Henrywood and Agarwal, Equation (12)

Average Error: 26.3 → 13.6

Time: 36.9s

Precision: binary64

Cost: 83532

Math

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

↓

\[\begin{array}{l} t_0 := \left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ t_1 := \left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot -0.5\right)\right)\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{-163}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 + \left(0.25 \cdot \frac{\frac{h}{d} \cdot \left(M \cdot D\right)}{d \cdot \frac{\ell}{M \cdot D}}\right) \cdot -0.5\right)\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{-227}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_1 \leq 10^{+243}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{\ell}} \cdot \left(1 + \left(\frac{h}{\ell} \cdot -0.5\right) \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right)}{\sqrt{\frac{h}{d}}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

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)))))
        (t_1
         (*
          (* (pow (/ d h) 0.5) (pow (/ d l) 0.5))
          (+ 1.0 (* (/ h l) (* (pow (/ (* M D) (* d 2.0)) 2.0) -0.5))))))
   (if (<= t_1 -5e-163)
     (*
      (* (sqrt (/ d h)) (/ 1.0 (sqrt (/ l d))))
      (+ 1.0 (* (* 0.25 (/ (* (/ h d) (* M D)) (* d (/ l (* M D))))) -0.5)))
     (if (<= t_1 5e-227)
       t_0
       (if (<= t_1 1e+243)
         (/
          (*
           (sqrt (/ d l))
           (+ 1.0 (* (* (/ h l) -0.5) (pow (* M (* 0.5 (/ D d))) 2.0))))
          (sqrt (/ h d)))
         t_0)))))

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))));
	double t_1 = (pow((d / h), 0.5) * pow((d / l), 0.5)) * (1.0 + ((h / l) * (pow(((M * D) / (d * 2.0)), 2.0) * -0.5)));
	double tmp;
	if (t_1 <= -5e-163) {
		tmp = (sqrt((d / h)) * (1.0 / sqrt((l / d)))) * (1.0 + ((0.25 * (((h / d) * (M * D)) / (d * (l / (M * D))))) * -0.5));
	} else if (t_1 <= 5e-227) {
		tmp = t_0;
	} else if (t_1 <= 1e+243) {
		tmp = (sqrt((d / l)) * (1.0 + (((h / l) * -0.5) * pow((M * (0.5 * (D / d))), 2.0)))) / sqrt((h / d));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    code = (((d / h) ** (1.0d0 / 2.0d0)) * ((d / l) ** (1.0d0 / 2.0d0))) * (1.0d0 - (((1.0d0 / 2.0d0) * (((m * d_1) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))
end function

↓

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = abs((d / sqrt((h * l))))
    t_1 = (((d / h) ** 0.5d0) * ((d / l) ** 0.5d0)) * (1.0d0 + ((h / l) * ((((m * d_1) / (d * 2.0d0)) ** 2.0d0) * (-0.5d0))))
    if (t_1 <= (-5d-163)) then
        tmp = (sqrt((d / h)) * (1.0d0 / sqrt((l / d)))) * (1.0d0 + ((0.25d0 * (((h / d) * (m * d_1)) / (d * (l / (m * d_1))))) * (-0.5d0)))
    else if (t_1 <= 5d-227) then
        tmp = t_0
    else if (t_1 <= 1d+243) then
        tmp = (sqrt((d / l)) * (1.0d0 + (((h / l) * (-0.5d0)) * ((m * (0.5d0 * (d_1 / d))) ** 2.0d0)))) / sqrt((h / d))
    else
        tmp = t_0
    end if
    code = tmp
end function

