Henrywood and Agarwal, Equation (12)

Average Error: 26.6 → 14.9

Time: 48.6s

Precision: binary64

Cost: 33800

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

Math

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

↓

\[\begin{array}{l} t_0 := \sqrt{-d}\\ t_1 := \frac{t_0}{\sqrt{-\ell}}\\ t_2 := \mathsf{fma}\left({\left(M \cdot \frac{\frac{D}{d}}{2}\right)}^{2}, \frac{h}{\ell} \cdot -0.5, 1\right)\\ t_3 := {\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2}\\ \mathbf{if}\;\ell \leq -1 \cdot 10^{+188}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(t_1 \cdot t_2\right)\\ \mathbf{elif}\;\ell \leq -1 \cdot 10^{-78}:\\ \;\;\;\;\frac{t_0}{\sqrt{-h}} \cdot \left(t_2 \cdot \sqrt{\frac{d}{\ell}}\right)\\ \mathbf{elif}\;\ell \leq 0:\\ \;\;\;\;\left(t_1 \cdot {\left(\frac{d}{h}\right)}^{0.5}\right) \cdot \left(1 - \frac{t_3 \cdot \left(h \cdot 0.5\right)}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{t_3}{\ell} \cdot \left(h \cdot -0.5\right)}{\frac{\sqrt{h} \cdot \sqrt{\ell}}{d}}\\ \end{array} \]

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)))
        (t_1 (/ t_0 (sqrt (- l))))
        (t_2 (fma (pow (* M (/ (/ D d) 2.0)) 2.0) (* (/ h l) -0.5) 1.0))
        (t_3 (pow (* M (* D (/ 0.5 d))) 2.0)))
   (if (<= l -1e+188)
     (* (sqrt (/ d h)) (* t_1 t_2))
     (if (<= l -1e-78)
       (* (/ t_0 (sqrt (- h))) (* t_2 (sqrt (/ d l))))
       (if (<= l 0.0)
         (* (* t_1 (pow (/ d h) 0.5)) (- 1.0 (/ (* t_3 (* h 0.5)) l)))
         (/ (+ 1.0 (* (/ t_3 l) (* h -0.5))) (/ (* (sqrt h) (sqrt l)) d)))))))

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);
	double t_1 = t_0 / sqrt(-l);
	double t_2 = fma(pow((M * ((D / d) / 2.0)), 2.0), ((h / l) * -0.5), 1.0);
	double t_3 = pow((M * (D * (0.5 / d))), 2.0);
	double tmp;
	if (l <= -1e+188) {
		tmp = sqrt((d / h)) * (t_1 * t_2);
	} else if (l <= -1e-78) {
		tmp = (t_0 / sqrt(-h)) * (t_2 * sqrt((d / l)));
	} else if (l <= 0.0) {
		tmp = (t_1 * pow((d / h), 0.5)) * (1.0 - ((t_3 * (h * 0.5)) / l));
	} else {
		tmp = (1.0 + ((t_3 / l) * (h * -0.5))) / ((sqrt(h) * sqrt(l)) / d);
	}
	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))
	t_1 = Float64(t_0 / sqrt(Float64(-l)))
	t_2 = fma((Float64(M * Float64(Float64(D / d) / 2.0)) ^ 2.0), Float64(Float64(h / l) * -0.5), 1.0)
	t_3 = Float64(M * Float64(D * Float64(0.5 / d))) ^ 2.0
	tmp = 0.0
	if (l <= -1e+188)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(t_1 * t_2));
	elseif (l <= -1e-78)
		tmp = Float64(Float64(t_0 / sqrt(Float64(-h))) * Float64(t_2 * sqrt(Float64(d / l))));
	elseif (l <= 0.0)
		tmp = Float64(Float64(t_1 * (Float64(d / h) ^ 0.5)) * Float64(1.0 - Float64(Float64(t_3 * Float64(h * 0.5)) / l)));
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(t_3 / l) * Float64(h * -0.5))) / Float64(Float64(sqrt(h) * sqrt(l)) / d));
	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[(-d)], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Power[N[(M * N[(N[(D / d), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$3 = N[Power[N[(M * N[(D * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[l, -1e+188], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(t$95$1 * t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -1e-78], N[(N[(t$95$0 / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[(t$95$2 * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 0.0], N[(N[(t$95$1 * N[Power[N[(d / h), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(t$95$3 * N[(h * 0.5), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(N[(t$95$3 / l), $MachinePrecision] * N[(h * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -1e188

