Henrywood and Agarwal, Equation (12)

Percentage Accurate: 65.8% → 78.6%

Time: 25.7s

Alternatives: 26

Speedup: 1.4×

Specification

Math

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

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)))))

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)));
}

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

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)));
}

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)))

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

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

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]

Initial Program: 65.8% accurate, 1.0× speedup

Math

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

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)))))

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)));
}

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

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)));
}

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)))

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

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

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]

Alternative 1: 78.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := {\left(M\_m \cdot \frac{D}{d}\right)}^{2}\\ t_1 := \sqrt{-d}\\ t_2 := \frac{t\_1}{\sqrt{-h}} \cdot \left(\sqrt{d \cdot \frac{1}{\ell}} \cdot \left(1 + \frac{{\left(D \cdot \frac{M\_m}{d}\right)}^{2} \cdot \left(h \cdot -0.125\right)}{\ell}\right)\right)\\ t_3 := 1 - h \cdot \left(0.125 \cdot \frac{t\_0}{\ell}\right)\\ \mathbf{if}\;h \leq -2 \cdot 10^{+143}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;h \leq -8.5 \cdot 10^{-86}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\frac{t\_1}{\sqrt{-\ell}} \cdot t\_3\right)\\ \mathbf{elif}\;h \leq -4 \cdot 10^{-310}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;h \leq 6 \cdot 10^{+125}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(t\_0 \cdot -0.125\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(t\_3 \cdot \sqrt{\frac{d}{\ell}}\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (let* ((t_0 (pow (* M_m (/ D d)) 2.0))
        (t_1 (sqrt (- d)))
        (t_2
         (*
          (/ t_1 (sqrt (- h)))
          (*
           (sqrt (* d (/ 1.0 l)))
           (+ 1.0 (/ (* (pow (* D (/ M_m d)) 2.0) (* h -0.125)) l)))))
        (t_3 (- 1.0 (* h (* 0.125 (/ t_0 l))))))
   (if (<= h -2e+143)
     t_2
     (if (<= h -8.5e-86)
       (* (sqrt (/ d h)) (* (/ t_1 (sqrt (- l))) t_3))
       (if (<= h -4e-310)
         t_2
         (if (<= h 6e+125)
           (* (/ d (sqrt (* l h))) (+ 1.0 (* (/ h l) (* t_0 -0.125))))
           (* (/ (sqrt d) (sqrt h)) (* t_3 (sqrt (/ d l))))))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double t_0 = pow((M_m * (D / d)), 2.0);
	double t_1 = sqrt(-d);
	double t_2 = (t_1 / sqrt(-h)) * (sqrt((d * (1.0 / l))) * (1.0 + ((pow((D * (M_m / d)), 2.0) * (h * -0.125)) / l)));
	double t_3 = 1.0 - (h * (0.125 * (t_0 / l)));
	double tmp;
	if (h <= -2e+143) {
		tmp = t_2;
	} else if (h <= -8.5e-86) {
		tmp = sqrt((d / h)) * ((t_1 / sqrt(-l)) * t_3);
	} else if (h <= -4e-310) {
		tmp = t_2;
	} else if (h <= 6e+125) {
		tmp = (d / sqrt((l * h))) * (1.0 + ((h / l) * (t_0 * -0.125)));
	} else {
		tmp = (sqrt(d) / sqrt(h)) * (t_3 * sqrt((d / l)));
	}
	return tmp;
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = (m_m * (d_1 / d)) ** 2.0d0
    t_1 = sqrt(-d)
    t_2 = (t_1 / sqrt(-h)) * (sqrt((d * (1.0d0 / l))) * (1.0d0 + ((((d_1 * (m_m / d)) ** 2.0d0) * (h * (-0.125d0))) / l)))
    t_3 = 1.0d0 - (h * (0.125d0 * (t_0 / l)))
    if (h <= (-2d+143)) then
        tmp = t_2
    else if (h <= (-8.5d-86)) then
        tmp = sqrt((d / h)) * ((t_1 / sqrt(-l)) * t_3)
    else if (h <= (-4d-310)) then
        tmp = t_2
    else if (h <= 6d+125) then
        tmp = (d / sqrt((l * h))) * (1.0d0 + ((h / l) * (t_0 * (-0.125d0))))
    else
        tmp = (sqrt(d) / sqrt(h)) * (t_3 * sqrt((d / l)))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double t_0 = Math.pow((M_m * (D / d)), 2.0);
	double t_1 = Math.sqrt(-d);
	double t_2 = (t_1 / Math.sqrt(-h)) * (Math.sqrt((d * (1.0 / l))) * (1.0 + ((Math.pow((D * (M_m / d)), 2.0) * (h * -0.125)) / l)));
	double t_3 = 1.0 - (h * (0.125 * (t_0 / l)));
	double tmp;
	if (h <= -2e+143) {
		tmp = t_2;
	} else if (h <= -8.5e-86) {
		tmp = Math.sqrt((d / h)) * ((t_1 / Math.sqrt(-l)) * t_3);
	} else if (h <= -4e-310) {
		tmp = t_2;
	} else if (h <= 6e+125) {
		tmp = (d / Math.sqrt((l * h))) * (1.0 + ((h / l) * (t_0 * -0.125)));
	} else {
		tmp = (Math.sqrt(d) / Math.sqrt(h)) * (t_3 * Math.sqrt((d / l)));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	t_0 = math.pow((M_m * (D / d)), 2.0)
	t_1 = math.sqrt(-d)
	t_2 = (t_1 / math.sqrt(-h)) * (math.sqrt((d * (1.0 / l))) * (1.0 + ((math.pow((D * (M_m / d)), 2.0) * (h * -0.125)) / l)))
	t_3 = 1.0 - (h * (0.125 * (t_0 / l)))
	tmp = 0
	if h <= -2e+143:
		tmp = t_2
	elif h <= -8.5e-86:
		tmp = math.sqrt((d / h)) * ((t_1 / math.sqrt(-l)) * t_3)
	elif h <= -4e-310:
		tmp = t_2
	elif h <= 6e+125:
		tmp = (d / math.sqrt((l * h))) * (1.0 + ((h / l) * (t_0 * -0.125)))
	else:
		tmp = (math.sqrt(d) / math.sqrt(h)) * (t_3 * math.sqrt((d / l)))
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	t_0 = Float64(M_m * Float64(D / d)) ^ 2.0
	t_1 = sqrt(Float64(-d))
	t_2 = Float64(Float64(t_1 / sqrt(Float64(-h))) * Float64(sqrt(Float64(d * Float64(1.0 / l))) * Float64(1.0 + Float64(Float64((Float64(D * Float64(M_m / d)) ^ 2.0) * Float64(h * -0.125)) / l))))
	t_3 = Float64(1.0 - Float64(h * Float64(0.125 * Float64(t_0 / l))))
	tmp = 0.0
	if (h <= -2e+143)
		tmp = t_2;
	elseif (h <= -8.5e-86)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(Float64(t_1 / sqrt(Float64(-l))) * t_3));
	elseif (h <= -4e-310)
		tmp = t_2;
	elseif (h <= 6e+125)
		tmp = Float64(Float64(d / sqrt(Float64(l * h))) * Float64(1.0 + Float64(Float64(h / l) * Float64(t_0 * -0.125))));
	else
		tmp = Float64(Float64(sqrt(d) / sqrt(h)) * Float64(t_3 * sqrt(Float64(d / l))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp_2 = code(d, h, l, M_m, D)
	t_0 = (M_m * (D / d)) ^ 2.0;
	t_1 = sqrt(-d);
	t_2 = (t_1 / sqrt(-h)) * (sqrt((d * (1.0 / l))) * (1.0 + ((((D * (M_m / d)) ^ 2.0) * (h * -0.125)) / l)));
	t_3 = 1.0 - (h * (0.125 * (t_0 / l)));
	tmp = 0.0;
	if (h <= -2e+143)
		tmp = t_2;
	elseif (h <= -8.5e-86)
		tmp = sqrt((d / h)) * ((t_1 / sqrt(-l)) * t_3);
	elseif (h <= -4e-310)
		tmp = t_2;
	elseif (h <= 6e+125)
		tmp = (d / sqrt((l * h))) * (1.0 + ((h / l) * (t_0 * -0.125)));
	else
		tmp = (sqrt(d) / sqrt(h)) * (t_3 * sqrt((d / l)));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := Block[{t$95$0 = N[Power[N[(M$95$m * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[(-d)], $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$1 / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(d * N[(1.0 / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(N[(N[Power[N[(D * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h * -0.125), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(1.0 - N[(h * N[(0.125 * N[(t$95$0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[h, -2e+143], t$95$2, If[LessEqual[h, -8.5e-86], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[(t$95$1 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, -4e-310], t$95$2, If[LessEqual[h, 6e+125], N[(N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(t$95$0 * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(t$95$3 * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if h < -2e143 or -8.499999999999999e-86 < h < -3.999999999999988e-310

   1. Initial program 54.6%

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

   3. Simplified 54.6%

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

   5. Taylor expanded in h around -inf 46.3%

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

      1. associate-*r* 46.3%

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

      2. neg-mul-1 46.3%

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

      3. sub-neg 46.3%

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

      4. distribute-lft-in 46.3%

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

   7. Simplified 60.9%

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

      1. div-inv 60.9%

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

   9. Applied egg-rr 60.9%

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

       1. pow1 60.9%

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

       2. associate-*r* 60.9%

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

   11. Applied egg-rr 60.9%

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

       1. unpow1 60.9%

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

       2. *-commutative 60.9%

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

       3. unpow2 60.9%

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

       4. *-commutative 60.9%

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

       5. *-commutative 60.9%

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

       6. swap-sqr 50.3%

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

       7. times-frac 46.4%

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

       8. unpow2 46.4%

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

       9. unpow2 46.4%

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

       10. unpow2 46.4%

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

       11. associate-*r/ 43.7%

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

       12. times-frac 46.3%

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

       13. associate-/r* 48.5%

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

       14. associate-*l/ 48.5%

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

   13. Simplified 60.9%

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

       1. frac-2neg 54.6%

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

       2. sqrt-div 73.4%

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

   15. Applied egg-rr 79.7%

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

   if -2e143 < h < -8.499999999999999e-86

   1. Initial program 63.0%

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

   3. Simplified 66.1%

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

   5. Taylor expanded in h around -inf 48.6%

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

      1. associate-*r* 48.6%

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

      2. neg-mul-1 48.6%

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

      3. sub-neg 48.6%

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

      4. distribute-lft-in 48.6%

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

   7. Simplified 71.5%

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

      1. frac-2neg 65.9%

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

      2. sqrt-div 76.8%

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

   9. Applied egg-rr 88.1%

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

   if -3.999999999999988e-310 < h < 6.0000000000000003e125

   1. Initial program 75.6%

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

   3. Simplified 76.6%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. pow1 76.6%

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

      2. sqrt-unprod 67.6%

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

   6. Applied egg-rr 67.6%

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

      1. unpow1 67.6%

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

   8. Simplified 67.6%

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

   9. Taylor expanded in M around 0 40.4%

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

       1. *-commutative 40.4%

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

       2. associate-*r* 41.4%

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

       3. times-frac 43.6%

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

       4. *-commutative 43.6%

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

       5. associate-/l* 43.6%

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

       6. unpow2 43.6%

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

       7. unpow2 43.6%

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

       8. unpow2 43.6%

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

       9. times-frac 49.2%

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

       10. swap-sqr 67.6%

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

       11. unpow2 67.6%

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

       12. associate-*r/ 66.5%

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

       13. *-commutative 66.5%

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

       14. associate-/l* 67.6%

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

   11. Simplified 67.6%

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

       1. pow1 67.6%

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

       2. *-commutative 67.6%

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

       3. associate-*l* 67.6%

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

       4. frac-times 59.1%

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

       5. sqrt-div 64.7%

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

       6. sqrt-unprod 91.3%

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

       7. add-sqr-sqrt 91.6%

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

   13. Applied egg-rr 91.6%

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

       1. unpow1 91.6%

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

       2. *-commutative 91.6%

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

       3. sub-neg 91.6%

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

       4. distribute-rgt-neg-in 91.6%

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

       5. *-commutative 91.6%

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

       6. distribute-lft-neg-in 91.6%

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

       7. metadata-eval 91.6%

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

       8. associate-*r* 91.6%

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

       9. *-commutative 91.6%

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

       10. *-commutative 91.6%

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

       11. associate-*r/ 90.5%

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

       12. *-commutative 90.5%

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

       13. associate-/l* 91.6%

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

   15. Simplified 91.6%

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

   if 6.0000000000000003e125 < h

   1. Initial program 53.2%

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

   3. Simplified 53.2%

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

   5. Taylor expanded in h around -inf 44.2%

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

      1. associate-*r* 44.2%

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

      2. neg-mul-1 44.2%

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

      3. sub-neg 44.2%

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

      4. distribute-lft-in 44.2%

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

   7. Simplified 61.8%

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

      1. sqrt-div 81.8%

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

   9. Applied egg-rr 81.8%

      \[\leadsto \color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\left(-h\right) \cdot \left(0.125 \cdot \frac{{\left(M \cdot \frac{D}{d}\right)}^{2}}{\ell}\right) + 1\right)\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -2 \cdot 10^{+143}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(\sqrt{d \cdot \frac{1}{\ell}} \cdot \left(1 + \frac{{\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \left(h \cdot -0.125\right)}{\ell}\right)\right)\\ \mathbf{elif}\;h \leq -8.5 \cdot 10^{-86}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \left(1 - h \cdot \left(0.125 \cdot \frac{{\left(M \cdot \frac{D}{d}\right)}^{2}}{\ell}\right)\right)\right)\\ \mathbf{elif}\;h \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(\sqrt{d \cdot \frac{1}{\ell}} \cdot \left(1 + \frac{{\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \left(h \cdot -0.125\right)}{\ell}\right)\right)\\ \mathbf{elif}\;h \leq 6 \cdot 10^{+125}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(M \cdot \frac{D}{d}\right)}^{2} \cdot -0.125\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\left(1 - h \cdot \left(0.125 \cdot \frac{{\left(M \cdot \frac{D}{d}\right)}^{2}}{\ell}\right)\right) \cdot \sqrt{\frac{d}{\ell}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 75.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{D \cdot M\_m}{d \cdot 2}\right)}^{2}\right)\right) \leq 2 \cdot 10^{+299}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\left(1 - h \cdot \left(0.125 \cdot \frac{{\left(M\_m \cdot \frac{D}{d}\right)}^{2}}{\ell}\right)\right) \cdot \sqrt{\frac{d}{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (if (<=
      (*
       (* (pow (/ d h) 0.5) (pow (/ d l) 0.5))
       (- 1.0 (* (/ h l) (* 0.5 (pow (/ (* D M_m) (* d 2.0)) 2.0)))))
      2e+299)
   (*
    (sqrt (/ d h))
    (* (- 1.0 (* h (* 0.125 (/ (pow (* M_m (/ D d)) 2.0) l)))) (sqrt (/ d l))))
   (fabs (/ d (sqrt (* l h))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (((pow((d / h), 0.5) * pow((d / l), 0.5)) * (1.0 - ((h / l) * (0.5 * pow(((D * M_m) / (d * 2.0)), 2.0))))) <= 2e+299) {
		tmp = sqrt((d / h)) * ((1.0 - (h * (0.125 * (pow((M_m * (D / d)), 2.0) / l)))) * sqrt((d / l)));
	} else {
		tmp = fabs((d / sqrt((l * h))));
	}
	return tmp;
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (((((d / h) ** 0.5d0) * ((d / l) ** 0.5d0)) * (1.0d0 - ((h / l) * (0.5d0 * (((d_1 * m_m) / (d * 2.0d0)) ** 2.0d0))))) <= 2d+299) then
        tmp = sqrt((d / h)) * ((1.0d0 - (h * (0.125d0 * (((m_m * (d_1 / d)) ** 2.0d0) / l)))) * sqrt((d / l)))
    else
        tmp = abs((d / sqrt((l * h))))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (((Math.pow((d / h), 0.5) * Math.pow((d / l), 0.5)) * (1.0 - ((h / l) * (0.5 * Math.pow(((D * M_m) / (d * 2.0)), 2.0))))) <= 2e+299) {
		tmp = Math.sqrt((d / h)) * ((1.0 - (h * (0.125 * (Math.pow((M_m * (D / d)), 2.0) / l)))) * Math.sqrt((d / l)));
	} else {
		tmp = Math.abs((d / Math.sqrt((l * h))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	tmp = 0
	if ((math.pow((d / h), 0.5) * math.pow((d / l), 0.5)) * (1.0 - ((h / l) * (0.5 * math.pow(((D * M_m) / (d * 2.0)), 2.0))))) <= 2e+299:
		tmp = math.sqrt((d / h)) * ((1.0 - (h * (0.125 * (math.pow((M_m * (D / d)), 2.0) / l)))) * math.sqrt((d / l)))
	else:
		tmp = math.fabs((d / math.sqrt((l * h))))
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	tmp = 0.0
	if (Float64(Float64((Float64(d / h) ^ 0.5) * (Float64(d / l) ^ 0.5)) * Float64(1.0 - Float64(Float64(h / l) * Float64(0.5 * (Float64(Float64(D * M_m) / Float64(d * 2.0)) ^ 2.0))))) <= 2e+299)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(Float64(1.0 - Float64(h * Float64(0.125 * Float64((Float64(M_m * Float64(D / d)) ^ 2.0) / l)))) * sqrt(Float64(d / l))));
	else
		tmp = abs(Float64(d / sqrt(Float64(l * h))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp_2 = code(d, h, l, M_m, D)
	tmp = 0.0;
	if (((((d / h) ^ 0.5) * ((d / l) ^ 0.5)) * (1.0 - ((h / l) * (0.5 * (((D * M_m) / (d * 2.0)) ^ 2.0))))) <= 2e+299)
		tmp = sqrt((d / h)) * ((1.0 - (h * (0.125 * (((M_m * (D / d)) ^ 2.0) / l)))) * sqrt((d / l)));
	else
		tmp = abs((d / sqrt((l * h))));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := If[LessEqual[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[(0.5 * N[Power[N[(N[(D * M$95$m), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e+299], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[(1.0 - N[(h * N[(0.125 * N[(N[Power[N[(M$95$m * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Abs[N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 2.0000000000000001e299

   1. Initial program 84.5%

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

   3. Simplified 86.1%

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

   5. Taylor expanded in h around -inf 60.7%

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

      1. associate-*r* 60.7%

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

      2. neg-mul-1 60.7%

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

      3. sub-neg 60.7%

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

      4. distribute-lft-in 60.7%

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

   7. Simplified 87.4%

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

   if 2.0000000000000001e299 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 17.2%

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

   3. Simplified 17.2%

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

   5. Taylor expanded in d around inf 34.2%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. *-un-lft-identity 34.2%

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

      2. inv-pow 34.2%

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

      3. sqrt-pow1 34.2%

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

      4. metadata-eval 34.2%

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

   7. Applied egg-rr 34.2%

      \[\leadsto d \cdot \color{blue}{\left(1 \cdot {\left(h \cdot \ell\right)}^{-0.5}\right)} \]
   8. Step-by-step derivation

      1. *-lft-identity 34.2%

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

   9. Simplified 34.2%

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

   10. Taylor expanded in d around 0 34.2%

       \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   11. Step-by-step derivation

       1. *-commutative 34.2%

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

       2. associate-/r* 34.3%

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

   12. Simplified 34.3%

       \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}} \]
   13. Step-by-step derivation

       1. add-sqr-sqrt 33.7%

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

       2. sqrt-unprod 34.7%

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

       3. swap-sqr 32.2%

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

       4. sqr-neg 32.2%

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

       5. add-sqr-sqrt 32.2%

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

       6. associate-/l/ 32.3%

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

       7. div-inv 32.2%

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

       8. sqr-neg 32.2%

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

       9. frac-times 26.8%

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

       10. sqrt-prod 26.7%

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

       11. frac-2neg 26.7%

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

       12. sqrt-undiv 18.9%

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

       13. add-sqr-sqrt 18.9%

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

       14. sqrt-prod 11.8%

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

       15. rem-sqrt-square 18.9%

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

   14. Applied egg-rr 59.0%

       \[\leadsto \color{blue}{\left|\frac{d}{\sqrt{h \cdot \ell}}\right|} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}\right)\right) \leq 2 \cdot 10^{+299}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\left(1 - h \cdot \left(0.125 \cdot \frac{{\left(M \cdot \frac{D}{d}\right)}^{2}}{\ell}\right)\right) \cdot \sqrt{\frac{d}{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := 1 - h \cdot \left(0.125 \cdot \frac{{\left(M\_m \cdot \frac{D}{d}\right)}^{2}}{\ell}\right)\\ t_1 := \sqrt{-d}\\ t_2 := \frac{t\_1}{\sqrt{-\ell}}\\ \mathbf{if}\;\ell \leq -8 \cdot 10^{-79}:\\ \;\;\;\;t\_2 \cdot \left(\frac{t\_1}{\sqrt{-h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(D \cdot \frac{\frac{M\_m}{2}}{d}\right)}^{2} \cdot -0.5\right)\right)\right)\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(t\_2 \cdot t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(t\_0 \cdot \sqrt{\frac{d}{\ell}}\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (let* ((t_0 (- 1.0 (* h (* 0.125 (/ (pow (* M_m (/ D d)) 2.0) l)))))
        (t_1 (sqrt (- d)))
        (t_2 (/ t_1 (sqrt (- l)))))
   (if (<= l -8e-79)
     (*
      t_2
      (*
       (/ t_1 (sqrt (- h)))
       (+ 1.0 (* (/ h l) (* (pow (* D (/ (/ M_m 2.0) d)) 2.0) -0.5)))))
     (if (<= l -2e-310)
       (* (sqrt (/ d h)) (* t_2 t_0))
       (* (/ (sqrt d) (sqrt h)) (* t_0 (sqrt (/ d l))))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double t_0 = 1.0 - (h * (0.125 * (pow((M_m * (D / d)), 2.0) / l)));
	double t_1 = sqrt(-d);
	double t_2 = t_1 / sqrt(-l);
	double tmp;
	if (l <= -8e-79) {
		tmp = t_2 * ((t_1 / sqrt(-h)) * (1.0 + ((h / l) * (pow((D * ((M_m / 2.0) / d)), 2.0) * -0.5))));
	} else if (l <= -2e-310) {
		tmp = sqrt((d / h)) * (t_2 * t_0);
	} else {
		tmp = (sqrt(d) / sqrt(h)) * (t_0 * sqrt((d / l)));
	}
	return tmp;
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = 1.0d0 - (h * (0.125d0 * (((m_m * (d_1 / d)) ** 2.0d0) / l)))
    t_1 = sqrt(-d)
    t_2 = t_1 / sqrt(-l)
    if (l <= (-8d-79)) then
        tmp = t_2 * ((t_1 / sqrt(-h)) * (1.0d0 + ((h / l) * (((d_1 * ((m_m / 2.0d0) / d)) ** 2.0d0) * (-0.5d0)))))
    else if (l <= (-2d-310)) then
        tmp = sqrt((d / h)) * (t_2 * t_0)
    else
        tmp = (sqrt(d) / sqrt(h)) * (t_0 * sqrt((d / l)))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double t_0 = 1.0 - (h * (0.125 * (Math.pow((M_m * (D / d)), 2.0) / l)));
	double t_1 = Math.sqrt(-d);
	double t_2 = t_1 / Math.sqrt(-l);
	double tmp;
	if (l <= -8e-79) {
		tmp = t_2 * ((t_1 / Math.sqrt(-h)) * (1.0 + ((h / l) * (Math.pow((D * ((M_m / 2.0) / d)), 2.0) * -0.5))));
	} else if (l <= -2e-310) {
		tmp = Math.sqrt((d / h)) * (t_2 * t_0);
	} else {
		tmp = (Math.sqrt(d) / Math.sqrt(h)) * (t_0 * Math.sqrt((d / l)));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	t_0 = 1.0 - (h * (0.125 * (math.pow((M_m * (D / d)), 2.0) / l)))
	t_1 = math.sqrt(-d)
	t_2 = t_1 / math.sqrt(-l)
	tmp = 0
	if l <= -8e-79:
		tmp = t_2 * ((t_1 / math.sqrt(-h)) * (1.0 + ((h / l) * (math.pow((D * ((M_m / 2.0) / d)), 2.0) * -0.5))))
	elif l <= -2e-310:
		tmp = math.sqrt((d / h)) * (t_2 * t_0)
	else:
		tmp = (math.sqrt(d) / math.sqrt(h)) * (t_0 * math.sqrt((d / l)))
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	t_0 = Float64(1.0 - Float64(h * Float64(0.125 * Float64((Float64(M_m * Float64(D / d)) ^ 2.0) / l))))
	t_1 = sqrt(Float64(-d))
	t_2 = Float64(t_1 / sqrt(Float64(-l)))
	tmp = 0.0
	if (l <= -8e-79)
		tmp = Float64(t_2 * Float64(Float64(t_1 / sqrt(Float64(-h))) * Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(D * Float64(Float64(M_m / 2.0) / d)) ^ 2.0) * -0.5)))));
	elseif (l <= -2e-310)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(t_2 * t_0));
	else
		tmp = Float64(Float64(sqrt(d) / sqrt(h)) * Float64(t_0 * sqrt(Float64(d / l))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp_2 = code(d, h, l, M_m, D)
	t_0 = 1.0 - (h * (0.125 * (((M_m * (D / d)) ^ 2.0) / l)));
	t_1 = sqrt(-d);
	t_2 = t_1 / sqrt(-l);
	tmp = 0.0;
	if (l <= -8e-79)
		tmp = t_2 * ((t_1 / sqrt(-h)) * (1.0 + ((h / l) * (((D * ((M_m / 2.0) / d)) ^ 2.0) * -0.5))));
	elseif (l <= -2e-310)
		tmp = sqrt((d / h)) * (t_2 * t_0);
	else
		tmp = (sqrt(d) / sqrt(h)) * (t_0 * sqrt((d / l)));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := Block[{t$95$0 = N[(1.0 - N[(h * N[(0.125 * N[(N[Power[N[(M$95$m * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[(-d)], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -8e-79], N[(t$95$2 * N[(N[(t$95$1 / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(D * N[(N[(M$95$m / 2.0), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -2e-310], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(t$95$2 * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -8e-79

   1. Initial program 51.0%

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

   3. Simplified 52.2%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(D \cdot \frac{\frac{M}{2}}{d}\right)}^{2} \cdot -0.5\right)\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. frac-2neg 52.2%

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

      2. sqrt-div 68.9%

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

   6. Applied egg-rr 68.9%

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\color{blue}{\frac{\sqrt{-d}}{\sqrt{-h}}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(D \cdot \frac{\frac{M}{2}}{d}\right)}^{2} \cdot -0.5\right)\right)\right) \]
   7. Step-by-step derivation