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))));
	double t_1 = (Math.pow((d / h), 0.5) * Math.pow((d / l), 0.5)) * (1.0 + ((h / l) * (Math.pow(((M * D) / (d * 2.0)), 2.0) * -0.5)));
	double tmp;
	if (t_1 <= -5e-163) {
		tmp = (Math.sqrt((d / h)) * (1.0 / Math.sqrt((l / d)))) * (1.0 + ((0.25 * (((h / d) * (M * D)) / (d * (l / (M * D))))) * -0.5));
	} else if (t_1 <= 5e-227) {
		tmp = t_0;
	} else if (t_1 <= 1e+243) {
		tmp = (Math.sqrt((d / l)) * (1.0 + (((h / l) * -0.5) * Math.pow((M * (0.5 * (D / d))), 2.0)))) / Math.sqrt((h / d));
	} else {
		tmp = t_0;
	}
	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))))
	t_1 = (math.pow((d / h), 0.5) * math.pow((d / l), 0.5)) * (1.0 + ((h / l) * (math.pow(((M * D) / (d * 2.0)), 2.0) * -0.5)))
	tmp = 0
	if t_1 <= -5e-163:
		tmp = (math.sqrt((d / h)) * (1.0 / math.sqrt((l / d)))) * (1.0 + ((0.25 * (((h / d) * (M * D)) / (d * (l / (M * D))))) * -0.5))
	elif t_1 <= 5e-227:
		tmp = t_0
	elif t_1 <= 1e+243:
		tmp = (math.sqrt((d / l)) * (1.0 + (((h / l) * -0.5) * math.pow((M * (0.5 * (D / d))), 2.0)))) / math.sqrt((h / d))
	else:
		tmp = t_0
	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 = abs(Float64(d / sqrt(Float64(h * l))))
	t_1 = Float64(Float64((Float64(d / h) ^ 0.5) * (Float64(d / l) ^ 0.5)) * Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(Float64(M * D) / Float64(d * 2.0)) ^ 2.0) * -0.5))))
	tmp = 0.0
	if (t_1 <= -5e-163)
		tmp = Float64(Float64(sqrt(Float64(d / h)) * Float64(1.0 / sqrt(Float64(l / d)))) * Float64(1.0 + Float64(Float64(0.25 * Float64(Float64(Float64(h / d) * Float64(M * D)) / Float64(d * Float64(l / Float64(M * D))))) * -0.5)));
	elseif (t_1 <= 5e-227)
		tmp = t_0;
	elseif (t_1 <= 1e+243)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * Float64(1.0 + Float64(Float64(Float64(h / l) * -0.5) * (Float64(M * Float64(0.5 * Float64(D / d))) ^ 2.0)))) / sqrt(Float64(h / d)));
	else
		tmp = t_0;
	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))));
	t_1 = (((d / h) ^ 0.5) * ((d / l) ^ 0.5)) * (1.0 + ((h / l) * ((((M * D) / (d * 2.0)) ^ 2.0) * -0.5)));
	tmp = 0.0;
	if (t_1 <= -5e-163)
		tmp = (sqrt((d / h)) * (1.0 / sqrt((l / d)))) * (1.0 + ((0.25 * (((h / d) * (M * D)) / (d * (l / (M * D))))) * -0.5));
	elseif (t_1 <= 5e-227)
		tmp = t_0;
	elseif (t_1 <= 1e+243)
		tmp = (sqrt((d / l)) * (1.0 + (((h / l) * -0.5) * ((M * (0.5 * (D / d))) ^ 2.0)))) / sqrt((h / d));
	else
		tmp = t_0;
	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[Abs[N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Power[N[(d / h), $MachinePrecision], 0.5], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision] * 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]}, If[LessEqual[t$95$1, -5e-163], 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[(N[(0.25 * N[(N[(N[(h / d), $MachinePrecision] * N[(M * D), $MachinePrecision]), $MachinePrecision] / N[(d * N[(l / N[(M * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e-227], t$95$0, If[LessEqual[t$95$1, 1e+243], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision] * N[Power[N[(M * N[(0.5 * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(h / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 3 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)))) < -4.99999999999999977e-163

