t_0 := {\left(\frac{d}{\ell}\right)}^{0.5}\\ t_1 := 1 - \frac{{\left(\left(D \cdot M\right) \cdot \frac{0.5}{d}\right)}^{2} \cdot \left(h \cdot 0.5\right)}{\ell}\\ t_2 := \sqrt{-d}\\ t_3 := t_1 \cdot \left(\frac{t_2}{\sqrt{-h}} \cdot t_0\right)\\ t_4 := \frac{t_2}{\sqrt{-\ell}}\\ \mathbf{if}\;\ell \leq -1 \cdot 10^{+211}:\\ \;\;\;\;\left(t_4 \cdot {\left(\frac{d}{h}\right)}^{0.5}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{D}{\frac{\frac{d}{0.5}}{M}}\right)}^{2}\right)\right)\\ \mathbf{elif}\;\ell \leq -1 \cdot 10^{+20}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;\ell \leq -1 \cdot 10^{-76}:\\ \;\;\;\;\mathsf{fma}\left(D \cdot \frac{M \cdot \left(D \cdot M\right)}{d}, \sqrt{\frac{h}{{\ell}^{3}}} \cdot 0.125, \left(-d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\right)\\ \mathbf{elif}\;\ell \leq -1 \cdot 10^{-102}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;\ell \leq -1.35 \cdot 10^{-308}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(t_4 \cdot \mathsf{fma}\left(-0.125, h \cdot \left(M \cdot \frac{D}{\frac{\ell}{\frac{D \cdot M}{d \cdot d}}}\right), 1\right)\right)\\ \mathbf{elif}\;\ell \leq 10^{-175}:\\ \;\;\;\;\left(\frac{1}{\sqrt{\frac{h}{d}}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot t_1\\ \mathbf{elif}\;\ell \leq 10^{+135}:\\ \;\;\;\;\left(\left(\sqrt{d} \cdot \sqrt{\frac{1}{h}}\right) \cdot t_0\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}\\ \end{array} \]

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 (pow (/ d l) 0.5))
        (t_1 (- 1.0 (/ (* (pow (* (* D M) (/ 0.5 d)) 2.0) (* h 0.5)) l)))
        (t_2 (sqrt (- d)))
        (t_3 (* t_1 (* (/ t_2 (sqrt (- h))) t_0)))
        (t_4 (/ t_2 (sqrt (- l)))))
   (if (<= l -1e+211)
     (*
      (* t_4 (pow (/ d h) 0.5))
      (- 1.0 (* (/ h l) (* 0.5 (pow (/ D (/ (/ d 0.5) M)) 2.0)))))
     (if (<= l -1e+20)
       t_3
       (if (<= l -1e-76)
         (fma
          (* D (/ (* M (* D M)) d))
          (* (sqrt (/ h (pow l 3.0))) 0.125)
          (* (- d) (sqrt (/ (/ 1.0 h) l))))
         (if (<= l -1e-102)
           t_3
           (if (<= l -1.35e-308)
             (*
              (sqrt (/ d h))
              (*
               t_4
               (fma -0.125 (* h (* M (/ D (/ l (/ (* D M) (* d d)))))) 1.0)))
             (if (<= l 1e-175)
               (* (* (/ 1.0 (sqrt (/ h d))) (/ (sqrt d) (sqrt l))) t_1)
               (if (<= l 1e+135)
                 (*
                  (* (* (sqrt d) (sqrt (/ 1.0 h))) t_0)
                  (+ 1.0 (* (/ h l) (* -0.5 (pow (/ (* D M) (* d 2.0)) 2.0)))))
                 (* d (/ (sqrt (/ 1.0 l)) (sqrt h))))))))))))

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 = pow((d / l), 0.5);
	double t_1 = 1.0 - ((pow(((D * M) * (0.5 / d)), 2.0) * (h * 0.5)) / l);
	double t_2 = sqrt(-d);
	double t_3 = t_1 * ((t_2 / sqrt(-h)) * t_0);
	double t_4 = t_2 / sqrt(-l);
	double tmp;
	if (l <= -1e+211) {
		tmp = (t_4 * pow((d / h), 0.5)) * (1.0 - ((h / l) * (0.5 * pow((D / ((d / 0.5) / M)), 2.0))));
	} else if (l <= -1e+20) {
		tmp = t_3;
	} else if (l <= -1e-76) {
		tmp = fma((D * ((M * (D * M)) / d)), (sqrt((h / pow(l, 3.0))) * 0.125), (-d * sqrt(((1.0 / h) / l))));
	} else if (l <= -1e-102) {
		tmp = t_3;
	} else if (l <= -1.35e-308) {
		tmp = sqrt((d / h)) * (t_4 * fma(-0.125, (h * (M * (D / (l / ((D * M) / (d * d)))))), 1.0));
	} else if (l <= 1e-175) {
		tmp = ((1.0 / sqrt((h / d))) * (sqrt(d) / sqrt(l))) * t_1;
	} else if (l <= 1e+135) {
		tmp = ((sqrt(d) * sqrt((1.0 / h))) * t_0) * (1.0 + ((h / l) * (-0.5 * pow(((D * M) / (d * 2.0)), 2.0))));
	} else {
		tmp = d * (sqrt((1.0 / l)) / sqrt(h));
	}
	return tmp;
}