   1. Initial program 30.5

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

   2. Simplified 30.1

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

   3. Applied egg-rr 23.8

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

   if -1e188 < l < -9.99999999999999999e-79

   1. Initial program 21.6

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

   2. Simplified 22.1

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

   3. Applied egg-rr 14.6

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

   if -9.99999999999999999e-79 < l < 0.0

   1. Initial program 30.9

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

   2. Applied egg-rr 25.8

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

   3. Applied egg-rr 12.3

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

   if 0.0 < l

   1. Initial program 26.9

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

   2. Applied egg-rr 26.5

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

   3. Applied egg-rr 26.5

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

   4. Applied egg-rr 26.4

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

   5. Applied egg-rr 13.2

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

4. Final simplification 14.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1 \cdot 10^{+188}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \mathsf{fma}\left({\left(M \cdot \frac{\frac{D}{d}}{2}\right)}^{2}, \frac{h}{\ell} \cdot -0.5, 1\right)\right)\\ \mathbf{elif}\;\ell \leq -1 \cdot 10^{-78}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(\mathsf{fma}\left({\left(M \cdot \frac{\frac{D}{d}}{2}\right)}^{2}, \frac{h}{\ell} \cdot -0.5, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right)\\ \mathbf{elif}\;\ell \leq 0:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot {\left(\frac{d}{h}\right)}^{0.5}\right) \cdot \left(1 - \frac{{\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot \left(h \cdot 0.5\right)}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{{\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2}}{\ell} \cdot \left(h \cdot -0.5\right)}{\frac{\sqrt{h} \cdot \sqrt{\ell}}{d}}\\ \end{array} \]

Alternatives

Alternative 1
Error	15.1
Cost	33668

\[\begin{array}{l} t_0 := \sqrt{-d}\\ t_1 := {\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2}\\ t_2 := 1 - \frac{t_1 \cdot \left(h \cdot 0.5\right)}{\ell}\\ t_3 := \frac{t_0}{\sqrt{-\ell}}\\ \mathbf{if}\;\ell \leq -1 \cdot 10^{+188}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(t_3 \cdot \mathsf{fma}\left({\left(M \cdot \frac{\frac{D}{d}}{2}\right)}^{2}, \frac{h}{\ell} \cdot -0.5, 1\right)\right)\\ \mathbf{elif}\;\ell \leq -1 \cdot 10^{-82}:\\ \;\;\;\;t_2 \cdot \left(\frac{t_0}{\sqrt{-h}} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right)\\ \mathbf{elif}\;\ell \leq 0:\\ \;\;\;\;\left(t_3 \cdot {\left(\frac{d}{h}\right)}^{0.5}\right) \cdot t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{t_1}{\ell} \cdot \left(h \cdot -0.5\right)}{\frac{\sqrt{h} \cdot \sqrt{\ell}}{d}}\\ \end{array} \]

Alternative 2
Error	15.1
Cost	27724

\[\begin{array}{l} t_0 := \sqrt{-d}\\ t_1 := {\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2}\\ t_2 := 1 - \frac{t_1 \cdot \left(h \cdot 0.5\right)}{\ell}\\ t_3 := \frac{t_0}{\sqrt{-\ell}}\\ \mathbf{if}\;\ell \leq -1 \cdot 10^{+188}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot t_3\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right)\right)\\ \mathbf{elif}\;\ell \leq -1 \cdot 10^{-82}:\\ \;\;\;\;t_2 \cdot \left(\frac{t_0}{\sqrt{-h}} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right)\\ \mathbf{elif}\;\ell \leq 0:\\ \;\;\;\;\left(t_3 \cdot {\left(\frac{d}{h}\right)}^{0.5}\right) \cdot t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{t_1}{\ell} \cdot \left(h \cdot -0.5\right)}{\frac{\sqrt{h} \cdot \sqrt{\ell}}{d}}\\ \end{array} \]

Alternative 3
Error	17.5
Cost	27660

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ t_1 := 1 + \frac{{\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2}}{\ell} \cdot \left(h \cdot -0.5\right)\\ \mathbf{if}\;d \leq -9.5 \cdot 10^{+242}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{\ell \cdot h}}\right)\\ \mathbf{elif}\;d \leq -1 \cdot 10^{-65}:\\ \;\;\;\;\frac{t_0 \cdot t_1}{\sqrt{\frac{\ell}{d}}}\\ \mathbf{elif}\;d \leq -6.6 \cdot 10^{-292}:\\ \;\;\;\;\left(t_0 \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{\frac{\sqrt{h} \cdot \sqrt{\ell}}{d}}\\ \end{array} \]