   1. Initial program 29.2

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

   2. Simplified 29.9

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
      Proof
      (*.f64 (*.f64 (sqrt.f64 (/.f64 d h)) (sqrt.f64 (/.f64 d l))) (-.f64 1 (*.f64 1/2 (*.f64 (pow.f64 (*.f64 (/.f64 M 2) (/.f64 D d)) 2) (/.f64 h l))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (Rewrite<= unpow1/2_binary64 (pow.f64 (/.f64 d h) 1/2)) (sqrt.f64 (/.f64 d l))) (-.f64 1 (*.f64 1/2 (*.f64 (pow.f64 (*.f64 (/.f64 M 2) (/.f64 D d)) 2) (/.f64 h l))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (pow.f64 (/.f64 d h) (Rewrite<= metadata-eval (/.f64 1 2))) (sqrt.f64 (/.f64 d l))) (-.f64 1 (*.f64 1/2 (*.f64 (pow.f64 (*.f64 (/.f64 M 2) (/.f64 D d)) 2) (/.f64 h l))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (Rewrite<= unpow1/2_binary64 (pow.f64 (/.f64 d l) 1/2))) (-.f64 1 (*.f64 1/2 (*.f64 (pow.f64 (*.f64 (/.f64 M 2) (/.f64 D d)) 2) (/.f64 h l))))): 1 points increase in error, 1 points decrease in error
      (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (Rewrite<= metadata-eval (/.f64 1 2)))) (-.f64 1 (*.f64 1/2 (*.f64 (pow.f64 (*.f64 (/.f64 M 2) (/.f64 D d)) 2) (/.f64 h l))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (-.f64 1 (*.f64 (Rewrite<= metadata-eval (/.f64 1 2)) (*.f64 (pow.f64 (*.f64 (/.f64 M 2) (/.f64 D d)) 2) (/.f64 h l))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (-.f64 1 (*.f64 (/.f64 1 2) (*.f64 (pow.f64 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 M D) (*.f64 2 d))) 2) (/.f64 h l))))): 2 points increase in error, 7 points decrease in error
      (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (-.f64 1 (Rewrite<= associate-*l*_binary64 (*.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

   3. Taylor expanded in M around 0 56.1

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

   4. Simplified 41.1

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(0.25 \cdot \left(\frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{\ell} \cdot \frac{h}{d \cdot d}\right)\right)}\right) \]
      Proof
      (*.f64 1/4 (*.f64 (/.f64 (*.f64 (*.f64 D M) (*.f64 D M)) l) (/.f64 h (*.f64 d d)))): 0 points increase in error, 0 points decrease in error
      (*.f64 1/4 (*.f64 (/.f64 (Rewrite<= unswap-sqr_binary64 (*.f64 (*.f64 D D) (*.f64 M M))) l) (/.f64 h (*.f64 d d)))): 40 points increase in error, 5 points decrease in error
      (*.f64 1/4 (*.f64 (/.f64 (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 D 2)) (*.f64 M M)) l) (/.f64 h (*.f64 d d)))): 0 points increase in error, 0 points decrease in error
      (*.f64 1/4 (*.f64 (/.f64 (*.f64 (pow.f64 D 2) (Rewrite<= unpow2_binary64 (pow.f64 M 2))) l) (/.f64 h (*.f64 d d)))): 0 points increase in error, 0 points decrease in error
      (*.f64 1/4 (*.f64 (/.f64 (*.f64 (pow.f64 D 2) (pow.f64 M 2)) l) (/.f64 h (Rewrite<= unpow2_binary64 (pow.f64 d 2))))): 0 points increase in error, 0 points decrease in error
      (*.f64 1/4 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (*.f64 (pow.f64 D 2) (pow.f64 M 2)) h) (*.f64 l (pow.f64 d 2))))): 15 points increase in error, 5 points decrease in error
      (*.f64 1/4 (/.f64 (Rewrite<= associate-*r*_binary64 (*.f64 (pow.f64 D 2) (*.f64 (pow.f64 M 2) h))) (*.f64 l (pow.f64 d 2)))): 8 points increase in error, 4 points decrease in error
      (*.f64 1/4 (/.f64 (*.f64 (pow.f64 D 2) (*.f64 (pow.f64 M 2) h)) (Rewrite=> *-commutative_binary64 (*.f64 (pow.f64 d 2) l)))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 1/4 (*.f64 (pow.f64 D 2) (*.f64 (pow.f64 M 2) h))) (*.f64 (pow.f64 d 2) l))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 1/4 (*.f64 (pow.f64 D 2) (*.f64 (pow.f64 M 2) h))) (Rewrite<= *-commutative_binary64 (*.f64 l (pow.f64 d 2)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*r/_binary64 (*.f64 1/4 (/.f64 (*.f64 (pow.f64 D 2) (*.f64 (pow.f64 M 2) h)) (*.f64 l (pow.f64 d 2))))): 0 points increase in error, 0 points decrease in error