      1. frac-2neg 68.9%

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

      2. sqrt-div 77.1%

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

   8. Applied egg-rr 77.1%

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

   if -8e-79 < l < -1.999999999999994e-310

   1. Initial program 70.4%

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

   3. Simplified 72.0%

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

   5. Taylor expanded in h around -inf 67.5%

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

      1. associate-*r* 67.5%

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

      2. neg-mul-1 67.5%

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

      3. sub-neg 67.5%

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

      4. distribute-lft-in 67.5%

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

   7. Simplified 85.4%

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

      1. frac-2neg 72.0%

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

      2. sqrt-div 78.5%

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

   9. Applied egg-rr 98.5%

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

   if -1.999999999999994e-310 < l

   1. Initial program 69.9%

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

   3. Simplified 70.7%

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

   5. Taylor expanded in h around -inf 47.5%

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

      1. associate-*r* 47.5%

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

      2. neg-mul-1 47.5%

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

      3. sub-neg 47.5%

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

      4. distribute-lft-in 47.5%

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

   7. Simplified 72.9%

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

      1. sqrt-div 85.3%

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

   9. Applied egg-rr 85.3%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -8 \cdot 10^{-79}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(D \cdot \frac{\frac{M}{2}}{d}\right)}^{2} \cdot -0.5\right)\right)\right)\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \left(1 - h \cdot \left(0.125 \cdot \frac{{\left(M \cdot \frac{D}{d}\right)}^{2}}{\ell}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\left(1 - h \cdot \left(0.125 \cdot \frac{{\left(M \cdot \frac{D}{d}\right)}^{2}}{\ell}\right)\right) \cdot \sqrt{\frac{d}{\ell}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := \sqrt{-d}\\ t_2 := \frac{t\_1}{\sqrt{-h}} \cdot \left(t\_0 \cdot \left(1 + \frac{{\left(D \cdot \frac{M\_m}{d}\right)}^{2} \cdot \left(h \cdot -0.125\right)}{\ell}\right)\right)\\ t_3 := {\left(M\_m \cdot \frac{D}{d}\right)}^{2}\\ t_4 := 1 - h \cdot \left(0.125 \cdot \frac{t\_3}{\ell}\right)\\ \mathbf{if}\;h \leq -2.2 \cdot 10^{+143}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;h \leq -1.4 \cdot 10^{-71}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\frac{t\_1}{\sqrt{-\ell}} \cdot t\_4\right)\\ \mathbf{elif}\;h \leq -4 \cdot 10^{-310}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;h \leq 4.8 \cdot 10^{+127}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(t\_3 \cdot -0.125\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(t\_4 \cdot t\_0\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (let* ((t_0 (sqrt (/ d l)))
        (t_1 (sqrt (- d)))
        (t_2
         (*
          (/ t_1 (sqrt (- h)))
          (* t_0 (+ 1.0 (/ (* (pow (* D (/ M_m d)) 2.0) (* h -0.125)) l)))))
        (t_3 (pow (* M_m (/ D d)) 2.0))
        (t_4 (- 1.0 (* h (* 0.125 (/ t_3 l))))))
   (if (<= h -2.2e+143)
     t_2
     (if (<= h -1.4e-71)
       (* (sqrt (/ d h)) (* (/ t_1 (sqrt (- l))) t_4))
       (if (<= h -4e-310)
         t_2
         (if (<= h 4.8e+127)
           (* (/ d (sqrt (* l h))) (+ 1.0 (* (/ h l) (* t_3 -0.125))))
           (* (/ (sqrt d) (sqrt h)) (* t_4 t_0))))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double t_0 = sqrt((d / l));
	double t_1 = sqrt(-d);
	double t_2 = (t_1 / sqrt(-h)) * (t_0 * (1.0 + ((pow((D * (M_m / d)), 2.0) * (h * -0.125)) / l)));
	double t_3 = pow((M_m * (D / d)), 2.0);
	double t_4 = 1.0 - (h * (0.125 * (t_3 / l)));
	double tmp;
	if (h <= -2.2e+143) {
		tmp = t_2;
	} else if (h <= -1.4e-71) {
		tmp = sqrt((d / h)) * ((t_1 / sqrt(-l)) * t_4);
	} else if (h <= -4e-310) {
		tmp = t_2;
	} else if (h <= 4.8e+127) {
		tmp = (d / sqrt((l * h))) * (1.0 + ((h / l) * (t_3 * -0.125)));
	} else {
		tmp = (sqrt(d) / sqrt(h)) * (t_4 * t_0);
	}
	return tmp;
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_0 = sqrt((d / l))
    t_1 = sqrt(-d)
    t_2 = (t_1 / sqrt(-h)) * (t_0 * (1.0d0 + ((((d_1 * (m_m / d)) ** 2.0d0) * (h * (-0.125d0))) / l)))
    t_3 = (m_m * (d_1 / d)) ** 2.0d0
    t_4 = 1.0d0 - (h * (0.125d0 * (t_3 / l)))
    if (h <= (-2.2d+143)) then
        tmp = t_2
    else if (h <= (-1.4d-71)) then
        tmp = sqrt((d / h)) * ((t_1 / sqrt(-l)) * t_4)
    else if (h <= (-4d-310)) then
        tmp = t_2
    else if (h <= 4.8d+127) then
        tmp = (d / sqrt((l * h))) * (1.0d0 + ((h / l) * (t_3 * (-0.125d0))))
    else
        tmp = (sqrt(d) / sqrt(h)) * (t_4 * t_0)
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double t_0 = Math.sqrt((d / l));
	double t_1 = Math.sqrt(-d);
	double t_2 = (t_1 / Math.sqrt(-h)) * (t_0 * (1.0 + ((Math.pow((D * (M_m / d)), 2.0) * (h * -0.125)) / l)));
	double t_3 = Math.pow((M_m * (D / d)), 2.0);
	double t_4 = 1.0 - (h * (0.125 * (t_3 / l)));
	double tmp;
	if (h <= -2.2e+143) {
		tmp = t_2;
	} else if (h <= -1.4e-71) {
		tmp = Math.sqrt((d / h)) * ((t_1 / Math.sqrt(-l)) * t_4);
	} else if (h <= -4e-310) {
		tmp = t_2;
	} else if (h <= 4.8e+127) {
		tmp = (d / Math.sqrt((l * h))) * (1.0 + ((h / l) * (t_3 * -0.125)));
	} else {
		tmp = (Math.sqrt(d) / Math.sqrt(h)) * (t_4 * t_0);
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	t_0 = math.sqrt((d / l))
	t_1 = math.sqrt(-d)
	t_2 = (t_1 / math.sqrt(-h)) * (t_0 * (1.0 + ((math.pow((D * (M_m / d)), 2.0) * (h * -0.125)) / l)))
	t_3 = math.pow((M_m * (D / d)), 2.0)
	t_4 = 1.0 - (h * (0.125 * (t_3 / l)))
	tmp = 0
	if h <= -2.2e+143:
		tmp = t_2
	elif h <= -1.4e-71:
		tmp = math.sqrt((d / h)) * ((t_1 / math.sqrt(-l)) * t_4)
	elif h <= -4e-310:
		tmp = t_2
	elif h <= 4.8e+127:
		tmp = (d / math.sqrt((l * h))) * (1.0 + ((h / l) * (t_3 * -0.125)))
	else:
		tmp = (math.sqrt(d) / math.sqrt(h)) * (t_4 * t_0)
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	t_0 = sqrt(Float64(d / l))
	t_1 = sqrt(Float64(-d))
	t_2 = Float64(Float64(t_1 / sqrt(Float64(-h))) * Float64(t_0 * Float64(1.0 + Float64(Float64((Float64(D * Float64(M_m / d)) ^ 2.0) * Float64(h * -0.125)) / l))))
	t_3 = Float64(M_m * Float64(D / d)) ^ 2.0
	t_4 = Float64(1.0 - Float64(h * Float64(0.125 * Float64(t_3 / l))))
	tmp = 0.0
	if (h <= -2.2e+143)
		tmp = t_2;
	elseif (h <= -1.4e-71)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(Float64(t_1 / sqrt(Float64(-l))) * t_4));
	elseif (h <= -4e-310)
		tmp = t_2;
	elseif (h <= 4.8e+127)
		tmp = Float64(Float64(d / sqrt(Float64(l * h))) * Float64(1.0 + Float64(Float64(h / l) * Float64(t_3 * -0.125))));
	else
		tmp = Float64(Float64(sqrt(d) / sqrt(h)) * Float64(t_4 * t_0));
	end
	return tmp
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp_2 = code(d, h, l, M_m, D)
	t_0 = sqrt((d / l));
	t_1 = sqrt(-d);
	t_2 = (t_1 / sqrt(-h)) * (t_0 * (1.0 + ((((D * (M_m / d)) ^ 2.0) * (h * -0.125)) / l)));
	t_3 = (M_m * (D / d)) ^ 2.0;
	t_4 = 1.0 - (h * (0.125 * (t_3 / l)));
	tmp = 0.0;
	if (h <= -2.2e+143)
		tmp = t_2;
	elseif (h <= -1.4e-71)
		tmp = sqrt((d / h)) * ((t_1 / sqrt(-l)) * t_4);
	elseif (h <= -4e-310)
		tmp = t_2;
	elseif (h <= 4.8e+127)
		tmp = (d / sqrt((l * h))) * (1.0 + ((h / l) * (t_3 * -0.125)));
	else
		tmp = (sqrt(d) / sqrt(h)) * (t_4 * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := Block[{t$95$0 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[(-d)], $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$1 / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * N[(1.0 + N[(N[(N[Power[N[(D * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h * -0.125), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Power[N[(M$95$m * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$4 = N[(1.0 - N[(h * N[(0.125 * N[(t$95$3 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[h, -2.2e+143], t$95$2, If[LessEqual[h, -1.4e-71], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[(t$95$1 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] * t$95$4), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, -4e-310], t$95$2, If[LessEqual[h, 4.8e+127], N[(N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(t$95$3 * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(t$95$4 * t$95$0), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if h < -2.20000000000000014e143 or -1.4e-71 < h < -3.999999999999988e-310

   1. Initial program 56.9%

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

   3. Simplified 56.9%

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

   5. Taylor expanded in h around -inf 49.0%

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

      1. associate-*r* 49.0%

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

      2. neg-mul-1 49.0%

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

      3. sub-neg 49.0%

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

      4. distribute-lft-in 49.0%

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

   7. Simplified 62.9%

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

      1. pow1 62.9%

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

      2. associate-*r* 62.9%

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

   9. Applied egg-rr 62.9%

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

       1. unpow1 62.9%

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

       2. *-commutative 62.9%

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

       3. unpow2 62.9%

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

       4. *-commutative 62.9%

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

       5. *-commutative 62.9%

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

       6. swap-sqr 52.9%

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

       7. times-frac 49.2%

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

       8. unpow2 49.2%

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

       9. unpow2 49.2%

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

       10. unpow2 49.2%

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

       11. associate-*r/ 46.5%

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

       12. times-frac 49.0%

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

       13. associate-/r* 51.2%

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

       14. associate-*l/ 51.2%

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

   11. Simplified 62.8%

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

       1. frac-2neg 56.9%

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

       2. sqrt-div 74.7%

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

   13. Applied egg-rr 80.7%

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

   if -2.20000000000000014e143 < h < -1.4e-71

   1. Initial program 60.4%

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

   3. Simplified 63.7%

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

   5. Taylor expanded in h around -inf 44.9%

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

      1. associate-*r* 44.9%

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

      2. neg-mul-1 44.9%

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

      3. sub-neg 44.9%

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

      4. distribute-lft-in 44.9%

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

   7. Simplified 69.5%

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

      1. frac-2neg 63.5%

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

      2. sqrt-div 75.2%

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

   9. Applied egg-rr 87.4%

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

   if -3.999999999999988e-310 < h < 4.8000000000000004e127

   1. Initial program 75.6%

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

   3. Simplified 76.6%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. pow1 76.6%

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

      2. sqrt-unprod 67.6%

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

   6. Applied egg-rr 67.6%

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

      1. unpow1 67.6%

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

   8. Simplified 67.6%

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

   9. Taylor expanded in M around 0 40.4%

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

       1. *-commutative 40.4%

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

       2. associate-*r* 41.4%

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

       3. times-frac 43.6%

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

       4. *-commutative 43.6%

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

       5. associate-/l* 43.6%

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

       6. unpow2 43.6%

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

       7. unpow2 43.6%

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

       8. unpow2 43.6%

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

       9. times-frac 49.2%

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

       10. swap-sqr 67.6%

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

       11. unpow2 67.6%

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

       12. associate-*r/ 66.5%

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

       13. *-commutative 66.5%

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

       14. associate-/l* 67.6%

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

   11. Simplified 67.6%

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

       1. pow1 67.6%

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

       2. *-commutative 67.6%

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

       3. associate-*l* 67.6%

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

       4. frac-times 59.1%

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

       5. sqrt-div 64.7%

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

       6. sqrt-unprod 91.3%

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

       7. add-sqr-sqrt 91.6%

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

   13. Applied egg-rr 91.6%

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

       1. unpow1 91.6%

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

       2. *-commutative 91.6%

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

       3. sub-neg 91.6%

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

       4. distribute-rgt-neg-in 91.6%

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

       5. *-commutative 91.6%

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

       6. distribute-lft-neg-in 91.6%

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

       7. metadata-eval 91.6%

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

       8. associate-*r* 91.6%

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

       9. *-commutative 91.6%

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

       10. *-commutative 91.6%

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

       11. associate-*r/ 90.5%

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

       12. *-commutative 90.5%

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

       13. associate-/l* 91.6%

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

   15. Simplified 91.6%

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

   if 4.8000000000000004e127 < h

   1. Initial program 53.2%

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

   3. Simplified 53.2%

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

   5. Taylor expanded in h around -inf 44.2%

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

      1. associate-*r* 44.2%

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

      2. neg-mul-1 44.2%

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

      3. sub-neg 44.2%

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

      4. distribute-lft-in 44.2%

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

   7. Simplified 61.8%

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

      1. sqrt-div 81.8%

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

   9. Applied egg-rr 81.8%

      \[\leadsto \color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\left(-h\right) \cdot \left(0.125 \cdot \frac{{\left(M \cdot \frac{D}{d}\right)}^{2}}{\ell}\right) + 1\right)\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -2.2 \cdot 10^{+143}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \frac{{\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \left(h \cdot -0.125\right)}{\ell}\right)\right)\\ \mathbf{elif}\;h \leq -1.4 \cdot 10^{-71}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \left(1 - h \cdot \left(0.125 \cdot \frac{{\left(M \cdot \frac{D}{d}\right)}^{2}}{\ell}\right)\right)\right)\\ \mathbf{elif}\;h \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \frac{{\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \left(h \cdot -0.125\right)}{\ell}\right)\right)\\ \mathbf{elif}\;h \leq 4.8 \cdot 10^{+127}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(M \cdot \frac{D}{d}\right)}^{2} \cdot -0.125\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\left(1 - h \cdot \left(0.125 \cdot \frac{{\left(M \cdot \frac{D}{d}\right)}^{2}}{\ell}\right)\right) \cdot \sqrt{\frac{d}{\ell}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := {\left(M\_m \cdot \frac{D}{d}\right)}^{2}\\ t_1 := \sqrt{-d}\\ t_2 := {\left(D \cdot \frac{M\_m}{d}\right)}^{2}\\ t_3 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;h \leq -7 \cdot 10^{-76}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\frac{t\_1}{\sqrt{-\ell}} \cdot \left(1 + \frac{t\_2 \cdot \left(h \cdot -0.125\right)}{\ell}\right)\right)\\ \mathbf{elif}\;h \leq -4.5 \cdot 10^{-303}:\\ \;\;\;\;\left(\frac{t\_1}{\sqrt{-h}} \cdot t\_3\right) \cdot \left(1 - 0.125 \cdot \left(\frac{h}{\ell} \cdot t\_2\right)\right)\\ \mathbf{elif}\;h \leq 6 \cdot 10^{+125}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(t\_0 \cdot -0.125\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\left(1 - h \cdot \left(0.125 \cdot \frac{t\_0}{\ell}\right)\right) \cdot t\_3\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (let* ((t_0 (pow (* M_m (/ D d)) 2.0))
        (t_1 (sqrt (- d)))
        (t_2 (pow (* D (/ M_m d)) 2.0))
        (t_3 (sqrt (/ d l))))
   (if (<= h -7e-76)
     (*
      (sqrt (/ d h))
      (* (/ t_1 (sqrt (- l))) (+ 1.0 (/ (* t_2 (* h -0.125)) l))))
     (if (<= h -4.5e-303)
       (* (* (/ t_1 (sqrt (- h))) t_3) (- 1.0 (* 0.125 (* (/ h l) t_2))))
       (if (<= h 6e+125)
         (* (/ d (sqrt (* l h))) (+ 1.0 (* (/ h l) (* t_0 -0.125))))
         (*
          (/ (sqrt d) (sqrt h))
          (* (- 1.0 (* h (* 0.125 (/ t_0 l)))) t_3)))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double t_0 = pow((M_m * (D / d)), 2.0);
	double t_1 = sqrt(-d);
	double t_2 = pow((D * (M_m / d)), 2.0);
	double t_3 = sqrt((d / l));
	double tmp;
	if (h <= -7e-76) {
		tmp = sqrt((d / h)) * ((t_1 / sqrt(-l)) * (1.0 + ((t_2 * (h * -0.125)) / l)));
	} else if (h <= -4.5e-303) {
		tmp = ((t_1 / sqrt(-h)) * t_3) * (1.0 - (0.125 * ((h / l) * t_2)));
	} else if (h <= 6e+125) {
		tmp = (d / sqrt((l * h))) * (1.0 + ((h / l) * (t_0 * -0.125)));
	} else {
		tmp = (sqrt(d) / sqrt(h)) * ((1.0 - (h * (0.125 * (t_0 / l)))) * t_3);
	}
	return tmp;
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = (m_m * (d_1 / d)) ** 2.0d0
    t_1 = sqrt(-d)
    t_2 = (d_1 * (m_m / d)) ** 2.0d0
    t_3 = sqrt((d / l))
    if (h <= (-7d-76)) then
        tmp = sqrt((d / h)) * ((t_1 / sqrt(-l)) * (1.0d0 + ((t_2 * (h * (-0.125d0))) / l)))
    else if (h <= (-4.5d-303)) then
        tmp = ((t_1 / sqrt(-h)) * t_3) * (1.0d0 - (0.125d0 * ((h / l) * t_2)))
    else if (h <= 6d+125) then
        tmp = (d / sqrt((l * h))) * (1.0d0 + ((h / l) * (t_0 * (-0.125d0))))
    else
        tmp = (sqrt(d) / sqrt(h)) * ((1.0d0 - (h * (0.125d0 * (t_0 / l)))) * t_3)
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double t_0 = Math.pow((M_m * (D / d)), 2.0);
	double t_1 = Math.sqrt(-d);
	double t_2 = Math.pow((D * (M_m / d)), 2.0);
	double t_3 = Math.sqrt((d / l));
	double tmp;
	if (h <= -7e-76) {
		tmp = Math.sqrt((d / h)) * ((t_1 / Math.sqrt(-l)) * (1.0 + ((t_2 * (h * -0.125)) / l)));
	} else if (h <= -4.5e-303) {
		tmp = ((t_1 / Math.sqrt(-h)) * t_3) * (1.0 - (0.125 * ((h / l) * t_2)));
	} else if (h <= 6e+125) {
		tmp = (d / Math.sqrt((l * h))) * (1.0 + ((h / l) * (t_0 * -0.125)));
	} else {
		tmp = (Math.sqrt(d) / Math.sqrt(h)) * ((1.0 - (h * (0.125 * (t_0 / l)))) * t_3);
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	t_0 = math.pow((M_m * (D / d)), 2.0)
	t_1 = math.sqrt(-d)
	t_2 = math.pow((D * (M_m / d)), 2.0)
	t_3 = math.sqrt((d / l))
	tmp = 0
	if h <= -7e-76:
		tmp = math.sqrt((d / h)) * ((t_1 / math.sqrt(-l)) * (1.0 + ((t_2 * (h * -0.125)) / l)))
	elif h <= -4.5e-303:
		tmp = ((t_1 / math.sqrt(-h)) * t_3) * (1.0 - (0.125 * ((h / l) * t_2)))
	elif h <= 6e+125:
		tmp = (d / math.sqrt((l * h))) * (1.0 + ((h / l) * (t_0 * -0.125)))
	else:
		tmp = (math.sqrt(d) / math.sqrt(h)) * ((1.0 - (h * (0.125 * (t_0 / l)))) * t_3)
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	t_0 = Float64(M_m * Float64(D / d)) ^ 2.0
	t_1 = sqrt(Float64(-d))
	t_2 = Float64(D * Float64(M_m / d)) ^ 2.0
	t_3 = sqrt(Float64(d / l))
	tmp = 0.0
	if (h <= -7e-76)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(Float64(t_1 / sqrt(Float64(-l))) * Float64(1.0 + Float64(Float64(t_2 * Float64(h * -0.125)) / l))));
	elseif (h <= -4.5e-303)
		tmp = Float64(Float64(Float64(t_1 / sqrt(Float64(-h))) * t_3) * Float64(1.0 - Float64(0.125 * Float64(Float64(h / l) * t_2))));
	elseif (h <= 6e+125)
		tmp = Float64(Float64(d / sqrt(Float64(l * h))) * Float64(1.0 + Float64(Float64(h / l) * Float64(t_0 * -0.125))));
	else
		tmp = Float64(Float64(sqrt(d) / sqrt(h)) * Float64(Float64(1.0 - Float64(h * Float64(0.125 * Float64(t_0 / l)))) * t_3));
	end
	return tmp
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp_2 = code(d, h, l, M_m, D)
	t_0 = (M_m * (D / d)) ^ 2.0;
	t_1 = sqrt(-d);
	t_2 = (D * (M_m / d)) ^ 2.0;
	t_3 = sqrt((d / l));
	tmp = 0.0;
	if (h <= -7e-76)
		tmp = sqrt((d / h)) * ((t_1 / sqrt(-l)) * (1.0 + ((t_2 * (h * -0.125)) / l)));
	elseif (h <= -4.5e-303)
		tmp = ((t_1 / sqrt(-h)) * t_3) * (1.0 - (0.125 * ((h / l) * t_2)));
	elseif (h <= 6e+125)
		tmp = (d / sqrt((l * h))) * (1.0 + ((h / l) * (t_0 * -0.125)));
	else
		tmp = (sqrt(d) / sqrt(h)) * ((1.0 - (h * (0.125 * (t_0 / l)))) * t_3);
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := Block[{t$95$0 = N[Power[N[(M$95$m * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[(-d)], $MachinePrecision]}, Block[{t$95$2 = N[Power[N[(D * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[h, -7e-76], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[(t$95$1 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(t$95$2 * N[(h * -0.125), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, -4.5e-303], N[(N[(N[(t$95$1 / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision] * N[(1.0 - N[(0.125 * N[(N[(h / l), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, 6e+125], N[(N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(t$95$0 * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - N[(h * N[(0.125 * N[(t$95$0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if h < -6.99999999999999995e-76

   1. Initial program 52.2%

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

   3. Simplified 54.1%

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

   5. Taylor expanded in h around -inf 42.0%

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

      1. associate-*r* 42.0%

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

      2. neg-mul-1 42.0%

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

      3. sub-neg 42.0%

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

      4. distribute-lft-in 42.0%

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

   7. Simplified 62.4%

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

      1. pow1 62.4%

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

      2. associate-*r* 62.4%

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

   9. Applied egg-rr 62.4%

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

       1. unpow1 62.4%

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

       2. *-commutative 62.4%

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

       3. unpow2 62.4%

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

       4. *-commutative 62.4%

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

       5. *-commutative 62.4%

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

       6. swap-sqr 54.1%

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

       7. times-frac 45.3%

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

       8. unpow2 45.3%

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

       9. unpow2 45.3%

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

       10. unpow2 45.3%

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

       11. associate-*r/ 43.0%

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

       12. times-frac 42.0%

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

       13. associate-/r* 47.0%

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

       14. associate-*l/ 47.0%

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

   11. Simplified 62.4%

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

       1. frac-2neg 62.8%

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

       2. sqrt-div 73.6%

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

   13. Applied egg-rr 72.4%

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

   if -6.99999999999999995e-76 < h < -4.5000000000000001e-303

   1. Initial program 74.6%

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

   3. Simplified 74.6%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. frac-2neg 74.6%

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

      2. sqrt-div 89.2%

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

   6. Applied egg-rr 89.2%

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

   7. Taylor expanded in M around 0 68.1%

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

      1. *-commutative 51.2%

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

      2. associate-*r* 51.2%

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

      3. times-frac 51.5%

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

      4. *-commutative 51.5%

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

      5. associate-/l* 51.5%

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

      6. unpow2 51.5%

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

      7. unpow2 51.5%

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

      8. unpow2 51.5%

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

      9. times-frac 54.6%

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

      10. swap-sqr 62.6%

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

      11. unpow2 62.6%

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

      12. associate-*r/ 62.6%

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

      13. *-commutative 62.6%

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

      14. associate-/l* 62.6%

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

   9. Simplified 89.2%

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

   if -4.5000000000000001e-303 < h < 6.0000000000000003e125

   1. Initial program 74.8%

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

   3. Simplified 75.8%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. pow1 75.8%

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

      2. sqrt-unprod 66.9%

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

   6. Applied egg-rr 66.9%

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

      1. unpow1 66.9%

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

   8. Simplified 66.9%

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

   9. Taylor expanded in M around 0 40.1%

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

       1. *-commutative 40.1%

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

       2. associate-*r* 41.2%

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

       3. times-frac 43.2%

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

       4. *-commutative 43.2%

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

       5. associate-/l* 43.1%

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

       6. unpow2 43.1%

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

       7. unpow2 43.1%

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

       8. unpow2 43.1%

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

       9. times-frac 48.6%

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

       10. swap-sqr 66.9%

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

       11. unpow2 66.9%

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

       12. associate-*r/ 65.8%

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

       13. *-commutative 65.8%

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

       14. associate-/l* 66.9%

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

   11. Simplified 66.9%

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

       1. pow1 66.9%

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

       2. *-commutative 66.9%

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

       3. associate-*l* 66.9%

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

       4. frac-times 58.5%

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

       5. sqrt-div 64.0%

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

       6. sqrt-unprod 90.3%

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

       7. add-sqr-sqrt 90.6%

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

   13. Applied egg-rr 90.6%

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

       1. unpow1 90.6%

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

       2. *-commutative 90.6%

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

       3. sub-neg 90.6%

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

       4. distribute-rgt-neg-in 90.6%

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

       5. *-commutative 90.6%

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

       6. distribute-lft-neg-in 90.6%

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

       7. metadata-eval 90.6%

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

       8. associate-*r* 90.6%

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

       9. *-commutative 90.6%

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

       10. *-commutative 90.6%

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

       11. associate-*r/ 89.5%

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

       12. *-commutative 89.5%

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

       13. associate-/l* 90.6%

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

   15. Simplified 90.6%

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

   if 6.0000000000000003e125 < h

   1. Initial program 53.2%

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

   3. Simplified 53.2%

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

   5. Taylor expanded in h around -inf 44.2%

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

      1. associate-*r* 44.2%

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

      2. neg-mul-1 44.2%

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

      3. sub-neg 44.2%

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

      4. distribute-lft-in 44.2%

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

   7. Simplified 61.8%

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

      1. sqrt-div 81.8%

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

   9. Applied egg-rr 81.8%

      \[\leadsto \color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\left(-h\right) \cdot \left(0.125 \cdot \frac{{\left(M \cdot \frac{D}{d}\right)}^{2}}{\ell}\right) + 1\right)\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -7 \cdot 10^{-76}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \left(1 + \frac{{\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \left(h \cdot -0.125\right)}{\ell}\right)\right)\\ \mathbf{elif}\;h \leq -4.5 \cdot 10^{-303}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.125 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}\right)\right)\\ \mathbf{elif}\;h \leq 6 \cdot 10^{+125}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(M \cdot \frac{D}{d}\right)}^{2} \cdot -0.125\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\left(1 - h \cdot \left(0.125 \cdot \frac{{\left(M \cdot \frac{D}{d}\right)}^{2}}{\ell}\right)\right) \cdot \sqrt{\frac{d}{\ell}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := 1 + \frac{{\left(D \cdot \frac{M\_m}{d}\right)}^{2} \cdot \left(h \cdot -0.125\right)}{\ell}\\ t_2 := \sqrt{-d}\\ \mathbf{if}\;\ell \leq -1.55 \cdot 10^{-184}:\\ \;\;\;\;\frac{t\_2}{\sqrt{-h}} \cdot \left(t\_0 \cdot t\_1\right)\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\frac{t\_2}{\sqrt{-\ell}} \cdot t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\left(1 - h \cdot \left(0.125 \cdot \frac{{\left(M\_m \cdot \frac{D}{d}\right)}^{2}}{\ell}\right)\right) \cdot t\_0\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (let* ((t_0 (sqrt (/ d l)))
        (t_1 (+ 1.0 (/ (* (pow (* D (/ M_m d)) 2.0) (* h -0.125)) l)))
        (t_2 (sqrt (- d))))
   (if (<= l -1.55e-184)
     (* (/ t_2 (sqrt (- h))) (* t_0 t_1))
     (if (<= l -2e-310)
       (* (sqrt (/ d h)) (* (/ t_2 (sqrt (- l))) t_1))
       (*
        (/ (sqrt d) (sqrt h))
        (* (- 1.0 (* h (* 0.125 (/ (pow (* M_m (/ D d)) 2.0) l)))) t_0))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double t_0 = sqrt((d / l));
	double t_1 = 1.0 + ((pow((D * (M_m / d)), 2.0) * (h * -0.125)) / l);
	double t_2 = sqrt(-d);
	double tmp;
	if (l <= -1.55e-184) {
		tmp = (t_2 / sqrt(-h)) * (t_0 * t_1);
	} else if (l <= -2e-310) {
		tmp = sqrt((d / h)) * ((t_2 / sqrt(-l)) * t_1);
	} else {
		tmp = (sqrt(d) / sqrt(h)) * ((1.0 - (h * (0.125 * (pow((M_m * (D / d)), 2.0) / l)))) * t_0);
	}
	return tmp;
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = sqrt((d / l))
    t_1 = 1.0d0 + ((((d_1 * (m_m / d)) ** 2.0d0) * (h * (-0.125d0))) / l)
    t_2 = sqrt(-d)
    if (l <= (-1.55d-184)) then
        tmp = (t_2 / sqrt(-h)) * (t_0 * t_1)
    else if (l <= (-2d-310)) then
        tmp = sqrt((d / h)) * ((t_2 / sqrt(-l)) * t_1)
    else
        tmp = (sqrt(d) / sqrt(h)) * ((1.0d0 - (h * (0.125d0 * (((m_m * (d_1 / d)) ** 2.0d0) / l)))) * t_0)
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double t_0 = Math.sqrt((d / l));
	double t_1 = 1.0 + ((Math.pow((D * (M_m / d)), 2.0) * (h * -0.125)) / l);
	double t_2 = Math.sqrt(-d);
	double tmp;
	if (l <= -1.55e-184) {
		tmp = (t_2 / Math.sqrt(-h)) * (t_0 * t_1);
	} else if (l <= -2e-310) {
		tmp = Math.sqrt((d / h)) * ((t_2 / Math.sqrt(-l)) * t_1);
	} else {
		tmp = (Math.sqrt(d) / Math.sqrt(h)) * ((1.0 - (h * (0.125 * (Math.pow((M_m * (D / d)), 2.0) / l)))) * t_0);
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	t_0 = math.sqrt((d / l))
	t_1 = 1.0 + ((math.pow((D * (M_m / d)), 2.0) * (h * -0.125)) / l)
	t_2 = math.sqrt(-d)
	tmp = 0
	if l <= -1.55e-184:
		tmp = (t_2 / math.sqrt(-h)) * (t_0 * t_1)
	elif l <= -2e-310:
		tmp = math.sqrt((d / h)) * ((t_2 / math.sqrt(-l)) * t_1)
	else:
		tmp = (math.sqrt(d) / math.sqrt(h)) * ((1.0 - (h * (0.125 * (math.pow((M_m * (D / d)), 2.0) / l)))) * t_0)
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	t_0 = sqrt(Float64(d / l))
	t_1 = Float64(1.0 + Float64(Float64((Float64(D * Float64(M_m / d)) ^ 2.0) * Float64(h * -0.125)) / l))
	t_2 = sqrt(Float64(-d))
	tmp = 0.0
	if (l <= -1.55e-184)
		tmp = Float64(Float64(t_2 / sqrt(Float64(-h))) * Float64(t_0 * t_1));
	elseif (l <= -2e-310)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(Float64(t_2 / sqrt(Float64(-l))) * t_1));
	else
		tmp = Float64(Float64(sqrt(d) / sqrt(h)) * Float64(Float64(1.0 - Float64(h * Float64(0.125 * Float64((Float64(M_m * Float64(D / d)) ^ 2.0) / l)))) * t_0));
	end
	return tmp
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp_2 = code(d, h, l, M_m, D)
	t_0 = sqrt((d / l));
	t_1 = 1.0 + ((((D * (M_m / d)) ^ 2.0) * (h * -0.125)) / l);
	t_2 = sqrt(-d);
	tmp = 0.0;
	if (l <= -1.55e-184)
		tmp = (t_2 / sqrt(-h)) * (t_0 * t_1);
	elseif (l <= -2e-310)
		tmp = sqrt((d / h)) * ((t_2 / sqrt(-l)) * t_1);
	else
		tmp = (sqrt(d) / sqrt(h)) * ((1.0 - (h * (0.125 * (((M_m * (D / d)) ^ 2.0) / l)))) * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := Block[{t$95$0 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + N[(N[(N[Power[N[(D * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h * -0.125), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[(-d)], $MachinePrecision]}, If[LessEqual[l, -1.55e-184], N[(N[(t$95$2 / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -2e-310], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[(t$95$2 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - N[(h * N[(0.125 * N[(N[Power[N[(M$95$m * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.5500000000000001e-184

