?

Average Error: 26.8 → 17.7
Time: 51.9s
Precision: binary64
Cost: 27592

?

\[ \begin{array}{c}[M, D] = \mathsf{sort}([M, D])\\ \end{array} \]
\[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
↓
\[\begin{array}{l} t_0 := {\left(\frac{0.5}{d} \cdot \left(M \cdot D\right)\right)}^{2}\\ t_1 := \sqrt{-d}\\ \mathbf{if}\;d \leq -4.2 \cdot 10^{+96}:\\ \;\;\;\;\left(\frac{t_1}{\sqrt{-h}} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 + \frac{\left(h \cdot t_0\right) \cdot -0.5}{\ell}\right)\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-309}:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot \frac{t_1}{\sqrt{-\ell}}\right) \cdot \left(1 - \frac{0.5 \cdot t_0}{\frac{\ell}{h}}\right)\\ \mathbf{elif}\;d \leq 6.8 \cdot 10^{+154}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.25 \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}, \frac{h}{\ell} \cdot -0.5, 1\right) \cdot \frac{d}{\sqrt{h}}}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]
(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 (* (/ 0.5 d) (* M D)) 2.0)) (t_1 (sqrt (- d))))
   (if (<= d -4.2e+96)
     (*
      (* (/ t_1 (sqrt (- h))) (pow (/ d l) 0.5))
      (+ 1.0 (/ (* (* h t_0) -0.5) l)))
     (if (<= d -5e-309)
       (*
        (* (pow (/ d h) 0.5) (/ t_1 (sqrt (- l))))
        (- 1.0 (/ (* 0.5 t_0) (/ l h))))
       (if (<= d 6.8e+154)
         (/
          (*
           (fma (* 0.25 (pow (* M (/ D d)) 2.0)) (* (/ h l) -0.5) 1.0)
           (/ d (sqrt h)))
          (sqrt l))
         (/ d (* (sqrt h) (sqrt l))))))))
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(((0.5 / d) * (M * D)), 2.0);
	double t_1 = sqrt(-d);
	double tmp;
	if (d <= -4.2e+96) {
		tmp = ((t_1 / sqrt(-h)) * pow((d / l), 0.5)) * (1.0 + (((h * t_0) * -0.5) / l));
	} else if (d <= -5e-309) {
		tmp = (pow((d / h), 0.5) * (t_1 / sqrt(-l))) * (1.0 - ((0.5 * t_0) / (l / h)));
	} else if (d <= 6.8e+154) {
		tmp = (fma((0.25 * pow((M * (D / d)), 2.0)), ((h / l) * -0.5), 1.0) * (d / sqrt(h))) / sqrt(l);
	} else {
		tmp = d / (sqrt(h) * sqrt(l));
	}
	return tmp;
}

function code(d, h, l, M, D)
	return Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
end
↓
function code(d, h, l, M, D)
	t_0 = Float64(Float64(0.5 / d) * Float64(M * D)) ^ 2.0
	t_1 = sqrt(Float64(-d))
	tmp = 0.0
	if (d <= -4.2e+96)
		tmp = Float64(Float64(Float64(t_1 / sqrt(Float64(-h))) * (Float64(d / l) ^ 0.5)) * Float64(1.0 + Float64(Float64(Float64(h * t_0) * -0.5) / l)));
	elseif (d <= -5e-309)
		tmp = Float64(Float64((Float64(d / h) ^ 0.5) * Float64(t_1 / sqrt(Float64(-l)))) * Float64(1.0 - Float64(Float64(0.5 * t_0) / Float64(l / h))));
	elseif (d <= 6.8e+154)
		tmp = Float64(Float64(fma(Float64(0.25 * (Float64(M * Float64(D / d)) ^ 2.0)), Float64(Float64(h / l) * -0.5), 1.0) * Float64(d / sqrt(h))) / sqrt(l));
	else
		tmp = Float64(d / Float64(sqrt(h) * sqrt(l)));
	end
	return tmp
end

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[(N[(0.5 / d), $MachinePrecision] * N[(M * D), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[(-d)], $MachinePrecision]}, If[LessEqual[d, -4.2e+96], N[(N[(N[(t$95$1 / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(N[(h * t$95$0), $MachinePrecision] * -0.5), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -5e-309], N[(N[(N[Power[N[(d / h), $MachinePrecision], 0.5], $MachinePrecision] * N[(t$95$1 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(0.5 * t$95$0), $MachinePrecision] / N[(l / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 6.8e+154], N[(N[(N[(N[(0.25 * N[Power[N[(M * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision] + 1.0), $MachinePrecision] * N[(d / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

