Henrywood and Agarwal, Equation (12)

Average Error: 26.1 → 15.3

Time: 58.8s

Precision: binary64

Cost: 104208

Math

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

↓

\[\begin{array}{l} t_0 := \left|\frac{d}{\sqrt{h \cdot \ell}}\right| \cdot \left(1 + \frac{{\left(M \cdot \frac{D}{d}\right)}^{2} \cdot -0.125}{\frac{\ell}{h}}\right)\\ t_1 := {\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\\ t_2 := t_1 \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot -0.5\right)\right)\\ t_3 := \frac{M}{2} \cdot \frac{D}{d}\\ \mathbf{if}\;t_2 \leq -1 \cdot 10^{-68}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - {\left(\frac{1}{\sqrt{\frac{\ell}{h}}} \cdot \left(t_3 \cdot \sqrt{0.5}\right)\right)}^{2}\right)\\ \mathbf{elif}\;t_2 \leq 10^{-228}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_2 \leq 2 \cdot 10^{+272}:\\ \;\;\;\;t_1 \cdot \left(1 + \frac{{t_3}^{2} \cdot \left(h \cdot -0.5\right)}{\ell}\right)\\ \mathbf{elif}\;t_2 \leq \infty:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-d\right)\\ \end{array} \]

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
         (*
          (fabs (/ d (sqrt (* h l))))
          (+ 1.0 (/ (* (pow (* M (/ D d)) 2.0) -0.125) (/ l h)))))
        (t_1 (* (pow (/ d h) 0.5) (pow (/ d l) 0.5)))
        (t_2
         (* t_1 (+ 1.0 (* (/ h l) (* (pow (/ (* M D) (* d 2.0)) 2.0) -0.5)))))
        (t_3 (* (/ M 2.0) (/ D d))))
   (if (<= t_2 -1e-68)
     (*
      (* (sqrt (/ d h)) (/ 1.0 (sqrt (/ l d))))
      (- 1.0 (pow (* (/ 1.0 (sqrt (/ l h))) (* t_3 (sqrt 0.5))) 2.0)))
     (if (<= t_2 1e-228)
       t_0
       (if (<= t_2 2e+272)
         (* t_1 (+ 1.0 (/ (* (pow t_3 2.0) (* h -0.5)) l)))
         (if (<= t_2 INFINITY) t_0 (* (sqrt (/ 1.0 (* h 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 = fabs((d / sqrt((h * l)))) * (1.0 + ((pow((M * (D / d)), 2.0) * -0.125) / (l / h)));
	double t_1 = pow((d / h), 0.5) * pow((d / l), 0.5);
	double t_2 = t_1 * (1.0 + ((h / l) * (pow(((M * D) / (d * 2.0)), 2.0) * -0.5)));
	double t_3 = (M / 2.0) * (D / d);
	double tmp;
	if (t_2 <= -1e-68) {
		tmp = (sqrt((d / h)) * (1.0 / sqrt((l / d)))) * (1.0 - pow(((1.0 / sqrt((l / h))) * (t_3 * sqrt(0.5))), 2.0));
	} else if (t_2 <= 1e-228) {
		tmp = t_0;
	} else if (t_2 <= 2e+272) {
		tmp = t_1 * (1.0 + ((pow(t_3, 2.0) * (h * -0.5)) / l));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = t_0;
	} else {
		tmp = sqrt((1.0 / (h * l))) * -d;
	}
	return tmp;
}

Java

public static double code(double d, double h, double l, double M, double D) {
	return (Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * Math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
}

↓

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.abs((d / Math.sqrt((h * l)))) * (1.0 + ((Math.pow((M * (D / d)), 2.0) * -0.125) / (l / h)));
	double t_1 = Math.pow((d / h), 0.5) * Math.pow((d / l), 0.5);
	double t_2 = t_1 * (1.0 + ((h / l) * (Math.pow(((M * D) / (d * 2.0)), 2.0) * -0.5)));
	double t_3 = (M / 2.0) * (D / d);
	double tmp;
	if (t_2 <= -1e-68) {
		tmp = (Math.sqrt((d / h)) * (1.0 / Math.sqrt((l / d)))) * (1.0 - Math.pow(((1.0 / Math.sqrt((l / h))) * (t_3 * Math.sqrt(0.5))), 2.0));
	} else if (t_2 <= 1e-228) {
		tmp = t_0;
	} else if (t_2 <= 2e+272) {
		tmp = t_1 * (1.0 + ((Math.pow(t_3, 2.0) * (h * -0.5)) / l));
	} else if (t_2 <= Double.POSITIVE_INFINITY) {
		tmp = t_0;
	} else {
		tmp = Math.sqrt((1.0 / (h * l))) * -d;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	return (math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))