Julia

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

↓

function code(d, h, l, M, D)
	t_0 = Float64(d / l) ^ 0.5
	t_1 = Float64(1.0 - Float64(Float64((Float64(Float64(D * M) * Float64(0.5 / d)) ^ 2.0) * Float64(h * 0.5)) / l))
	t_2 = sqrt(Float64(-d))
	t_3 = Float64(t_1 * Float64(Float64(t_2 / sqrt(Float64(-h))) * t_0))
	t_4 = Float64(t_2 / sqrt(Float64(-l)))
	tmp = 0.0
	if (l <= -1e+211)
		tmp = Float64(Float64(t_4 * (Float64(d / h) ^ 0.5)) * Float64(1.0 - Float64(Float64(h / l) * Float64(0.5 * (Float64(D / Float64(Float64(d / 0.5) / M)) ^ 2.0)))));
	elseif (l <= -1e+20)
		tmp = t_3;
	elseif (l <= -1e-76)
		tmp = fma(Float64(D * Float64(Float64(M * Float64(D * M)) / d)), Float64(sqrt(Float64(h / (l ^ 3.0))) * 0.125), Float64(Float64(-d) * sqrt(Float64(Float64(1.0 / h) / l))));
	elseif (l <= -1e-102)
		tmp = t_3;
	elseif (l <= -1.35e-308)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(t_4 * fma(-0.125, Float64(h * Float64(M * Float64(D / Float64(l / Float64(Float64(D * M) / Float64(d * d)))))), 1.0)));
	elseif (l <= 1e-175)
		tmp = Float64(Float64(Float64(1.0 / sqrt(Float64(h / d))) * Float64(sqrt(d) / sqrt(l))) * t_1);
	elseif (l <= 1e+135)
		tmp = Float64(Float64(Float64(sqrt(d) * sqrt(Float64(1.0 / h))) * t_0) * Float64(1.0 + Float64(Float64(h / l) * Float64(-0.5 * (Float64(Float64(D * M) / Float64(d * 2.0)) ^ 2.0)))));
	else
		tmp = Float64(d * Float64(sqrt(Float64(1.0 / l)) / sqrt(h)));
	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[Power[N[(d / l), $MachinePrecision], 0.5], $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - N[(N[(N[Power[N[(N[(D * M), $MachinePrecision] * N[(0.5 / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h * 0.5), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[(-d)], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 * N[(N[(t$95$2 / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$2 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -1e+211], N[(N[(t$95$4 * N[Power[N[(d / h), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(0.5 * N[Power[N[(D / N[(N[(d / 0.5), $MachinePrecision] / M), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -1e+20], t$95$3, If[LessEqual[l, -1e-76], N[(N[(D * N[(N[(M * N[(D * M), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(h / N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.125), $MachinePrecision] + N[((-d) * N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -1e-102], t$95$3, If[LessEqual[l, -1.35e-308], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(t$95$4 * N[(-0.125 * N[(h * N[(M * N[(D / N[(l / N[(N[(D * M), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1e-175], N[(N[(N[(1.0 / N[Sqrt[N[(h / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[l, 1e+135], N[(N[(N[(N[Sqrt[d], $MachinePrecision] * N[Sqrt[N[(1.0 / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$0), $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], N[(d * N[(N[Sqrt[N[(1.0 / l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]]

Derivation

1. Split input into 7 regimes

2. if l < -9.9999999999999996e210

   1. Initial program 32.8

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

   2. Applied egg-rr 25.3

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

   3. Taylor expanded in M around 0 25.3

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

   4. Simplified 25.4

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\frac{D}{\frac{\frac{d}{0.5}}{M}}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      Proof
      (/.f64 D (/.f64 (/.f64 d 1/2) M)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 D M) (/.f64 d 1/2))): 44 points increase in error, 58 points decrease in error
      (Rewrite=> associate-/r/_binary64 (*.f64 (/.f64 (*.f64 D M) d) 1/2)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= *-commutative_binary64 (*.f64 1/2 (/.f64 (*.f64 D M) d))): 0 points increase in error, 0 points decrease in error

   if -9.9999999999999996e210 < l < -1e20 or -9.99999999999999927e-77 < l < -9.99999999999999933e-103

   1. Initial program 24.7

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

   2. Applied egg-rr 25.4

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

   3. Applied egg-rr 18.7

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

   if -1e20 < l < -9.99999999999999927e-77

   1. Initial program 17.4

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

   2. Applied egg-rr 15.6

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

   3. Applied egg-rr 15.3

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

   4. Taylor expanded in d around -inf 29.0

      \[\leadsto \color{blue}{0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) + -1 \cdot \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \]