   1. Initial program 55.7%

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

   3. Simplified 57.5%

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

   5. Taylor expanded in h around -inf 42.1%

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

      1. associate-*r* 42.1%

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

      2. neg-mul-1 42.1%

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

      3. sub-neg 42.1%

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

      4. distribute-lft-in 42.1%

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

   7. Simplified 61.2%

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

      1. pow1 61.2%

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

      2. associate-*r* 61.2%

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

   9. Applied egg-rr 61.2%

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

       1. unpow1 61.2%

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

       2. *-commutative 61.2%

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

       3. unpow2 61.2%

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

       4. *-commutative 61.2%

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

       5. *-commutative 61.2%

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

       6. swap-sqr 52.4%

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

       7. times-frac 43.2%

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

       8. unpow2 43.2%

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

       9. unpow2 43.2%

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

       10. unpow2 43.2%

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

       11. associate-*r/ 42.2%

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

       12. times-frac 42.1%

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

       13. associate-/r* 44.8%

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

       14. associate-*l/ 44.8%

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

   11. Simplified 61.2%

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

       1. frac-2neg 57.5%

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

       2. sqrt-div 71.2%

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

   13. Applied egg-rr 74.9%

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

   if -1.5500000000000001e-184 < l < -1.999999999999994e-310

   1. Initial program 66.4%

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

   3. Simplified 66.4%

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

   5. Taylor expanded in h around -inf 63.1%

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

      1. associate-*r* 63.1%

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

      2. neg-mul-1 63.1%

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

      3. sub-neg 63.1%

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

      4. distribute-lft-in 63.1%

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

   7. Simplified 79.3%

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

      1. pow1 79.3%

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

      2. associate-*r* 79.3%

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

   9. Applied egg-rr 79.3%

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

       1. unpow1 79.3%

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

       2. *-commutative 79.3%

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

       3. unpow2 79.3%

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

       4. *-commutative 79.3%

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

       5. *-commutative 79.3%

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

       6. swap-sqr 73.1%

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

       7. times-frac 72.4%

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

       8. unpow2 72.4%

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

       9. unpow2 72.4%

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

       10. unpow2 72.4%

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

       11. associate-*r/ 62.7%

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

       12. times-frac 63.1%

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

       13. associate-/r* 72.4%

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

       14. associate-*l/ 72.4%

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

   11. Simplified 79.2%

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

       1. frac-2neg 66.4%

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

       2. sqrt-div 76.4%

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

   13. Applied egg-rr 98.0%

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

   if -1.999999999999994e-310 < l

   1. Initial program 69.9%

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

   3. Simplified 70.7%

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

   5. Taylor expanded in h around -inf 47.5%

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

      1. associate-*r* 47.5%

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

      2. neg-mul-1 47.5%

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

      3. sub-neg 47.5%

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

      4. distribute-lft-in 47.5%

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

   7. Simplified 72.9%

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

      1. sqrt-div 85.3%

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

   9. Applied egg-rr 85.3%

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

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.55 \cdot 10^{-184}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \frac{{\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \left(h \cdot -0.125\right)}{\ell}\right)\right)\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \left(1 + \frac{{\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \left(h \cdot -0.125\right)}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\left(1 - h \cdot \left(0.125 \cdot \frac{{\left(M \cdot \frac{D}{d}\right)}^{2}}{\ell}\right)\right) \cdot \sqrt{\frac{d}{\ell}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := {\left(M\_m \cdot \frac{D}{d}\right)}^{2}\\ \mathbf{if}\;h \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \left(1 + \frac{{\left(D \cdot \frac{M\_m}{d}\right)}^{2} \cdot \left(h \cdot -0.125\right)}{\ell}\right)\right)\\ \mathbf{elif}\;h \leq 6.4 \cdot 10^{+125}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(t\_0 \cdot -0.125\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\left(1 - h \cdot \left(0.125 \cdot \frac{t\_0}{\ell}\right)\right) \cdot \sqrt{\frac{d}{\ell}}\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (let* ((t_0 (pow (* M_m (/ D d)) 2.0)))
   (if (<= h -4e-310)
     (*
      (sqrt (/ d h))
      (*
       (/ (sqrt (- d)) (sqrt (- l)))
       (+ 1.0 (/ (* (pow (* D (/ M_m d)) 2.0) (* h -0.125)) l))))
     (if (<= h 6.4e+125)
       (* (/ d (sqrt (* l h))) (+ 1.0 (* (/ h l) (* t_0 -0.125))))
       (*
        (/ (sqrt d) (sqrt h))
        (* (- 1.0 (* h (* 0.125 (/ t_0 l)))) (sqrt (/ d l))))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double t_0 = pow((M_m * (D / d)), 2.0);
	double tmp;
	if (h <= -4e-310) {
		tmp = sqrt((d / h)) * ((sqrt(-d) / sqrt(-l)) * (1.0 + ((pow((D * (M_m / d)), 2.0) * (h * -0.125)) / l)));
	} else if (h <= 6.4e+125) {
		tmp = (d / sqrt((l * h))) * (1.0 + ((h / l) * (t_0 * -0.125)));
	} else {
		tmp = (sqrt(d) / sqrt(h)) * ((1.0 - (h * (0.125 * (t_0 / l)))) * sqrt((d / l)));
	}
	return tmp;
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (m_m * (d_1 / d)) ** 2.0d0
    if (h <= (-4d-310)) then
        tmp = sqrt((d / h)) * ((sqrt(-d) / sqrt(-l)) * (1.0d0 + ((((d_1 * (m_m / d)) ** 2.0d0) * (h * (-0.125d0))) / l)))
    else if (h <= 6.4d+125) then
        tmp = (d / sqrt((l * h))) * (1.0d0 + ((h / l) * (t_0 * (-0.125d0))))
    else
        tmp = (sqrt(d) / sqrt(h)) * ((1.0d0 - (h * (0.125d0 * (t_0 / l)))) * sqrt((d / l)))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double t_0 = Math.pow((M_m * (D / d)), 2.0);
	double tmp;
	if (h <= -4e-310) {
		tmp = Math.sqrt((d / h)) * ((Math.sqrt(-d) / Math.sqrt(-l)) * (1.0 + ((Math.pow((D * (M_m / d)), 2.0) * (h * -0.125)) / l)));
	} else if (h <= 6.4e+125) {
		tmp = (d / Math.sqrt((l * h))) * (1.0 + ((h / l) * (t_0 * -0.125)));
	} else {
		tmp = (Math.sqrt(d) / Math.sqrt(h)) * ((1.0 - (h * (0.125 * (t_0 / l)))) * Math.sqrt((d / l)));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	t_0 = math.pow((M_m * (D / d)), 2.0)
	tmp = 0
	if h <= -4e-310:
		tmp = math.sqrt((d / h)) * ((math.sqrt(-d) / math.sqrt(-l)) * (1.0 + ((math.pow((D * (M_m / d)), 2.0) * (h * -0.125)) / l)))
	elif h <= 6.4e+125:
		tmp = (d / math.sqrt((l * h))) * (1.0 + ((h / l) * (t_0 * -0.125)))
	else:
		tmp = (math.sqrt(d) / math.sqrt(h)) * ((1.0 - (h * (0.125 * (t_0 / l)))) * math.sqrt((d / l)))
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	t_0 = Float64(M_m * Float64(D / d)) ^ 2.0
	tmp = 0.0
	if (h <= -4e-310)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-l))) * Float64(1.0 + Float64(Float64((Float64(D * Float64(M_m / d)) ^ 2.0) * Float64(h * -0.125)) / l))));
	elseif (h <= 6.4e+125)
		tmp = Float64(Float64(d / sqrt(Float64(l * h))) * Float64(1.0 + Float64(Float64(h / l) * Float64(t_0 * -0.125))));
	else
		tmp = Float64(Float64(sqrt(d) / sqrt(h)) * Float64(Float64(1.0 - Float64(h * Float64(0.125 * Float64(t_0 / l)))) * sqrt(Float64(d / l))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp_2 = code(d, h, l, M_m, D)
	t_0 = (M_m * (D / d)) ^ 2.0;
	tmp = 0.0;
	if (h <= -4e-310)
		tmp = sqrt((d / h)) * ((sqrt(-d) / sqrt(-l)) * (1.0 + ((((D * (M_m / d)) ^ 2.0) * (h * -0.125)) / l)));
	elseif (h <= 6.4e+125)
		tmp = (d / sqrt((l * h))) * (1.0 + ((h / l) * (t_0 * -0.125)));
	else
		tmp = (sqrt(d) / sqrt(h)) * ((1.0 - (h * (0.125 * (t_0 / l)))) * sqrt((d / l)));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := Block[{t$95$0 = N[Power[N[(M$95$m * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[h, -4e-310], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(N[Power[N[(D * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h * -0.125), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, 6.4e+125], N[(N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(t$95$0 * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - N[(h * N[(0.125 * N[(t$95$0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if h < -3.999999999999988e-310

   1. Initial program 58.3%

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

   3. Simplified 59.7%

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

   5. Taylor expanded in h around -inf 47.3%

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

      1. associate-*r* 47.3%

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

      2. neg-mul-1 47.3%

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

      3. sub-neg 47.3%

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

      4. distribute-lft-in 47.3%

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

   7. Simplified 65.6%

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

      1. pow1 65.6%

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

      2. associate-*r* 65.6%

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

   9. Applied egg-rr 65.6%

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

       1. unpow1 65.6%

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

       2. *-commutative 65.6%

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

       3. unpow2 65.6%

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

       4. *-commutative 65.6%

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

       5. *-commutative 65.6%

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

       6. swap-sqr 57.5%

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

       7. times-frac 50.4%

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

       8. unpow2 50.4%

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

       9. unpow2 50.4%

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

       10. unpow2 50.4%

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

       11. associate-*r/ 47.3%

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

       12. times-frac 47.3%

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

       13. associate-/r* 51.6%

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

       14. associate-*l/ 51.6%

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

   11. Simplified 65.6%

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

       1. frac-2neg 70.1%

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

       2. sqrt-div 77.6%

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

   13. Applied egg-rr 73.4%

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

   if -3.999999999999988e-310 < h < 6.39999999999999967e125

   1. Initial program 75.6%

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

   3. Simplified 76.6%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. pow1 76.6%

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

      2. sqrt-unprod 67.6%

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

   6. Applied egg-rr 67.6%

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

      1. unpow1 67.6%

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

   8. Simplified 67.6%

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

   9. Taylor expanded in M around 0 40.4%

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

       1. *-commutative 40.4%

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

       2. associate-*r* 41.4%

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

       3. times-frac 43.6%

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

       4. *-commutative 43.6%

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

       5. associate-/l* 43.6%

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

       6. unpow2 43.6%

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

       7. unpow2 43.6%

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

       8. unpow2 43.6%

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

       9. times-frac 49.2%

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

       10. swap-sqr 67.6%

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

       11. unpow2 67.6%

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

       12. associate-*r/ 66.5%

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

       13. *-commutative 66.5%

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

       14. associate-/l* 67.6%

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

   11. Simplified 67.6%

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

       1. pow1 67.6%

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

       2. *-commutative 67.6%

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

       3. associate-*l* 67.6%

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

       4. frac-times 59.1%

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

       5. sqrt-div 64.7%

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

       6. sqrt-unprod 91.3%

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

       7. add-sqr-sqrt 91.6%

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

   13. Applied egg-rr 91.6%

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

       1. unpow1 91.6%

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

       2. *-commutative 91.6%

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

       3. sub-neg 91.6%

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

       4. distribute-rgt-neg-in 91.6%

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

       5. *-commutative 91.6%

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

       6. distribute-lft-neg-in 91.6%

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

       7. metadata-eval 91.6%

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

       8. associate-*r* 91.6%

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

       9. *-commutative 91.6%

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

       10. *-commutative 91.6%

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

       11. associate-*r/ 90.5%

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

       12. *-commutative 90.5%

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

       13. associate-/l* 91.6%

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

   15. Simplified 91.6%

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

   if 6.39999999999999967e125 < h

   1. Initial program 53.2%

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

   3. Simplified 53.2%

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

   5. Taylor expanded in h around -inf 44.2%

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

      1. associate-*r* 44.2%

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

      2. neg-mul-1 44.2%

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

      3. sub-neg 44.2%

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

      4. distribute-lft-in 44.2%

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

   7. Simplified 61.8%

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

      1. sqrt-div 81.8%

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

   9. Applied egg-rr 81.8%

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

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \left(1 + \frac{{\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \left(h \cdot -0.125\right)}{\ell}\right)\right)\\ \mathbf{elif}\;h \leq 6.4 \cdot 10^{+125}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(M \cdot \frac{D}{d}\right)}^{2} \cdot -0.125\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\left(1 - h \cdot \left(0.125 \cdot \frac{{\left(M \cdot \frac{D}{d}\right)}^{2}}{\ell}\right)\right) \cdot \sqrt{\frac{d}{\ell}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 72.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := d \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ t_1 := {\left(D \cdot \frac{M\_m}{d}\right)}^{2}\\ t_2 := \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \frac{t\_1 \cdot \left(h \cdot -0.125\right)}{\ell}\right)\right)\\ \mathbf{if}\;d \leq -3.6 \cdot 10^{+159}:\\ \;\;\;\;t\_0 \cdot \left(-1 + 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M\_m}{2} \cdot \frac{D}{d}\right)}^{2}\right)\right)\\ \mathbf{elif}\;d \leq -4.4 \cdot 10^{-106}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;d \leq -8.5 \cdot 10^{-308}:\\ \;\;\;\;t\_0 \cdot \left(0.125 \cdot \left(\frac{h}{\ell} \cdot t\_1\right) + -1\right)\\ \mathbf{elif}\;d \leq 6.6 \cdot 10^{-155}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left(t\_1 \cdot 0.25, \frac{h}{\ell} \cdot -0.5, 1\right)}{\sqrt{\ell \cdot h}}\\ \mathbf{elif}\;d \leq 4.1 \cdot 10^{+113}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (let* ((t_0 (* d (sqrt (/ 1.0 (* l h)))))
        (t_1 (pow (* D (/ M_m d)) 2.0))
        (t_2
         (*
          (sqrt (/ d h))
          (* (sqrt (/ d l)) (+ 1.0 (/ (* t_1 (* h -0.125)) l))))))
   (if (<= d -3.6e+159)
     (* t_0 (+ -1.0 (* 0.5 (* (/ h l) (pow (* (/ M_m 2.0) (/ D d)) 2.0)))))
     (if (<= d -4.4e-106)
       t_2
       (if (<= d -8.5e-308)
         (* t_0 (+ (* 0.125 (* (/ h l) t_1)) -1.0))
         (if (<= d 6.6e-155)
           (* d (/ (fma (* t_1 0.25) (* (/ h l) -0.5) 1.0) (sqrt (* l h))))
           (if (<= d 4.1e+113) t_2 (* d (/ (sqrt (/ 1.0 h)) (sqrt l))))))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double t_0 = d * sqrt((1.0 / (l * h)));
	double t_1 = pow((D * (M_m / d)), 2.0);
	double t_2 = sqrt((d / h)) * (sqrt((d / l)) * (1.0 + ((t_1 * (h * -0.125)) / l)));
	double tmp;
	if (d <= -3.6e+159) {
		tmp = t_0 * (-1.0 + (0.5 * ((h / l) * pow(((M_m / 2.0) * (D / d)), 2.0))));
	} else if (d <= -4.4e-106) {
		tmp = t_2;
	} else if (d <= -8.5e-308) {
		tmp = t_0 * ((0.125 * ((h / l) * t_1)) + -1.0);
	} else if (d <= 6.6e-155) {
		tmp = d * (fma((t_1 * 0.25), ((h / l) * -0.5), 1.0) / sqrt((l * h)));
	} else if (d <= 4.1e+113) {
		tmp = t_2;
	} else {
		tmp = d * (sqrt((1.0 / h)) / sqrt(l));
	}
	return tmp;
}

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	t_0 = Float64(d * sqrt(Float64(1.0 / Float64(l * h))))
	t_1 = Float64(D * Float64(M_m / d)) ^ 2.0
	t_2 = Float64(sqrt(Float64(d / h)) * Float64(sqrt(Float64(d / l)) * Float64(1.0 + Float64(Float64(t_1 * Float64(h * -0.125)) / l))))
	tmp = 0.0
	if (d <= -3.6e+159)
		tmp = Float64(t_0 * Float64(-1.0 + Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(M_m / 2.0) * Float64(D / d)) ^ 2.0)))));
	elseif (d <= -4.4e-106)
		tmp = t_2;
	elseif (d <= -8.5e-308)
		tmp = Float64(t_0 * Float64(Float64(0.125 * Float64(Float64(h / l) * t_1)) + -1.0));
	elseif (d <= 6.6e-155)
		tmp = Float64(d * Float64(fma(Float64(t_1 * 0.25), Float64(Float64(h / l) * -0.5), 1.0) / sqrt(Float64(l * h))));
	elseif (d <= 4.1e+113)
		tmp = t_2;
	else
		tmp = Float64(d * Float64(sqrt(Float64(1.0 / h)) / sqrt(l)));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := Block[{t$95$0 = N[(d * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(D * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(N[(t$95$1 * N[(h * -0.125), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -3.6e+159], N[(t$95$0 * N[(-1.0 + N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M$95$m / 2.0), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -4.4e-106], t$95$2, If[LessEqual[d, -8.5e-308], N[(t$95$0 * N[(N[(0.125 * N[(N[(h / l), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 6.6e-155], N[(d * N[(N[(N[(t$95$1 * 0.25), $MachinePrecision] * N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision] + 1.0), $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 4.1e+113], t$95$2, N[(d * N[(N[Sqrt[N[(1.0 / h), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if d < -3.60000000000000037e159

   1. Initial program 55.0%

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

   3. Simplified 59.4%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. pow1 59.4%

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

      2. sqrt-unprod 51.4%

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

   6. Applied egg-rr 51.4%

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

      1. unpow1 51.4%

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

      2. associate-*l/ 43.1%

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

      3. associate-*r/ 26.4%

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

      4. unpow2 26.4%

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

   8. Simplified 26.4%

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

   9. Taylor expanded in d around -inf 82.5%

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

   if -3.60000000000000037e159 < d < -4.39999999999999989e-106 or 6.59999999999999972e-155 < d < 4.09999999999999993e113

   1. Initial program 78.0%

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

   3. Simplified 78.0%

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

   5. Taylor expanded in h around -inf 60.8%

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

      1. associate-*r* 60.8%

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

      2. neg-mul-1 60.8%

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

      3. sub-neg 60.8%

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

      4. distribute-lft-in 60.8%

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

   7. Simplified 86.1%

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

      1. pow1 86.1%

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

      2. associate-*r* 86.1%

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

   9. Applied egg-rr 86.1%

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

       1. unpow1 86.1%

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

       2. *-commutative 86.1%

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

       3. unpow2 86.1%

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

       4. *-commutative 86.1%

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

       5. *-commutative 86.1%

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

       6. swap-sqr 68.5%

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

       7. times-frac 65.2%

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

       8. unpow2 65.2%

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

       9. unpow2 65.2%

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

       10. unpow2 65.2%

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

       11. associate-*r/ 62.1%

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

       12. times-frac 60.8%

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

       13. associate-/r* 67.2%

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

       14. associate-*l/ 66.5%

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

   11. Simplified 85.3%

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

   if -4.39999999999999989e-106 < d < -8.49999999999999972e-308

   1. Initial program 40.1%

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

   3. Simplified 42.1%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. pow1 42.1%

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

      2. sqrt-unprod 32.2%

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

   6. Applied egg-rr 32.2%

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

      1. unpow1 32.2%

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

   8. Simplified 32.2%

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

   9. Taylor expanded in M around 0 22.6%

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

       1. *-commutative 22.6%

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

       2. associate-*r* 22.6%

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

       3. times-frac 22.6%

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

       4. *-commutative 22.6%

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

       5. associate-/l* 22.6%

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

       6. unpow2 22.6%

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

       7. unpow2 22.6%

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

       8. unpow2 22.6%

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

       9. times-frac 29.5%

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

       10. swap-sqr 32.2%

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

       11. unpow2 32.2%

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

       12. associate-*r/ 32.2%

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

       13. *-commutative 32.2%

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

       14. associate-/l* 32.2%

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

   11. Simplified 32.2%

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

   12. Taylor expanded in d around -inf 56.9%

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

   if -8.49999999999999972e-308 < d < 6.59999999999999972e-155

   1. Initial program 44.3%

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

   3. Simplified 44.3%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. pow1 44.3%

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

      2. sqrt-unprod 32.8%

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

   6. Applied egg-rr 32.8%

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

      1. unpow1 32.8%

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

      2. associate-*l/ 28.7%

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

      3. associate-*r/ 9.5%

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

      4. unpow2 9.5%

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

   8. Simplified 9.5%

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

      1. pow1 9.5%

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

      2. associate-/l/ 9.5%

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

      3. sqrt-div 12.0%

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

      4. unpow2 12.0%

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

      5. sqrt-prod 74.7%

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

      6. add-sqr-sqrt 75.0%

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

      7. cancel-sign-sub-inv 75.0%

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

      8. metadata-eval 75.0%

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

      9. *-commutative 75.0%

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

      10. div-inv 75.0%

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

      11. metadata-eval 75.0%

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

   10. Applied egg-rr 75.0%

       \[\leadsto \color{blue}{{\left(\frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 + -0.5 \cdot \left({\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)}^{1}} \]
   11. Step-by-step derivation

       1. unpow1 75.0%

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

       2. associate-*l/ 77.5%

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

       3. associate-/l* 77.5%

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

   12. Simplified 77.4%

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

   if 4.09999999999999993e113 < d

   1. Initial program 61.4%

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

   3. Simplified 63.9%

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

   5. Taylor expanded in d around inf 84.9%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. associate-/r* 85.7%

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

      2. sqrt-div 89.6%

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

   7. Applied egg-rr 89.6%

      \[\leadsto d \cdot \color{blue}{\frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -3.6 \cdot 10^{+159}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(-1 + 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}\right)\right)\\ \mathbf{elif}\;d \leq -4.4 \cdot 10^{-106}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \frac{{\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \left(h \cdot -0.125\right)}{\ell}\right)\right)\\ \mathbf{elif}\;d \leq -8.5 \cdot 10^{-308}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(0.125 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}\right) + -1\right)\\ \mathbf{elif}\;d \leq 6.6 \cdot 10^{-155}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot 0.25, \frac{h}{\ell} \cdot -0.5, 1\right)}{\sqrt{\ell \cdot h}}\\ \mathbf{elif}\;d \leq 4.1 \cdot 10^{+113}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \frac{{\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \left(h \cdot -0.125\right)}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 74.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M\_m}{2} \cdot \frac{D}{d}\right)}^{2}\right)\\ t_1 := d \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ t_2 := {\left(D \cdot \frac{M\_m}{d}\right)}^{2}\\ \mathbf{if}\;d \leq -8.6 \cdot 10^{+158}:\\ \;\;\;\;t\_1 \cdot \left(-1 + t\_0\right)\\ \mathbf{elif}\;d \leq -8.8 \cdot 10^{-106}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \frac{t\_2 \cdot \left(h \cdot -0.125\right)}{\ell}\right)\right)\\ \mathbf{elif}\;d \leq -8.5 \cdot 10^{-308}:\\ \;\;\;\;t\_1 \cdot \left(0.125 \cdot \left(\frac{h}{\ell} \cdot t\_2\right) + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - t\_0\right) \cdot \frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (let* ((t_0 (* 0.5 (* (/ h l) (pow (* (/ M_m 2.0) (/ D d)) 2.0))))
        (t_1 (* d (sqrt (/ 1.0 (* l h)))))
        (t_2 (pow (* D (/ M_m d)) 2.0)))
   (if (<= d -8.6e+158)
     (* t_1 (+ -1.0 t_0))
     (if (<= d -8.8e-106)
       (* (sqrt (/ d h)) (* (sqrt (/ d l)) (+ 1.0 (/ (* t_2 (* h -0.125)) l))))
       (if (<= d -8.5e-308)
         (* t_1 (+ (* 0.125 (* (/ h l) t_2)) -1.0))
         (* (- 1.0 t_0) (/ (/ d (sqrt h)) (sqrt l))))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double t_0 = 0.5 * ((h / l) * pow(((M_m / 2.0) * (D / d)), 2.0));
	double t_1 = d * sqrt((1.0 / (l * h)));
	double t_2 = pow((D * (M_m / d)), 2.0);
	double tmp;
	if (d <= -8.6e+158) {
		tmp = t_1 * (-1.0 + t_0);
	} else if (d <= -8.8e-106) {
		tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0 + ((t_2 * (h * -0.125)) / l)));
	} else if (d <= -8.5e-308) {
		tmp = t_1 * ((0.125 * ((h / l) * t_2)) + -1.0);
	} else {
		tmp = (1.0 - t_0) * ((d / sqrt(h)) / sqrt(l));
	}
	return tmp;
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = 0.5d0 * ((h / l) * (((m_m / 2.0d0) * (d_1 / d)) ** 2.0d0))
    t_1 = d * sqrt((1.0d0 / (l * h)))
    t_2 = (d_1 * (m_m / d)) ** 2.0d0
    if (d <= (-8.6d+158)) then
        tmp = t_1 * ((-1.0d0) + t_0)
    else if (d <= (-8.8d-106)) then
        tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0d0 + ((t_2 * (h * (-0.125d0))) / l)))
    else if (d <= (-8.5d-308)) then
        tmp = t_1 * ((0.125d0 * ((h / l) * t_2)) + (-1.0d0))
    else
        tmp = (1.0d0 - t_0) * ((d / sqrt(h)) / sqrt(l))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double t_0 = 0.5 * ((h / l) * Math.pow(((M_m / 2.0) * (D / d)), 2.0));
	double t_1 = d * Math.sqrt((1.0 / (l * h)));
	double t_2 = Math.pow((D * (M_m / d)), 2.0);
	double tmp;
	if (d <= -8.6e+158) {
		tmp = t_1 * (-1.0 + t_0);
	} else if (d <= -8.8e-106) {
		tmp = Math.sqrt((d / h)) * (Math.sqrt((d / l)) * (1.0 + ((t_2 * (h * -0.125)) / l)));
	} else if (d <= -8.5e-308) {
		tmp = t_1 * ((0.125 * ((h / l) * t_2)) + -1.0);
	} else {
		tmp = (1.0 - t_0) * ((d / Math.sqrt(h)) / Math.sqrt(l));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	t_0 = 0.5 * ((h / l) * math.pow(((M_m / 2.0) * (D / d)), 2.0))
	t_1 = d * math.sqrt((1.0 / (l * h)))
	t_2 = math.pow((D * (M_m / d)), 2.0)
	tmp = 0
	if d <= -8.6e+158:
		tmp = t_1 * (-1.0 + t_0)
	elif d <= -8.8e-106:
		tmp = math.sqrt((d / h)) * (math.sqrt((d / l)) * (1.0 + ((t_2 * (h * -0.125)) / l)))
	elif d <= -8.5e-308:
		tmp = t_1 * ((0.125 * ((h / l) * t_2)) + -1.0)
	else:
		tmp = (1.0 - t_0) * ((d / math.sqrt(h)) / math.sqrt(l))
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	t_0 = Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(M_m / 2.0) * Float64(D / d)) ^ 2.0)))
	t_1 = Float64(d * sqrt(Float64(1.0 / Float64(l * h))))
	t_2 = Float64(D * Float64(M_m / d)) ^ 2.0
	tmp = 0.0
	if (d <= -8.6e+158)
		tmp = Float64(t_1 * Float64(-1.0 + t_0));
	elseif (d <= -8.8e-106)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(sqrt(Float64(d / l)) * Float64(1.0 + Float64(Float64(t_2 * Float64(h * -0.125)) / l))));
	elseif (d <= -8.5e-308)
		tmp = Float64(t_1 * Float64(Float64(0.125 * Float64(Float64(h / l) * t_2)) + -1.0));
	else
		tmp = Float64(Float64(1.0 - t_0) * Float64(Float64(d / sqrt(h)) / sqrt(l)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp_2 = code(d, h, l, M_m, D)
	t_0 = 0.5 * ((h / l) * (((M_m / 2.0) * (D / d)) ^ 2.0));
	t_1 = d * sqrt((1.0 / (l * h)));
	t_2 = (D * (M_m / d)) ^ 2.0;
	tmp = 0.0;
	if (d <= -8.6e+158)
		tmp = t_1 * (-1.0 + t_0);
	elseif (d <= -8.8e-106)
		tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0 + ((t_2 * (h * -0.125)) / l)));
	elseif (d <= -8.5e-308)
		tmp = t_1 * ((0.125 * ((h / l) * t_2)) + -1.0);
	else
		tmp = (1.0 - t_0) * ((d / sqrt(h)) / sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := Block[{t$95$0 = N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M$95$m / 2.0), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(d * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[(D * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[d, -8.6e+158], N[(t$95$1 * N[(-1.0 + t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -8.8e-106], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(N[(t$95$2 * N[(h * -0.125), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -8.5e-308], N[(t$95$1 * N[(N[(0.125 * N[(N[(h / l), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - t$95$0), $MachinePrecision] * N[(N[(d / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -8.6e158