↓

def code(d, h, l, M, D):
	t_0 = math.fabs((d / math.sqrt((h * l)))) * (1.0 + ((math.pow((M * (D / d)), 2.0) * -0.125) / (l / h)))
	t_1 = math.pow((d / h), 0.5) * math.pow((d / l), 0.5)
	t_2 = t_1 * (1.0 + ((h / l) * (math.pow(((M * D) / (d * 2.0)), 2.0) * -0.5)))
	t_3 = (M / 2.0) * (D / d)
	tmp = 0
	if t_2 <= -1e-68:
		tmp = (math.sqrt((d / h)) * (1.0 / math.sqrt((l / d)))) * (1.0 - math.pow(((1.0 / math.sqrt((l / h))) * (t_3 * math.sqrt(0.5))), 2.0))
	elif t_2 <= 1e-228:
		tmp = t_0
	elif t_2 <= 2e+272:
		tmp = t_1 * (1.0 + ((math.pow(t_3, 2.0) * (h * -0.5)) / l))
	elif t_2 <= math.inf:
		tmp = t_0
	else:
		tmp = math.sqrt((1.0 / (h * 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 = Float64(abs(Float64(d / sqrt(Float64(h * l)))) * Float64(1.0 + Float64(Float64((Float64(M * Float64(D / d)) ^ 2.0) * -0.125) / Float64(l / h))))
	t_1 = Float64((Float64(d / h) ^ 0.5) * (Float64(d / l) ^ 0.5))
	t_2 = Float64(t_1 * Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(Float64(M * D) / Float64(d * 2.0)) ^ 2.0) * -0.5))))
	t_3 = Float64(Float64(M / 2.0) * Float64(D / d))
	tmp = 0.0
	if (t_2 <= -1e-68)
		tmp = Float64(Float64(sqrt(Float64(d / h)) * Float64(1.0 / sqrt(Float64(l / d)))) * Float64(1.0 - (Float64(Float64(1.0 / sqrt(Float64(l / h))) * Float64(t_3 * sqrt(0.5))) ^ 2.0)));
	elseif (t_2 <= 1e-228)
		tmp = t_0;
	elseif (t_2 <= 2e+272)
		tmp = Float64(t_1 * Float64(1.0 + Float64(Float64((t_3 ^ 2.0) * Float64(h * -0.5)) / l)));
	elseif (t_2 <= Inf)
		tmp = t_0;
	else
		tmp = Float64(sqrt(Float64(1.0 / Float64(h * l))) * Float64(-d));
	end
	return tmp
end

MATLAB

function tmp = code(d, h, l, M, D)
	tmp = (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((M * D) / (2.0 * d)) ^ 2.0)) * (h / l)));
end

↓

function tmp_2 = code(d, h, l, M, D)
	t_0 = abs((d / sqrt((h * l)))) * (1.0 + ((((M * (D / d)) ^ 2.0) * -0.125) / (l / h)));
	t_1 = ((d / h) ^ 0.5) * ((d / l) ^ 0.5);
	t_2 = t_1 * (1.0 + ((h / l) * ((((M * D) / (d * 2.0)) ^ 2.0) * -0.5)));
	t_3 = (M / 2.0) * (D / d);
	tmp = 0.0;
	if (t_2 <= -1e-68)
		tmp = (sqrt((d / h)) * (1.0 / sqrt((l / d)))) * (1.0 - (((1.0 / sqrt((l / h))) * (t_3 * sqrt(0.5))) ^ 2.0));
	elseif (t_2 <= 1e-228)
		tmp = t_0;
	elseif (t_2 <= 2e+272)
		tmp = t_1 * (1.0 + (((t_3 ^ 2.0) * (h * -0.5)) / l));
	elseif (t_2 <= Inf)
		tmp = t_0;
	else
		tmp = sqrt((1.0 / (h * l))) * -d;
	end
	tmp_2 = 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[(N[Abs[N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(N[(N[Power[N[(M * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * -0.125), $MachinePrecision] / N[(l / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[N[(d / h), $MachinePrecision], 0.5], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(M / 2.0), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e-68], N[(N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(1.0 / N[Sqrt[N[(l / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[Power[N[(N[(1.0 / N[Sqrt[N[(l / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(t$95$3 * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e-228], t$95$0, If[LessEqual[t$95$2, 2e+272], N[(t$95$1 * N[(1.0 + N[(N[(N[Power[t$95$3, 2.0], $MachinePrecision] * N[(h * -0.5), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], t$95$0, N[(N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-d)), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (-.f64 1 (*.f64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)) (/.f64 h l)))) < -1.00000000000000007e-68