   5. Simplified 17.6

      \[\leadsto \color{blue}{\mathsf{fma}\left(D \cdot \frac{M \cdot \left(D \cdot M\right)}{d}, \sqrt{\frac{h}{{\ell}^{3}}} \cdot 0.125, \sqrt{\frac{\frac{1}{h}}{\ell}} \cdot \left(-d\right)\right)} \]
      Proof
      (fma.f64 (*.f64 D (/.f64 (*.f64 M (*.f64 D M)) d)) (*.f64 (sqrt.f64 (/.f64 h (pow.f64 l 3))) 1/8) (*.f64 (sqrt.f64 (/.f64 (/.f64 1 h) l)) (neg.f64 d))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (*.f64 D (/.f64 (Rewrite=> *-commutative_binary64 (*.f64 (*.f64 D M) M)) d)) (*.f64 (sqrt.f64 (/.f64 h (pow.f64 l 3))) 1/8) (*.f64 (sqrt.f64 (/.f64 (/.f64 1 h) l)) (neg.f64 d))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (*.f64 D (/.f64 (Rewrite<= associate-*r*_binary64 (*.f64 D (*.f64 M M))) d)) (*.f64 (sqrt.f64 (/.f64 h (pow.f64 l 3))) 1/8) (*.f64 (sqrt.f64 (/.f64 (/.f64 1 h) l)) (neg.f64 d))): 8 points increase in error, 4 points decrease in error
      (fma.f64 (*.f64 D (/.f64 (*.f64 D (Rewrite<= unpow2_binary64 (pow.f64 M 2))) d)) (*.f64 (sqrt.f64 (/.f64 h (pow.f64 l 3))) 1/8) (*.f64 (sqrt.f64 (/.f64 (/.f64 1 h) l)) (neg.f64 d))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (*.f64 D (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 D d) (pow.f64 M 2)))) (*.f64 (sqrt.f64 (/.f64 h (pow.f64 l 3))) 1/8) (*.f64 (sqrt.f64 (/.f64 (/.f64 1 h) l)) (neg.f64 d))): 14 points increase in error, 2 points decrease in error
      (fma.f64 (*.f64 D (Rewrite<= associate-/r/_binary64 (/.f64 D (/.f64 d (pow.f64 M 2))))) (*.f64 (sqrt.f64 (/.f64 h (pow.f64 l 3))) 1/8) (*.f64 (sqrt.f64 (/.f64 (/.f64 1 h) l)) (neg.f64 d))): 6 points increase in error, 14 points decrease in error
      (fma.f64 (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 D D) (/.f64 d (pow.f64 M 2)))) (*.f64 (sqrt.f64 (/.f64 h (pow.f64 l 3))) 1/8) (*.f64 (sqrt.f64 (/.f64 (/.f64 1 h) l)) (neg.f64 d))): 30 points increase in error, 2 points decrease in error
      (fma.f64 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 D 2)) (/.f64 d (pow.f64 M 2))) (*.f64 (sqrt.f64 (/.f64 h (pow.f64 l 3))) 1/8) (*.f64 (sqrt.f64 (/.f64 (/.f64 1 h) l)) (neg.f64 d))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (pow.f64 D 2) (pow.f64 M 2)) d)) (*.f64 (sqrt.f64 (/.f64 h (pow.f64 l 3))) 1/8) (*.f64 (sqrt.f64 (/.f64 (/.f64 1 h) l)) (neg.f64 d))): 3 points increase in error, 7 points decrease in error
      (fma.f64 (/.f64 (*.f64 (pow.f64 D 2) (pow.f64 M 2)) d) (*.f64 (sqrt.f64 (/.f64 h (pow.f64 l 3))) 1/8) (*.f64 (sqrt.f64 (Rewrite=> associate-/l/_binary64 (/.f64 1 (*.f64 l h)))) (neg.f64 d))): 9 points increase in error, 9 points decrease in error
      (fma.f64 (/.f64 (*.f64 (pow.f64 D 2) (pow.f64 M 2)) d) (*.f64 (sqrt.f64 (/.f64 h (pow.f64 l 3))) 1/8) (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (sqrt.f64 (/.f64 1 (*.f64 l h))) d)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (/.f64 (*.f64 (pow.f64 D 2) (pow.f64 M 2)) d) (*.f64 (sqrt.f64 (/.f64 h (pow.f64 l 3))) 1/8) (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 d (sqrt.f64 (/.f64 1 (*.f64 l h))))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (/.f64 (*.f64 (pow.f64 D 2) (pow.f64 M 2)) d) (*.f64 (sqrt.f64 (/.f64 h (pow.f64 l 3))) 1/8) (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (*.f64 d (sqrt.f64 (/.f64 1 (*.f64 l h))))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (/.f64 (*.f64 (pow.f64 D 2) (pow.f64 M 2)) d) (*.f64 (sqrt.f64 (/.f64 h (pow.f64 l 3))) 1/8)) (*.f64 -1 (*.f64 d (sqrt.f64 (/.f64 1 (*.f64 l h))))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (/.f64 (*.f64 (pow.f64 D 2) (pow.f64 M 2)) d) (sqrt.f64 (/.f64 h (pow.f64 l 3)))) 1/8)) (*.f64 -1 (*.f64 d (sqrt.f64 (/.f64 1 (*.f64 l h)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 1/8 (*.f64 (/.f64 (*.f64 (pow.f64 D 2) (pow.f64 M 2)) d) (sqrt.f64 (/.f64 h (pow.f64 l 3)))))) (*.f64 -1 (*.f64 d (sqrt.f64 (/.f64 1 (*.f64 l h)))))): 0 points increase in error, 0 points decrease in error

   if -9.99999999999999933e-103 < l < -1.3499999999999998e-308

   1. Initial program 33.4

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

   2. Simplified 34.0

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(\frac{\frac{D}{d}}{\frac{2}{M}}\right)}^{2}, \frac{h}{\ell} \cdot -0.5, 1\right)\right)} \]
      Proof
      (*.f64 (sqrt.f64 (/.f64 d h)) (*.f64 (sqrt.f64 (/.f64 d l)) (fma.f64 (pow.f64 (/.f64 (/.f64 D d) (/.f64 2 M)) 2) (*.f64 (/.f64 h l) -1/2) 1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= unpow1/2_binary64 (pow.f64 (/.f64 d h) 1/2)) (*.f64 (sqrt.f64 (/.f64 d l)) (fma.f64 (pow.f64 (/.f64 (/.f64 D d) (/.f64 2 M)) 2) (*.f64 (/.f64 h l) -1/2) 1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (pow.f64 (/.f64 d h) (Rewrite<= metadata-eval (/.f64 1 2))) (*.f64 (sqrt.f64 (/.f64 d l)) (fma.f64 (pow.f64 (/.f64 (/.f64 D d) (/.f64 2 M)) 2) (*.f64 (/.f64 h l) -1/2) 1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (*.f64 (Rewrite<= unpow1/2_binary64 (pow.f64 (/.f64 d l) 1/2)) (fma.f64 (pow.f64 (/.f64 (/.f64 D d) (/.f64 2 M)) 2) (*.f64 (/.f64 h l) -1/2) 1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (*.f64 (pow.f64 (/.f64 d l) (Rewrite<= metadata-eval (/.f64 1 2))) (fma.f64 (pow.f64 (/.f64 (/.f64 D d) (/.f64 2 M)) 2) (*.f64 (/.f64 h l) -1/2) 1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (*.f64 (pow.f64 (/.f64 d l) (/.f64 1 2)) (fma.f64 (pow.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (/.f64 D d) M) 2)) 2) (*.f64 (/.f64 h l) -1/2) 1))): 0 points increase in error, 5 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))): 3 points increase in error, 9 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))))): 2 points increase in error, 2 points decrease in error