\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{0.5}{d} \cdot \left(M \cdot D\right)\right)}^{2}\\
t_1 := \sqrt{-d}\\
\mathbf{if}\;d \leq -4.2 \cdot 10^{+96}:\\
\;\;\;\;\left(\frac{t_1}{\sqrt{-h}} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 + \frac{\left(h \cdot t_0\right) \cdot -0.5}{\ell}\right)\\

\mathbf{elif}\;d \leq -5 \cdot 10^{-309}:\\
\;\;\;\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot \frac{t_1}{\sqrt{-\ell}}\right) \cdot \left(1 - \frac{0.5 \cdot t_0}{\frac{\ell}{h}}\right)\\

\mathbf{elif}\;d \leq 6.8 \cdot 10^{+154}:\\
\;\;\;\;\frac{\mathsf{fma}\left(0.25 \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}, \frac{h}{\ell} \cdot -0.5, 1\right) \cdot \frac{d}{\sqrt{h}}}{\sqrt{\ell}}\\

\mathbf{else}:\\
\;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\


\end{array}

Error?

Derivation?

  1. Split input into 4 regimes

  2. if d < -4.2000000000000002e96

    1. Initial program 27.6

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

    2. Applied egg-rr26.8

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

    3. Applied egg-rr13.3

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

    if -4.2000000000000002e96 < d < -4.9999999999999995e-309

    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. Applied egg-rr26.6

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

    3. Applied egg-rr22.3

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

    if -4.9999999999999995e-309 < d < 6.79999999999999948e154

    1. Initial program 25.6

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

    2. Simplified26.5

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

      [Start]25.6

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

      metadata-eval [=>]25.6

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

      unpow1/2 [=>]25.6

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

      metadata-eval [=>]25.6

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

      unpow1/2 [=>]25.6

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

      associate-*l* [=>]25.6

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

      metadata-eval [=>]25.6

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

      times-frac [=>]26.5

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

    3. Applied egg-rr19.4

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

    4. Simplified19.5

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

      [Start]19.4

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

      *-lft-identity [<=]19.4

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

      *-commutative [<=]19.4

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

      distribute-rgt-in [<=]19.4

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

      associate-/r* [=>]19.5

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

      +-commutative [=>]19.5

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

      *-commutative [=>]19.5

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

      fma-def [=>]19.5

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

      *-commutative [=>]19.5

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

      associate-*r* [=>]19.5

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

      *-commutative [=>]19.5

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

    5. Applied egg-rr19.0

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.25 \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}, \frac{h}{\ell} \cdot -0.5, 1\right) \cdot \frac{d}{\sqrt{h}}}{\sqrt{\ell}}} \]

    if 6.79999999999999948e154 < d

    1. Initial program 29.1

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

    2. Taylor expanded in d around inf 17.2

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

    3. Simplified16.7

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

      [Start]17.2

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

      *-commutative [=>]17.2

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

      associate-/l/ [<=]16.7

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

    4. Applied egg-rr14.4

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

    5. Simplified6.6

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

      [Start]14.4

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

      associate-/l/ [=>]6.6

      \[ \color{blue}{\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}} \]
  3. Recombined 4 regimes into one program.

  4. Final simplification17.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -4.2 \cdot 10^{+96}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 + \frac{\left(h \cdot {\left(\frac{0.5}{d} \cdot \left(M \cdot D\right)\right)}^{2}\right) \cdot -0.5}{\ell}\right)\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-309}:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\right) \cdot \left(1 - \frac{0.5 \cdot {\left(\frac{0.5}{d} \cdot \left(M \cdot D\right)\right)}^{2}}{\frac{\ell}{h}}\right)\\ \mathbf{elif}\;d \leq 6.8 \cdot 10^{+154}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.25 \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}, \frac{h}{\ell} \cdot -0.5, 1\right) \cdot \frac{d}{\sqrt{h}}}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternatives

Alternative 1
Error17.5
Cost27660
\[\begin{array}{l} t_0 := \sqrt{-d}\\ t_1 := 1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\\ t_2 := \sqrt{\frac{d}{h}}\\ t_3 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;h \leq -1.4 \cdot 10^{+145}:\\ \;\;\;\;\left(t_2 \cdot t_3\right) \cdot \left(1 + -0.5 \cdot \left(h \cdot \frac{{\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}}{\ell}\right)\right)\\ \mathbf{elif}\;h \leq -3.3 \cdot 10^{-130}:\\ \;\;\;\;\left(\frac{t_0}{\sqrt{-\ell}} \cdot t_2\right) \cdot t_1\\ \mathbf{elif}\;h \leq -1 \cdot 10^{-310}:\\ \;\;\;\;t_1 \cdot \left(\frac{t_0}{\sqrt{-h}} \cdot t_3\right)\\ \mathbf{elif}\;h \leq 2.5 \cdot 10^{+221}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 + \left(\frac{h}{\ell} \cdot -0.5\right) \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}}\\ \end{array} \]
Alternative 2
Error17.4
Cost27660
\[\begin{array}{l} t_0 := \sqrt{-d}\\ t_1 := \sqrt{\frac{d}{h}}\\ t_2 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;h \leq -6 \cdot 10^{+144}:\\ \;\;\;\;\left(t_1 \cdot t_2\right) \cdot \left(1 + -0.5 \cdot \left(h \cdot \frac{{\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}}{\ell}\right)\right)\\ \mathbf{elif}\;h \leq -3.3 \cdot 10^{-130}:\\ \;\;\;\;\left(\frac{t_0}{\sqrt{-\ell}} \cdot t_1\right) \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right)\\ \mathbf{elif}\;h \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\left(1 - \frac{0.5 \cdot {\left(\frac{0.5}{d} \cdot \left(M \cdot D\right)\right)}^{2}}{\frac{\ell}{h}}\right) \cdot \left(\frac{t_0}{\sqrt{-h}} \cdot t_2\right)\\ \mathbf{elif}\;h \leq 6 \cdot 10^{+221}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 + \left(\frac{h}{\ell} \cdot -0.5\right) \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}}\\ \end{array} \]
Alternative 3
Error17.2
Cost27660
\[\begin{array}{l} t_0 := {\left(\frac{0.5}{d} \cdot \left(M \cdot D\right)\right)}^{2}\\ t_1 := \sqrt{-d}\\ t_2 := \frac{t_1}{\sqrt{-h}}\\ \mathbf{if}\;h \leq -3.8 \cdot 10^{+167}:\\ \;\;\;\;\left(t_2 \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 + \frac{\left(h \cdot t_0\right) \cdot -0.5}{\ell}\right)\\ \mathbf{elif}\;h \leq -3.9 \cdot 10^{-130}:\\ \;\;\;\;\left(\frac{t_1}{\sqrt{-\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right)\\ \mathbf{elif}\;h \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\left(1 - \frac{0.5 \cdot t_0}{\frac{\ell}{h}}\right) \cdot \left(t_2 \cdot \sqrt{\frac{d}{\ell}}\right)\\ \mathbf{elif}\;h \leq 7.6 \cdot 10^{+217}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 + \left(\frac{h}{\ell} \cdot -0.5\right) \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}}\\ \end{array} \]
Alternative 4
Error18.8
Cost27528
\[\begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ t_1 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;h \leq -4.4 \cdot 10^{+144}:\\ \;\;\;\;\left(t_0 \cdot t_1\right) \cdot \left(1 + -0.5 \cdot \left(h \cdot \frac{{\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}}{\ell}\right)\right)\\ \mathbf{elif}\;h \leq -9.4 \cdot 10^{-87}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot t_0\right) \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right)\\ \mathbf{elif}\;h \leq -2.6 \cdot 10^{-89}:\\ \;\;\;\;t_0 \cdot \left(t_1 \cdot \mathsf{fma}\left(-0.5, h \cdot \frac{{\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2}}{\ell}, 1\right)\right)\\ \mathbf{elif}\;h \leq -1.35 \cdot 10^{-302}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{elif}\;h \leq 4.1 \cdot 10^{+223}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 + \left(\frac{h}{\ell} \cdot -0.5\right) \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}}\\ \end{array} \]
Alternative 5
Error20.7
Cost21268