   1. Initial program 30.1

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

   2. Applied egg-rr 23.4

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

   3. Applied egg-rr 23.6

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

   4. Applied egg-rr 23.6

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

   5. Applied egg-rr 23.6

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

   if -1.00000000000000007e-68 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (-.f64 1 (*.f64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)) (/.f64 h l)))) < 1.00000000000000003e-228 or 2.0000000000000001e272 < (*.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)))) < +inf.0

   1. Initial program 45.1

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

   2. Applied egg-rr 45.8

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

   3. Applied egg-rr 52.4

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

   4. Applied egg-rr 52.4

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

   5. Applied egg-rr 12.0

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

   if 1.00000000000000003e-228 < (*.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)))) < 2.0000000000000001e272

   1. Initial program 0.7

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

   2. Applied egg-rr 0.9

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

   if +inf.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (-.f64 1 (*.f64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)) (/.f64 h l))))

   1. Initial program 64.0

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

   2. Applied egg-rr 62.1

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

   3. Applied egg-rr 62.1

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

   4. Taylor expanded in d around -inf 50.4

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

   5. Simplified 50.4

      \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-d\right)} \]
      Proof
      (*.f64 (sqrt.f64 (/.f64 1 (*.f64 h l))) (neg.f64 d)): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 (/.f64 1 (Rewrite<= *-commutative_binary64 (*.f64 l h)))) (neg.f64 d)): 0 points increase in error, 0 points decrease in error
      (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
      (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 d (sqrt.f64 (/.f64 1 (*.f64 l h)))))): 0 points increase in error, 0 points decrease in error
      (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
3. Recombined 4 regimes into one program.

4. Final simplification 15.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot -0.5\right)\right) \leq -1 \cdot 10^{-68}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - {\left(\frac{1}{\sqrt{\frac{\ell}{h}}} \cdot \left(\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \sqrt{0.5}\right)\right)}^{2}\right)\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot -0.5\right)\right) \leq 10^{-228}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right| \cdot \left(1 + \frac{{\left(M \cdot \frac{D}{d}\right)}^{2} \cdot -0.125}{\frac{\ell}{h}}\right)\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot -0.5\right)\right) \leq 2 \cdot 10^{+272}:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 + \frac{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(h \cdot -0.5\right)}{\ell}\right)\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot -0.5\right)\right) \leq \infty:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right| \cdot \left(1 + \frac{{\left(M \cdot \frac{D}{d}\right)}^{2} \cdot -0.125}{\frac{\ell}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-d\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	15.6
Cost	125140

\[\begin{array}{l} t_0 := 1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot -0.5\right)\\ t_1 := {\left(\frac{d}{h}\right)}^{0.5}\\ t_2 := t_1 \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\\ t_3 := t_2 \cdot t_0\\ t_4 := \left|\frac{d}{\sqrt{h \cdot \ell}}\right| \cdot \left(1 + \frac{{\left(M \cdot \frac{D}{d}\right)}^{2} \cdot -0.125}{\frac{\ell}{h}}\right)\\ \mathbf{if}\;t_3 \leq -2 \cdot 10^{+279}:\\ \;\;\;\;\left(t_1 \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 + \left(M \cdot \left(M \cdot \frac{\frac{D}{\frac{d}{D}}}{\frac{\ell}{\frac{h}{d}}}\right)\right) \cdot -0.125\right)\\ \mathbf{elif}\;t_3 \leq -1 \cdot 10^{-68}:\\ \;\;\;\;t_0 \cdot \left(t_1 \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right)\\ \mathbf{elif}\;t_3 \leq 10^{-228}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;t_3 \leq 2 \cdot 10^{+272}:\\ \;\;\;\;t_2 \cdot \left(1 + \frac{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(h \cdot -0.5\right)}{\ell}\right)\\ \mathbf{elif}\;t_3 \leq \infty:\\ \;\;\;\;t_4\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-d\right)\\ \end{array} \]