   3. Taylor expanded in D around 0 47.5

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

   4. Simplified 38.2

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\mathsf{fma}\left(-0.125, h \cdot \left(\frac{D}{\frac{\ell}{M} \cdot \frac{d \cdot d}{D}} \cdot M\right), 1\right)}\right) \]
      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, 4 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, 6 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): 8 points increase in error, 11 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): 4 points increase in error, 4 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): 6 points increase in error, 3 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): 13 points increase in error, 1 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): 20 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): 3 points increase in error, 3 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): 6 points increase in error, 5 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): 2 points increase in error, 2 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

   5. Applied egg-rr 36.8

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(-0.125, h \cdot \left(\frac{D}{\color{blue}{\frac{\ell}{\frac{M \cdot D}{d \cdot d}}}} \cdot M\right), 1\right)\right) \]

   6. Applied egg-rr 21.9

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{\frac{\sqrt{-d}}{\sqrt{-\ell}}} \cdot \mathsf{fma}\left(-0.125, h \cdot \left(\frac{D}{\frac{\ell}{\frac{M \cdot D}{d \cdot d}}} \cdot M\right), 1\right)\right) \]

   if -1.3499999999999998e-308 < l < 1e-175

   1. Initial program 35.3

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

   2. Applied egg-rr 29.1

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

   3. Applied egg-rr 29.4

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

   4. Applied egg-rr 13.5

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

   if 1e-175 < l < 9.99999999999999962e134

   1. Initial program 21.4

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

   2. Applied egg-rr 15.8

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

   if 9.99999999999999962e134 < l

   1. Initial program 30.2

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

   2. Taylor expanded in d around inf 28.2

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

   3. Applied egg-rr 17.0

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}} \cdot d \]
3. Recombined 7 regimes into one program.

4. Final simplification 18.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1 \cdot 10^{+211}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot {\left(\frac{d}{h}\right)}^{0.5}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{D}{\frac{\frac{d}{0.5}}{M}}\right)}^{2}\right)\right)\\ \mathbf{elif}\;\ell \leq -1 \cdot 10^{+20}:\\ \;\;\;\;\left(1 - \frac{{\left(\left(D \cdot M\right) \cdot \frac{0.5}{d}\right)}^{2} \cdot \left(h \cdot 0.5\right)}{\ell}\right) \cdot \left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right)\\ \mathbf{elif}\;\ell \leq -1 \cdot 10^{-76}:\\ \;\;\;\;\mathsf{fma}\left(D \cdot \frac{M \cdot \left(D \cdot M\right)}{d}, \sqrt{\frac{h}{{\ell}^{3}}} \cdot 0.125, \left(-d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\right)\\ \mathbf{elif}\;\ell \leq -1 \cdot 10^{-102}:\\ \;\;\;\;\left(1 - \frac{{\left(\left(D \cdot M\right) \cdot \frac{0.5}{d}\right)}^{2} \cdot \left(h \cdot 0.5\right)}{\ell}\right) \cdot \left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right)\\ \mathbf{elif}\;\ell \leq -1.35 \cdot 10^{-308}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \mathsf{fma}\left(-0.125, h \cdot \left(M \cdot \frac{D}{\frac{\ell}{\frac{D \cdot M}{d \cdot d}}}\right), 1\right)\right)\\ \mathbf{elif}\;\ell \leq 10^{-175}:\\ \;\;\;\;\left(\frac{1}{\sqrt{\frac{h}{d}}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \left(1 - \frac{{\left(\left(D \cdot M\right) \cdot \frac{0.5}{d}\right)}^{2} \cdot \left(h \cdot 0.5\right)}{\ell}\right)\\ \mathbf{elif}\;\ell \leq 10^{+135}:\\ \;\;\;\;\left(\left(\sqrt{d} \cdot \sqrt{\frac{1}{h}}\right) \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}\\ \end{array} \]

Alternatives

Alternative 1
Error	20.5
Cost	83532

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := {\left(\frac{d}{h}\right)}^{0.5}\\ t_2 := \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{0.5} \cdot t_1\right)\\ \mathbf{if}\;t_2 \leq -2 \cdot 10^{-203}:\\ \;\;\;\;\left(1 - \frac{{\left(\left(D \cdot M\right) \cdot \frac{0.5}{d}\right)}^{2} \cdot \left(h \cdot 0.5\right)}{\ell}\right) \cdot \left(t_0 \cdot t_1\right)\\ \mathbf{elif}\;t_2 \leq 4 \cdot 10^{-307}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{\ell \cdot h}}\right)\\ \mathbf{elif}\;t_2 \leq 5 \cdot 10^{+278}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(t_0 \cdot \left(1 + \left(\frac{h}{\ell} \cdot -0.5\right) \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \end{array} \]