Alternative 4
Error	16.9
Cost	27460

\[\begin{array}{l} t_0 := {\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2}\\ \mathbf{if}\;d \leq -1.08 \cdot 10^{-131}:\\ \;\;\;\;\left(1 - \frac{t_0 \cdot \left(h \cdot 0.5\right)}{\ell}\right) \cdot \left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right)\\ \mathbf{elif}\;d \leq -6.8 \cdot 10^{-292}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{h}}{\ell}}\right)\\ \mathbf{elif}\;d \leq 10^{-295}:\\ \;\;\;\;\left(D \cdot \left(D \cdot \frac{M}{\frac{d}{M}}\right)\right) \cdot \left(-0.125 \cdot {\left(\frac{\sqrt[3]{\sqrt{h}}}{\sqrt{\ell}}\right)}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{t_0}{\ell} \cdot \left(h \cdot -0.5\right)}{\frac{\sqrt{h} \cdot \sqrt{\ell}}{d}}\\ \end{array} \]

Alternative 5
Error	18.4
Cost	21384

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ t_1 := M \cdot \frac{D}{d}\\ t_2 := 1 + \frac{{\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2}}{\ell} \cdot \left(h \cdot -0.5\right)\\ \mathbf{if}\;h \leq -5 \cdot 10^{-127}:\\ \;\;\;\;\frac{t_0 \cdot t_2}{\sqrt{\frac{\ell}{d}}}\\ \mathbf{elif}\;h \leq -1 \cdot 10^{-160}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - 0.125 \cdot \left(\left(D \cdot \frac{D}{\ell}\right) \cdot \frac{h}{\frac{d \cdot d}{M \cdot M}}\right)\right)\\ \mathbf{elif}\;h \leq -1 \cdot 10^{-208}:\\ \;\;\;\;t_0 \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(t_1 \cdot t_1, h \cdot \frac{-0.125}{\ell}, 1\right)\right)\\ \mathbf{elif}\;h \leq 0:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{h}}{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t_2}{\frac{\sqrt{h} \cdot \sqrt{\ell}}{d}}\\ \end{array} \]

Alternative 6
Error	18.0
Cost	21196

\[\begin{array}{l} t_0 := M \cdot \frac{D}{d}\\ t_1 := d \cdot \left(-\sqrt{\frac{\frac{1}{h}}{\ell}}\right)\\ t_2 := \sqrt{\frac{d}{h}}\\ t_3 := 1 + \frac{{\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2}}{\ell} \cdot \left(h \cdot -0.5\right)\\ \mathbf{if}\;h \leq -3.3 \cdot 10^{-35}:\\ \;\;\;\;\frac{t_2 \cdot t_3}{\sqrt{\frac{\ell}{d}}}\\ \mathbf{elif}\;h \leq -1 \cdot 10^{-160}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;h \leq -1 \cdot 10^{-208}:\\ \;\;\;\;t_2 \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(t_0 \cdot t_0, h \cdot \frac{-0.125}{\ell}, 1\right)\right)\\ \mathbf{elif}\;h \leq 0:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{t_3}{\frac{\sqrt{h} \cdot \sqrt{\ell}}{d}}\\ \end{array} \]

Alternative 7
Error	18.0
Cost	21136

\[\begin{array}{l} t_0 := d \cdot \left(-\sqrt{\frac{\frac{1}{h}}{\ell}}\right)\\ t_1 := 1 + \frac{{\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2}}{\ell} \cdot \left(h \cdot -0.5\right)\\ t_2 := \frac{\sqrt{\frac{d}{h}} \cdot t_1}{\sqrt{\frac{\ell}{d}}}\\ \mathbf{if}\;h \leq -3.3 \cdot 10^{-35}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;h \leq -1 \cdot 10^{-160}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;h \leq -1 \cdot 10^{-208}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;h \leq 0:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{\frac{\sqrt{h} \cdot \sqrt{\ell}}{d}}\\ \end{array} \]