   5. Applied egg-rr 20.5

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

   6. Applied egg-rr 20.9

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

   if -4.99999999999999977e-163 < (*.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)))) < 4.99999999999999961e-227 or 1.0000000000000001e243 < (*.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 53.1

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

   2. Simplified 53.4

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
      Proof
      (*.f64 (*.f64 (sqrt.f64 (/.f64 d h)) (sqrt.f64 (/.f64 d l))) (-.f64 1 (*.f64 1/2 (*.f64 (pow.f64 (*.f64 (/.f64 M 2) (/.f64 D d)) 2) (/.f64 h l))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (Rewrite<= unpow1/2_binary64 (pow.f64 (/.f64 d h) 1/2)) (sqrt.f64 (/.f64 d l))) (-.f64 1 (*.f64 1/2 (*.f64 (pow.f64 (*.f64 (/.f64 M 2) (/.f64 D d)) 2) (/.f64 h l))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (pow.f64 (/.f64 d h) (Rewrite<= metadata-eval (/.f64 1 2))) (sqrt.f64 (/.f64 d l))) (-.f64 1 (*.f64 1/2 (*.f64 (pow.f64 (*.f64 (/.f64 M 2) (/.f64 D d)) 2) (/.f64 h l))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (Rewrite<= unpow1/2_binary64 (pow.f64 (/.f64 d l) 1/2))) (-.f64 1 (*.f64 1/2 (*.f64 (pow.f64 (*.f64 (/.f64 M 2) (/.f64 D d)) 2) (/.f64 h l))))): 1 points increase in error, 1 points decrease in error
      (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (Rewrite<= metadata-eval (/.f64 1 2)))) (-.f64 1 (*.f64 1/2 (*.f64 (pow.f64 (*.f64 (/.f64 M 2) (/.f64 D d)) 2) (/.f64 h l))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (-.f64 1 (*.f64 (Rewrite<= metadata-eval (/.f64 1 2)) (*.f64 (pow.f64 (*.f64 (/.f64 M 2) (/.f64 D d)) 2) (/.f64 h l))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (-.f64 1 (*.f64 (/.f64 1 2) (*.f64 (pow.f64 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 M D) (*.f64 2 d))) 2) (/.f64 h l))))): 2 points increase in error, 7 points decrease in error
      (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (-.f64 1 (Rewrite<= associate-*l*_binary64 (*.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

   3. Taylor expanded in d around inf 40.2

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

   4. Simplified 40.2

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}} \]
      Proof
      (*.f64 d (sqrt.f64 (/.f64 (/.f64 1 l) h))): 0 points increase in error, 0 points decrease in error
      (*.f64 d (sqrt.f64 (Rewrite<= associate-/r*_binary64 (/.f64 1 (*.f64 l h))))): 11 points increase in error, 16 points decrease in error
      (Rewrite<= *-commutative_binary64 (*.f64 (sqrt.f64 (/.f64 1 (*.f64 l h))) d)): 0 points increase in error, 0 points decrease in error