Alternative 2
Error	15.3
Cost	104208

\[\begin{array}{l} t_0 := \frac{M}{2} \cdot \frac{D}{d}\\ t_1 := {\left(\frac{d}{h}\right)}^{0.5}\\ t_2 := t_1 \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\\ t_3 := t_2 \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot -0.5\right)\right)\\ t_4 := \left|\frac{d}{\sqrt{h \cdot \ell}}\right| \cdot \left(1 + \frac{{\left(M \cdot \frac{D}{d}\right)}^{2} \cdot -0.125}{\frac{\ell}{h}}\right)\\ \mathbf{if}\;t_3 \leq -1 \cdot 10^{-68}:\\ \;\;\;\;\left(t_1 \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\left(t_0 \cdot \sqrt{0.5}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right)\\ \mathbf{elif}\;t_3 \leq 10^{-228}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;t_3 \leq 2 \cdot 10^{+272}:\\ \;\;\;\;t_2 \cdot \left(1 + \frac{{t_0}^{2} \cdot \left(h \cdot -0.5\right)}{\ell}\right)\\ \mathbf{elif}\;t_3 \leq \infty:\\ \;\;\;\;t_4\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-d\right)\\ \end{array} \]

Alternative 3
Error	22.7
Cost	21592

\[\begin{array}{l} t_0 := {\left(\frac{d}{h}\right)}^{0.5}\\ t_1 := \sqrt{\frac{d}{\ell}}\\ t_2 := \sqrt{\frac{1}{h \cdot \ell}}\\ t_3 := \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \mathbf{if}\;d \leq -2.25 \cdot 10^{+26}:\\ \;\;\;\;t_2 \cdot \left(-d\right)\\ \mathbf{elif}\;d \leq -1 \cdot 10^{-90}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(t_1 \cdot \mathsf{fma}\left(-0.125, h \cdot \left(M \cdot \frac{D}{\frac{\ell}{M} \cdot \frac{d \cdot d}{D}}\right), 1\right)\right)\\ \mathbf{elif}\;d \leq -2.85 \cdot 10^{-151}:\\ \;\;\;\;\left(1 + \left(M \cdot \left(M \cdot \frac{\frac{D}{\frac{d}{D}}}{\frac{\ell}{\frac{h}{d}}}\right)\right) \cdot -0.125\right) \cdot \left(t_0 \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\right)\\ \mathbf{elif}\;d \leq -1.9 \cdot 10^{-297}:\\ \;\;\;\;\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\left(D \cdot D\right) \cdot \left(\frac{0.125}{d} \cdot \left(M \cdot M\right)\right)\right) - d \cdot t_2\\ \mathbf{elif}\;d \leq 2.7 \cdot 10^{-230}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;d \leq 10^{+145}:\\ \;\;\;\;\left(1 + \frac{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(h \cdot -0.5\right)}{\ell}\right) \cdot \left(t_0 \cdot t_1\right)\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 4
Error	22.5
Cost	21528

\[\begin{array}{l} t_0 := \frac{\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M}{2 \cdot \frac{d}{D}}\right)}^{2} \cdot -0.5\right)\right)}{\sqrt{\frac{\ell}{d}}}\\ t_1 := \sqrt{\frac{1}{h \cdot \ell}}\\ t_2 := {\left(M \cdot \frac{D}{d}\right)}^{2}\\ \mathbf{if}\;\ell \leq -2.3 \cdot 10^{+244}:\\ \;\;\;\;t_1 \cdot \left(-d\right)\\ \mathbf{elif}\;\ell \leq -3.2 \cdot 10^{+169}:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 + \left(M \cdot \left(M \cdot \frac{\frac{D}{\frac{d}{D}}}{\frac{\ell}{\frac{h}{d}}}\right)\right) \cdot -0.125\right)\\ \mathbf{elif}\;\ell \leq -1 \cdot 10^{-115}:\\ \;\;\;\;\left(d \cdot t_1\right) \cdot \left(\frac{t_2 \cdot 0.125}{\frac{\ell}{h}} + -1\right)\\ \mathbf{elif}\;\ell \leq 10^{-135}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq 2 \cdot 10^{+63}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 + \frac{t_2 \cdot -0.125}{\frac{\ell}{h}}\right)\\ \mathbf{elif}\;\ell \leq 10^{+122}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