   1. Initial program 55.0%

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

   3. Simplified 59.4%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. pow1 59.4%

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

      2. sqrt-unprod 51.4%

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

   6. Applied egg-rr 51.4%

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

      1. unpow1 51.4%

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

      2. associate-*l/ 43.1%

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

      3. associate-*r/ 26.4%

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

      4. unpow2 26.4%

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

   8. Simplified 26.4%

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

   9. Taylor expanded in d around -inf 82.5%

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

   if -8.6e158 < d < -8.79999999999999977e-106

   1. Initial program 71.4%

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

   3. Simplified 71.4%

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

   5. Taylor expanded in h around -inf 62.2%

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

      1. associate-*r* 62.2%

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

      2. neg-mul-1 62.2%

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

      3. sub-neg 62.2%

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

      4. distribute-lft-in 62.2%

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

   7. Simplified 83.0%

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

      1. pow1 83.0%

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

      2. associate-*r* 83.0%

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

   9. Applied egg-rr 83.0%

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

       1. unpow1 83.0%

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

       2. *-commutative 83.0%

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

       3. unpow2 83.0%

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

       4. *-commutative 83.0%

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

       5. *-commutative 83.0%

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

       6. swap-sqr 71.6%

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

       7. times-frac 68.3%

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

       8. unpow2 68.3%

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

       9. unpow2 68.3%

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

       10. unpow2 68.3%

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

       11. associate-*r/ 63.8%

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

       12. times-frac 62.2%

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

       13. associate-/r* 70.6%

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

       14. associate-*l/ 70.6%

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

   11. Simplified 83.0%

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

   if -8.79999999999999977e-106 < d < -8.49999999999999972e-308

   1. Initial program 40.1%

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

   3. Simplified 42.1%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. pow1 42.1%

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

      2. sqrt-unprod 32.2%

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

   6. Applied egg-rr 32.2%

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

      1. unpow1 32.2%

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

   8. Simplified 32.2%

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

   9. Taylor expanded in M around 0 22.6%

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

       1. *-commutative 22.6%

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

       2. associate-*r* 22.6%

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

       3. times-frac 22.6%

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

       4. *-commutative 22.6%

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

       5. associate-/l* 22.6%

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

       6. unpow2 22.6%

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

       7. unpow2 22.6%

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

       8. unpow2 22.6%

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

       9. times-frac 29.5%

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

       10. swap-sqr 32.2%

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

       11. unpow2 32.2%

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

       12. associate-*r/ 32.2%

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

       13. *-commutative 32.2%

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

       14. associate-/l* 32.2%

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

   11. Simplified 32.2%

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

   12. Taylor expanded in d around -inf 56.9%

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

   if -8.49999999999999972e-308 < d

   1. Initial program 69.3%

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

   3. Simplified 70.1%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. pow1 70.1%

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

      2. sqrt-unprod 59.7%

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

   6. Applied egg-rr 59.7%

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

      1. unpow1 59.7%

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

      2. associate-*l/ 59.6%

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

      3. associate-*r/ 50.3%

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

      4. unpow2 50.3%

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

   8. Simplified 50.3%

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

      1. sqrt-div 56.3%

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

      2. sqrt-div 60.5%

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

      3. unpow2 60.5%

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

      4. sqrt-prod 83.6%

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

      5. add-sqr-sqrt 83.8%

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

   10. Applied egg-rr 83.8%

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

       1. associate-/l/ 86.0%

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

       2. associate-/r* 81.9%

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

   12. Simplified 81.9%

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

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -8.6 \cdot 10^{+158}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(-1 + 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}\right)\right)\\ \mathbf{elif}\;d \leq -8.8 \cdot 10^{-106}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \frac{{\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \left(h \cdot -0.125\right)}{\ell}\right)\right)\\ \mathbf{elif}\;d \leq -8.5 \cdot 10^{-308}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(0.125 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}\right) + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}\right)\right) \cdot \frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 69.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;h \leq -9.2 \cdot 10^{+130}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \frac{h \cdot {\left(\frac{D}{d} \cdot \left(M\_m \cdot 0.5\right)\right)}^{2}}{\ell}\right)\\ \mathbf{elif}\;h \leq -4.5 \cdot 10^{-303}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(-1 + 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M\_m}{2} \cdot \frac{D}{d}\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left({\left(D \cdot \frac{M\_m}{d}\right)}^{2} \cdot 0.25, \frac{h}{\ell} \cdot -0.5, 1\right)}{\sqrt{\ell \cdot h}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (if (<= h -9.2e+130)
   (*
    (sqrt (* (/ d h) (/ d l)))
    (- 1.0 (* 0.5 (/ (* h (pow (* (/ D d) (* M_m 0.5)) 2.0)) l))))
   (if (<= h -4.5e-303)
     (*
      (* d (sqrt (/ 1.0 (* l h))))
      (+ -1.0 (* 0.5 (* (/ h l) (pow (* (/ M_m 2.0) (/ D d)) 2.0)))))
     (*
      d
      (/
       (fma (* (pow (* D (/ M_m d)) 2.0) 0.25) (* (/ h l) -0.5) 1.0)
       (sqrt (* l h)))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (h <= -9.2e+130) {
		tmp = sqrt(((d / h) * (d / l))) * (1.0 - (0.5 * ((h * pow(((D / d) * (M_m * 0.5)), 2.0)) / l)));
	} else if (h <= -4.5e-303) {
		tmp = (d * sqrt((1.0 / (l * h)))) * (-1.0 + (0.5 * ((h / l) * pow(((M_m / 2.0) * (D / d)), 2.0))));
	} else {
		tmp = d * (fma((pow((D * (M_m / d)), 2.0) * 0.25), ((h / l) * -0.5), 1.0) / sqrt((l * h)));
	}
	return tmp;
}

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	tmp = 0.0
	if (h <= -9.2e+130)
		tmp = Float64(sqrt(Float64(Float64(d / h) * Float64(d / l))) * Float64(1.0 - Float64(0.5 * Float64(Float64(h * (Float64(Float64(D / d) * Float64(M_m * 0.5)) ^ 2.0)) / l))));
	elseif (h <= -4.5e-303)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(l * h)))) * Float64(-1.0 + Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(M_m / 2.0) * Float64(D / d)) ^ 2.0)))));
	else
		tmp = Float64(d * Float64(fma(Float64((Float64(D * Float64(M_m / d)) ^ 2.0) * 0.25), Float64(Float64(h / l) * -0.5), 1.0) / sqrt(Float64(l * h))));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := If[LessEqual[h, -9.2e+130], N[(N[Sqrt[N[(N[(d / h), $MachinePrecision] * N[(d / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(h * N[Power[N[(N[(D / d), $MachinePrecision] * N[(M$95$m * 0.5), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, -4.5e-303], N[(N[(d * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-1.0 + N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M$95$m / 2.0), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[(N[(N[Power[N[(D * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * 0.25), $MachinePrecision] * N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision] + 1.0), $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if h < -9.20000000000000085e130

   1. Initial program 40.1%

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

   3. Simplified 40.1%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. pow1 40.1%

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

      2. sqrt-unprod 28.6%

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

   6. Applied egg-rr 28.6%

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

      1. unpow1 28.6%

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

   8. Simplified 28.6%

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

      1. associate-*r/ 43.9%

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

      2. *-commutative 43.9%

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

      3. div-inv 43.9%

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

      4. metadata-eval 43.9%

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

   10. Applied egg-rr 43.9%

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

   if -9.20000000000000085e130 < h < -4.5000000000000001e-303

   1. Initial program 67.8%

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

   3. Simplified 69.9%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. pow1 69.9%

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

      2. sqrt-unprod 56.9%

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

   6. Applied egg-rr 56.9%

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

      1. unpow1 56.9%

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

      2. associate-*l/ 55.8%

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

      3. associate-*r/ 44.5%

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

      4. unpow2 44.5%

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

   8. Simplified 44.5%

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

   9. Taylor expanded in d around -inf 75.3%

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

   if -4.5000000000000001e-303 < h

   1. Initial program 69.3%

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

   3. Simplified 70.1%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. pow1 70.1%

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

      2. sqrt-unprod 59.7%

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

   6. Applied egg-rr 59.7%

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

      1. unpow1 59.7%

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

      2. associate-*l/ 59.6%

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

      3. associate-*r/ 50.3%

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

      4. unpow2 50.3%

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

   8. Simplified 50.3%

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

      1. pow1 50.3%

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

      2. associate-/l/ 50.3%

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

      3. sqrt-div 54.5%

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

      4. unpow2 54.5%

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

      5. sqrt-prod 78.3%

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

      6. add-sqr-sqrt 78.5%

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

      7. cancel-sign-sub-inv 78.5%

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

      8. metadata-eval 78.5%

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

      9. *-commutative 78.5%

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

      10. div-inv 78.5%

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

      11. metadata-eval 78.5%

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

   10. Applied egg-rr 78.5%

       \[\leadsto \color{blue}{{\left(\frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 + -0.5 \cdot \left({\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)}^{1}} \]
   11. Step-by-step derivation

       1. unpow1 78.5%

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

       2. associate-*l/ 79.0%

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

       3. associate-/l* 79.0%

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

   12. Simplified 78.9%

       \[\leadsto \color{blue}{d \cdot \frac{\mathsf{fma}\left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot 0.25, \frac{h}{\ell} \cdot -0.5, 1\right)}{\sqrt{h \cdot \ell}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -9.2 \cdot 10^{+130}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \frac{h \cdot {\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2}}{\ell}\right)\\ \mathbf{elif}\;h \leq -4.5 \cdot 10^{-303}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(-1 + 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot 0.25, \frac{h}{\ell} \cdot -0.5, 1\right)}{\sqrt{\ell \cdot h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 57.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := \frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{if}\;d \leq -7.2 \cdot 10^{+118}:\\ \;\;\;\;\left|t\_0\right|\\ \mathbf{elif}\;d \leq -1.5 \cdot 10^{-77}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}}\\ \mathbf{elif}\;d \leq -4 \cdot 10^{-311}:\\ \;\;\;\;d \cdot {\left(-1 + \mathsf{fma}\left(h, \ell, 1\right)\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(M\_m \cdot \frac{D}{d}\right)}^{2} \cdot -0.125\right)\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (let* ((t_0 (/ d (sqrt (* l h)))))
   (if (<= d -7.2e+118)
     (fabs t_0)
     (if (<= d -1.5e-77)
       (/ (sqrt (/ d l)) (sqrt (/ h d)))
       (if (<= d -4e-311)
         (* d (pow (+ -1.0 (fma h l 1.0)) -0.5))
         (* t_0 (+ 1.0 (* (/ h l) (* (pow (* M_m (/ D d)) 2.0) -0.125)))))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double t_0 = d / sqrt((l * h));
	double tmp;
	if (d <= -7.2e+118) {
		tmp = fabs(t_0);
	} else if (d <= -1.5e-77) {
		tmp = sqrt((d / l)) / sqrt((h / d));
	} else if (d <= -4e-311) {
		tmp = d * pow((-1.0 + fma(h, l, 1.0)), -0.5);
	} else {
		tmp = t_0 * (1.0 + ((h / l) * (pow((M_m * (D / d)), 2.0) * -0.125)));
	}
	return tmp;
}

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	t_0 = Float64(d / sqrt(Float64(l * h)))
	tmp = 0.0
	if (d <= -7.2e+118)
		tmp = abs(t_0);
	elseif (d <= -1.5e-77)
		tmp = Float64(sqrt(Float64(d / l)) / sqrt(Float64(h / d)));
	elseif (d <= -4e-311)
		tmp = Float64(d * (Float64(-1.0 + fma(h, l, 1.0)) ^ -0.5));
	else
		tmp = Float64(t_0 * Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(M_m * Float64(D / d)) ^ 2.0) * -0.125))));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := Block[{t$95$0 = N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -7.2e+118], N[Abs[t$95$0], $MachinePrecision], If[LessEqual[d, -1.5e-77], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(h / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -4e-311], N[(d * N[Power[N[(-1.0 + N[(h * l + 1.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(M$95$m * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -7.2e118

   1. Initial program 54.0%

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

   3. Simplified 57.2%

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

   5. Taylor expanded in d around inf 1.2%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. *-un-lft-identity 1.2%

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

      2. inv-pow 1.2%

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

      3. sqrt-pow1 1.2%

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

      4. metadata-eval 1.2%

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

   7. Applied egg-rr 1.2%

      \[\leadsto d \cdot \color{blue}{\left(1 \cdot {\left(h \cdot \ell\right)}^{-0.5}\right)} \]
   8. Step-by-step derivation

      1. *-lft-identity 1.2%

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

   9. Simplified 1.2%

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

   10. Taylor expanded in d around 0 1.2%

       \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   11. Step-by-step derivation

       1. *-commutative 1.2%

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

       2. associate-/r* 1.3%

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

   12. Simplified 1.3%

       \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}} \]
   13. Step-by-step derivation

       1. add-sqr-sqrt 0.2%

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

       2. sqrt-unprod 35.3%

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

       3. swap-sqr 22.7%

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

       4. sqr-neg 22.7%

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

       5. add-sqr-sqrt 22.8%

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

       6. associate-/l/ 22.8%

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

       7. div-inv 22.8%

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

       8. sqr-neg 22.8%

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

       9. frac-times 41.3%

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

       10. sqrt-prod 47.2%

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

       11. frac-2neg 47.2%

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

       12. sqrt-undiv 64.9%

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

       13. add-sqr-sqrt 64.8%

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

       14. sqrt-prod 41.0%

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

       15. rem-sqrt-square 64.9%

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

   14. Applied egg-rr 72.7%

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

   if -7.2e118 < d < -1.50000000000000008e-77

   1. Initial program 72.7%

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

   3. Simplified 72.7%

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

   5. Taylor expanded in d around inf 6.3%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. *-un-lft-identity 6.3%

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

      2. inv-pow 6.3%

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

      3. sqrt-pow1 6.3%

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

      4. metadata-eval 6.3%

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

   7. Applied egg-rr 6.3%

      \[\leadsto d \cdot \color{blue}{\left(1 \cdot {\left(h \cdot \ell\right)}^{-0.5}\right)} \]
   8. Step-by-step derivation

      1. *-lft-identity 6.3%

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

   9. Simplified 6.3%

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

   10. Taylor expanded in d around 0 6.3%

       \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   11. Step-by-step derivation

       1. *-commutative 6.3%

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

       2. associate-/r* 6.3%

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

   12. Simplified 6.3%

       \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}} \]
   13. Step-by-step derivation

       1. add-sqr-sqrt 0.6%

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

       2. sqrt-unprod 40.8%

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

       3. swap-sqr 40.6%

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

       4. sqr-neg 40.6%

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

       5. add-sqr-sqrt 40.7%

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

       6. associate-/l/ 40.7%

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

       7. div-inv 40.7%

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

       8. sqr-neg 40.7%

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

       9. frac-times 40.6%

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

       10. sqrt-prod 56.7%

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

       11. frac-2neg 56.7%

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

       12. sqrt-undiv 56.7%

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

       13. *-commutative 56.7%

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

       14. clear-num 56.6%

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

       15. un-div-inv 56.7%

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

       16. sqrt-undiv 56.8%

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

       17. frac-2neg 56.8%

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

   14. Applied egg-rr 56.8%

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

   if -1.50000000000000008e-77 < d < -3.99999999999979e-311

   1. Initial program 45.9%

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

   3. Simplified 47.6%

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

   5. Taylor expanded in d around inf 14.0%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. *-un-lft-identity 14.0%

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

      2. inv-pow 14.0%

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

      3. sqrt-pow1 14.0%

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

      4. metadata-eval 14.0%

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

   7. Applied egg-rr 14.0%

      \[\leadsto d \cdot \color{blue}{\left(1 \cdot {\left(h \cdot \ell\right)}^{-0.5}\right)} \]
   8. Step-by-step derivation

      1. *-lft-identity 14.0%

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

   9. Simplified 14.0%

      \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]
   10. Step-by-step derivation

       1. expm1-log1p-u 14.0%

          \[\leadsto d \cdot {\color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(h \cdot \ell\right)\right)\right)}}^{-0.5} \]

       2. expm1-undefine 34.2%

          \[\leadsto d \cdot {\color{blue}{\left(e^{\mathsf{log1p}\left(h \cdot \ell\right)} - 1\right)}}^{-0.5} \]

   11. Applied egg-rr 34.2%

       \[\leadsto d \cdot {\color{blue}{\left(e^{\mathsf{log1p}\left(h \cdot \ell\right)} - 1\right)}}^{-0.5} \]
   12. Step-by-step derivation

       1. sub-neg 34.2%

          \[\leadsto d \cdot {\color{blue}{\left(e^{\mathsf{log1p}\left(h \cdot \ell\right)} + \left(-1\right)\right)}}^{-0.5} \]

       2. metadata-eval 34.2%

          \[\leadsto d \cdot {\left(e^{\mathsf{log1p}\left(h \cdot \ell\right)} + \color{blue}{-1}\right)}^{-0.5} \]

       3. +-commutative 34.2%

          \[\leadsto d \cdot {\color{blue}{\left(-1 + e^{\mathsf{log1p}\left(h \cdot \ell\right)}\right)}}^{-0.5} \]

       4. log1p-undefine 34.2%

          \[\leadsto d \cdot {\left(-1 + e^{\color{blue}{\log \left(1 + h \cdot \ell\right)}}\right)}^{-0.5} \]

       5. rem-exp-log 34.2%

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

       6. +-commutative 34.2%

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

       7. fma-define 34.2%

          \[\leadsto d \cdot {\left(-1 + \color{blue}{\mathsf{fma}\left(h, \ell, 1\right)}\right)}^{-0.5} \]

   13. Simplified 34.2%

       \[\leadsto d \cdot {\color{blue}{\left(-1 + \mathsf{fma}\left(h, \ell, 1\right)\right)}}^{-0.5} \]

   if -3.99999999999979e-311 < d

   1. Initial program 69.9%

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

   3. Simplified 70.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. pow1 70.7%

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

      2. sqrt-unprod 60.1%

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

   6. Applied egg-rr 60.1%

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

      1. unpow1 60.1%

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

   8. Simplified 60.1%

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

   9. Taylor expanded in M around 0 37.5%

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

       1. *-commutative 37.5%

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

       2. associate-*r* 38.3%

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

       3. times-frac 39.0%

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

       4. *-commutative 39.0%

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

       5. associate-/l* 38.9%

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

       6. unpow2 38.9%

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

       7. unpow2 38.9%

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

       8. unpow2 38.9%

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

       9. times-frac 44.7%

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

       10. swap-sqr 60.1%

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

       11. unpow2 60.1%

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

       12. associate-*r/ 59.3%

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

       13. *-commutative 59.3%

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

       14. associate-/l* 60.2%

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

   11. Simplified 60.2%

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

       1. pow1 60.2%

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

       2. *-commutative 60.2%

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

       3. associate-*l* 60.2%

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

       4. frac-times 50.8%

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

       5. sqrt-div 54.9%

          \[\leadsto {\left(\left(1 - {\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot 0.125\right)\right) \cdot \color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}}\right)}^{1} \]

       6. sqrt-unprod 78.9%

          \[\leadsto {\left(\left(1 - {\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot 0.125\right)\right) \cdot \frac{\color{blue}{\sqrt{d} \cdot \sqrt{d}}}{\sqrt{h \cdot \ell}}\right)}^{1} \]

       7. add-sqr-sqrt 79.2%

          \[\leadsto {\left(\left(1 - {\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot 0.125\right)\right) \cdot \frac{\color{blue}{d}}{\sqrt{h \cdot \ell}}\right)}^{1} \]

   13. Applied egg-rr 79.2%

       \[\leadsto \color{blue}{{\left(\left(1 - {\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot 0.125\right)\right) \cdot \frac{d}{\sqrt{h \cdot \ell}}\right)}^{1}} \]
   14. Step-by-step derivation

       1. unpow1 79.2%

          \[\leadsto \color{blue}{\left(1 - {\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot 0.125\right)\right) \cdot \frac{d}{\sqrt{h \cdot \ell}}} \]

       2. *-commutative 79.2%

          \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 - {\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot 0.125\right)\right)} \]

       3. sub-neg 79.2%

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \color{blue}{\left(1 + \left(-{\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot 0.125\right)\right)\right)} \]

       4. distribute-rgt-neg-in 79.2%

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 + \color{blue}{{\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \left(-\frac{h}{\ell} \cdot 0.125\right)}\right) \]

       5. *-commutative 79.2%

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 + {\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \left(-\color{blue}{0.125 \cdot \frac{h}{\ell}}\right)\right) \]

       6. distribute-lft-neg-in 79.2%

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 + {\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \color{blue}{\left(\left(-0.125\right) \cdot \frac{h}{\ell}\right)}\right) \]

       7. metadata-eval 79.2%

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 + {\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \left(\color{blue}{-0.125} \cdot \frac{h}{\ell}\right)\right) \]

       8. associate-*r* 79.1%

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 + \color{blue}{\left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot -0.125\right) \cdot \frac{h}{\ell}}\right) \]

       9. *-commutative 79.1%

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 + \color{blue}{\frac{h}{\ell} \cdot \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot -0.125\right)}\right) \]

       10. *-commutative 79.1%

           \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 + \frac{h}{\ell} \cdot \color{blue}{\left(-0.125 \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}\right)}\right) \]

       11. associate-*r/ 78.3%

           \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.125 \cdot {\color{blue}{\left(\frac{D \cdot M}{d}\right)}}^{2}\right)\right) \]

       12. *-commutative 78.3%

           \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.125 \cdot {\left(\frac{\color{blue}{M \cdot D}}{d}\right)}^{2}\right)\right) \]

       13. associate-/l* 79.2%

           \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.125 \cdot {\color{blue}{\left(M \cdot \frac{D}{d}\right)}}^{2}\right)\right) \]

   15. Simplified 79.2%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.125 \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 65.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -7.2 \cdot 10^{+118}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \mathbf{elif}\;d \leq -1.5 \cdot 10^{-77}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}}\\ \mathbf{elif}\;d \leq -4 \cdot 10^{-311}:\\ \;\;\;\;d \cdot {\left(-1 + \mathsf{fma}\left(h, \ell, 1\right)\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(M \cdot \frac{D}{d}\right)}^{2} \cdot -0.125\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 68.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;h \leq -1.3 \cdot 10^{+131}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \frac{h \cdot {\left(\frac{D}{d} \cdot \left(M\_m \cdot 0.5\right)\right)}^{2}}{\ell}\right)\\ \mathbf{elif}\;h \leq -4.5 \cdot 10^{-303}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(-1 + 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M\_m}{2} \cdot \frac{D}{d}\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(M\_m \cdot \frac{D}{d}\right)}^{2} \cdot -0.125\right)\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (if (<= h -1.3e+131)
   (*
    (sqrt (* (/ d h) (/ d l)))
    (- 1.0 (* 0.5 (/ (* h (pow (* (/ D d) (* M_m 0.5)) 2.0)) l))))
   (if (<= h -4.5e-303)
     (*
      (* d (sqrt (/ 1.0 (* l h))))
      (+ -1.0 (* 0.5 (* (/ h l) (pow (* (/ M_m 2.0) (/ D d)) 2.0)))))
     (*
      (/ d (sqrt (* l h)))
      (+ 1.0 (* (/ h l) (* (pow (* M_m (/ D d)) 2.0) -0.125)))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (h <= -1.3e+131) {
		tmp = sqrt(((d / h) * (d / l))) * (1.0 - (0.5 * ((h * pow(((D / d) * (M_m * 0.5)), 2.0)) / l)));
	} else if (h <= -4.5e-303) {
		tmp = (d * sqrt((1.0 / (l * h)))) * (-1.0 + (0.5 * ((h / l) * pow(((M_m / 2.0) * (D / d)), 2.0))));
	} else {
		tmp = (d / sqrt((l * h))) * (1.0 + ((h / l) * (pow((M_m * (D / d)), 2.0) * -0.125)));
	}
	return tmp;
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (h <= (-1.3d+131)) then
        tmp = sqrt(((d / h) * (d / l))) * (1.0d0 - (0.5d0 * ((h * (((d_1 / d) * (m_m * 0.5d0)) ** 2.0d0)) / l)))
    else if (h <= (-4.5d-303)) then
        tmp = (d * sqrt((1.0d0 / (l * h)))) * ((-1.0d0) + (0.5d0 * ((h / l) * (((m_m / 2.0d0) * (d_1 / d)) ** 2.0d0))))
    else
        tmp = (d / sqrt((l * h))) * (1.0d0 + ((h / l) * (((m_m * (d_1 / d)) ** 2.0d0) * (-0.125d0))))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (h <= -1.3e+131) {
		tmp = Math.sqrt(((d / h) * (d / l))) * (1.0 - (0.5 * ((h * Math.pow(((D / d) * (M_m * 0.5)), 2.0)) / l)));
	} else if (h <= -4.5e-303) {
		tmp = (d * Math.sqrt((1.0 / (l * h)))) * (-1.0 + (0.5 * ((h / l) * Math.pow(((M_m / 2.0) * (D / d)), 2.0))));
	} else {
		tmp = (d / Math.sqrt((l * h))) * (1.0 + ((h / l) * (Math.pow((M_m * (D / d)), 2.0) * -0.125)));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	tmp = 0
	if h <= -1.3e+131:
		tmp = math.sqrt(((d / h) * (d / l))) * (1.0 - (0.5 * ((h * math.pow(((D / d) * (M_m * 0.5)), 2.0)) / l)))
	elif h <= -4.5e-303:
		tmp = (d * math.sqrt((1.0 / (l * h)))) * (-1.0 + (0.5 * ((h / l) * math.pow(((M_m / 2.0) * (D / d)), 2.0))))
	else:
		tmp = (d / math.sqrt((l * h))) * (1.0 + ((h / l) * (math.pow((M_m * (D / d)), 2.0) * -0.125)))
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	tmp = 0.0
	if (h <= -1.3e+131)
		tmp = Float64(sqrt(Float64(Float64(d / h) * Float64(d / l))) * Float64(1.0 - Float64(0.5 * Float64(Float64(h * (Float64(Float64(D / d) * Float64(M_m * 0.5)) ^ 2.0)) / l))));
	elseif (h <= -4.5e-303)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(l * h)))) * Float64(-1.0 + Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(M_m / 2.0) * Float64(D / d)) ^ 2.0)))));
	else
		tmp = Float64(Float64(d / sqrt(Float64(l * h))) * Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(M_m * Float64(D / d)) ^ 2.0) * -0.125))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp_2 = code(d, h, l, M_m, D)
	tmp = 0.0;
	if (h <= -1.3e+131)
		tmp = sqrt(((d / h) * (d / l))) * (1.0 - (0.5 * ((h * (((D / d) * (M_m * 0.5)) ^ 2.0)) / l)));
	elseif (h <= -4.5e-303)
		tmp = (d * sqrt((1.0 / (l * h)))) * (-1.0 + (0.5 * ((h / l) * (((M_m / 2.0) * (D / d)) ^ 2.0))));
	else
		tmp = (d / sqrt((l * h))) * (1.0 + ((h / l) * (((M_m * (D / d)) ^ 2.0) * -0.125)));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := If[LessEqual[h, -1.3e+131], N[(N[Sqrt[N[(N[(d / h), $MachinePrecision] * N[(d / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(h * N[Power[N[(N[(D / d), $MachinePrecision] * N[(M$95$m * 0.5), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, -4.5e-303], N[(N[(d * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-1.0 + N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M$95$m / 2.0), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(M$95$m * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if h < -1.3e131