Alternative 2
Error	18.1
Cost	33932

\[\begin{array}{l} t_0 := \sqrt{-d}\\ t_1 := \frac{t_0}{\sqrt{-h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(\frac{\frac{D}{d}}{\frac{2}{M}}\right)}^{2}, \frac{h}{\ell} \cdot -0.5, 1\right)\right)\\ \mathbf{if}\;\ell \leq -1 \cdot 10^{+20}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq -1 \cdot 10^{-76}:\\ \;\;\;\;\mathsf{fma}\left(D \cdot \frac{M \cdot \left(D \cdot M\right)}{d}, \sqrt{\frac{h}{{\ell}^{3}}} \cdot 0.125, \left(-d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\right)\\ \mathbf{elif}\;\ell \leq -1 \cdot 10^{-102}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq -1.35 \cdot 10^{-308}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\frac{t_0}{\sqrt{-\ell}} \cdot \mathsf{fma}\left(-0.125, h \cdot \left(M \cdot \frac{D}{\frac{\ell}{\frac{D \cdot M}{d \cdot d}}}\right), 1\right)\right)\\ \mathbf{elif}\;\ell \leq 10^{-175}:\\ \;\;\;\;\left(\frac{1}{\sqrt{\frac{h}{d}}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \left(1 - \frac{{\left(\left(D \cdot M\right) \cdot \frac{0.5}{d}\right)}^{2} \cdot \left(h \cdot 0.5\right)}{\ell}\right)\\ \mathbf{elif}\;\ell \leq 10^{+135}:\\ \;\;\;\;\left(\left(\sqrt{d} \cdot \sqrt{\frac{1}{h}}\right) \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}\\ \end{array} \]

Alternative 3
Error	22.6
Cost	28124

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ t_1 := \sqrt{\frac{\ell}{d}}\\ \mathbf{if}\;h \leq -1 \cdot 10^{+135}:\\ \;\;\;\;\frac{t_0 \cdot \left(1 + \frac{{\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2}}{\ell} \cdot \left(h \cdot -0.5\right)\right)}{t_1}\\ \mathbf{elif}\;h \leq -1 \cdot 10^{+40}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot {\left(\frac{d}{h}\right)}^{0.5}\right) \cdot \left(1 - 0.125 \cdot \left(M \cdot \left(M \cdot \frac{\frac{D}{\frac{d}{D}}}{\frac{\ell}{\frac{h}{d}}}\right)\right)\right)\\ \mathbf{elif}\;h \leq -1 \cdot 10^{-47}:\\ \;\;\;\;\left(1 - \frac{{\left(\left(D \cdot M\right) \cdot \frac{0.5}{d}\right)}^{2} \cdot \left(h \cdot 0.5\right)}{\ell}\right) \cdot \left(t_0 \cdot \frac{1}{t_1}\right)\\ \mathbf{elif}\;h \leq 0:\\ \;\;\;\;\mathsf{fma}\left(D \cdot \frac{M \cdot \left(D \cdot M\right)}{d}, \sqrt{\frac{h}{{\ell}^{3}}} \cdot 0.125, \left(-d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\right)\\ \mathbf{elif}\;h \leq 10^{-150}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}\\ \mathbf{elif}\;h \leq 10^{+85}:\\ \;\;\;\;\frac{\sqrt{d} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right)\right)}{\sqrt{h}}\\ \mathbf{elif}\;h \leq 10^{+250}:\\ \;\;\;\;t_0 \cdot \left(\mathsf{fma}\left(-0.125, h \cdot \left(M \cdot \frac{D}{\frac{\ell}{\frac{D \cdot M}{d \cdot d}}}\right), 1\right) \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{h}}{{\ell}^{1.5}} \cdot \left(M \cdot \left(-0.125 \cdot \left(D \cdot \frac{D}{\frac{d}{M}}\right)\right)\right)\\ \end{array} \]

Alternative 4
Error	18.6
Cost	28060

\[\begin{array}{l} t_0 := 1 - \frac{{\left(\left(D \cdot M\right) \cdot \frac{0.5}{d}\right)}^{2} \cdot \left(h \cdot 0.5\right)}{\ell}\\ t_1 := \sqrt{-d}\\ t_2 := t_0 \cdot \left(\frac{t_1}{\sqrt{-h}} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right)\\ t_3 := \frac{t_1}{\sqrt{-\ell}}\\ \mathbf{if}\;\ell \leq -1 \cdot 10^{+211}:\\ \;\;\;\;\left(t_3 \cdot {\left(\frac{d}{h}\right)}^{0.5}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{D}{\frac{\frac{d}{0.5}}{M}}\right)}^{2}\right)\right)\\ \mathbf{elif}\;\ell \leq -1 \cdot 10^{+20}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq -1 \cdot 10^{-76}:\\ \;\;\;\;\mathsf{fma}\left(D \cdot \frac{M \cdot \left(D \cdot M\right)}{d}, \sqrt{\frac{h}{{\ell}^{3}}} \cdot 0.125, \left(-d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\right)\\ \mathbf{elif}\;\ell \leq -1 \cdot 10^{-102}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq -1.35 \cdot 10^{-308}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(t_3 \cdot \mathsf{fma}\left(-0.125, h \cdot \left(M \cdot \frac{D}{\frac{\ell}{\frac{D \cdot M}{d \cdot d}}}\right), 1\right)\right)\\ \mathbf{elif}\;\ell \leq 10^{-175}:\\ \;\;\;\;\left(\frac{1}{\sqrt{\frac{h}{d}}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot t_0\\ \mathbf{elif}\;\ell \leq 10^{+135}:\\ \;\;\;\;\frac{\sqrt{d} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right)\right)}{\sqrt{h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}\\ \end{array} \]