   5. Applied egg-rr 51.0

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

   6. Simplified 24.1

      \[\leadsto \color{blue}{\left|\frac{d}{\sqrt{\ell \cdot h}}\right|} \]
      Proof
      (fabs.f64 (/.f64 d (sqrt.f64 (*.f64 l h)))): 0 points increase in error, 0 points decrease in error
      (fabs.f64 (/.f64 (Rewrite<= *-lft-identity_binary64 (*.f64 1 d)) (sqrt.f64 (*.f64 l h)))): 0 points increase in error, 0 points decrease in error
      (fabs.f64 (/.f64 (*.f64 1 d) (Rewrite<= unpow1/2_binary64 (pow.f64 (*.f64 l h) 1/2)))): 0 points increase in error, 0 points decrease in error
      (fabs.f64 (/.f64 (*.f64 1 d) (Rewrite<= exp-to-pow_binary64 (exp.f64 (*.f64 (log.f64 (*.f64 l h)) 1/2))))): 161 points increase in error, 25 points decrease in error
      (fabs.f64 (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 1 (exp.f64 (*.f64 (log.f64 (*.f64 l h)) 1/2))) d))): 30 points increase in error, 24 points decrease in error
      (fabs.f64 (*.f64 (Rewrite<= exp-neg_binary64 (exp.f64 (neg.f64 (*.f64 (log.f64 (*.f64 l h)) 1/2)))) d)): 29 points increase in error, 15 points decrease in error
      (fabs.f64 (*.f64 (exp.f64 (Rewrite<= distribute-lft-neg-out_binary64 (*.f64 (neg.f64 (log.f64 (*.f64 l h))) 1/2))) d)): 0 points increase in error, 0 points decrease in error
      (fabs.f64 (*.f64 (Rewrite=> exp-prod_binary64 (pow.f64 (exp.f64 (neg.f64 (log.f64 (*.f64 l h)))) 1/2)) d)): 9 points increase in error, 8 points decrease in error
      (fabs.f64 (*.f64 (Rewrite=> unpow1/2_binary64 (sqrt.f64 (exp.f64 (neg.f64 (log.f64 (*.f64 l h)))))) d)): 0 points increase in error, 0 points decrease in error
      (fabs.f64 (*.f64 (sqrt.f64 (Rewrite=> exp-neg_binary64 (/.f64 1 (exp.f64 (log.f64 (*.f64 l h)))))) d)): 10 points increase in error, 14 points decrease in error
      (fabs.f64 (*.f64 (sqrt.f64 (/.f64 1 (Rewrite=> rem-exp-log_binary64 (*.f64 l h)))) d)): 24 points increase in error, 158 points decrease in error
      (Rewrite<= rem-sqrt-square_binary64 (sqrt.f64 (*.f64 (*.f64 (sqrt.f64 (/.f64 1 (*.f64 l h))) d) (*.f64 (sqrt.f64 (/.f64 1 (*.f64 l h))) d)))): 60 points increase in error, 23 points decrease in error
      (sqrt.f64 (Rewrite<= unswap-sqr_binary64 (*.f64 (*.f64 (sqrt.f64 (/.f64 1 (*.f64 l h))) (sqrt.f64 (/.f64 1 (*.f64 l h)))) (*.f64 d d)))): 45 points increase in error, 11 points decrease in error
      (sqrt.f64 (*.f64 (Rewrite=> rem-square-sqrt_binary64 (/.f64 1 (*.f64 l h))) (*.f64 d d))): 6 points increase in error, 17 points decrease in error

   if 4.99999999999999961e-227 < (*.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)))) < 1.0000000000000001e243

   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. Simplified 1.0

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
      Proof
      (*.f64 (*.f64 (sqrt.f64 (/.f64 d h)) (sqrt.f64 (/.f64 d l))) (-.f64 1 (*.f64 1/2 (*.f64 (pow.f64 (*.f64 (/.f64 M 2) (/.f64 D d)) 2) (/.f64 h l))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (Rewrite<= unpow1/2_binary64 (pow.f64 (/.f64 d h) 1/2)) (sqrt.f64 (/.f64 d l))) (-.f64 1 (*.f64 1/2 (*.f64 (pow.f64 (*.f64 (/.f64 M 2) (/.f64 D d)) 2) (/.f64 h l))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (pow.f64 (/.f64 d h) (Rewrite<= metadata-eval (/.f64 1 2))) (sqrt.f64 (/.f64 d l))) (-.f64 1 (*.f64 1/2 (*.f64 (pow.f64 (*.f64 (/.f64 M 2) (/.f64 D d)) 2) (/.f64 h l))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (Rewrite<= unpow1/2_binary64 (pow.f64 (/.f64 d l) 1/2))) (-.f64 1 (*.f64 1/2 (*.f64 (pow.f64 (*.f64 (/.f64 M 2) (/.f64 D d)) 2) (/.f64 h l))))): 1 points increase in error, 1 points decrease in error
      (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (Rewrite<= metadata-eval (/.f64 1 2)))) (-.f64 1 (*.f64 1/2 (*.f64 (pow.f64 (*.f64 (/.f64 M 2) (/.f64 D d)) 2) (/.f64 h l))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (-.f64 1 (*.f64 (Rewrite<= metadata-eval (/.f64 1 2)) (*.f64 (pow.f64 (*.f64 (/.f64 M 2) (/.f64 D d)) 2) (/.f64 h l))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (-.f64 1 (*.f64 (/.f64 1 2) (*.f64 (pow.f64 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 M D) (*.f64 2 d))) 2) (/.f64 h l))))): 2 points increase in error, 7 points decrease in error
      (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (-.f64 1 (Rewrite<= associate-*l*_binary64 (*.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