   1. Initial program 40.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 40.1%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. pow1 40.1%

         \[\leadsto \color{blue}{{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)}^{1}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      2. sqrt-unprod 28.6%

         \[\leadsto {\color{blue}{\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right)}}^{1} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   6. Applied egg-rr 28.6%

      \[\leadsto \color{blue}{{\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right)}^{1}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
   7. Step-by-step derivation

      1. unpow1 28.6%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   8. Simplified 28.6%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
   9. Step-by-step derivation

      1. associate-*r/ 43.9%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot h}{\ell}}\right) \]

      2. *-commutative 43.9%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \frac{{\color{blue}{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}}^{2} \cdot h}{\ell}\right) \]

      3. div-inv 43.9%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \frac{{\left(\frac{D}{d} \cdot \color{blue}{\left(M \cdot \frac{1}{2}\right)}\right)}^{2} \cdot h}{\ell}\right) \]

      4. metadata-eval 43.9%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \frac{{\left(\frac{D}{d} \cdot \left(M \cdot \color{blue}{0.5}\right)\right)}^{2} \cdot h}{\ell}\right) \]

   10. Applied egg-rr 43.9%

       \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{{\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2} \cdot h}{\ell}}\right) \]

   if -1.3e131 < h < -4.5000000000000001e-303

   1. Initial program 67.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 69.9%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. pow1 69.9%

         \[\leadsto \color{blue}{{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)}^{1}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      2. sqrt-unprod 56.9%

         \[\leadsto {\color{blue}{\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right)}}^{1} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   6. Applied egg-rr 56.9%

      \[\leadsto \color{blue}{{\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right)}^{1}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
   7. Step-by-step derivation

      1. unpow1 56.9%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      2. associate-*l/ 55.8%

         \[\leadsto \sqrt{\color{blue}{\frac{d \cdot \frac{d}{\ell}}{h}}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      3. associate-*r/ 44.5%

         \[\leadsto \sqrt{\frac{\color{blue}{\frac{d \cdot d}{\ell}}}{h}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      4. unpow2 44.5%

         \[\leadsto \sqrt{\frac{\frac{\color{blue}{{d}^{2}}}{\ell}}{h}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   8. Simplified 44.5%

      \[\leadsto \color{blue}{\sqrt{\frac{\frac{{d}^{2}}{\ell}}{h}}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   9. Taylor expanded in d around -inf 75.3%

      \[\leadsto \color{blue}{\left(-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)\right)} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   if -4.5000000000000001e-303 < h

   1. Initial program 69.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 70.1%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. pow1 70.1%

         \[\leadsto \color{blue}{{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)}^{1}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      2. sqrt-unprod 59.7%

         \[\leadsto {\color{blue}{\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right)}}^{1} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   6. Applied egg-rr 59.7%

      \[\leadsto \color{blue}{{\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right)}^{1}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
   7. Step-by-step derivation

      1. unpow1 59.7%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   8. Simplified 59.7%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   9. Taylor expanded in M around 0 37.3%

      \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \color{blue}{0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}\right) \]
   10. Step-by-step derivation

       1. *-commutative 37.3%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell} \cdot 0.125}\right) \]

       2. associate-*r* 38.1%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{\color{blue}{\left({D}^{2} \cdot {M}^{2}\right) \cdot h}}{{d}^{2} \cdot \ell} \cdot 0.125\right) \]

       3. times-frac 38.7%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \color{blue}{\left(\frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}} \cdot \frac{h}{\ell}\right)} \cdot 0.125\right) \]

       4. *-commutative 38.7%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\frac{\color{blue}{{M}^{2} \cdot {D}^{2}}}{{d}^{2}} \cdot \frac{h}{\ell}\right) \cdot 0.125\right) \]

       5. associate-/l* 38.6%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\color{blue}{\left({M}^{2} \cdot \frac{{D}^{2}}{{d}^{2}}\right)} \cdot \frac{h}{\ell}\right) \cdot 0.125\right) \]

       6. unpow2 38.6%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\left(\color{blue}{\left(M \cdot M\right)} \cdot \frac{{D}^{2}}{{d}^{2}}\right) \cdot \frac{h}{\ell}\right) \cdot 0.125\right) \]

       7. unpow2 38.6%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\left(\left(M \cdot M\right) \cdot \frac{\color{blue}{D \cdot D}}{{d}^{2}}\right) \cdot \frac{h}{\ell}\right) \cdot 0.125\right) \]

       8. unpow2 38.6%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\left(\left(M \cdot M\right) \cdot \frac{D \cdot D}{\color{blue}{d \cdot d}}\right) \cdot \frac{h}{\ell}\right) \cdot 0.125\right) \]

       9. times-frac 44.4%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\left(\left(M \cdot M\right) \cdot \color{blue}{\left(\frac{D}{d} \cdot \frac{D}{d}\right)}\right) \cdot \frac{h}{\ell}\right) \cdot 0.125\right) \]

       10. swap-sqr 59.7%

           \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\color{blue}{\left(\left(M \cdot \frac{D}{d}\right) \cdot \left(M \cdot \frac{D}{d}\right)\right)} \cdot \frac{h}{\ell}\right) \cdot 0.125\right) \]

       11. unpow2 59.7%

           \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\color{blue}{{\left(M \cdot \frac{D}{d}\right)}^{2}} \cdot \frac{h}{\ell}\right) \cdot 0.125\right) \]

       12. associate-*r/ 58.9%

           \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \left({\color{blue}{\left(\frac{M \cdot D}{d}\right)}}^{2} \cdot \frac{h}{\ell}\right) \cdot 0.125\right) \]

       13. *-commutative 58.9%

           \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \left({\left(\frac{\color{blue}{D \cdot M}}{d}\right)}^{2} \cdot \frac{h}{\ell}\right) \cdot 0.125\right) \]

       14. associate-/l* 59.7%

           \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \left({\color{blue}{\left(D \cdot \frac{M}{d}\right)}}^{2} \cdot \frac{h}{\ell}\right) \cdot 0.125\right) \]

   11. Simplified 59.7%

       \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \color{blue}{\left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \frac{h}{\ell}\right) \cdot 0.125}\right) \]
   12. Step-by-step derivation

       1. pow1 59.7%

          \[\leadsto \color{blue}{{\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \frac{h}{\ell}\right) \cdot 0.125\right)\right)}^{1}} \]

       2. *-commutative 59.7%

          \[\leadsto {\color{blue}{\left(\left(1 - \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \frac{h}{\ell}\right) \cdot 0.125\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right)}}^{1} \]

       3. associate-*l* 59.7%

          \[\leadsto {\left(\left(1 - \color{blue}{{\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot 0.125\right)}\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right)}^{1} \]

       4. frac-times 50.4%

          \[\leadsto {\left(\left(1 - {\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot 0.125\right)\right) \cdot \sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}}\right)}^{1} \]

       5. sqrt-div 54.5%

          \[\leadsto {\left(\left(1 - {\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot 0.125\right)\right) \cdot \color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}}\right)}^{1} \]

       6. sqrt-unprod 78.3%

          \[\leadsto {\left(\left(1 - {\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot 0.125\right)\right) \cdot \frac{\color{blue}{\sqrt{d} \cdot \sqrt{d}}}{\sqrt{h \cdot \ell}}\right)}^{1} \]

       7. add-sqr-sqrt 78.6%

          \[\leadsto {\left(\left(1 - {\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot 0.125\right)\right) \cdot \frac{\color{blue}{d}}{\sqrt{h \cdot \ell}}\right)}^{1} \]

   13. Applied egg-rr 78.6%

       \[\leadsto \color{blue}{{\left(\left(1 - {\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot 0.125\right)\right) \cdot \frac{d}{\sqrt{h \cdot \ell}}\right)}^{1}} \]
   14. Step-by-step derivation

       1. unpow1 78.6%

          \[\leadsto \color{blue}{\left(1 - {\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot 0.125\right)\right) \cdot \frac{d}{\sqrt{h \cdot \ell}}} \]

       2. *-commutative 78.6%

          \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 - {\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot 0.125\right)\right)} \]

       3. sub-neg 78.6%

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \color{blue}{\left(1 + \left(-{\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot 0.125\right)\right)\right)} \]

       4. distribute-rgt-neg-in 78.6%

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 + \color{blue}{{\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \left(-\frac{h}{\ell} \cdot 0.125\right)}\right) \]

       5. *-commutative 78.6%

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 + {\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \left(-\color{blue}{0.125 \cdot \frac{h}{\ell}}\right)\right) \]

       6. distribute-lft-neg-in 78.6%

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 + {\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \color{blue}{\left(\left(-0.125\right) \cdot \frac{h}{\ell}\right)}\right) \]

       7. metadata-eval 78.6%

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 + {\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \left(\color{blue}{-0.125} \cdot \frac{h}{\ell}\right)\right) \]

       8. associate-*r* 78.5%

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 + \color{blue}{\left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot -0.125\right) \cdot \frac{h}{\ell}}\right) \]

       9. *-commutative 78.5%

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 + \color{blue}{\frac{h}{\ell} \cdot \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot -0.125\right)}\right) \]

       10. *-commutative 78.5%

           \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 + \frac{h}{\ell} \cdot \color{blue}{\left(-0.125 \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}\right)}\right) \]

       11. associate-*r/ 77.7%

           \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.125 \cdot {\color{blue}{\left(\frac{D \cdot M}{d}\right)}}^{2}\right)\right) \]

       12. *-commutative 77.7%

           \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.125 \cdot {\left(\frac{\color{blue}{M \cdot D}}{d}\right)}^{2}\right)\right) \]

       13. associate-/l* 78.5%

           \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.125 \cdot {\color{blue}{\left(M \cdot \frac{D}{d}\right)}}^{2}\right)\right) \]

   15. Simplified 78.5%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.125 \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 71.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -1.3 \cdot 10^{+131}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \frac{h \cdot {\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2}}{\ell}\right)\\ \mathbf{elif}\;h \leq -4.5 \cdot 10^{-303}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(-1 + 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(M \cdot \frac{D}{d}\right)}^{2} \cdot -0.125\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 63.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := \frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{if}\;d \leq -1.86 \cdot 10^{+155}:\\ \;\;\;\;\left|t\_0\right|\\ \mathbf{elif}\;d \leq -5.3 \cdot 10^{-285}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \frac{h \cdot {\left(\frac{D}{d} \cdot \left(M\_m \cdot 0.5\right)\right)}^{2}}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(M\_m \cdot \frac{D}{d}\right)}^{2} \cdot -0.125\right)\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (let* ((t_0 (/ d (sqrt (* l h)))))
   (if (<= d -1.86e+155)
     (fabs t_0)
     (if (<= d -5.3e-285)
       (*
        (sqrt (* (/ d h) (/ d l)))
        (- 1.0 (* 0.5 (/ (* h (pow (* (/ D d) (* M_m 0.5)) 2.0)) l))))
       (* t_0 (+ 1.0 (* (/ h l) (* (pow (* M_m (/ D d)) 2.0) -0.125))))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double t_0 = d / sqrt((l * h));
	double tmp;
	if (d <= -1.86e+155) {
		tmp = fabs(t_0);
	} else if (d <= -5.3e-285) {
		tmp = sqrt(((d / h) * (d / l))) * (1.0 - (0.5 * ((h * pow(((D / d) * (M_m * 0.5)), 2.0)) / l)));
	} else {
		tmp = t_0 * (1.0 + ((h / l) * (pow((M_m * (D / d)), 2.0) * -0.125)));
	}
	return tmp;
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = d / sqrt((l * h))
    if (d <= (-1.86d+155)) then
        tmp = abs(t_0)
    else if (d <= (-5.3d-285)) then
        tmp = sqrt(((d / h) * (d / l))) * (1.0d0 - (0.5d0 * ((h * (((d_1 / d) * (m_m * 0.5d0)) ** 2.0d0)) / l)))
    else
        tmp = t_0 * (1.0d0 + ((h / l) * (((m_m * (d_1 / d)) ** 2.0d0) * (-0.125d0))))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double t_0 = d / Math.sqrt((l * h));
	double tmp;
	if (d <= -1.86e+155) {
		tmp = Math.abs(t_0);
	} else if (d <= -5.3e-285) {
		tmp = Math.sqrt(((d / h) * (d / l))) * (1.0 - (0.5 * ((h * Math.pow(((D / d) * (M_m * 0.5)), 2.0)) / l)));
	} else {
		tmp = t_0 * (1.0 + ((h / l) * (Math.pow((M_m * (D / d)), 2.0) * -0.125)));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	t_0 = d / math.sqrt((l * h))
	tmp = 0
	if d <= -1.86e+155:
		tmp = math.fabs(t_0)
	elif d <= -5.3e-285:
		tmp = math.sqrt(((d / h) * (d / l))) * (1.0 - (0.5 * ((h * math.pow(((D / d) * (M_m * 0.5)), 2.0)) / l)))
	else:
		tmp = t_0 * (1.0 + ((h / l) * (math.pow((M_m * (D / d)), 2.0) * -0.125)))
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	t_0 = Float64(d / sqrt(Float64(l * h)))
	tmp = 0.0
	if (d <= -1.86e+155)
		tmp = abs(t_0);
	elseif (d <= -5.3e-285)
		tmp = Float64(sqrt(Float64(Float64(d / h) * Float64(d / l))) * Float64(1.0 - Float64(0.5 * Float64(Float64(h * (Float64(Float64(D / d) * Float64(M_m * 0.5)) ^ 2.0)) / l))));
	else
		tmp = Float64(t_0 * Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(M_m * Float64(D / d)) ^ 2.0) * -0.125))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp_2 = code(d, h, l, M_m, D)
	t_0 = d / sqrt((l * h));
	tmp = 0.0;
	if (d <= -1.86e+155)
		tmp = abs(t_0);
	elseif (d <= -5.3e-285)
		tmp = sqrt(((d / h) * (d / l))) * (1.0 - (0.5 * ((h * (((D / d) * (M_m * 0.5)) ^ 2.0)) / l)));
	else
		tmp = t_0 * (1.0 + ((h / l) * (((M_m * (D / d)) ^ 2.0) * -0.125)));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := Block[{t$95$0 = N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -1.86e+155], N[Abs[t$95$0], $MachinePrecision], If[LessEqual[d, -5.3e-285], N[(N[Sqrt[N[(N[(d / h), $MachinePrecision] * N[(d / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(h * N[Power[N[(N[(D / d), $MachinePrecision] * N[(M$95$m * 0.5), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(M$95$m * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.86000000000000008e155

   1. Initial program 52.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 56.8%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 0.9%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. *-un-lft-identity 0.9%

         \[\leadsto d \cdot \color{blue}{\left(1 \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. inv-pow 0.9%

         \[\leadsto d \cdot \left(1 \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}}\right) \]

      3. sqrt-pow1 0.9%

         \[\leadsto d \cdot \left(1 \cdot \color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}}\right) \]

      4. metadata-eval 0.9%

         \[\leadsto d \cdot \left(1 \cdot {\left(h \cdot \ell\right)}^{\color{blue}{-0.5}}\right) \]

   7. Applied egg-rr 0.9%

      \[\leadsto d \cdot \color{blue}{\left(1 \cdot {\left(h \cdot \ell\right)}^{-0.5}\right)} \]
   8. Step-by-step derivation

      1. *-lft-identity 0.9%

         \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

   9. Simplified 0.9%

      \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

   10. Taylor expanded in d around 0 0.9%

       \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   11. Step-by-step derivation

       1. *-commutative 0.9%

          \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

       2. associate-/r* 0.9%

          \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \]

   12. Simplified 0.9%

       \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}} \]
   13. Step-by-step derivation

       1. add-sqr-sqrt 0.3%

          \[\leadsto \color{blue}{\sqrt{d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}} \cdot \sqrt{d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}}} \]

       2. sqrt-unprod 32.9%

          \[\leadsto \color{blue}{\sqrt{\left(d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right) \cdot \left(d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right)}} \]

       3. swap-sqr 16.6%

          \[\leadsto \sqrt{\color{blue}{\left(d \cdot d\right) \cdot \left(\sqrt{\frac{\frac{1}{\ell}}{h}} \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right)}} \]

       4. sqr-neg 16.6%

          \[\leadsto \sqrt{\color{blue}{\left(\left(-d\right) \cdot \left(-d\right)\right)} \cdot \left(\sqrt{\frac{\frac{1}{\ell}}{h}} \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right)} \]

       5. add-sqr-sqrt 16.6%

          \[\leadsto \sqrt{\left(\left(-d\right) \cdot \left(-d\right)\right) \cdot \color{blue}{\frac{\frac{1}{\ell}}{h}}} \]

       6. associate-/l/ 16.6%

          \[\leadsto \sqrt{\left(\left(-d\right) \cdot \left(-d\right)\right) \cdot \color{blue}{\frac{1}{h \cdot \ell}}} \]

       7. div-inv 16.6%

          \[\leadsto \sqrt{\color{blue}{\frac{\left(-d\right) \cdot \left(-d\right)}{h \cdot \ell}}} \]

       8. sqr-neg 16.6%

          \[\leadsto \sqrt{\frac{\color{blue}{d \cdot d}}{h \cdot \ell}} \]

       9. frac-times 40.6%

          \[\leadsto \sqrt{\color{blue}{\frac{d}{h} \cdot \frac{d}{\ell}}} \]

       10. sqrt-prod 48.2%

           \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}} \]

       11. frac-2neg 48.2%

           \[\leadsto \sqrt{\color{blue}{\frac{-d}{-h}}} \cdot \sqrt{\frac{d}{\ell}} \]

       12. sqrt-undiv 67.2%

           \[\leadsto \color{blue}{\frac{\sqrt{-d}}{\sqrt{-h}}} \cdot \sqrt{\frac{d}{\ell}} \]

       13. add-sqr-sqrt 67.2%

           \[\leadsto \color{blue}{\sqrt{\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}} \cdot \sqrt{\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}}} \]

       14. sqrt-prod 40.3%

           \[\leadsto \color{blue}{\sqrt{\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\right)}} \]

       15. rem-sqrt-square 67.2%

           \[\leadsto \color{blue}{\left|\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\right|} \]

   14. Applied egg-rr 77.1%

       \[\leadsto \color{blue}{\left|\frac{d}{\sqrt{h \cdot \ell}}\right|} \]

   if -1.86000000000000008e155 < d < -5.3e-285

   1. Initial program 62.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 63.3%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. pow1 63.3%

         \[\leadsto \color{blue}{{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)}^{1}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      2. sqrt-unprod 49.1%

         \[\leadsto {\color{blue}{\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right)}}^{1} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   6. Applied egg-rr 49.1%

      \[\leadsto \color{blue}{{\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right)}^{1}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
   7. Step-by-step derivation

      1. unpow1 49.1%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   8. Simplified 49.1%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
   9. Step-by-step derivation

      1. associate-*r/ 56.7%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot h}{\ell}}\right) \]

      2. *-commutative 56.7%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \frac{{\color{blue}{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}}^{2} \cdot h}{\ell}\right) \]

      3. div-inv 56.7%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \frac{{\left(\frac{D}{d} \cdot \color{blue}{\left(M \cdot \frac{1}{2}\right)}\right)}^{2} \cdot h}{\ell}\right) \]

      4. metadata-eval 56.7%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \frac{{\left(\frac{D}{d} \cdot \left(M \cdot \color{blue}{0.5}\right)\right)}^{2} \cdot h}{\ell}\right) \]

   10. Applied egg-rr 56.7%

       \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{{\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2} \cdot h}{\ell}}\right) \]

   if -5.3e-285 < d

   1. Initial program 67.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 68.0%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. pow1 68.0%

         \[\leadsto \color{blue}{{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)}^{1}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      2. sqrt-unprod 57.9%

         \[\leadsto {\color{blue}{\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right)}}^{1} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   6. Applied egg-rr 57.9%

      \[\leadsto \color{blue}{{\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right)}^{1}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
   7. Step-by-step derivation

      1. unpow1 57.9%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   8. Simplified 57.9%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   9. Taylor expanded in M around 0 36.1%

      \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \color{blue}{0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}\right) \]
   10. Step-by-step derivation

       1. *-commutative 36.1%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell} \cdot 0.125}\right) \]

       2. associate-*r* 36.9%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{\color{blue}{\left({D}^{2} \cdot {M}^{2}\right) \cdot h}}{{d}^{2} \cdot \ell} \cdot 0.125\right) \]

       3. times-frac 37.5%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \color{blue}{\left(\frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}} \cdot \frac{h}{\ell}\right)} \cdot 0.125\right) \]

       4. *-commutative 37.5%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\frac{\color{blue}{{M}^{2} \cdot {D}^{2}}}{{d}^{2}} \cdot \frac{h}{\ell}\right) \cdot 0.125\right) \]

       5. associate-/l* 37.5%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\color{blue}{\left({M}^{2} \cdot \frac{{D}^{2}}{{d}^{2}}\right)} \cdot \frac{h}{\ell}\right) \cdot 0.125\right) \]

       6. unpow2 37.5%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\left(\color{blue}{\left(M \cdot M\right)} \cdot \frac{{D}^{2}}{{d}^{2}}\right) \cdot \frac{h}{\ell}\right) \cdot 0.125\right) \]

       7. unpow2 37.5%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\left(\left(M \cdot M\right) \cdot \frac{\color{blue}{D \cdot D}}{{d}^{2}}\right) \cdot \frac{h}{\ell}\right) \cdot 0.125\right) \]

       8. unpow2 37.5%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\left(\left(M \cdot M\right) \cdot \frac{D \cdot D}{\color{blue}{d \cdot d}}\right) \cdot \frac{h}{\ell}\right) \cdot 0.125\right) \]

       9. times-frac 43.1%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\left(\left(M \cdot M\right) \cdot \color{blue}{\left(\frac{D}{d} \cdot \frac{D}{d}\right)}\right) \cdot \frac{h}{\ell}\right) \cdot 0.125\right) \]

       10. swap-sqr 57.9%

           \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\color{blue}{\left(\left(M \cdot \frac{D}{d}\right) \cdot \left(M \cdot \frac{D}{d}\right)\right)} \cdot \frac{h}{\ell}\right) \cdot 0.125\right) \]

       11. unpow2 57.9%

           \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\color{blue}{{\left(M \cdot \frac{D}{d}\right)}^{2}} \cdot \frac{h}{\ell}\right) \cdot 0.125\right) \]

       12. associate-*r/ 57.1%

           \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \left({\color{blue}{\left(\frac{M \cdot D}{d}\right)}}^{2} \cdot \frac{h}{\ell}\right) \cdot 0.125\right) \]

       13. *-commutative 57.1%

           \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \left({\left(\frac{\color{blue}{D \cdot M}}{d}\right)}^{2} \cdot \frac{h}{\ell}\right) \cdot 0.125\right) \]

       14. associate-/l* 57.9%

           \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \left({\color{blue}{\left(D \cdot \frac{M}{d}\right)}}^{2} \cdot \frac{h}{\ell}\right) \cdot 0.125\right) \]

   11. Simplified 57.9%

       \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \color{blue}{\left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \frac{h}{\ell}\right) \cdot 0.125}\right) \]
   12. Step-by-step derivation

       1. pow1 57.9%

          \[\leadsto \color{blue}{{\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \frac{h}{\ell}\right) \cdot 0.125\right)\right)}^{1}} \]

       2. *-commutative 57.9%

          \[\leadsto {\color{blue}{\left(\left(1 - \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \frac{h}{\ell}\right) \cdot 0.125\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right)}}^{1} \]

       3. associate-*l* 57.9%

          \[\leadsto {\left(\left(1 - \color{blue}{{\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot 0.125\right)}\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right)}^{1} \]

       4. frac-times 48.9%

          \[\leadsto {\left(\left(1 - {\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot 0.125\right)\right) \cdot \sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}}\right)}^{1} \]

       5. sqrt-div 52.9%

          \[\leadsto {\left(\left(1 - {\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot 0.125\right)\right) \cdot \color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}}\right)}^{1} \]

       6. sqrt-unprod 75.9%

          \[\leadsto {\left(\left(1 - {\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot 0.125\right)\right) \cdot \frac{\color{blue}{\sqrt{d} \cdot \sqrt{d}}}{\sqrt{h \cdot \ell}}\right)}^{1} \]

       7. add-sqr-sqrt 76.2%

          \[\leadsto {\left(\left(1 - {\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot 0.125\right)\right) \cdot \frac{\color{blue}{d}}{\sqrt{h \cdot \ell}}\right)}^{1} \]

   13. Applied egg-rr 76.2%

       \[\leadsto \color{blue}{{\left(\left(1 - {\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot 0.125\right)\right) \cdot \frac{d}{\sqrt{h \cdot \ell}}\right)}^{1}} \]
   14. Step-by-step derivation

       1. unpow1 76.2%

          \[\leadsto \color{blue}{\left(1 - {\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot 0.125\right)\right) \cdot \frac{d}{\sqrt{h \cdot \ell}}} \]

       2. *-commutative 76.2%

          \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 - {\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot 0.125\right)\right)} \]

       3. sub-neg 76.2%

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \color{blue}{\left(1 + \left(-{\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot 0.125\right)\right)\right)} \]

       4. distribute-rgt-neg-in 76.2%

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 + \color{blue}{{\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \left(-\frac{h}{\ell} \cdot 0.125\right)}\right) \]

       5. *-commutative 76.2%

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 + {\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \left(-\color{blue}{0.125 \cdot \frac{h}{\ell}}\right)\right) \]

       6. distribute-lft-neg-in 76.2%

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 + {\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \color{blue}{\left(\left(-0.125\right) \cdot \frac{h}{\ell}\right)}\right) \]

       7. metadata-eval 76.2%

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 + {\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \left(\color{blue}{-0.125} \cdot \frac{h}{\ell}\right)\right) \]

       8. associate-*r* 76.2%

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 + \color{blue}{\left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot -0.125\right) \cdot \frac{h}{\ell}}\right) \]

       9. *-commutative 76.2%

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 + \color{blue}{\frac{h}{\ell} \cdot \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot -0.125\right)}\right) \]

       10. *-commutative 76.2%

           \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 + \frac{h}{\ell} \cdot \color{blue}{\left(-0.125 \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}\right)}\right) \]