Alternative 5
Error	18.5
Cost	28052

\[\begin{array}{l} t_0 := \sqrt{-d}\\ t_1 := \frac{t_0}{\sqrt{-\ell}}\\ t_2 := 1 - \frac{{\left(\left(D \cdot M\right) \cdot \frac{0.5}{d}\right)}^{2} \cdot \left(h \cdot 0.5\right)}{\ell}\\ t_3 := t_2 \cdot \left(\frac{t_0}{\sqrt{-h}} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right)\\ t_4 := {\left(\frac{d}{h}\right)}^{0.5}\\ \mathbf{if}\;h \leq -1 \cdot 10^{+135}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;h \leq -1 \cdot 10^{+49}:\\ \;\;\;\;\left(t_1 \cdot t_4\right) \cdot \left(1 - 0.125 \cdot \left(M \cdot \left(M \cdot \frac{\frac{D}{\frac{d}{D}}}{\frac{\ell}{\frac{h}{d}}}\right)\right)\right)\\ \mathbf{elif}\;h \leq -0.013:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(t_1 \cdot \mathsf{fma}\left(-0.125, h \cdot \left(M \cdot \frac{D}{\frac{\ell}{\frac{D \cdot M}{d \cdot d}}}\right), 1\right)\right)\\ \mathbf{elif}\;h \leq -1 \cdot 10^{-151}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{\ell \cdot h}}\right)\\ \mathbf{elif}\;h \leq 0:\\ \;\;\;\;t_3\\ \mathbf{elif}\;h \leq 10^{-140}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot \left(\frac{\sqrt{d}}{\sqrt{\ell}} \cdot t_4\right)\\ \end{array} \]

Alternative 6
Error	17.7
Cost	27988

\[\begin{array}{l} t_0 := {\left(\frac{d}{\ell}\right)}^{0.5}\\ t_1 := 1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}\right)\\ t_2 := \sqrt{-d}\\ t_3 := \frac{t_2}{\sqrt{-\ell}}\\ \mathbf{if}\;\ell \leq -1 \cdot 10^{+211}:\\ \;\;\;\;\left(t_3 \cdot {\left(\frac{d}{h}\right)}^{0.5}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{D}{\frac{\frac{d}{0.5}}{M}}\right)}^{2}\right)\right)\\ \mathbf{elif}\;\ell \leq -1 \cdot 10^{-102}:\\ \;\;\;\;t_1 \cdot \left(\frac{t_2}{\sqrt{-h}} \cdot t_0\right)\\ \mathbf{elif}\;\ell \leq -1.35 \cdot 10^{-308}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(t_3 \cdot \mathsf{fma}\left(-0.125, h \cdot \left(M \cdot \frac{D}{\frac{\ell}{\frac{D \cdot M}{d \cdot d}}}\right), 1\right)\right)\\ \mathbf{elif}\;\ell \leq 10^{-175}:\\ \;\;\;\;\left(\frac{1}{\sqrt{\frac{h}{d}}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \left(1 - \frac{{\left(\left(D \cdot M\right) \cdot \frac{0.5}{d}\right)}^{2} \cdot \left(h \cdot 0.5\right)}{\ell}\right)\\ \mathbf{elif}\;\ell \leq 10^{+135}:\\ \;\;\;\;\left(\left(\sqrt{d} \cdot \sqrt{\frac{1}{h}}\right) \cdot t_0\right) \cdot t_1\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}\\ \end{array} \]

Alternative 7
Error	18.9
Cost	27928

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ t_1 := \frac{\sqrt{-d}}{\sqrt{-\ell}}\\ \mathbf{if}\;\ell \leq -1 \cdot 10^{+101}:\\ \;\;\;\;\left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}\right)\right) \cdot \left(t_0 \cdot t_1\right)\\ \mathbf{elif}\;\ell \leq -1 \cdot 10^{-85}:\\ \;\;\;\;\mathsf{fma}\left(D \cdot \frac{M \cdot \left(D \cdot M\right)}{d}, \sqrt{\frac{h}{{\ell}^{3}}} \cdot 0.125, \left(-d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\right)\\ \mathbf{elif}\;\ell \leq -1 \cdot 10^{-120}:\\ \;\;\;\;\frac{t_0 \cdot \left(1 + \frac{{\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2}}{\ell} \cdot \left(h \cdot -0.5\right)\right)}{\sqrt{\frac{\ell}{d}}}\\ \mathbf{elif}\;\ell \leq -1.35 \cdot 10^{-308}:\\ \;\;\;\;t_0 \cdot \left(t_1 \cdot \mathsf{fma}\left(-0.125, h \cdot \left(M \cdot \frac{D}{\frac{\ell}{\frac{D \cdot M}{d \cdot d}}}\right), 1\right)\right)\\ \mathbf{elif}\;\ell \leq 10^{-175}:\\ \;\;\;\;\left(\frac{1}{\sqrt{\frac{h}{d}}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \left(1 - \frac{{\left(\left(D \cdot M\right) \cdot \frac{0.5}{d}\right)}^{2} \cdot \left(h \cdot 0.5\right)}{\ell}\right)\\ \mathbf{elif}\;\ell \leq 10^{+135}:\\ \;\;\;\;\frac{\sqrt{d} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right)\right)}{\sqrt{h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}\\ \end{array} \]

Alternative 8
Error	18.8
Cost	27928

\[\begin{array}{l} t_0 := 1 - \frac{{\left(\left(D \cdot M\right) \cdot \frac{0.5}{d}\right)}^{2} \cdot \left(h \cdot 0.5\right)}{\ell}\\ t_1 := \sqrt{\frac{d}{h}}\\ t_2 := \sqrt{-d}\\ t_3 := \frac{t_2}{\sqrt{-\ell}}\\ \mathbf{if}\;\ell \leq -1 \cdot 10^{+101}:\\ \;\;\;\;\left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}\right)\right) \cdot \left(t_1 \cdot t_3\right)\\ \mathbf{elif}\;\ell \leq -1 \cdot 10^{-76}:\\ \;\;\;\;\mathsf{fma}\left(D \cdot \frac{M \cdot \left(D \cdot M\right)}{d}, \sqrt{\frac{h}{{\ell}^{3}}} \cdot 0.125, \left(-d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\right)\\ \mathbf{elif}\;\ell \leq -1 \cdot 10^{-102}:\\ \;\;\;\;t_0 \cdot \left(\frac{t_2}{\sqrt{-h}} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right)\\ \mathbf{elif}\;\ell \leq -1.35 \cdot 10^{-308}:\\ \;\;\;\;t_1 \cdot \left(t_3 \cdot \mathsf{fma}\left(-0.125, h \cdot \left(M \cdot \frac{D}{\frac{\ell}{\frac{D \cdot M}{d \cdot d}}}\right), 1\right)\right)\\ \mathbf{elif}\;\ell \leq 10^{-175}:\\ \;\;\;\;\left(\frac{1}{\sqrt{\frac{h}{d}}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot t_0\\ \mathbf{elif}\;\ell \leq 10^{+135}:\\ \;\;\;\;\frac{\sqrt{d} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right)\right)}{\sqrt{h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}\\ \end{array} \]