   3. Applied egg-rr 1.1

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

   4. Applied egg-rr 1.0

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

4. Final simplification 13.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot -0.5\right)\right) \leq -5 \cdot 10^{-163}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 + \left(0.25 \cdot \frac{\frac{h}{d} \cdot \left(M \cdot D\right)}{d \cdot \frac{\ell}{M \cdot D}}\right) \cdot -0.5\right)\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot -0.5\right)\right) \leq 5 \cdot 10^{-227}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot -0.5\right)\right) \leq 10^{+243}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{\ell}} \cdot \left(1 + \left(\frac{h}{\ell} \cdot -0.5\right) \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right)}{\sqrt{\frac{h}{d}}}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \end{array} \]

Alternatives

Alternative 1
Error	20.5
Cost	21400

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right)\right)\\ \mathbf{if}\;\ell \leq -4 \cdot 10^{+238}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq -7.4 \cdot 10^{-57}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(-1 - \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{0.5 \cdot \left(M \cdot D\right)}{d}\right)}^{2}\right)\right)\\ \mathbf{elif}\;\ell \leq -6.5 \cdot 10^{-180}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq -4.8 \cdot 10^{-261}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \mathbf{elif}\;\ell \leq -1 \cdot 10^{-310}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq 3.35 \cdot 10^{-130}:\\ \;\;\;\;\frac{{\ell}^{-0.5}}{\frac{\sqrt{h}}{d}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \left(\frac{h}{\ell} \cdot -0.5\right) \cdot {\left(M \cdot \frac{0.5 \cdot D}{d}\right)}^{2}\right) \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 2
Error	17.5
Cost	21136

\[\begin{array}{l} t_0 := \frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \mathbf{if}\;d \leq -1.2 \cdot 10^{+124}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \mathbf{elif}\;d \leq -1 \cdot 10^{-126}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 + 0.5 \cdot \left(\frac{h \cdot \left(M \cdot D\right)}{\frac{\ell}{M \cdot D} \cdot \left(d \cdot d\right)} \cdot -0.25\right)\right)\\ \mathbf{elif}\;d \leq -2.3 \cdot 10^{-302}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(-1 - \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{0.5 \cdot \left(M \cdot D\right)}{d}\right)}^{2}\right)\right)\\ \mathbf{elif}\;d \leq 2 \cdot 10^{+193}:\\ \;\;\;\;\left(1 + -0.125 \cdot \frac{h}{d \cdot \frac{d}{\frac{{\left(M \cdot D\right)}^{2}}{\ell}}}\right) \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	21.4
Cost	15184

\[\begin{array}{l} t_0 := \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 + 0.5 \cdot \left(\left(\frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{\ell} \cdot \frac{h}{d \cdot d}\right) \cdot -0.25\right)\right)\\ t_1 := \left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \mathbf{if}\;d \leq -1.25 \cdot 10^{+117}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq -2.2 \cdot 10^{-91}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 6.5 \cdot 10^{-118}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq 2.7 \cdot 10^{+56}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 4
Error	20.7
Cost	15184