       11. associate-*r/ 75.4%

           \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.125 \cdot {\color{blue}{\left(\frac{D \cdot M}{d}\right)}}^{2}\right)\right) \]

       12. *-commutative 75.4%

           \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.125 \cdot {\left(\frac{\color{blue}{M \cdot D}}{d}\right)}^{2}\right)\right) \]

       13. associate-/l* 76.2%

           \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.125 \cdot {\color{blue}{\left(M \cdot \frac{D}{d}\right)}}^{2}\right)\right) \]

   15. Simplified 76.2%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.125 \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.86 \cdot 10^{+155}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \mathbf{elif}\;d \leq -5.3 \cdot 10^{-285}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \frac{h \cdot {\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2}}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(M \cdot \frac{D}{d}\right)}^{2} \cdot -0.125\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 68.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;h \leq -8.8 \cdot 10^{+129}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \frac{h \cdot {\left(\frac{D}{d} \cdot \left(M\_m \cdot 0.5\right)\right)}^{2}}{\ell}\right)\\ \mathbf{elif}\;h \leq -4.5 \cdot 10^{-303}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(0.125 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M\_m}{d}\right)}^{2}\right) + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(M\_m \cdot \frac{D}{d}\right)}^{2} \cdot -0.125\right)\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (if (<= h -8.8e+129)
   (*
    (sqrt (* (/ d h) (/ d l)))
    (- 1.0 (* 0.5 (/ (* h (pow (* (/ D d) (* M_m 0.5)) 2.0)) l))))
   (if (<= h -4.5e-303)
     (*
      (* d (sqrt (/ 1.0 (* l h))))
      (+ (* 0.125 (* (/ h l) (pow (* D (/ M_m d)) 2.0))) -1.0))
     (*
      (/ d (sqrt (* l h)))
      (+ 1.0 (* (/ h l) (* (pow (* M_m (/ D d)) 2.0) -0.125)))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (h <= -8.8e+129) {
		tmp = sqrt(((d / h) * (d / l))) * (1.0 - (0.5 * ((h * pow(((D / d) * (M_m * 0.5)), 2.0)) / l)));
	} else if (h <= -4.5e-303) {
		tmp = (d * sqrt((1.0 / (l * h)))) * ((0.125 * ((h / l) * pow((D * (M_m / d)), 2.0))) + -1.0);
	} else {
		tmp = (d / sqrt((l * h))) * (1.0 + ((h / l) * (pow((M_m * (D / d)), 2.0) * -0.125)));
	}
	return tmp;
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (h <= (-8.8d+129)) then
        tmp = sqrt(((d / h) * (d / l))) * (1.0d0 - (0.5d0 * ((h * (((d_1 / d) * (m_m * 0.5d0)) ** 2.0d0)) / l)))
    else if (h <= (-4.5d-303)) then
        tmp = (d * sqrt((1.0d0 / (l * h)))) * ((0.125d0 * ((h / l) * ((d_1 * (m_m / d)) ** 2.0d0))) + (-1.0d0))
    else
        tmp = (d / sqrt((l * h))) * (1.0d0 + ((h / l) * (((m_m * (d_1 / d)) ** 2.0d0) * (-0.125d0))))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (h <= -8.8e+129) {
		tmp = Math.sqrt(((d / h) * (d / l))) * (1.0 - (0.5 * ((h * Math.pow(((D / d) * (M_m * 0.5)), 2.0)) / l)));
	} else if (h <= -4.5e-303) {
		tmp = (d * Math.sqrt((1.0 / (l * h)))) * ((0.125 * ((h / l) * Math.pow((D * (M_m / d)), 2.0))) + -1.0);
	} else {
		tmp = (d / Math.sqrt((l * h))) * (1.0 + ((h / l) * (Math.pow((M_m * (D / d)), 2.0) * -0.125)));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	tmp = 0
	if h <= -8.8e+129:
		tmp = math.sqrt(((d / h) * (d / l))) * (1.0 - (0.5 * ((h * math.pow(((D / d) * (M_m * 0.5)), 2.0)) / l)))
	elif h <= -4.5e-303:
		tmp = (d * math.sqrt((1.0 / (l * h)))) * ((0.125 * ((h / l) * math.pow((D * (M_m / d)), 2.0))) + -1.0)
	else:
		tmp = (d / math.sqrt((l * h))) * (1.0 + ((h / l) * (math.pow((M_m * (D / d)), 2.0) * -0.125)))
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	tmp = 0.0
	if (h <= -8.8e+129)
		tmp = Float64(sqrt(Float64(Float64(d / h) * Float64(d / l))) * Float64(1.0 - Float64(0.5 * Float64(Float64(h * (Float64(Float64(D / d) * Float64(M_m * 0.5)) ^ 2.0)) / l))));
	elseif (h <= -4.5e-303)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(l * h)))) * Float64(Float64(0.125 * Float64(Float64(h / l) * (Float64(D * Float64(M_m / d)) ^ 2.0))) + -1.0));
	else
		tmp = Float64(Float64(d / sqrt(Float64(l * h))) * Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(M_m * Float64(D / d)) ^ 2.0) * -0.125))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp_2 = code(d, h, l, M_m, D)
	tmp = 0.0;
	if (h <= -8.8e+129)
		tmp = sqrt(((d / h) * (d / l))) * (1.0 - (0.5 * ((h * (((D / d) * (M_m * 0.5)) ^ 2.0)) / l)));
	elseif (h <= -4.5e-303)
		tmp = (d * sqrt((1.0 / (l * h)))) * ((0.125 * ((h / l) * ((D * (M_m / d)) ^ 2.0))) + -1.0);
	else
		tmp = (d / sqrt((l * h))) * (1.0 + ((h / l) * (((M_m * (D / d)) ^ 2.0) * -0.125)));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := If[LessEqual[h, -8.8e+129], N[(N[Sqrt[N[(N[(d / h), $MachinePrecision] * N[(d / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(h * N[Power[N[(N[(D / d), $MachinePrecision] * N[(M$95$m * 0.5), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, -4.5e-303], N[(N[(d * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(0.125 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(D * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(M$95$m * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if h < -8.7999999999999997e129

   1. Initial program 40.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 40.1%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. pow1 40.1%

         \[\leadsto \color{blue}{{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)}^{1}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      2. sqrt-unprod 28.6%

         \[\leadsto {\color{blue}{\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right)}}^{1} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   6. Applied egg-rr 28.6%

      \[\leadsto \color{blue}{{\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right)}^{1}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
   7. Step-by-step derivation

      1. unpow1 28.6%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   8. Simplified 28.6%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
   9. Step-by-step derivation

      1. associate-*r/ 43.9%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot h}{\ell}}\right) \]

      2. *-commutative 43.9%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \frac{{\color{blue}{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}}^{2} \cdot h}{\ell}\right) \]

      3. div-inv 43.9%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \frac{{\left(\frac{D}{d} \cdot \color{blue}{\left(M \cdot \frac{1}{2}\right)}\right)}^{2} \cdot h}{\ell}\right) \]

      4. metadata-eval 43.9%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \frac{{\left(\frac{D}{d} \cdot \left(M \cdot \color{blue}{0.5}\right)\right)}^{2} \cdot h}{\ell}\right) \]

   10. Applied egg-rr 43.9%

       \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{{\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2} \cdot h}{\ell}}\right) \]

   if -8.7999999999999997e129 < h < -4.5000000000000001e-303

   1. Initial program 67.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 69.9%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. pow1 69.9%

         \[\leadsto \color{blue}{{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)}^{1}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      2. sqrt-unprod 56.9%

         \[\leadsto {\color{blue}{\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right)}}^{1} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   6. Applied egg-rr 56.9%

      \[\leadsto \color{blue}{{\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right)}^{1}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
   7. Step-by-step derivation

      1. unpow1 56.9%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   8. Simplified 56.9%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   9. Taylor expanded in M around 0 43.3%

      \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \color{blue}{0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}\right) \]
   10. Step-by-step derivation

       1. *-commutative 43.3%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell} \cdot 0.125}\right) \]

       2. associate-*r* 43.5%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{\color{blue}{\left({D}^{2} \cdot {M}^{2}\right) \cdot h}}{{d}^{2} \cdot \ell} \cdot 0.125\right) \]

       3. times-frac 44.7%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \color{blue}{\left(\frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}} \cdot \frac{h}{\ell}\right)} \cdot 0.125\right) \]

       4. *-commutative 44.7%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\frac{\color{blue}{{M}^{2} \cdot {D}^{2}}}{{d}^{2}} \cdot \frac{h}{\ell}\right) \cdot 0.125\right) \]

       5. associate-/l* 44.7%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\color{blue}{\left({M}^{2} \cdot \frac{{D}^{2}}{{d}^{2}}\right)} \cdot \frac{h}{\ell}\right) \cdot 0.125\right) \]

       6. unpow2 44.7%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\left(\color{blue}{\left(M \cdot M\right)} \cdot \frac{{D}^{2}}{{d}^{2}}\right) \cdot \frac{h}{\ell}\right) \cdot 0.125\right) \]

       7. unpow2 44.7%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\left(\left(M \cdot M\right) \cdot \frac{\color{blue}{D \cdot D}}{{d}^{2}}\right) \cdot \frac{h}{\ell}\right) \cdot 0.125\right) \]

       8. unpow2 44.7%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\left(\left(M \cdot M\right) \cdot \frac{D \cdot D}{\color{blue}{d \cdot d}}\right) \cdot \frac{h}{\ell}\right) \cdot 0.125\right) \]

       9. times-frac 51.1%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\left(\left(M \cdot M\right) \cdot \color{blue}{\left(\frac{D}{d} \cdot \frac{D}{d}\right)}\right) \cdot \frac{h}{\ell}\right) \cdot 0.125\right) \]

       10. swap-sqr 56.9%

           \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\color{blue}{\left(\left(M \cdot \frac{D}{d}\right) \cdot \left(M \cdot \frac{D}{d}\right)\right)} \cdot \frac{h}{\ell}\right) \cdot 0.125\right) \]

       11. unpow2 56.9%

           \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\color{blue}{{\left(M \cdot \frac{D}{d}\right)}^{2}} \cdot \frac{h}{\ell}\right) \cdot 0.125\right) \]

       12. associate-*r/ 55.8%

           \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \left({\color{blue}{\left(\frac{M \cdot D}{d}\right)}}^{2} \cdot \frac{h}{\ell}\right) \cdot 0.125\right) \]

       13. *-commutative 55.8%

           \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \left({\left(\frac{\color{blue}{D \cdot M}}{d}\right)}^{2} \cdot \frac{h}{\ell}\right) \cdot 0.125\right) \]

       14. associate-/l* 56.9%

           \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \left({\color{blue}{\left(D \cdot \frac{M}{d}\right)}}^{2} \cdot \frac{h}{\ell}\right) \cdot 0.125\right) \]

   11. Simplified 56.9%

       \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \color{blue}{\left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \frac{h}{\ell}\right) \cdot 0.125}\right) \]

   12. Taylor expanded in d around -inf 74.7%

       \[\leadsto \color{blue}{\left(-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)\right)} \cdot \left(1 - \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \frac{h}{\ell}\right) \cdot 0.125\right) \]

   if -4.5000000000000001e-303 < h

   1. Initial program 69.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 70.1%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. pow1 70.1%

         \[\leadsto \color{blue}{{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)}^{1}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      2. sqrt-unprod 59.7%

         \[\leadsto {\color{blue}{\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right)}}^{1} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   6. Applied egg-rr 59.7%

      \[\leadsto \color{blue}{{\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right)}^{1}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
   7. Step-by-step derivation

      1. unpow1 59.7%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   8. Simplified 59.7%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   9. Taylor expanded in M around 0 37.3%

      \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \color{blue}{0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}\right) \]
   10. Step-by-step derivation

       1. *-commutative 37.3%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell} \cdot 0.125}\right) \]

       2. associate-*r* 38.1%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{\color{blue}{\left({D}^{2} \cdot {M}^{2}\right) \cdot h}}{{d}^{2} \cdot \ell} \cdot 0.125\right) \]

       3. times-frac 38.7%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \color{blue}{\left(\frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}} \cdot \frac{h}{\ell}\right)} \cdot 0.125\right) \]

       4. *-commutative 38.7%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\frac{\color{blue}{{M}^{2} \cdot {D}^{2}}}{{d}^{2}} \cdot \frac{h}{\ell}\right) \cdot 0.125\right) \]

       5. associate-/l* 38.6%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\color{blue}{\left({M}^{2} \cdot \frac{{D}^{2}}{{d}^{2}}\right)} \cdot \frac{h}{\ell}\right) \cdot 0.125\right) \]

       6. unpow2 38.6%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\left(\color{blue}{\left(M \cdot M\right)} \cdot \frac{{D}^{2}}{{d}^{2}}\right) \cdot \frac{h}{\ell}\right) \cdot 0.125\right) \]

       7. unpow2 38.6%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\left(\left(M \cdot M\right) \cdot \frac{\color{blue}{D \cdot D}}{{d}^{2}}\right) \cdot \frac{h}{\ell}\right) \cdot 0.125\right) \]

       8. unpow2 38.6%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\left(\left(M \cdot M\right) \cdot \frac{D \cdot D}{\color{blue}{d \cdot d}}\right) \cdot \frac{h}{\ell}\right) \cdot 0.125\right) \]

       9. times-frac 44.4%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\left(\left(M \cdot M\right) \cdot \color{blue}{\left(\frac{D}{d} \cdot \frac{D}{d}\right)}\right) \cdot \frac{h}{\ell}\right) \cdot 0.125\right) \]

       10. swap-sqr 59.7%

           \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\color{blue}{\left(\left(M \cdot \frac{D}{d}\right) \cdot \left(M \cdot \frac{D}{d}\right)\right)} \cdot \frac{h}{\ell}\right) \cdot 0.125\right) \]

       11. unpow2 59.7%

           \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\color{blue}{{\left(M \cdot \frac{D}{d}\right)}^{2}} \cdot \frac{h}{\ell}\right) \cdot 0.125\right) \]

       12. associate-*r/ 58.9%

           \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \left({\color{blue}{\left(\frac{M \cdot D}{d}\right)}}^{2} \cdot \frac{h}{\ell}\right) \cdot 0.125\right) \]

       13. *-commutative 58.9%

           \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \left({\left(\frac{\color{blue}{D \cdot M}}{d}\right)}^{2} \cdot \frac{h}{\ell}\right) \cdot 0.125\right) \]

       14. associate-/l* 59.7%

           \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \left({\color{blue}{\left(D \cdot \frac{M}{d}\right)}}^{2} \cdot \frac{h}{\ell}\right) \cdot 0.125\right) \]

   11. Simplified 59.7%

       \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \color{blue}{\left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \frac{h}{\ell}\right) \cdot 0.125}\right) \]
   12. Step-by-step derivation

       1. pow1 59.7%

          \[\leadsto \color{blue}{{\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \frac{h}{\ell}\right) \cdot 0.125\right)\right)}^{1}} \]

       2. *-commutative 59.7%

          \[\leadsto {\color{blue}{\left(\left(1 - \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \frac{h}{\ell}\right) \cdot 0.125\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right)}}^{1} \]

       3. associate-*l* 59.7%

          \[\leadsto {\left(\left(1 - \color{blue}{{\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot 0.125\right)}\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right)}^{1} \]

       4. frac-times 50.4%

          \[\leadsto {\left(\left(1 - {\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot 0.125\right)\right) \cdot \sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}}\right)}^{1} \]

       5. sqrt-div 54.5%

          \[\leadsto {\left(\left(1 - {\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot 0.125\right)\right) \cdot \color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}}\right)}^{1} \]

       6. sqrt-unprod 78.3%

          \[\leadsto {\left(\left(1 - {\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot 0.125\right)\right) \cdot \frac{\color{blue}{\sqrt{d} \cdot \sqrt{d}}}{\sqrt{h \cdot \ell}}\right)}^{1} \]

       7. add-sqr-sqrt 78.6%

          \[\leadsto {\left(\left(1 - {\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot 0.125\right)\right) \cdot \frac{\color{blue}{d}}{\sqrt{h \cdot \ell}}\right)}^{1} \]

   13. Applied egg-rr 78.6%

       \[\leadsto \color{blue}{{\left(\left(1 - {\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot 0.125\right)\right) \cdot \frac{d}{\sqrt{h \cdot \ell}}\right)}^{1}} \]
   14. Step-by-step derivation

       1. unpow1 78.6%

          \[\leadsto \color{blue}{\left(1 - {\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot 0.125\right)\right) \cdot \frac{d}{\sqrt{h \cdot \ell}}} \]

       2. *-commutative 78.6%

          \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 - {\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot 0.125\right)\right)} \]

       3. sub-neg 78.6%

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \color{blue}{\left(1 + \left(-{\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot 0.125\right)\right)\right)} \]

       4. distribute-rgt-neg-in 78.6%

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 + \color{blue}{{\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \left(-\frac{h}{\ell} \cdot 0.125\right)}\right) \]

       5. *-commutative 78.6%

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 + {\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \left(-\color{blue}{0.125 \cdot \frac{h}{\ell}}\right)\right) \]

       6. distribute-lft-neg-in 78.6%

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 + {\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \color{blue}{\left(\left(-0.125\right) \cdot \frac{h}{\ell}\right)}\right) \]

       7. metadata-eval 78.6%

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 + {\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \left(\color{blue}{-0.125} \cdot \frac{h}{\ell}\right)\right) \]

       8. associate-*r* 78.5%

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 + \color{blue}{\left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot -0.125\right) \cdot \frac{h}{\ell}}\right) \]

       9. *-commutative 78.5%

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 + \color{blue}{\frac{h}{\ell} \cdot \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot -0.125\right)}\right) \]

       10. *-commutative 78.5%

           \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 + \frac{h}{\ell} \cdot \color{blue}{\left(-0.125 \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}\right)}\right) \]

       11. associate-*r/ 77.7%

           \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.125 \cdot {\color{blue}{\left(\frac{D \cdot M}{d}\right)}}^{2}\right)\right) \]

       12. *-commutative 77.7%

           \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.125 \cdot {\left(\frac{\color{blue}{M \cdot D}}{d}\right)}^{2}\right)\right) \]

       13. associate-/l* 78.5%

           \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.125 \cdot {\color{blue}{\left(M \cdot \frac{D}{d}\right)}}^{2}\right)\right) \]

   15. Simplified 78.5%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.125 \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -8.8 \cdot 10^{+129}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \frac{h \cdot {\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2}}{\ell}\right)\\ \mathbf{elif}\;h \leq -4.5 \cdot 10^{-303}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(0.125 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}\right) + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(M \cdot \frac{D}{d}\right)}^{2} \cdot -0.125\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 63.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := \frac{d}{\sqrt{\ell \cdot h}}\\ t_1 := {\left(M\_m \cdot \frac{D}{d}\right)}^{2}\\ \mathbf{if}\;d \leq -1.72 \cdot 10^{+155}:\\ \;\;\;\;\left|t\_0\right|\\ \mathbf{elif}\;d \leq -5.3 \cdot 10^{-285}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - 0.125 \cdot \left(h \cdot \frac{t\_1}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(1 + \frac{h}{\ell} \cdot \left(t\_1 \cdot -0.125\right)\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (let* ((t_0 (/ d (sqrt (* l h)))) (t_1 (pow (* M_m (/ D d)) 2.0)))
   (if (<= d -1.72e+155)
     (fabs t_0)
     (if (<= d -5.3e-285)
       (* (sqrt (* (/ d h) (/ d l))) (- 1.0 (* 0.125 (* h (/ t_1 l)))))
       (* t_0 (+ 1.0 (* (/ h l) (* t_1 -0.125))))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double t_0 = d / sqrt((l * h));
	double t_1 = pow((M_m * (D / d)), 2.0);
	double tmp;
	if (d <= -1.72e+155) {
		tmp = fabs(t_0);
	} else if (d <= -5.3e-285) {
		tmp = sqrt(((d / h) * (d / l))) * (1.0 - (0.125 * (h * (t_1 / l))));
	} else {
		tmp = t_0 * (1.0 + ((h / l) * (t_1 * -0.125)));
	}
	return tmp;
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = d / sqrt((l * h))
    t_1 = (m_m * (d_1 / d)) ** 2.0d0
    if (d <= (-1.72d+155)) then
        tmp = abs(t_0)
    else if (d <= (-5.3d-285)) then
        tmp = sqrt(((d / h) * (d / l))) * (1.0d0 - (0.125d0 * (h * (t_1 / l))))
    else
        tmp = t_0 * (1.0d0 + ((h / l) * (t_1 * (-0.125d0))))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double t_0 = d / Math.sqrt((l * h));
	double t_1 = Math.pow((M_m * (D / d)), 2.0);
	double tmp;
	if (d <= -1.72e+155) {
		tmp = Math.abs(t_0);
	} else if (d <= -5.3e-285) {
		tmp = Math.sqrt(((d / h) * (d / l))) * (1.0 - (0.125 * (h * (t_1 / l))));
	} else {
		tmp = t_0 * (1.0 + ((h / l) * (t_1 * -0.125)));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	t_0 = d / math.sqrt((l * h))
	t_1 = math.pow((M_m * (D / d)), 2.0)
	tmp = 0
	if d <= -1.72e+155:
		tmp = math.fabs(t_0)
	elif d <= -5.3e-285:
		tmp = math.sqrt(((d / h) * (d / l))) * (1.0 - (0.125 * (h * (t_1 / l))))
	else:
		tmp = t_0 * (1.0 + ((h / l) * (t_1 * -0.125)))
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	t_0 = Float64(d / sqrt(Float64(l * h)))
	t_1 = Float64(M_m * Float64(D / d)) ^ 2.0
	tmp = 0.0
	if (d <= -1.72e+155)
		tmp = abs(t_0);
	elseif (d <= -5.3e-285)
		tmp = Float64(sqrt(Float64(Float64(d / h) * Float64(d / l))) * Float64(1.0 - Float64(0.125 * Float64(h * Float64(t_1 / l)))));
	else
		tmp = Float64(t_0 * Float64(1.0 + Float64(Float64(h / l) * Float64(t_1 * -0.125))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp_2 = code(d, h, l, M_m, D)
	t_0 = d / sqrt((l * h));
	t_1 = (M_m * (D / d)) ^ 2.0;
	tmp = 0.0;
	if (d <= -1.72e+155)
		tmp = abs(t_0);
	elseif (d <= -5.3e-285)
		tmp = sqrt(((d / h) * (d / l))) * (1.0 - (0.125 * (h * (t_1 / l))));
	else
		tmp = t_0 * (1.0 + ((h / l) * (t_1 * -0.125)));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := Block[{t$95$0 = N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(M$95$m * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[d, -1.72e+155], N[Abs[t$95$0], $MachinePrecision], If[LessEqual[d, -5.3e-285], N[(N[Sqrt[N[(N[(d / h), $MachinePrecision] * N[(d / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(0.125 * N[(h * N[(t$95$1 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(t$95$1 * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.7199999999999999e155

   1. Initial program 52.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 56.8%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 0.9%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. *-un-lft-identity 0.9%

         \[\leadsto d \cdot \color{blue}{\left(1 \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. inv-pow 0.9%

         \[\leadsto d \cdot \left(1 \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}}\right) \]

      3. sqrt-pow1 0.9%

         \[\leadsto d \cdot \left(1 \cdot \color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}}\right) \]

      4. metadata-eval 0.9%

         \[\leadsto d \cdot \left(1 \cdot {\left(h \cdot \ell\right)}^{\color{blue}{-0.5}}\right) \]

   7. Applied egg-rr 0.9%

      \[\leadsto d \cdot \color{blue}{\left(1 \cdot {\left(h \cdot \ell\right)}^{-0.5}\right)} \]
   8. Step-by-step derivation

      1. *-lft-identity 0.9%

         \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

   9. Simplified 0.9%

      \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

   10. Taylor expanded in d around 0 0.9%

       \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   11. Step-by-step derivation

       1. *-commutative 0.9%

          \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

       2. associate-/r* 0.9%

          \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \]

   12. Simplified 0.9%

       \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}} \]
   13. Step-by-step derivation

       1. add-sqr-sqrt 0.3%

          \[\leadsto \color{blue}{\sqrt{d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}} \cdot \sqrt{d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}}} \]

       2. sqrt-unprod 32.9%

          \[\leadsto \color{blue}{\sqrt{\left(d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right) \cdot \left(d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right)}} \]

       3. swap-sqr 16.6%

          \[\leadsto \sqrt{\color{blue}{\left(d \cdot d\right) \cdot \left(\sqrt{\frac{\frac{1}{\ell}}{h}} \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right)}} \]

       4. sqr-neg 16.6%

          \[\leadsto \sqrt{\color{blue}{\left(\left(-d\right) \cdot \left(-d\right)\right)} \cdot \left(\sqrt{\frac{\frac{1}{\ell}}{h}} \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right)} \]

       5. add-sqr-sqrt 16.6%

          \[\leadsto \sqrt{\left(\left(-d\right) \cdot \left(-d\right)\right) \cdot \color{blue}{\frac{\frac{1}{\ell}}{h}}} \]

       6. associate-/l/ 16.6%

          \[\leadsto \sqrt{\left(\left(-d\right) \cdot \left(-d\right)\right) \cdot \color{blue}{\frac{1}{h \cdot \ell}}} \]

       7. div-inv 16.6%

          \[\leadsto \sqrt{\color{blue}{\frac{\left(-d\right) \cdot \left(-d\right)}{h \cdot \ell}}} \]

       8. sqr-neg 16.6%

          \[\leadsto \sqrt{\frac{\color{blue}{d \cdot d}}{h \cdot \ell}} \]

       9. frac-times 40.6%

          \[\leadsto \sqrt{\color{blue}{\frac{d}{h} \cdot \frac{d}{\ell}}} \]

       10. sqrt-prod 48.2%

           \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}} \]

       11. frac-2neg 48.2%

           \[\leadsto \sqrt{\color{blue}{\frac{-d}{-h}}} \cdot \sqrt{\frac{d}{\ell}} \]

       12. sqrt-undiv 67.2%

           \[\leadsto \color{blue}{\frac{\sqrt{-d}}{\sqrt{-h}}} \cdot \sqrt{\frac{d}{\ell}} \]

       13. add-sqr-sqrt 67.2%

           \[\leadsto \color{blue}{\sqrt{\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}} \cdot \sqrt{\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}}} \]

       14. sqrt-prod 40.3%

           \[\leadsto \color{blue}{\sqrt{\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\right)}} \]

       15. rem-sqrt-square 67.2%

           \[\leadsto \color{blue}{\left|\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\right|} \]

   14. Applied egg-rr 77.1%

       \[\leadsto \color{blue}{\left|\frac{d}{\sqrt{h \cdot \ell}}\right|} \]

   if -1.7199999999999999e155 < d < -5.3e-285

   1. Initial program 62.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 63.3%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. pow1 63.3%

         \[\leadsto \color{blue}{{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)}^{1}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      2. sqrt-unprod 49.1%

         \[\leadsto {\color{blue}{\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right)}}^{1} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   6. Applied egg-rr 49.1%

      \[\leadsto \color{blue}{{\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right)}^{1}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
   7. Step-by-step derivation

      1. unpow1 49.1%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   8. Simplified 49.1%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   9. Taylor expanded in M around 0 39.3%

      \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \color{blue}{0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}\right) \]
   10. Step-by-step derivation

       1. *-commutative 39.3%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell} \cdot 0.125}\right) \]

       2. associate-*r* 41.3%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{\color{blue}{\left({D}^{2} \cdot {M}^{2}\right) \cdot h}}{{d}^{2} \cdot \ell} \cdot 0.125\right) \]

       3. times-frac 38.4%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \color{blue}{\left(\frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}} \cdot \frac{h}{\ell}\right)} \cdot 0.125\right) \]

       4. *-commutative 38.4%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\frac{\color{blue}{{M}^{2} \cdot {D}^{2}}}{{d}^{2}} \cdot \frac{h}{\ell}\right) \cdot 0.125\right) \]

       5. associate-/l* 38.3%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\color{blue}{\left({M}^{2} \cdot \frac{{D}^{2}}{{d}^{2}}\right)} \cdot \frac{h}{\ell}\right) \cdot 0.125\right) \]

       6. unpow2 38.3%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\left(\color{blue}{\left(M \cdot M\right)} \cdot \frac{{D}^{2}}{{d}^{2}}\right) \cdot \frac{h}{\ell}\right) \cdot 0.125\right) \]

       7. unpow2 38.3%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\left(\left(M \cdot M\right) \cdot \frac{\color{blue}{D \cdot D}}{{d}^{2}}\right) \cdot \frac{h}{\ell}\right) \cdot 0.125\right) \]