Alternative 8
Error	17.9
Cost	21136

\[\begin{array}{l} t_0 := d \cdot \left(-\sqrt{\frac{\frac{1}{h}}{\ell}}\right)\\ t_1 := 1 + \frac{{\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2}}{\ell} \cdot \left(h \cdot -0.5\right)\\ \mathbf{if}\;h \leq -3.3 \cdot 10^{-35}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{h}} \cdot t_1}{\sqrt{\frac{\ell}{d}}}\\ \mathbf{elif}\;h \leq -1 \cdot 10^{-160}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;h \leq -1 \cdot 10^{-208}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{\ell}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right)}{\sqrt{\frac{h}{d}}}\\ \mathbf{elif}\;h \leq 0:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{\frac{\sqrt{h} \cdot \sqrt{\ell}}{d}}\\ \end{array} \]

Alternative 9
Error	18.7
Cost	21004

\[\begin{array}{l} \mathbf{if}\;d \leq -3 \cdot 10^{+75}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{\ell \cdot h}}\right)\\ \mathbf{elif}\;d \leq -1.08 \cdot 10^{-131}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - 0.125 \cdot \frac{\frac{D \cdot \left(h \cdot \frac{D}{\ell}\right)}{\frac{d}{M}}}{\frac{d}{M}}\right)\\ \mathbf{elif}\;d \leq 2.9 \cdot 10^{-292}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{h}}{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{{\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2}}{\ell} \cdot \left(h \cdot -0.5\right)}{\frac{\sqrt{h} \cdot \sqrt{\ell}}{d}}\\ \end{array} \]

Alternative 10
Error	22.7
Cost	15120

\[\begin{array}{l} t_0 := \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - 0.125 \cdot \frac{\frac{D \cdot \left(h \cdot \frac{D}{\ell}\right)}{\frac{d}{M}}}{\frac{d}{M}}\right)\\ \mathbf{if}\;d \leq -3 \cdot 10^{+75}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{\ell \cdot h}}\right)\\ \mathbf{elif}\;d \leq -1.08 \cdot 10^{-131}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 1.45 \cdot 10^{-263}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{h}}{\ell}}\right)\\ \mathbf{elif}\;d \leq 10^{+108}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \]

Alternative 11
Error	23.1
Cost	14856

\[\begin{array}{l} \mathbf{if}\;d \leq -8.8 \cdot 10^{+61}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{\ell \cdot h}}\right)\\ \mathbf{elif}\;d \leq -9.2 \cdot 10^{-99}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 + -0.125 \cdot \left(\left(D \cdot \frac{D}{\ell}\right) \cdot \frac{h}{\frac{d}{M} \cdot \frac{d}{M}}\right)\right)\\ \mathbf{elif}\;d \leq 2.9 \cdot 10^{-287}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{h}}{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \]

Alternative 12
Error	23.2
Cost	14792

\[\begin{array}{l} \mathbf{if}\;d \leq -8.8 \cdot 10^{+61}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{\ell \cdot h}}\right)\\ \mathbf{elif}\;d \leq -9.2 \cdot 10^{-99}:\\ \;\;\;\;\left(1 - 0.125 \cdot \left(\left(D \cdot \frac{D}{\ell}\right) \cdot \frac{h}{\frac{d \cdot d}{M \cdot M}}\right)\right) \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\\ \mathbf{elif}\;d \leq 2.9 \cdot 10^{-287}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{h}}{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \]

Alternative 13
Error	22.9
Cost	13644

\[\begin{array}{l} \mathbf{if}\;d \leq -6.1 \cdot 10^{+78}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{\ell \cdot h}}\right)\\ \mathbf{elif}\;d \leq -9.2 \cdot 10^{-99}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;d \leq 2.9 \cdot 10^{-287}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{h}}{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \]

Alternative 14
Error	23.7
Cost	13380

\[\begin{array}{l} \mathbf{if}\;d \leq 2.9 \cdot 10^{-287}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{h}}{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \]

Alternative 15
Error	23.7
Cost	13252

\[\begin{array}{l} \mathbf{if}\;d \leq 2.9 \cdot 10^{-287}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{h}}{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 16
Error	27.3
Cost	7044

\[\begin{array}{l} \mathbf{if}\;\ell \leq -6.1 \cdot 10^{-294}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{h}}{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}}\\ \end{array} \]

Alternative 17
Error	36.9
Cost	6980

\[\begin{array}{l} \mathbf{if}\;\ell \leq -1.8 \cdot 10^{-242}:\\ \;\;\;\;\sqrt{\frac{d \cdot d}{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}}\\ \end{array} \]

Alternative 18
Error	62.2
Cost	6720

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

Alternative 19
Error	43.6
Cost	6720

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

Reproduce

herbie shell --seed 2022284 
(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)))))