Alternative 9
Error	20.4
Cost	27924

\[\begin{array}{l} t_0 := 1 - \frac{{\left(\left(D \cdot M\right) \cdot \frac{0.5}{d}\right)}^{2} \cdot \left(h \cdot 0.5\right)}{\ell}\\ t_1 := \sqrt{\frac{d}{h}}\\ t_2 := \sqrt{\frac{\ell}{d}}\\ \mathbf{if}\;h \leq -1 \cdot 10^{+135}:\\ \;\;\;\;\frac{t_1 \cdot \left(1 + \frac{{\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2}}{\ell} \cdot \left(h \cdot -0.5\right)\right)}{t_2}\\ \mathbf{elif}\;h \leq -1 \cdot 10^{+40}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot {\left(\frac{d}{h}\right)}^{0.5}\right) \cdot \left(1 - 0.125 \cdot \left(M \cdot \left(M \cdot \frac{\frac{D}{\frac{d}{D}}}{\frac{\ell}{\frac{h}{d}}}\right)\right)\right)\\ \mathbf{elif}\;h \leq -1 \cdot 10^{-47}:\\ \;\;\;\;t_0 \cdot \left(t_1 \cdot \frac{1}{t_2}\right)\\ \mathbf{elif}\;h \leq 0:\\ \;\;\;\;\mathsf{fma}\left(D \cdot \frac{M \cdot \left(D \cdot M\right)}{d}, \sqrt{\frac{h}{{\ell}^{3}}} \cdot 0.125, \left(-d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\right)\\ \mathbf{elif}\;h \leq 10^{-140}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sqrt{\frac{h}{d}}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot t_0\\ \end{array} \]

Alternative 10
Error	22.3
Cost	27796

\[\begin{array}{l} t_0 := \left(-d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{if}\;\ell \leq -2.7 \cdot 10^{+190}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot {\left(\frac{d}{h}\right)}^{0.5}\right) \cdot \left(1 - 0.125 \cdot \left(M \cdot \left(M \cdot \frac{\frac{D}{\frac{d}{D}}}{\frac{\ell}{\frac{h}{d}}}\right)\right)\right)\\ \mathbf{elif}\;\ell \leq -6.8 \cdot 10^{-62}:\\ \;\;\;\;\mathsf{fma}\left(D \cdot \frac{M \cdot \left(D \cdot M\right)}{d}, \sqrt{\frac{h}{{\ell}^{3}}} \cdot 0.125, t_0\right)\\ \mathbf{elif}\;\ell \leq -1.35 \cdot 10^{-308}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq 3.7 \cdot 10^{-222}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \mathbf{elif}\;\ell \leq 10^{+135}:\\ \;\;\;\;\frac{\sqrt{d} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right)\right)}{\sqrt{h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}\\ \end{array} \]

Alternative 11
Error	20.2
Cost	27796

\[\begin{array}{l} t_0 := \frac{\sqrt{-d}}{\sqrt{-\ell}}\\ \mathbf{if}\;\ell \leq -2.7 \cdot 10^{+190}:\\ \;\;\;\;\left(t_0 \cdot {\left(\frac{d}{h}\right)}^{0.5}\right) \cdot \left(1 - 0.125 \cdot \left(M \cdot \left(M \cdot \frac{\frac{D}{\frac{d}{D}}}{\frac{\ell}{\frac{h}{d}}}\right)\right)\right)\\ \mathbf{elif}\;\ell \leq -1 \cdot 10^{-85}:\\ \;\;\;\;\mathsf{fma}\left(D \cdot \frac{M \cdot \left(D \cdot M\right)}{d}, \sqrt{\frac{h}{{\ell}^{3}}} \cdot 0.125, \left(-d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\right)\\ \mathbf{elif}\;\ell \leq -1.35 \cdot 10^{-308}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(t_0 \cdot \mathsf{fma}\left(-0.125, h \cdot \left(M \cdot \frac{D}{\frac{\ell}{\frac{D \cdot M}{d \cdot d}}}\right), 1\right)\right)\\ \mathbf{elif}\;\ell \leq 10^{-175}:\\ \;\;\;\;\left(\frac{1}{\sqrt{\frac{h}{d}}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \left(1 - \frac{{\left(\left(D \cdot M\right) \cdot \frac{0.5}{d}\right)}^{2} \cdot \left(h \cdot 0.5\right)}{\ell}\right)\\ \mathbf{elif}\;\ell \leq 10^{+135}:\\ \;\;\;\;\frac{\sqrt{d} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right)\right)}{\sqrt{h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}\\ \end{array} \]