\[\begin{array}{l} t_0 := \frac{h}{\ell} \cdot -0.5\\ t_1 := \sqrt{\frac{d}{h}}\\ t_2 := \sqrt{\frac{d}{\ell}}\\ t_3 := \left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{if}\;d \leq -4.7 \cdot 10^{+163}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;d \leq -9.5 \cdot 10^{-59}:\\ \;\;\;\;\left(t_1 \cdot t_2\right) \cdot \left(1 + -0.5 \cdot \left(0.25 \cdot \left(D \cdot \frac{M \cdot D}{\frac{\ell}{M} \cdot \frac{d \cdot d}{h}}\right)\right)\right)\\ \mathbf{elif}\;d \leq -1.9 \cdot 10^{-225}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-309}:\\ \;\;\;\;t_1 \cdot \left(t_2 \cdot \left(1 + t_0 \cdot {\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2}\right)\right)\\ \mathbf{elif}\;d \leq 8.8 \cdot 10^{+189}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 + t_0 \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{\ell}^{-0.5}}{\sqrt{h}}\\ \end{array} \]
Alternative 6
Error20.6
Cost21136
\[\begin{array}{l} t_0 := \left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{if}\;d \leq -1.3 \cdot 10^{+163}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq -4.6 \cdot 10^{-58}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 + -0.5 \cdot \left(0.25 \cdot \left(D \cdot \frac{M \cdot D}{\frac{\ell}{M} \cdot \frac{d \cdot d}{h}}\right)\right)\right)\\ \mathbf{elif}\;d \leq 1.04 \cdot 10^{-299}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 1.7 \cdot 10^{+190}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 + \left(\frac{h}{\ell} \cdot -0.5\right) \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{\ell}^{-0.5}}{\sqrt{h}}\\ \end{array} \]
Alternative 7
Error23.1
Cost21008
\[\begin{array}{l} \mathbf{if}\;h \leq -9.4 \cdot 10^{+58}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;h \leq -1.35 \cdot 10^{-302}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{elif}\;h \leq 1.05 \cdot 10^{-191}:\\ \;\;\;\;{h}^{-0.5} \cdot \frac{d}{\sqrt{\ell}}\\ \mathbf{elif}\;h \leq 7 \cdot 10^{+56}:\\ \;\;\;\;\frac{d \cdot \mathsf{fma}\left(0.25 \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}, \frac{h}{\ell} \cdot -0.5, 1\right)}{\sqrt{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}}\\ \end{array} \]
Alternative 8
Error23.0
Cost21008
\[\begin{array}{l} \mathbf{if}\;h \leq -2.3 \cdot 10^{+59}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;h \leq -1.35 \cdot 10^{-302}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{elif}\;h \leq 2.2 \cdot 10^{-196}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 + -0.125 \cdot \left(\frac{\frac{\frac{h}{d}}{d}}{\ell} \cdot {\left(M \cdot D\right)}^{2}\right)\right)\\ \mathbf{elif}\;h \leq 7.4 \cdot 10^{+55}:\\ \;\;\;\;\frac{d \cdot \mathsf{fma}\left(0.25 \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}, \frac{h}{\ell} \cdot -0.5, 1\right)}{\sqrt{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}}\\ \end{array} \]
Alternative 9
Error19.7
Cost21004
\[\begin{array}{l} \mathbf{if}\;h \leq -4 \cdot 10^{-17}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 + -0.5 \cdot \left(h \cdot \frac{{\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}}{\ell}\right)\right)\\ \mathbf{elif}\;h \leq -1.35 \cdot 10^{-302}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{elif}\;h \leq 2.8 \cdot 10^{+221}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 + \left(\frac{h}{\ell} \cdot -0.5\right) \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}}\\ \end{array} \]
Alternative 10
Error20.6
Cost21004
\[\begin{array}{l} \mathbf{if}\;h \leq -2.55 \cdot 10^{-89}:\\ \;\;\;\;\left(1 - \frac{0.5 \cdot {\left(\frac{0.5}{d} \cdot \left(M \cdot D\right)\right)}^{2}}{\frac{\ell}{h}}\right) \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\\ \mathbf{elif}\;h \leq -1.35 \cdot 10^{-302}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{elif}\;h \leq 1.08 \cdot 10^{+220}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 + \left(\frac{h}{\ell} \cdot -0.5\right) \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}}\\ \end{array} \]
Alternative 11
Error23.1
Cost20880
\[\begin{array}{l} \mathbf{if}\;h \leq -2.3 \cdot 10^{+59}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;h \leq -1.35 \cdot 10^{-302}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{elif}\;h \leq 1.1 \cdot 10^{-191}:\\ \;\;\;\;{h}^{-0.5} \cdot \frac{d}{\sqrt{\ell}}\\ \mathbf{elif}\;h \leq 7.8 \cdot 10^{+65}:\\ \;\;\;\;\mathsf{fma}\left({\left(D \cdot \frac{M}{d}\right)}^{2}, \frac{h}{\ell} \cdot -0.125, 1\right) \cdot \frac{d}{\sqrt{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}}\\ \end{array} \]