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ t_1 := \left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \mathbf{if}\;d \leq -1.85 \cdot 10^{+118}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq -4.8 \cdot 10^{-92}:\\ \;\;\;\;t_0 \cdot \left(1 + 0.5 \cdot \left(\left(\frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{\ell} \cdot \frac{h}{d \cdot d}\right) \cdot -0.25\right)\right)\\ \mathbf{elif}\;d \leq 4.2 \cdot 10^{-172}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq 6.8 \cdot 10^{+47}:\\ \;\;\;\;t_0 \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \frac{D \cdot \left(M \cdot \frac{h}{d}\right)}{d \cdot \frac{\ell}{M \cdot D}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 5
Error	19.6
Cost	15184

\[\begin{array}{l} t_0 := \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 + 0.5 \cdot \left(\frac{h \cdot \left(M \cdot D\right)}{\frac{\ell}{M \cdot D} \cdot \left(d \cdot d\right)} \cdot -0.25\right)\right)\\ t_1 := \left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \mathbf{if}\;d \leq -6 \cdot 10^{+120}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq -6 \cdot 10^{-99}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 1.85 \cdot 10^{-118}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq 2.8 \cdot 10^{+56}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 6
Error	22.9
Cost	14468

\[\begin{array}{l} t_0 := \frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \mathbf{if}\;\ell \leq -1.8 \cdot 10^{-151}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(-1 + \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2}\right)\right)\\ \mathbf{elif}\;\ell \leq -9.8 \cdot 10^{-286}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \mathbf{elif}\;\ell \leq 1.8 \cdot 10^{-54}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq 3.4 \cdot 10^{-26}:\\ \;\;\;\;\frac{M \cdot D}{\frac{d}{M \cdot D}} \cdot \left(-0.125 \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	22.9
Cost	14468

\[\begin{array}{l} t_0 := \frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \mathbf{if}\;\ell \leq -9.8 \cdot 10^{-151}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(-1 - \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{0.5 \cdot \left(M \cdot D\right)}{d}\right)}^{2}\right)\right)\\ \mathbf{elif}\;\ell \leq -9.8 \cdot 10^{-286}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \mathbf{elif}\;\ell \leq 1.8 \cdot 10^{-54}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq 3.5 \cdot 10^{-26}:\\ \;\;\;\;\frac{M \cdot D}{\frac{d}{M \cdot D}} \cdot \left(-0.125 \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	23.1
Cost	13516

\[\begin{array}{l} t_0 := \left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \mathbf{if}\;d \leq -1.1 \cdot 10^{+119}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq -4.2 \cdot 10^{-174}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;d \leq 2.5 \cdot 10^{-201}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 9
Error	27.7
Cost	13252

\[\begin{array}{l} \mathbf{if}\;\ell \leq 2.8 \cdot 10^{+251}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \end{array} \]

Alternative 10
Error	23.4
Cost	13252

\[\begin{array}{l} \mathbf{if}\;d \leq 3.2 \cdot 10^{-201}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 11
Error	33.4
Cost	7112

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{if}\;\ell \leq 4.4 \cdot 10^{-294}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq 1.66 \cdot 10^{+242}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 12
Error	33.4
Cost	7112

\[\begin{array}{l} \mathbf{if}\;\ell \leq -1.5 \cdot 10^{-296}:\\ \;\;\;\;\sqrt{\frac{d}{\frac{\ell}{\frac{d}{h}}}}\\ \mathbf{elif}\;\ell \leq 5.8 \cdot 10^{+248}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \end{array} \]

Alternative 13
Error	27.9
Cost	7112

\[\begin{array}{l} \mathbf{if}\;\ell \leq -9.8 \cdot 10^{-286}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{elif}\;\ell \leq 5.5 \cdot 10^{+242}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \end{array} \]

Alternative 14
Error	33.1
Cost	6980

\[\begin{array}{l} \mathbf{if}\;d \leq 3.8 \cdot 10^{-283}:\\ \;\;\;\;\sqrt{d \cdot \frac{\frac{d}{\ell}}{h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot {\left(h \cdot \ell\right)}^{-0.5}\\ \end{array} \]

Alternative 15
Error	43.7
Cost	6784

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

Alternative 16
Error	43.7
Cost	6720

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

Reproduce

herbie shell --seed 2022326 
(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)))))