       8. unpow2 38.3%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\left(\left(M \cdot M\right) \cdot \frac{D \cdot D}{\color{blue}{d \cdot d}}\right) \cdot \frac{h}{\ell}\right) \cdot 0.125\right) \]

       9. times-frac 43.0%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\left(\left(M \cdot M\right) \cdot \color{blue}{\left(\frac{D}{d} \cdot \frac{D}{d}\right)}\right) \cdot \frac{h}{\ell}\right) \cdot 0.125\right) \]

       10. swap-sqr 49.1%

           \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\color{blue}{\left(\left(M \cdot \frac{D}{d}\right) \cdot \left(M \cdot \frac{D}{d}\right)\right)} \cdot \frac{h}{\ell}\right) \cdot 0.125\right) \]

       11. unpow2 49.1%

           \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\color{blue}{{\left(M \cdot \frac{D}{d}\right)}^{2}} \cdot \frac{h}{\ell}\right) \cdot 0.125\right) \]

       12. associate-*r/ 49.1%

           \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \left({\color{blue}{\left(\frac{M \cdot D}{d}\right)}}^{2} \cdot \frac{h}{\ell}\right) \cdot 0.125\right) \]

       13. *-commutative 49.1%

           \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \left({\left(\frac{\color{blue}{D \cdot M}}{d}\right)}^{2} \cdot \frac{h}{\ell}\right) \cdot 0.125\right) \]

       14. associate-/l* 49.1%

           \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \left({\color{blue}{\left(D \cdot \frac{M}{d}\right)}}^{2} \cdot \frac{h}{\ell}\right) \cdot 0.125\right) \]

   11. Simplified 49.1%

       \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \color{blue}{\left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \frac{h}{\ell}\right) \cdot 0.125}\right) \]

   12. Taylor expanded in D around 0 39.3%

       \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}} \cdot 0.125\right) \]
   13. Step-by-step derivation

       1. associate-*r* 41.3%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{\color{blue}{\left({D}^{2} \cdot {M}^{2}\right) \cdot h}}{{d}^{2} \cdot \ell} \cdot 0.125\right) \]

       2. times-frac 38.4%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \color{blue}{\left(\frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}} \cdot \frac{h}{\ell}\right)} \cdot 0.125\right) \]

       3. associate-/l* 37.4%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\color{blue}{\left({D}^{2} \cdot \frac{{M}^{2}}{{d}^{2}}\right)} \cdot \frac{h}{\ell}\right) \cdot 0.125\right) \]

       4. unpow2 37.4%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\left(\color{blue}{\left(D \cdot D\right)} \cdot \frac{{M}^{2}}{{d}^{2}}\right) \cdot \frac{h}{\ell}\right) \cdot 0.125\right) \]

       5. unpow2 37.4%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\left(\left(D \cdot D\right) \cdot \frac{\color{blue}{M \cdot M}}{{d}^{2}}\right) \cdot \frac{h}{\ell}\right) \cdot 0.125\right) \]

       6. unpow2 37.4%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{\color{blue}{d \cdot d}}\right) \cdot \frac{h}{\ell}\right) \cdot 0.125\right) \]

       7. times-frac 40.1%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\left(\left(D \cdot D\right) \cdot \color{blue}{\left(\frac{M}{d} \cdot \frac{M}{d}\right)}\right) \cdot \frac{h}{\ell}\right) \cdot 0.125\right) \]

       8. swap-sqr 49.1%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\color{blue}{\left(\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)\right)} \cdot \frac{h}{\ell}\right) \cdot 0.125\right) \]

       9. unpow2 49.1%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\color{blue}{{\left(D \cdot \frac{M}{d}\right)}^{2}} \cdot \frac{h}{\ell}\right) \cdot 0.125\right) \]

       10. associate-*r/ 56.6%

           \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \color{blue}{\frac{{\left(D \cdot \frac{M}{d}\right)}^{2} \cdot h}{\ell}} \cdot 0.125\right) \]

       11. *-commutative 56.6%

           \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{\color{blue}{h \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}}}{\ell} \cdot 0.125\right) \]

       12. associate-/l* 56.6%

           \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \color{blue}{\left(h \cdot \frac{{\left(D \cdot \frac{M}{d}\right)}^{2}}{\ell}\right)} \cdot 0.125\right) \]

       13. associate-*r/ 56.6%

           \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(h \cdot \frac{{\color{blue}{\left(\frac{D \cdot M}{d}\right)}}^{2}}{\ell}\right) \cdot 0.125\right) \]

       14. *-commutative 56.6%

           \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(h \cdot \frac{{\left(\frac{\color{blue}{M \cdot D}}{d}\right)}^{2}}{\ell}\right) \cdot 0.125\right) \]

       15. associate-/l* 56.6%

           \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(h \cdot \frac{{\color{blue}{\left(M \cdot \frac{D}{d}\right)}}^{2}}{\ell}\right) \cdot 0.125\right) \]

   14. Simplified 56.6%

       \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \color{blue}{\left(h \cdot \frac{{\left(M \cdot \frac{D}{d}\right)}^{2}}{\ell}\right)} \cdot 0.125\right) \]

   if -5.3e-285 < d

   1. Initial program 67.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 68.0%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. pow1 68.0%

         \[\leadsto \color{blue}{{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)}^{1}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      2. sqrt-unprod 57.9%

         \[\leadsto {\color{blue}{\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right)}}^{1} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   6. Applied egg-rr 57.9%

      \[\leadsto \color{blue}{{\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right)}^{1}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
   7. Step-by-step derivation

      1. unpow1 57.9%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   8. Simplified 57.9%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   9. Taylor expanded in M around 0 36.1%

      \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \color{blue}{0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}\right) \]
   10. Step-by-step derivation

       1. *-commutative 36.1%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell} \cdot 0.125}\right) \]

       2. associate-*r* 36.9%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{\color{blue}{\left({D}^{2} \cdot {M}^{2}\right) \cdot h}}{{d}^{2} \cdot \ell} \cdot 0.125\right) \]

       3. times-frac 37.5%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \color{blue}{\left(\frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}} \cdot \frac{h}{\ell}\right)} \cdot 0.125\right) \]

       4. *-commutative 37.5%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\frac{\color{blue}{{M}^{2} \cdot {D}^{2}}}{{d}^{2}} \cdot \frac{h}{\ell}\right) \cdot 0.125\right) \]

       5. associate-/l* 37.5%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\color{blue}{\left({M}^{2} \cdot \frac{{D}^{2}}{{d}^{2}}\right)} \cdot \frac{h}{\ell}\right) \cdot 0.125\right) \]

       6. unpow2 37.5%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\left(\color{blue}{\left(M \cdot M\right)} \cdot \frac{{D}^{2}}{{d}^{2}}\right) \cdot \frac{h}{\ell}\right) \cdot 0.125\right) \]

       7. unpow2 37.5%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\left(\left(M \cdot M\right) \cdot \frac{\color{blue}{D \cdot D}}{{d}^{2}}\right) \cdot \frac{h}{\ell}\right) \cdot 0.125\right) \]

       8. unpow2 37.5%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\left(\left(M \cdot M\right) \cdot \frac{D \cdot D}{\color{blue}{d \cdot d}}\right) \cdot \frac{h}{\ell}\right) \cdot 0.125\right) \]

       9. times-frac 43.1%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\left(\left(M \cdot M\right) \cdot \color{blue}{\left(\frac{D}{d} \cdot \frac{D}{d}\right)}\right) \cdot \frac{h}{\ell}\right) \cdot 0.125\right) \]

       10. swap-sqr 57.9%

           \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\color{blue}{\left(\left(M \cdot \frac{D}{d}\right) \cdot \left(M \cdot \frac{D}{d}\right)\right)} \cdot \frac{h}{\ell}\right) \cdot 0.125\right) \]

       11. unpow2 57.9%

           \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\color{blue}{{\left(M \cdot \frac{D}{d}\right)}^{2}} \cdot \frac{h}{\ell}\right) \cdot 0.125\right) \]

       12. associate-*r/ 57.1%

           \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \left({\color{blue}{\left(\frac{M \cdot D}{d}\right)}}^{2} \cdot \frac{h}{\ell}\right) \cdot 0.125\right) \]

       13. *-commutative 57.1%

           \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \left({\left(\frac{\color{blue}{D \cdot M}}{d}\right)}^{2} \cdot \frac{h}{\ell}\right) \cdot 0.125\right) \]

       14. associate-/l* 57.9%

           \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \left({\color{blue}{\left(D \cdot \frac{M}{d}\right)}}^{2} \cdot \frac{h}{\ell}\right) \cdot 0.125\right) \]

   11. Simplified 57.9%

       \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \color{blue}{\left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \frac{h}{\ell}\right) \cdot 0.125}\right) \]
   12. Step-by-step derivation

       1. pow1 57.9%

          \[\leadsto \color{blue}{{\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \frac{h}{\ell}\right) \cdot 0.125\right)\right)}^{1}} \]

       2. *-commutative 57.9%

          \[\leadsto {\color{blue}{\left(\left(1 - \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \frac{h}{\ell}\right) \cdot 0.125\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right)}}^{1} \]

       3. associate-*l* 57.9%

          \[\leadsto {\left(\left(1 - \color{blue}{{\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot 0.125\right)}\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right)}^{1} \]

       4. frac-times 48.9%

          \[\leadsto {\left(\left(1 - {\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot 0.125\right)\right) \cdot \sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}}\right)}^{1} \]

       5. sqrt-div 52.9%

          \[\leadsto {\left(\left(1 - {\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot 0.125\right)\right) \cdot \color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}}\right)}^{1} \]

       6. sqrt-unprod 75.9%

          \[\leadsto {\left(\left(1 - {\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot 0.125\right)\right) \cdot \frac{\color{blue}{\sqrt{d} \cdot \sqrt{d}}}{\sqrt{h \cdot \ell}}\right)}^{1} \]

       7. add-sqr-sqrt 76.2%

          \[\leadsto {\left(\left(1 - {\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot 0.125\right)\right) \cdot \frac{\color{blue}{d}}{\sqrt{h \cdot \ell}}\right)}^{1} \]

   13. Applied egg-rr 76.2%

       \[\leadsto \color{blue}{{\left(\left(1 - {\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot 0.125\right)\right) \cdot \frac{d}{\sqrt{h \cdot \ell}}\right)}^{1}} \]
   14. Step-by-step derivation

       1. unpow1 76.2%

          \[\leadsto \color{blue}{\left(1 - {\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot 0.125\right)\right) \cdot \frac{d}{\sqrt{h \cdot \ell}}} \]

       2. *-commutative 76.2%

          \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 - {\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot 0.125\right)\right)} \]

       3. sub-neg 76.2%

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \color{blue}{\left(1 + \left(-{\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot 0.125\right)\right)\right)} \]

       4. distribute-rgt-neg-in 76.2%

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 + \color{blue}{{\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \left(-\frac{h}{\ell} \cdot 0.125\right)}\right) \]

       5. *-commutative 76.2%

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 + {\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \left(-\color{blue}{0.125 \cdot \frac{h}{\ell}}\right)\right) \]

       6. distribute-lft-neg-in 76.2%

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 + {\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \color{blue}{\left(\left(-0.125\right) \cdot \frac{h}{\ell}\right)}\right) \]

       7. metadata-eval 76.2%

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 + {\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \left(\color{blue}{-0.125} \cdot \frac{h}{\ell}\right)\right) \]

       8. associate-*r* 76.2%

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 + \color{blue}{\left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot -0.125\right) \cdot \frac{h}{\ell}}\right) \]

       9. *-commutative 76.2%

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 + \color{blue}{\frac{h}{\ell} \cdot \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot -0.125\right)}\right) \]

       10. *-commutative 76.2%

           \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 + \frac{h}{\ell} \cdot \color{blue}{\left(-0.125 \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}\right)}\right) \]

       11. associate-*r/ 75.4%

           \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.125 \cdot {\color{blue}{\left(\frac{D \cdot M}{d}\right)}}^{2}\right)\right) \]

       12. *-commutative 75.4%

           \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.125 \cdot {\left(\frac{\color{blue}{M \cdot D}}{d}\right)}^{2}\right)\right) \]

       13. associate-/l* 76.2%

           \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.125 \cdot {\color{blue}{\left(M \cdot \frac{D}{d}\right)}}^{2}\right)\right) \]

   15. Simplified 76.2%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.125 \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.72 \cdot 10^{+155}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \mathbf{elif}\;d \leq -5.3 \cdot 10^{-285}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - 0.125 \cdot \left(h \cdot \frac{{\left(M \cdot \frac{D}{d}\right)}^{2}}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(M \cdot \frac{D}{d}\right)}^{2} \cdot -0.125\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 44.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -1.4 \cdot 10^{+113}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \mathbf{elif}\;d \leq -6.4 \cdot 10^{-80}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}}\\ \mathbf{elif}\;d \leq -3 \cdot 10^{-233}:\\ \;\;\;\;d \cdot \sqrt[3]{{\left(\frac{\frac{1}{\ell}}{h}\right)}^{1.5}}\\ \mathbf{elif}\;d \leq 1.75 \cdot 10^{-153}:\\ \;\;\;\;d \cdot \left(-{\left(\ell \cdot h\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (if (<= d -1.4e+113)
   (fabs (/ d (sqrt (* l h))))
   (if (<= d -6.4e-80)
     (/ (sqrt (/ d l)) (sqrt (/ h d)))
     (if (<= d -3e-233)
       (* d (cbrt (pow (/ (/ 1.0 l) h) 1.5)))
       (if (<= d 1.75e-153)
         (* d (- (pow (* l h) -0.5)))
         (/ d (* (sqrt h) (sqrt l))))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (d <= -1.4e+113) {
		tmp = fabs((d / sqrt((l * h))));
	} else if (d <= -6.4e-80) {
		tmp = sqrt((d / l)) / sqrt((h / d));
	} else if (d <= -3e-233) {
		tmp = d * cbrt(pow(((1.0 / l) / h), 1.5));
	} else if (d <= 1.75e-153) {
		tmp = d * -pow((l * h), -0.5);
	} else {
		tmp = d / (sqrt(h) * sqrt(l));
	}
	return tmp;
}

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (d <= -1.4e+113) {
		tmp = Math.abs((d / Math.sqrt((l * h))));
	} else if (d <= -6.4e-80) {
		tmp = Math.sqrt((d / l)) / Math.sqrt((h / d));
	} else if (d <= -3e-233) {
		tmp = d * Math.cbrt(Math.pow(((1.0 / l) / h), 1.5));
	} else if (d <= 1.75e-153) {
		tmp = d * -Math.pow((l * h), -0.5);
	} else {
		tmp = d / (Math.sqrt(h) * Math.sqrt(l));
	}
	return tmp;
}

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	tmp = 0.0
	if (d <= -1.4e+113)
		tmp = abs(Float64(d / sqrt(Float64(l * h))));
	elseif (d <= -6.4e-80)
		tmp = Float64(sqrt(Float64(d / l)) / sqrt(Float64(h / d)));
	elseif (d <= -3e-233)
		tmp = Float64(d * cbrt((Float64(Float64(1.0 / l) / h) ^ 1.5)));
	elseif (d <= 1.75e-153)
		tmp = Float64(d * Float64(-(Float64(l * h) ^ -0.5)));
	else
		tmp = Float64(d / Float64(sqrt(h) * sqrt(l)));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := If[LessEqual[d, -1.4e+113], N[Abs[N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[d, -6.4e-80], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(h / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -3e-233], N[(d * N[Power[N[Power[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision], 1.5], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.75e-153], N[(d * (-N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision])), $MachinePrecision], N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if d < -1.39999999999999999e113

   1. Initial program 54.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 57.2%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 1.2%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. *-un-lft-identity 1.2%

         \[\leadsto d \cdot \color{blue}{\left(1 \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. inv-pow 1.2%

         \[\leadsto d \cdot \left(1 \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}}\right) \]

      3. sqrt-pow1 1.2%

         \[\leadsto d \cdot \left(1 \cdot \color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}}\right) \]

      4. metadata-eval 1.2%

         \[\leadsto d \cdot \left(1 \cdot {\left(h \cdot \ell\right)}^{\color{blue}{-0.5}}\right) \]

   7. Applied egg-rr 1.2%

      \[\leadsto d \cdot \color{blue}{\left(1 \cdot {\left(h \cdot \ell\right)}^{-0.5}\right)} \]
   8. Step-by-step derivation

      1. *-lft-identity 1.2%

         \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

   9. Simplified 1.2%

      \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

   10. Taylor expanded in d around 0 1.2%

       \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   11. Step-by-step derivation

       1. *-commutative 1.2%

          \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

       2. associate-/r* 1.3%

          \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \]

   12. Simplified 1.3%

       \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}} \]
   13. Step-by-step derivation

       1. add-sqr-sqrt 0.2%

          \[\leadsto \color{blue}{\sqrt{d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}} \cdot \sqrt{d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}}} \]

       2. sqrt-unprod 35.3%

          \[\leadsto \color{blue}{\sqrt{\left(d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right) \cdot \left(d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right)}} \]

       3. swap-sqr 22.7%

          \[\leadsto \sqrt{\color{blue}{\left(d \cdot d\right) \cdot \left(\sqrt{\frac{\frac{1}{\ell}}{h}} \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right)}} \]

       4. sqr-neg 22.7%

          \[\leadsto \sqrt{\color{blue}{\left(\left(-d\right) \cdot \left(-d\right)\right)} \cdot \left(\sqrt{\frac{\frac{1}{\ell}}{h}} \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right)} \]

       5. add-sqr-sqrt 22.8%

          \[\leadsto \sqrt{\left(\left(-d\right) \cdot \left(-d\right)\right) \cdot \color{blue}{\frac{\frac{1}{\ell}}{h}}} \]

       6. associate-/l/ 22.8%

          \[\leadsto \sqrt{\left(\left(-d\right) \cdot \left(-d\right)\right) \cdot \color{blue}{\frac{1}{h \cdot \ell}}} \]

       7. div-inv 22.8%

          \[\leadsto \sqrt{\color{blue}{\frac{\left(-d\right) \cdot \left(-d\right)}{h \cdot \ell}}} \]

       8. sqr-neg 22.8%

          \[\leadsto \sqrt{\frac{\color{blue}{d \cdot d}}{h \cdot \ell}} \]

       9. frac-times 41.3%

          \[\leadsto \sqrt{\color{blue}{\frac{d}{h} \cdot \frac{d}{\ell}}} \]

       10. sqrt-prod 47.2%

           \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}} \]

       11. frac-2neg 47.2%

           \[\leadsto \sqrt{\color{blue}{\frac{-d}{-h}}} \cdot \sqrt{\frac{d}{\ell}} \]

       12. sqrt-undiv 64.9%

           \[\leadsto \color{blue}{\frac{\sqrt{-d}}{\sqrt{-h}}} \cdot \sqrt{\frac{d}{\ell}} \]

       13. add-sqr-sqrt 64.8%

           \[\leadsto \color{blue}{\sqrt{\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}} \cdot \sqrt{\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}}} \]

       14. sqrt-prod 41.0%

           \[\leadsto \color{blue}{\sqrt{\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\right)}} \]

       15. rem-sqrt-square 64.9%

           \[\leadsto \color{blue}{\left|\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\right|} \]

   14. Applied egg-rr 72.7%

       \[\leadsto \color{blue}{\left|\frac{d}{\sqrt{h \cdot \ell}}\right|} \]

   if -1.39999999999999999e113 < d < -6.3999999999999998e-80

   1. Initial program 72.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 72.7%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 6.3%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. *-un-lft-identity 6.3%

         \[\leadsto d \cdot \color{blue}{\left(1 \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. inv-pow 6.3%

         \[\leadsto d \cdot \left(1 \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}}\right) \]

      3. sqrt-pow1 6.3%

         \[\leadsto d \cdot \left(1 \cdot \color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}}\right) \]

      4. metadata-eval 6.3%

         \[\leadsto d \cdot \left(1 \cdot {\left(h \cdot \ell\right)}^{\color{blue}{-0.5}}\right) \]

   7. Applied egg-rr 6.3%

      \[\leadsto d \cdot \color{blue}{\left(1 \cdot {\left(h \cdot \ell\right)}^{-0.5}\right)} \]
   8. Step-by-step derivation

      1. *-lft-identity 6.3%

         \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

   9. Simplified 6.3%

      \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

   10. Taylor expanded in d around 0 6.3%

       \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   11. Step-by-step derivation

       1. *-commutative 6.3%

          \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

       2. associate-/r* 6.3%

          \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \]

   12. Simplified 6.3%

       \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}} \]
   13. Step-by-step derivation

       1. add-sqr-sqrt 0.6%

          \[\leadsto \color{blue}{\sqrt{d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}} \cdot \sqrt{d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}}} \]

       2. sqrt-unprod 40.8%

          \[\leadsto \color{blue}{\sqrt{\left(d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right) \cdot \left(d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right)}} \]

       3. swap-sqr 40.6%

          \[\leadsto \sqrt{\color{blue}{\left(d \cdot d\right) \cdot \left(\sqrt{\frac{\frac{1}{\ell}}{h}} \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right)}} \]

       4. sqr-neg 40.6%

          \[\leadsto \sqrt{\color{blue}{\left(\left(-d\right) \cdot \left(-d\right)\right)} \cdot \left(\sqrt{\frac{\frac{1}{\ell}}{h}} \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right)} \]

       5. add-sqr-sqrt 40.7%

          \[\leadsto \sqrt{\left(\left(-d\right) \cdot \left(-d\right)\right) \cdot \color{blue}{\frac{\frac{1}{\ell}}{h}}} \]

       6. associate-/l/ 40.7%

          \[\leadsto \sqrt{\left(\left(-d\right) \cdot \left(-d\right)\right) \cdot \color{blue}{\frac{1}{h \cdot \ell}}} \]

       7. div-inv 40.7%

          \[\leadsto \sqrt{\color{blue}{\frac{\left(-d\right) \cdot \left(-d\right)}{h \cdot \ell}}} \]

       8. sqr-neg 40.7%

          \[\leadsto \sqrt{\frac{\color{blue}{d \cdot d}}{h \cdot \ell}} \]

       9. frac-times 40.6%

          \[\leadsto \sqrt{\color{blue}{\frac{d}{h} \cdot \frac{d}{\ell}}} \]

       10. sqrt-prod 56.7%

           \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}} \]

       11. frac-2neg 56.7%

           \[\leadsto \sqrt{\color{blue}{\frac{-d}{-h}}} \cdot \sqrt{\frac{d}{\ell}} \]

       12. sqrt-undiv 56.7%

           \[\leadsto \color{blue}{\frac{\sqrt{-d}}{\sqrt{-h}}} \cdot \sqrt{\frac{d}{\ell}} \]

       13. *-commutative 56.7%

           \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \frac{\sqrt{-d}}{\sqrt{-h}}} \]

       14. clear-num 56.6%

           \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \color{blue}{\frac{1}{\frac{\sqrt{-h}}{\sqrt{-d}}}} \]

       15. un-div-inv 56.7%

           \[\leadsto \color{blue}{\frac{\sqrt{\frac{d}{\ell}}}{\frac{\sqrt{-h}}{\sqrt{-d}}}} \]

       16. sqrt-undiv 56.8%

           \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\color{blue}{\sqrt{\frac{-h}{-d}}}} \]

       17. frac-2neg 56.8%

           \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\color{blue}{\frac{h}{d}}}} \]

   14. Applied egg-rr 56.8%

       \[\leadsto \color{blue}{\frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}}} \]

   if -6.3999999999999998e-80 < d < -2.99999999999999999e-233

   1. Initial program 54.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 54.5%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 16.7%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. *-un-lft-identity 16.7%

         \[\leadsto d \cdot \color{blue}{\left(1 \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. inv-pow 16.7%

         \[\leadsto d \cdot \left(1 \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}}\right) \]

      3. sqrt-pow1 16.7%

         \[\leadsto d \cdot \left(1 \cdot \color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}}\right) \]

      4. metadata-eval 16.7%

         \[\leadsto d \cdot \left(1 \cdot {\left(h \cdot \ell\right)}^{\color{blue}{-0.5}}\right) \]

   7. Applied egg-rr 16.7%

      \[\leadsto d \cdot \color{blue}{\left(1 \cdot {\left(h \cdot \ell\right)}^{-0.5}\right)} \]
   8. Step-by-step derivation

      1. *-lft-identity 16.7%

         \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

   9. Simplified 16.7%

      \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

   10. Taylor expanded in d around 0 16.7%

       \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   11. Step-by-step derivation

       1. *-commutative 16.7%

          \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

       2. associate-/r* 16.7%

          \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \]

   12. Simplified 16.7%

       \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}} \]
   13. Step-by-step derivation

       1. add-cbrt-cube 29.5%

          \[\leadsto d \cdot \color{blue}{\sqrt[3]{\left(\sqrt{\frac{\frac{1}{\ell}}{h}} \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}}} \]

       2. pow1/3 29.5%

          \[\leadsto d \cdot \color{blue}{{\left(\left(\sqrt{\frac{\frac{1}{\ell}}{h}} \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right)}^{0.3333333333333333}} \]

       3. add-sqr-sqrt 29.5%

          \[\leadsto d \cdot {\left(\color{blue}{\frac{\frac{1}{\ell}}{h}} \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right)}^{0.3333333333333333} \]

       4. pow1 29.5%

          \[\leadsto d \cdot {\left(\color{blue}{{\left(\frac{\frac{1}{\ell}}{h}\right)}^{1}} \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right)}^{0.3333333333333333} \]

       5. pow1/2 29.5%

          \[\leadsto d \cdot {\left({\left(\frac{\frac{1}{\ell}}{h}\right)}^{1} \cdot \color{blue}{{\left(\frac{\frac{1}{\ell}}{h}\right)}^{0.5}}\right)}^{0.3333333333333333} \]

       6. pow-prod-up 29.5%

          \[\leadsto d \cdot {\color{blue}{\left({\left(\frac{\frac{1}{\ell}}{h}\right)}^{\left(1 + 0.5\right)}\right)}}^{0.3333333333333333} \]

       7. div-inv 29.5%

          \[\leadsto d \cdot {\left({\color{blue}{\left(\frac{1}{\ell} \cdot \frac{1}{h}\right)}}^{\left(1 + 0.5\right)}\right)}^{0.3333333333333333} \]

       8. frac-times 29.5%

          \[\leadsto d \cdot {\left({\color{blue}{\left(\frac{1 \cdot 1}{\ell \cdot h}\right)}}^{\left(1 + 0.5\right)}\right)}^{0.3333333333333333} \]

       9. metadata-eval 29.5%

          \[\leadsto d \cdot {\left({\left(\frac{\color{blue}{1}}{\ell \cdot h}\right)}^{\left(1 + 0.5\right)}\right)}^{0.3333333333333333} \]

       10. metadata-eval 29.5%

           \[\leadsto d \cdot {\left({\left(\frac{1}{\ell \cdot h}\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333} \]

   14. Applied egg-rr 29.5%

       \[\leadsto d \cdot \color{blue}{{\left({\left(\frac{1}{\ell \cdot h}\right)}^{1.5}\right)}^{0.3333333333333333}} \]
   15. Step-by-step derivation

       1. unpow1/3 29.5%

          \[\leadsto d \cdot \color{blue}{\sqrt[3]{{\left(\frac{1}{\ell \cdot h}\right)}^{1.5}}} \]

       2. associate-/r* 29.5%

          \[\leadsto d \cdot \sqrt[3]{{\color{blue}{\left(\frac{\frac{1}{\ell}}{h}\right)}}^{1.5}} \]

   16. Simplified 29.5%

       \[\leadsto d \cdot \color{blue}{\sqrt[3]{{\left(\frac{\frac{1}{\ell}}{h}\right)}^{1.5}}} \]

   if -2.99999999999999999e-233 < d < 1.7499999999999999e-153

   1. Initial program 40.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 42.3%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 18.1%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

   6. Taylor expanded in h around -inf 0.0%

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   7. Step-by-step derivation

      1. associate-*l* 0.0%

         \[\leadsto \color{blue}{d \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. *-commutative 0.0%

         \[\leadsto \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot d} \]

      3. unpow2 0.0%

         \[\leadsto \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot d \]

      4. rem-square-sqrt 29.7%

         \[\leadsto \left(\color{blue}{-1} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot d \]

      5. *-commutative 29.7%

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot -1\right)} \cdot d \]

      6. associate-*r* 29.7%

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-1 \cdot d\right)} \]

      7. unpow1/2 29.7%

         \[\leadsto \color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{0.5}} \cdot \left(-1 \cdot d\right) \]

      8. rem-exp-log 29.3%

         \[\leadsto {\left(\frac{1}{\color{blue}{e^{\log \left(h \cdot \ell\right)}}}\right)}^{0.5} \cdot \left(-1 \cdot d\right) \]