Alternative 12
Error23.8
Cost15188
\[\begin{array}{l} t_0 := \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 + -0.125 \cdot \left(\frac{\frac{h}{d}}{d} \cdot \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{\ell}\right)\right)\\ t_1 := \left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{if}\;d \leq -2.05 \cdot 10^{+164}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq -8 \cdot 10^{-59}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 1.04 \cdot 10^{-299}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq 6.2 \cdot 10^{-139}:\\ \;\;\;\;\frac{D}{\frac{\frac{d}{M}}{M \cdot D}} \cdot \left(-0.125 \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)\\ \mathbf{elif}\;d \leq 60000:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]
Alternative 13
Error23.8
Cost15188
\[\begin{array}{l} t_0 := \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ t_1 := \left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{if}\;d \leq -1.1 \cdot 10^{+166}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq -6.6 \cdot 10^{-58}:\\ \;\;\;\;t_0 \cdot \left(1 + 0.5 \cdot \left(\left(h \cdot \left(D \cdot \left(\left(M \cdot D\right) \cdot \frac{M}{\ell \cdot \left(d \cdot d\right)}\right)\right)\right) \cdot -0.25\right)\right)\\ \mathbf{elif}\;d \leq 3.2 \cdot 10^{-299}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq 6.5 \cdot 10^{-139}:\\ \;\;\;\;\frac{D}{\frac{\frac{d}{M}}{M \cdot D}} \cdot \left(-0.125 \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)\\ \mathbf{elif}\;d \leq 63000:\\ \;\;\;\;t_0 \cdot \left(1 + -0.125 \cdot \left(\frac{\frac{h}{d}}{d} \cdot \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]
Alternative 14
Error23.8
Cost15188
\[\begin{array}{l} t_0 := \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ t_1 := \left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{if}\;d \leq -3.6 \cdot 10^{+164}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq -7.2 \cdot 10^{-59}:\\ \;\;\;\;t_0 \cdot \left(1 + -0.5 \cdot \left(0.25 \cdot \left(D \cdot \frac{M \cdot D}{\frac{\ell}{M} \cdot \frac{d \cdot d}{h}}\right)\right)\right)\\ \mathbf{elif}\;d \leq 1.2 \cdot 10^{-299}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq 6 \cdot 10^{-139}:\\ \;\;\;\;\frac{D}{\frac{\frac{d}{M}}{M \cdot D}} \cdot \left(-0.125 \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)\\ \mathbf{elif}\;d \leq 11500:\\ \;\;\;\;t_0 \cdot \left(1 + -0.125 \cdot \left(\frac{\frac{h}{d}}{d} \cdot \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]
Alternative 15
Error24.3
Cost13384
\[\begin{array}{l} \mathbf{if}\;h \leq -5 \cdot 10^{+91}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{elif}\;h \leq -1.35 \cdot 10^{-302}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]
Alternative 16
Error23.4
Cost13384
\[\begin{array}{l} \mathbf{if}\;h \leq -2.3 \cdot 10^{+59}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;h \leq -1.35 \cdot 10^{-302}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]
Alternative 17
Error28.1
Cost7176
\[\begin{array}{l} \mathbf{if}\;h \leq -4.8 \cdot 10^{+92}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{elif}\;h \leq -1.35 \cdot 10^{-302}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \end{array} \]
Alternative 18
Error34.9
Cost6980
\[\begin{array}{l} \mathbf{if}\;\ell \leq -1.7 \cdot 10^{-274}:\\ \;\;\;\;\sqrt{d \cdot \frac{d}{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}}\\ \end{array} \]
Alternative 19
Error33.5
Cost6980
\[\begin{array}{l} \mathbf{if}\;\ell \leq 1.5 \cdot 10^{-277}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}}\\ \end{array} \]
Alternative 20
Error33.5
Cost6980
\[\begin{array}{l} \mathbf{if}\;\ell \leq 1.05 \cdot 10^{-266}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \end{array} \]
Alternative 21
Error44.2
Cost6720
\[\frac{d}{\sqrt{h \cdot \ell}} \]

Error

Reproduce?

herbie shell --seed 2023027 
(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)))))