Alternative 5
Error	22.5
Cost	21460

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ t_2 := \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{if}\;d \leq -2.25 \cdot 10^{+26}:\\ \;\;\;\;t_2 \cdot \left(-d\right)\\ \mathbf{elif}\;d \leq -2.85 \cdot 10^{-151}:\\ \;\;\;\;\left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot -0.5\right)\right) \cdot \left(\sqrt{\frac{d}{h}} \cdot t_0\right)\\ \mathbf{elif}\;d \leq -1.9 \cdot 10^{-297}:\\ \;\;\;\;\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\left(D \cdot D\right) \cdot \left(\frac{0.125}{d} \cdot \left(M \cdot M\right)\right)\right) - d \cdot t_2\\ \mathbf{elif}\;d \leq 2.7 \cdot 10^{-230}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq 10^{+145}:\\ \;\;\;\;\left(1 + \frac{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(h \cdot -0.5\right)}{\ell}\right) \cdot \left({\left(\frac{d}{h}\right)}^{0.5} \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	22.9
Cost	21396

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ t_1 := \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{if}\;d \leq -3.1 \cdot 10^{+34}:\\ \;\;\;\;t_1 \cdot \left(-d\right)\\ \mathbf{elif}\;d \leq -2.85 \cdot 10^{-151}:\\ \;\;\;\;\frac{t_0 \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M}{2 \cdot \frac{d}{D}}\right)}^{2} \cdot -0.5\right)\right)}{\sqrt{\frac{\ell}{d}}}\\ \mathbf{elif}\;d \leq 3.7 \cdot 10^{-290}:\\ \;\;\;\;\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\left(D \cdot D\right) \cdot \left(\frac{0.125}{d} \cdot \left(M \cdot M\right)\right)\right) - d \cdot t_1\\ \mathbf{elif}\;d \leq 5.5 \cdot 10^{-101}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 + \frac{{\left(M \cdot \frac{D}{d}\right)}^{2} \cdot -0.125}{\frac{\ell}{h}}\right)\\ \mathbf{elif}\;d \leq 10^{+145}:\\ \;\;\;\;t_0 \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - {\left(\frac{D}{d} \cdot \left(0.5 \cdot M\right)\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

Alternative 7
Error	22.7
Cost	21396

\[\begin{array}{l} t_0 := \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot -0.5\right)\right) \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\\ t_1 := \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{if}\;d \leq -2.25 \cdot 10^{+26}:\\ \;\;\;\;t_1 \cdot \left(-d\right)\\ \mathbf{elif}\;d \leq -2.85 \cdot 10^{-151}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 3.7 \cdot 10^{-290}:\\ \;\;\;\;\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\left(D \cdot D\right) \cdot \left(\frac{0.125}{d} \cdot \left(M \cdot M\right)\right)\right) - d \cdot t_1\\ \mathbf{elif}\;d \leq 5.5 \cdot 10^{-101}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 + \frac{{\left(M \cdot \frac{D}{d}\right)}^{2} \cdot -0.125}{\frac{\ell}{h}}\right)\\ \mathbf{elif}\;d \leq 10^{+150}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

Alternative 8
Error	24.1
Cost	15136

\[\begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{0.5} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 + \left(M \cdot \left(M \cdot \frac{\frac{D}{\frac{d}{D}}}{\frac{\ell}{\frac{h}{d}}}\right)\right) \cdot -0.125\right)\\ t_1 := \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-d\right)\\ \mathbf{if}\;\ell \leq -2.3 \cdot 10^{+244}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq -3.2 \cdot 10^{+169}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq -2.5 \cdot 10^{+116}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{+18}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-199}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 9 \cdot 10^{-282}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + {\left(\frac{M \cdot D}{d}\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot -0.125\right)\right)\\ \mathbf{elif}\;\ell \leq 5.2 \cdot 10^{-104}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}\\ \mathbf{elif}\;\ell \leq 10^{+130}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 + \frac{{\left(M \cdot \frac{D}{d}\right)}^{2} \cdot -0.125}{\frac{\ell}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