Alternative 12
Error	22.8
Cost	27336

\[\begin{array}{l} t_0 := {\left(\frac{d}{h}\right)}^{0.5}\\ t_1 := \left(-d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{if}\;\ell \leq -2.7 \cdot 10^{+190}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot t_0\right) \cdot \left(1 - 0.125 \cdot \left(M \cdot \left(M \cdot \frac{\frac{D}{\frac{d}{D}}}{\frac{\ell}{\frac{h}{d}}}\right)\right)\right)\\ \mathbf{elif}\;\ell \leq -6.8 \cdot 10^{-62}:\\ \;\;\;\;\mathsf{fma}\left(D \cdot \frac{M \cdot \left(D \cdot M\right)}{d}, \sqrt{\frac{h}{{\ell}^{3}}} \cdot 0.125, t_1\right)\\ \mathbf{elif}\;\ell \leq -1.35 \cdot 10^{-308}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 1.7 \cdot 10^{-229}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \mathbf{elif}\;\ell \leq 8 \cdot 10^{+85}:\\ \;\;\;\;\left(1 - \frac{{\left(\left(D \cdot M\right) \cdot \frac{0.5}{d}\right)}^{2} \cdot \left(h \cdot 0.5\right)}{\ell}\right) \cdot \left(\sqrt{\frac{d}{\ell}} \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}\\ \end{array} \]

Alternative 13
Error	22.2
Cost	21328

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(-0.125, \frac{h \cdot \left(D \cdot M\right)}{\left(d \cdot d\right) \cdot \frac{\ell}{D \cdot M}}, 1\right)\right)\\ \mathbf{if}\;d \leq -1.02 \cdot 10^{+90}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{elif}\;d \leq -1.15 \cdot 10^{-55}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 3 \cdot 10^{-146}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{\ell \cdot h}}\right)\\ \mathbf{elif}\;d \leq 10^{+86}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}\\ \end{array} \]

Alternative 14
Error	21.8
Cost	21260

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ t_1 := \sqrt{\frac{\ell}{d}}\\ \mathbf{if}\;h \leq -1.5 \cdot 10^{+93}:\\ \;\;\;\;\frac{t_0 \cdot \left(1 + \frac{{\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2}}{\ell} \cdot \left(h \cdot -0.5\right)\right)}{t_1}\\ \mathbf{elif}\;h \leq 0:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{\ell \cdot h}}\right)\\ \mathbf{elif}\;h \leq 10^{-140}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \frac{{\left(\left(D \cdot M\right) \cdot \frac{0.5}{d}\right)}^{2} \cdot \left(h \cdot 0.5\right)}{\ell}\right) \cdot \left(t_0 \cdot \frac{1}{t_1}\right)\\ \end{array} \]

Alternative 15
Error	21.6
Cost	21132

\[\begin{array}{l} t_0 := \frac{\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{{\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2}}{\ell} \cdot \left(h \cdot -0.5\right)\right)}{\sqrt{\frac{\ell}{d}}}\\ \mathbf{if}\;h \leq -1.5 \cdot 10^{+93}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;h \leq 0:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{\ell \cdot h}}\right)\\ \mathbf{elif}\;h \leq 10^{-140}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 16
Error	22.0
Cost	21132

\[\begin{array}{l} t_0 := 1 + \frac{{\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2}}{\ell} \cdot \left(h \cdot -0.5\right)\\ \mathbf{if}\;h \leq -1.5 \cdot 10^{+93}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{h}} \cdot t_0}{\sqrt{\frac{\ell}{d}}}\\ \mathbf{elif}\;h \leq 0:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{\ell \cdot h}}\right)\\ \mathbf{elif}\;h \leq 10^{-155}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{\ell}} \cdot t_0}{\sqrt{\frac{h}{d}}}\\ \end{array} \]

Alternative 17
Error	23.3
Cost	15316

\[\begin{array}{l} t_0 := \left(1 - 0.125 \cdot \left(M \cdot \left(M \cdot \frac{\frac{D}{\frac{d}{D}}}{\frac{\ell}{\frac{h}{d}}}\right)\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{0.5} \cdot {\left(\frac{d}{h}\right)}^{0.5}\right)\\ \mathbf{if}\;d \leq -1.02 \cdot 10^{+90}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{elif}\;d \leq -1.15 \cdot 10^{-55}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 1.7 \cdot 10^{-270}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{\ell \cdot h}}\right)\\ \mathbf{elif}\;d \leq 0.00115:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \mathbf{elif}\;d \leq 10^{+86}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}\\ \end{array} \]

Alternative 18
Error	23.4
Cost	14984

\[\begin{array}{l} \mathbf{if}\;d \leq -1.02 \cdot 10^{+90}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{elif}\;d \leq -2.3 \cdot 10^{-40}:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{\left(M \cdot M\right) \cdot \left(\frac{h}{d} \cdot \frac{D \cdot 0.125}{\frac{d}{D}}\right)}{\ell}\right)\\ \mathbf{elif}\;d \leq 1.7 \cdot 10^{-270}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{\ell \cdot h}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}\\ \end{array} \]

Alternative 19
Error	22.9
Cost	13512

\[\begin{array}{l} \mathbf{if}\;h \leq -1.5 \cdot 10^{+93}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;h \leq 0:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{\ell \cdot h}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}\\ \end{array} \]

Alternative 20
Error	23.5
Cost	13380

\[\begin{array}{l} \mathbf{if}\;d \leq 1.7 \cdot 10^{-270}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}\\ \end{array} \]

Alternative 21
Error	23.5
Cost	13252

\[\begin{array}{l} \mathbf{if}\;d \leq 1.7 \cdot 10^{-270}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

Alternative 22
Error	27.2
Cost	7044

\[\begin{array}{l} \mathbf{if}\;\ell \leq 9 \cdot 10^{-308}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \end{array} \]

Alternative 23
Error	36.7
Cost	6980

\[\begin{array}{l} \mathbf{if}\;d \leq -1.1 \cdot 10^{-153}:\\ \;\;\;\;\sqrt{\frac{d \cdot d}{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \end{array} \]

Alternative 24
Error	43.7
Cost	6848

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

Alternative 25
Error	43.8
Cost	6784

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

Alternative 26
Error	43.8
Cost	6720

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

Reproduce

herbie shell --seed 2022302 
(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)))))