Alternative 9
Error	23.8
Cost	15136

\[\begin{array}{l} t_0 := \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ t_1 := \left({\left(\frac{d}{h}\right)}^{0.5} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 + \left(M \cdot \left(M \cdot \frac{\frac{D}{\frac{d}{D}}}{\frac{\ell}{\frac{h}{d}}}\right)\right) \cdot -0.125\right)\\ t_2 := \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-d\right)\\ t_3 := {\left(M \cdot \frac{D}{d}\right)}^{2} \cdot -0.125\\ \mathbf{if}\;\ell \leq -2.3 \cdot 10^{+244}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq -3.2 \cdot 10^{+169}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq -2.5 \cdot 10^{+116}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{+18}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq -3.4 \cdot 10^{-147}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq 4 \cdot 10^{-246}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \frac{h \cdot t_3}{\ell}\right)\\ \mathbf{elif}\;\ell \leq 2 \cdot 10^{-121}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq 10^{+130}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 + \frac{t_3}{\frac{\ell}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 10
Error	23.6
Cost	15120

\[\begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{0.5} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 + \left(M \cdot \left(M \cdot \frac{\frac{D}{\frac{d}{D}}}{\frac{\ell}{\frac{h}{d}}}\right)\right) \cdot -0.125\right)\\ t_1 := \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-d\right)\\ \mathbf{if}\;\ell \leq -2.3 \cdot 10^{+244}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq -3.2 \cdot 10^{+169}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq -2.5 \cdot 10^{+116}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{+18}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq -7.4 \cdot 10^{-298}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 2 \cdot 10^{-121}:\\ \;\;\;\;\frac{1}{\sqrt{\ell}} \cdot \frac{d}{\sqrt{h}}\\ \mathbf{elif}\;\ell \leq 10^{+130}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 + \frac{{\left(M \cdot \frac{D}{d}\right)}^{2} \cdot -0.125}{\frac{\ell}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

Alternative 11
Error	22.1
Cost	14872

\[\begin{array}{l} t_0 := \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ t_1 := \sqrt{\frac{1}{h \cdot \ell}}\\ t_2 := {\left(M \cdot \frac{D}{d}\right)}^{2}\\ t_3 := t_2 \cdot -0.125\\ \mathbf{if}\;\ell \leq -2.3 \cdot 10^{+244}:\\ \;\;\;\;t_1 \cdot \left(-d\right)\\ \mathbf{elif}\;\ell \leq -3.2 \cdot 10^{+169}:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 + \left(M \cdot \left(M \cdot \frac{\frac{D}{\frac{d}{D}}}{\frac{\ell}{\frac{h}{d}}}\right)\right) \cdot -0.125\right)\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-146}:\\ \;\;\;\;\left(d \cdot t_1\right) \cdot \left(\frac{t_2 \cdot 0.125}{\frac{\ell}{h}} + -1\right)\\ \mathbf{elif}\;\ell \leq 4 \cdot 10^{-246}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \frac{h \cdot t_3}{\ell}\right)\\ \mathbf{elif}\;\ell \leq 2 \cdot 10^{-121}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq 10^{+130}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 + \frac{t_3}{\frac{\ell}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 12
Error	23.4
Cost	14856

\[\begin{array}{l} t_0 := \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-d\right)\\ \mathbf{if}\;d \leq -9.5 \cdot 10^{+15}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq -2.85 \cdot 10^{-151}:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 + \left(M \cdot \left(M \cdot \frac{\frac{D}{\frac{d}{D}}}{\frac{\ell}{\frac{h}{d}}}\right)\right) \cdot -0.125\right)\\ \mathbf{elif}\;d \leq 1.4 \cdot 10^{-305}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

Alternative 13
Error	23.0
Cost	13516

\[\begin{array}{l} t_0 := \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-d\right)\\ \mathbf{if}\;d \leq -2.25 \cdot 10^{+26}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq -2.85 \cdot 10^{-151}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;d \leq 1.4 \cdot 10^{-305}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]

Alternative 14
Error	23.5
Cost	13252

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

Alternative 15
Error	27.5
Cost	7044

\[\begin{array}{l} \mathbf{if}\;\ell \leq -4.2 \cdot 10^{-286}:\\ \;\;\;\;\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-d\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \end{array} \]

Alternative 16
Error	36.6
Cost	6980

\[\begin{array}{l} \mathbf{if}\;\ell \leq -3.1 \cdot 10^{-263}:\\ \;\;\;\;\sqrt{\frac{d \cdot d}{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \end{array} \]

Alternative 17
Error	43.7
Cost	6848

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

Alternative 18
Error	43.7
Cost	6720

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

Reproduce

herbie shell --seed 2022289 
(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)))))