      9. exp-neg 29.3%

         \[\leadsto {\color{blue}{\left(e^{-\log \left(h \cdot \ell\right)}\right)}}^{0.5} \cdot \left(-1 \cdot d\right) \]

      10. exp-prod 29.3%

          \[\leadsto \color{blue}{e^{\left(-\log \left(h \cdot \ell\right)\right) \cdot 0.5}} \cdot \left(-1 \cdot d\right) \]

      11. distribute-lft-neg-out 29.3%

          \[\leadsto e^{\color{blue}{-\log \left(h \cdot \ell\right) \cdot 0.5}} \cdot \left(-1 \cdot d\right) \]

      12. distribute-rgt-neg-in 29.3%

          \[\leadsto e^{\color{blue}{\log \left(h \cdot \ell\right) \cdot \left(-0.5\right)}} \cdot \left(-1 \cdot d\right) \]

      13. metadata-eval 29.3%

          \[\leadsto e^{\log \left(h \cdot \ell\right) \cdot \color{blue}{-0.5}} \cdot \left(-1 \cdot d\right) \]

      14. exp-to-pow 29.7%

          \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \left(-1 \cdot d\right) \]

      15. neg-mul-1 29.7%

          \[\leadsto {\left(h \cdot \ell\right)}^{-0.5} \cdot \color{blue}{\left(-d\right)} \]

   8. Simplified 29.7%

      \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot \left(-d\right)} \]

   if 1.7499999999999999e-153 < d

   1. Initial program 75.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 76.3%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 64.9%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. *-un-lft-identity 64.9%

         \[\leadsto d \cdot \color{blue}{\left(1 \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. inv-pow 64.9%

         \[\leadsto d \cdot \left(1 \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}}\right) \]

      3. sqrt-pow1 64.9%

         \[\leadsto d \cdot \left(1 \cdot \color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}}\right) \]

      4. metadata-eval 64.9%

         \[\leadsto d \cdot \left(1 \cdot {\left(h \cdot \ell\right)}^{\color{blue}{-0.5}}\right) \]

   7. Applied egg-rr 64.9%

      \[\leadsto d \cdot \color{blue}{\left(1 \cdot {\left(h \cdot \ell\right)}^{-0.5}\right)} \]
   8. Step-by-step derivation

      1. *-lft-identity 64.9%

         \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

   9. Simplified 64.9%

      \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

   10. Taylor expanded in d around 0 64.9%

       \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   11. Step-by-step derivation

       1. *-commutative 64.9%

          \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

       2. associate-/r* 65.2%

          \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \]

   12. Simplified 65.2%

       \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}} \]
   13. Step-by-step derivation

       1. add-sqr-sqrt 65.0%

          \[\leadsto \color{blue}{\sqrt{d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}} \cdot \sqrt{d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}}} \]

       2. sqrt-unprod 50.2%

          \[\leadsto \color{blue}{\sqrt{\left(d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right) \cdot \left(d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right)}} \]

       3. swap-sqr 43.0%

          \[\leadsto \sqrt{\color{blue}{\left(d \cdot d\right) \cdot \left(\sqrt{\frac{\frac{1}{\ell}}{h}} \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right)}} \]

       4. sqr-neg 43.0%

          \[\leadsto \sqrt{\color{blue}{\left(\left(-d\right) \cdot \left(-d\right)\right)} \cdot \left(\sqrt{\frac{\frac{1}{\ell}}{h}} \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right)} \]

       5. add-sqr-sqrt 43.1%

          \[\leadsto \sqrt{\left(\left(-d\right) \cdot \left(-d\right)\right) \cdot \color{blue}{\frac{\frac{1}{\ell}}{h}}} \]

       6. associate-/l/ 43.1%

          \[\leadsto \sqrt{\left(\left(-d\right) \cdot \left(-d\right)\right) \cdot \color{blue}{\frac{1}{h \cdot \ell}}} \]

       7. div-inv 43.1%

          \[\leadsto \sqrt{\color{blue}{\frac{\left(-d\right) \cdot \left(-d\right)}{h \cdot \ell}}} \]

       8. sqr-neg 43.1%

          \[\leadsto \sqrt{\frac{\color{blue}{d \cdot d}}{h \cdot \ell}} \]

       9. frac-times 47.6%

          \[\leadsto \sqrt{\color{blue}{\frac{d}{h} \cdot \frac{d}{\ell}}} \]

       10. sqrt-prod 54.1%

           \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}} \]

       11. frac-2neg 54.1%

           \[\leadsto \sqrt{\color{blue}{\frac{-d}{-h}}} \cdot \sqrt{\frac{d}{\ell}} \]

       12. sqrt-undiv 0.0%

           \[\leadsto \color{blue}{\frac{\sqrt{-d}}{\sqrt{-h}}} \cdot \sqrt{\frac{d}{\ell}} \]

       13. *-commutative 0.0%

           \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \frac{\sqrt{-d}}{\sqrt{-h}}} \]

       14. sqrt-div 0.0%

           \[\leadsto \color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}} \cdot \frac{\sqrt{-d}}{\sqrt{-h}} \]

       15. frac-times 0.0%

           \[\leadsto \color{blue}{\frac{\sqrt{d} \cdot \sqrt{-d}}{\sqrt{\ell} \cdot \sqrt{-h}}} \]

   14. Applied egg-rr 69.6%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 55.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.4 \cdot 10^{+113}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \mathbf{elif}\;d \leq -6.4 \cdot 10^{-80}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}}\\ \mathbf{elif}\;d \leq -3 \cdot 10^{-233}:\\ \;\;\;\;d \cdot \sqrt[3]{{\left(\frac{\frac{1}{\ell}}{h}\right)}^{1.5}}\\ \mathbf{elif}\;d \leq 1.75 \cdot 10^{-153}:\\ \;\;\;\;d \cdot \left(-{\left(\ell \cdot h\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 45.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.22 \cdot 10^{+104}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}}\\ \mathbf{elif}\;\ell \leq 2.75 \cdot 10^{-229}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (if (<= l -1.22e+104)
   (/ (sqrt (/ d l)) (sqrt (/ h d)))
   (if (<= l 2.75e-229)
     (* (- d) (sqrt (/ 1.0 (* l h))))
     (* d (/ (sqrt (/ 1.0 h)) (sqrt l))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (l <= -1.22e+104) {
		tmp = sqrt((d / l)) / sqrt((h / d));
	} else if (l <= 2.75e-229) {
		tmp = -d * sqrt((1.0 / (l * h)));
	} else {
		tmp = d * (sqrt((1.0 / h)) / sqrt(l));
	}
	return tmp;
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (l <= (-1.22d+104)) then
        tmp = sqrt((d / l)) / sqrt((h / d))
    else if (l <= 2.75d-229) then
        tmp = -d * sqrt((1.0d0 / (l * h)))
    else
        tmp = d * (sqrt((1.0d0 / h)) / sqrt(l))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (l <= -1.22e+104) {
		tmp = Math.sqrt((d / l)) / Math.sqrt((h / d));
	} else if (l <= 2.75e-229) {
		tmp = -d * Math.sqrt((1.0 / (l * h)));
	} else {
		tmp = d * (Math.sqrt((1.0 / h)) / Math.sqrt(l));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	tmp = 0
	if l <= -1.22e+104:
		tmp = math.sqrt((d / l)) / math.sqrt((h / d))
	elif l <= 2.75e-229:
		tmp = -d * math.sqrt((1.0 / (l * h)))
	else:
		tmp = d * (math.sqrt((1.0 / h)) / math.sqrt(l))
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	tmp = 0.0
	if (l <= -1.22e+104)
		tmp = Float64(sqrt(Float64(d / l)) / sqrt(Float64(h / d)));
	elseif (l <= 2.75e-229)
		tmp = Float64(Float64(-d) * sqrt(Float64(1.0 / Float64(l * h))));
	else
		tmp = Float64(d * Float64(sqrt(Float64(1.0 / h)) / sqrt(l)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp_2 = code(d, h, l, M_m, D)
	tmp = 0.0;
	if (l <= -1.22e+104)
		tmp = sqrt((d / l)) / sqrt((h / d));
	elseif (l <= 2.75e-229)
		tmp = -d * sqrt((1.0 / (l * h)));
	else
		tmp = d * (sqrt((1.0 / h)) / sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := If[LessEqual[l, -1.22e+104], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(h / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 2.75e-229], N[((-d) * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Sqrt[N[(1.0 / h), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.22e104

   1. Initial program 43.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 46.2%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 3.6%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. *-un-lft-identity 3.6%

         \[\leadsto d \cdot \color{blue}{\left(1 \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. inv-pow 3.6%

         \[\leadsto d \cdot \left(1 \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}}\right) \]

      3. sqrt-pow1 3.6%

         \[\leadsto d \cdot \left(1 \cdot \color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}}\right) \]

      4. metadata-eval 3.6%

         \[\leadsto d \cdot \left(1 \cdot {\left(h \cdot \ell\right)}^{\color{blue}{-0.5}}\right) \]

   7. Applied egg-rr 3.6%

      \[\leadsto d \cdot \color{blue}{\left(1 \cdot {\left(h \cdot \ell\right)}^{-0.5}\right)} \]
   8. Step-by-step derivation

      1. *-lft-identity 3.6%

         \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

   9. Simplified 3.6%

      \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

   10. Taylor expanded in d around 0 3.6%

       \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   11. Step-by-step derivation

       1. *-commutative 3.6%

          \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

       2. associate-/r* 3.7%

          \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \]

   12. Simplified 3.7%

       \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}} \]
   13. Step-by-step derivation

       1. add-sqr-sqrt 2.6%

          \[\leadsto \color{blue}{\sqrt{d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}} \cdot \sqrt{d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}}} \]

       2. sqrt-unprod 19.7%

          \[\leadsto \color{blue}{\sqrt{\left(d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right) \cdot \left(d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right)}} \]

       3. swap-sqr 12.8%

          \[\leadsto \sqrt{\color{blue}{\left(d \cdot d\right) \cdot \left(\sqrt{\frac{\frac{1}{\ell}}{h}} \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right)}} \]

       4. sqr-neg 12.8%

          \[\leadsto \sqrt{\color{blue}{\left(\left(-d\right) \cdot \left(-d\right)\right)} \cdot \left(\sqrt{\frac{\frac{1}{\ell}}{h}} \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right)} \]

       5. add-sqr-sqrt 12.9%

          \[\leadsto \sqrt{\left(\left(-d\right) \cdot \left(-d\right)\right) \cdot \color{blue}{\frac{\frac{1}{\ell}}{h}}} \]

       6. associate-/l/ 12.8%

          \[\leadsto \sqrt{\left(\left(-d\right) \cdot \left(-d\right)\right) \cdot \color{blue}{\frac{1}{h \cdot \ell}}} \]

       7. div-inv 12.9%

          \[\leadsto \sqrt{\color{blue}{\frac{\left(-d\right) \cdot \left(-d\right)}{h \cdot \ell}}} \]

       8. sqr-neg 12.9%

          \[\leadsto \sqrt{\frac{\color{blue}{d \cdot d}}{h \cdot \ell}} \]

       9. frac-times 26.2%

          \[\leadsto \sqrt{\color{blue}{\frac{d}{h} \cdot \frac{d}{\ell}}} \]

       10. sqrt-prod 36.7%

           \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}} \]

       11. frac-2neg 36.7%

           \[\leadsto \sqrt{\color{blue}{\frac{-d}{-h}}} \cdot \sqrt{\frac{d}{\ell}} \]

       12. sqrt-undiv 40.9%

           \[\leadsto \color{blue}{\frac{\sqrt{-d}}{\sqrt{-h}}} \cdot \sqrt{\frac{d}{\ell}} \]

       13. *-commutative 40.9%

           \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \frac{\sqrt{-d}}{\sqrt{-h}}} \]

       14. clear-num 40.9%

           \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \color{blue}{\frac{1}{\frac{\sqrt{-h}}{\sqrt{-d}}}} \]

       15. un-div-inv 40.9%

           \[\leadsto \color{blue}{\frac{\sqrt{\frac{d}{\ell}}}{\frac{\sqrt{-h}}{\sqrt{-d}}}} \]

       16. sqrt-undiv 36.7%

           \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\color{blue}{\sqrt{\frac{-h}{-d}}}} \]

       17. frac-2neg 36.7%

           \[\leadsto \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\color{blue}{\frac{h}{d}}}} \]

   14. Applied egg-rr 36.7%

       \[\leadsto \color{blue}{\frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}}} \]

   if -1.22e104 < l < 2.7500000000000001e-229

   1. Initial program 66.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 66.8%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in l around -inf 0.0%

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. *-commutative 0.0%

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]

      2. *-commutative 0.0%

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot d\right)} \]

      3. unpow2 0.0%

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right) \]

      4. rem-square-sqrt 44.4%

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(\color{blue}{-1} \cdot d\right) \]

      5. neg-mul-1 44.4%

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(-d\right)} \]

   7. Simplified 44.4%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-d\right)} \]

   if 2.7500000000000001e-229 < l

   1. Initial program 70.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 71.1%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 62.9%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. associate-/r* 63.2%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \]

      2. sqrt-div 66.5%

         \[\leadsto d \cdot \color{blue}{\frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}} \]

   7. Applied egg-rr 66.5%

      \[\leadsto d \cdot \color{blue}{\frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 52.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.22 \cdot 10^{+104}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}}\\ \mathbf{elif}\;\ell \leq 2.75 \cdot 10^{-229}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 45.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -2.1 \cdot 10^{+107}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;\ell \leq 1.42 \cdot 10^{-229}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (if (<= l -2.1e+107)
   (* (sqrt (/ d h)) (sqrt (/ d l)))
   (if (<= l 1.42e-229)
     (* (- d) (sqrt (/ 1.0 (* l h))))
     (* d (/ (sqrt (/ 1.0 h)) (sqrt l))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (l <= -2.1e+107) {
		tmp = sqrt((d / h)) * sqrt((d / l));
	} else if (l <= 1.42e-229) {
		tmp = -d * sqrt((1.0 / (l * h)));
	} else {
		tmp = d * (sqrt((1.0 / h)) / sqrt(l));
	}
	return tmp;
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (l <= (-2.1d+107)) then
        tmp = sqrt((d / h)) * sqrt((d / l))
    else if (l <= 1.42d-229) then
        tmp = -d * sqrt((1.0d0 / (l * h)))
    else
        tmp = d * (sqrt((1.0d0 / h)) / sqrt(l))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (l <= -2.1e+107) {
		tmp = Math.sqrt((d / h)) * Math.sqrt((d / l));
	} else if (l <= 1.42e-229) {
		tmp = -d * Math.sqrt((1.0 / (l * h)));
	} else {
		tmp = d * (Math.sqrt((1.0 / h)) / Math.sqrt(l));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	tmp = 0
	if l <= -2.1e+107:
		tmp = math.sqrt((d / h)) * math.sqrt((d / l))
	elif l <= 1.42e-229:
		tmp = -d * math.sqrt((1.0 / (l * h)))
	else:
		tmp = d * (math.sqrt((1.0 / h)) / math.sqrt(l))
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	tmp = 0.0
	if (l <= -2.1e+107)
		tmp = Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l)));
	elseif (l <= 1.42e-229)
		tmp = Float64(Float64(-d) * sqrt(Float64(1.0 / Float64(l * h))));
	else
		tmp = Float64(d * Float64(sqrt(Float64(1.0 / h)) / sqrt(l)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp_2 = code(d, h, l, M_m, D)
	tmp = 0.0;
	if (l <= -2.1e+107)
		tmp = sqrt((d / h)) * sqrt((d / l));
	elseif (l <= 1.42e-229)
		tmp = -d * sqrt((1.0 / (l * h)));
	else
		tmp = d * (sqrt((1.0 / h)) / sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := If[LessEqual[l, -2.1e+107], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.42e-229], N[((-d) * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Sqrt[N[(1.0 / h), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -2.1e107

   1. Initial program 43.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 46.2%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 36.7%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]

   if -2.1e107 < l < 1.42000000000000004e-229

   1. Initial program 66.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 66.8%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in l around -inf 0.0%

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. *-commutative 0.0%

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]

      2. *-commutative 0.0%

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot d\right)} \]

      3. unpow2 0.0%

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right) \]

      4. rem-square-sqrt 44.4%

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(\color{blue}{-1} \cdot d\right) \]

      5. neg-mul-1 44.4%

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(-d\right)} \]

   7. Simplified 44.4%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-d\right)} \]

   if 1.42000000000000004e-229 < l

   1. Initial program 70.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 71.1%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 62.9%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. associate-/r* 63.2%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \]

      2. sqrt-div 66.5%

         \[\leadsto d \cdot \color{blue}{\frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}} \]

   7. Applied egg-rr 66.5%

      \[\leadsto d \cdot \color{blue}{\frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 52.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.1 \cdot 10^{+107}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;\ell \leq 1.42 \cdot 10^{-229}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 43.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;M\_m \leq 4 \cdot 10^{-86}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \mathbf{else}:\\ \;\;\;\;d \cdot {\left(-1 + \mathsf{fma}\left(h, \ell, 1\right)\right)}^{-0.5}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (if (<= M_m 4e-86)
   (fabs (/ d (sqrt (* l h))))
   (* d (pow (+ -1.0 (fma h l 1.0)) -0.5))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (M_m <= 4e-86) {
		tmp = fabs((d / sqrt((l * h))));
	} else {
		tmp = d * pow((-1.0 + fma(h, l, 1.0)), -0.5);
	}
	return tmp;
}

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	tmp = 0.0
	if (M_m <= 4e-86)
		tmp = abs(Float64(d / sqrt(Float64(l * h))));
	else
		tmp = Float64(d * (Float64(-1.0 + fma(h, l, 1.0)) ^ -0.5));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := If[LessEqual[M$95$m, 4e-86], N[Abs[N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(d * N[Power[N[(-1.0 + N[(h * l + 1.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 4.00000000000000034e-86

   1. Initial program 61.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 62.8%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 35.9%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. *-un-lft-identity 35.9%

         \[\leadsto d \cdot \color{blue}{\left(1 \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. inv-pow 35.9%

         \[\leadsto d \cdot \left(1 \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}}\right) \]

      3. sqrt-pow1 35.9%

         \[\leadsto d \cdot \left(1 \cdot \color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}}\right) \]

      4. metadata-eval 35.9%

         \[\leadsto d \cdot \left(1 \cdot {\left(h \cdot \ell\right)}^{\color{blue}{-0.5}}\right) \]

   7. Applied egg-rr 35.9%

      \[\leadsto d \cdot \color{blue}{\left(1 \cdot {\left(h \cdot \ell\right)}^{-0.5}\right)} \]
   8. Step-by-step derivation

      1. *-lft-identity 35.9%

         \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

   9. Simplified 35.9%

      \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

   10. Taylor expanded in d around 0 35.9%

       \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   11. Step-by-step derivation

       1. *-commutative 35.9%

          \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

       2. associate-/r* 35.9%

          \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \]

   12. Simplified 35.9%

       \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}} \]
   13. Step-by-step derivation

       1. add-sqr-sqrt 32.1%

          \[\leadsto \color{blue}{\sqrt{d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}} \cdot \sqrt{d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}}} \]

       2. sqrt-unprod 37.6%

          \[\leadsto \color{blue}{\sqrt{\left(d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right) \cdot \left(d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right)}} \]

       3. swap-sqr 33.7%

          \[\leadsto \sqrt{\color{blue}{\left(d \cdot d\right) \cdot \left(\sqrt{\frac{\frac{1}{\ell}}{h}} \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right)}} \]

       4. sqr-neg 33.7%

          \[\leadsto \sqrt{\color{blue}{\left(\left(-d\right) \cdot \left(-d\right)\right)} \cdot \left(\sqrt{\frac{\frac{1}{\ell}}{h}} \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right)} \]

       5. add-sqr-sqrt 33.8%

          \[\leadsto \sqrt{\left(\left(-d\right) \cdot \left(-d\right)\right) \cdot \color{blue}{\frac{\frac{1}{\ell}}{h}}} \]

       6. associate-/l/ 33.8%

          \[\leadsto \sqrt{\left(\left(-d\right) \cdot \left(-d\right)\right) \cdot \color{blue}{\frac{1}{h \cdot \ell}}} \]

       7. div-inv 33.8%

          \[\leadsto \sqrt{\color{blue}{\frac{\left(-d\right) \cdot \left(-d\right)}{h \cdot \ell}}} \]

       8. sqr-neg 33.8%

          \[\leadsto \sqrt{\frac{\color{blue}{d \cdot d}}{h \cdot \ell}} \]

       9. frac-times 35.7%

          \[\leadsto \sqrt{\color{blue}{\frac{d}{h} \cdot \frac{d}{\ell}}} \]

       10. sqrt-prod 42.1%

           \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}} \]

       11. frac-2neg 42.1%

           \[\leadsto \sqrt{\color{blue}{\frac{-d}{-h}}} \cdot \sqrt{\frac{d}{\ell}} \]

       12. sqrt-undiv 20.0%

           \[\leadsto \color{blue}{\frac{\sqrt{-d}}{\sqrt{-h}}} \cdot \sqrt{\frac{d}{\ell}} \]

       13. add-sqr-sqrt 19.9%

           \[\leadsto \color{blue}{\sqrt{\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}} \cdot \sqrt{\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}}} \]

       14. sqrt-prod 14.1%

           \[\leadsto \color{blue}{\sqrt{\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\right)}} \]

       15. rem-sqrt-square 20.0%

           \[\leadsto \color{blue}{\left|\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\right|} \]

   14. Applied egg-rr 51.2%

       \[\leadsto \color{blue}{\left|\frac{d}{\sqrt{h \cdot \ell}}\right|} \]

   if 4.00000000000000034e-86 < M

   1. Initial program 68.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 69.9%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 22.5%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. *-un-lft-identity 22.5%

         \[\leadsto d \cdot \color{blue}{\left(1 \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. inv-pow 22.5%

         \[\leadsto d \cdot \left(1 \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}}\right) \]

      3. sqrt-pow1 22.5%

         \[\leadsto d \cdot \left(1 \cdot \color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}}\right) \]

      4. metadata-eval 22.5%

         \[\leadsto d \cdot \left(1 \cdot {\left(h \cdot \ell\right)}^{\color{blue}{-0.5}}\right) \]

   7. Applied egg-rr 22.5%

      \[\leadsto d \cdot \color{blue}{\left(1 \cdot {\left(h \cdot \ell\right)}^{-0.5}\right)} \]
   8. Step-by-step derivation

      1. *-lft-identity 22.5%

         \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

   9. Simplified 22.5%

      \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]
   10. Step-by-step derivation

       1. expm1-log1p-u 22.0%

          \[\leadsto d \cdot {\color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(h \cdot \ell\right)\right)\right)}}^{-0.5} \]

       2. expm1-undefine 23.4%

          \[\leadsto d \cdot {\color{blue}{\left(e^{\mathsf{log1p}\left(h \cdot \ell\right)} - 1\right)}}^{-0.5} \]

   11. Applied egg-rr 23.4%

       \[\leadsto d \cdot {\color{blue}{\left(e^{\mathsf{log1p}\left(h \cdot \ell\right)} - 1\right)}}^{-0.5} \]
   12. Step-by-step derivation

       1. sub-neg 23.4%

          \[\leadsto d \cdot {\color{blue}{\left(e^{\mathsf{log1p}\left(h \cdot \ell\right)} + \left(-1\right)\right)}}^{-0.5} \]

       2. metadata-eval 23.4%

          \[\leadsto d \cdot {\left(e^{\mathsf{log1p}\left(h \cdot \ell\right)} + \color{blue}{-1}\right)}^{-0.5} \]

       3. +-commutative 23.4%

          \[\leadsto d \cdot {\color{blue}{\left(-1 + e^{\mathsf{log1p}\left(h \cdot \ell\right)}\right)}}^{-0.5} \]

       4. log1p-undefine 23.4%

          \[\leadsto d \cdot {\left(-1 + e^{\color{blue}{\log \left(1 + h \cdot \ell\right)}}\right)}^{-0.5} \]

       5. rem-exp-log 23.8%

          \[\leadsto d \cdot {\left(-1 + \color{blue}{\left(1 + h \cdot \ell\right)}\right)}^{-0.5} \]

       6. +-commutative 23.8%

          \[\leadsto d \cdot {\left(-1 + \color{blue}{\left(h \cdot \ell + 1\right)}\right)}^{-0.5} \]

       7. fma-define 23.8%

          \[\leadsto d \cdot {\left(-1 + \color{blue}{\mathsf{fma}\left(h, \ell, 1\right)}\right)}^{-0.5} \]

   13. Simplified 23.8%

       \[\leadsto d \cdot {\color{blue}{\left(-1 + \mathsf{fma}\left(h, \ell, 1\right)\right)}}^{-0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 42.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 4 \cdot 10^{-86}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \mathbf{else}:\\ \;\;\;\;d \cdot {\left(-1 + \mathsf{fma}\left(h, \ell, 1\right)\right)}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 45.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 2.45 \cdot 10^{-228}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (if (<= l 2.45e-228)
   (* (- d) (sqrt (/ 1.0 (* l h))))
   (* d (/ (sqrt (/ 1.0 h)) (sqrt l)))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (l <= 2.45e-228) {
		tmp = -d * sqrt((1.0 / (l * h)));
	} else {
		tmp = d * (sqrt((1.0 / h)) / sqrt(l));
	}
	return tmp;
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (l <= 2.45d-228) then
        tmp = -d * sqrt((1.0d0 / (l * h)))
    else
        tmp = d * (sqrt((1.0d0 / h)) / sqrt(l))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (l <= 2.45e-228) {
		tmp = -d * Math.sqrt((1.0 / (l * h)));
	} else {
		tmp = d * (Math.sqrt((1.0 / h)) / Math.sqrt(l));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	tmp = 0
	if l <= 2.45e-228:
		tmp = -d * math.sqrt((1.0 / (l * h)))
	else:
		tmp = d * (math.sqrt((1.0 / h)) / math.sqrt(l))
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	tmp = 0.0
	if (l <= 2.45e-228)
		tmp = Float64(Float64(-d) * sqrt(Float64(1.0 / Float64(l * h))));
	else
		tmp = Float64(d * Float64(sqrt(Float64(1.0 / h)) / sqrt(l)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp_2 = code(d, h, l, M_m, D)
	tmp = 0.0;
	if (l <= 2.45e-228)
		tmp = -d * sqrt((1.0 / (l * h)));
	else
		tmp = d * (sqrt((1.0 / h)) / sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := If[LessEqual[l, 2.45e-228], N[((-d) * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Sqrt[N[(1.0 / h), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 2.44999999999999994e-228

   1. Initial program 59.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 61.0%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in l around -inf 0.0%

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. *-commutative 0.0%

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]

      2. *-commutative 0.0%

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot d\right)} \]

      3. unpow2 0.0%

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right) \]

      4. rem-square-sqrt 39.2%

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(\color{blue}{-1} \cdot d\right) \]

      5. neg-mul-1 39.2%

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(-d\right)} \]

   7. Simplified 39.2%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-d\right)} \]

   if 2.44999999999999994e-228 < l

   1. Initial program 70.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 71.1%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 62.9%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. associate-/r* 63.2%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \]

      2. sqrt-div 66.5%

         \[\leadsto d \cdot \color{blue}{\frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}} \]

   7. Applied egg-rr 66.5%

      \[\leadsto d \cdot \color{blue}{\frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 50.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2.45 \cdot 10^{-228}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 45.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.52 \cdot 10^{-229}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (if (<= l 1.52e-229)
   (* (- d) (sqrt (/ 1.0 (* l h))))
   (/ d (* (sqrt h) (sqrt l)))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (l <= 1.52e-229) {
		tmp = -d * sqrt((1.0 / (l * h)));
	} else {
		tmp = d / (sqrt(h) * sqrt(l));
	}
	return tmp;
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (l <= 1.52d-229) then
        tmp = -d * sqrt((1.0d0 / (l * h)))
    else
        tmp = d / (sqrt(h) * sqrt(l))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (l <= 1.52e-229) {
		tmp = -d * Math.sqrt((1.0 / (l * h)));
	} else {
		tmp = d / (Math.sqrt(h) * Math.sqrt(l));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	tmp = 0
	if l <= 1.52e-229:
		tmp = -d * math.sqrt((1.0 / (l * h)))
	else:
		tmp = d / (math.sqrt(h) * math.sqrt(l))
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	tmp = 0.0
	if (l <= 1.52e-229)
		tmp = Float64(Float64(-d) * sqrt(Float64(1.0 / Float64(l * h))));
	else
		tmp = Float64(d / Float64(sqrt(h) * sqrt(l)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp_2 = code(d, h, l, M_m, D)
	tmp = 0.0;
	if (l <= 1.52e-229)
		tmp = -d * sqrt((1.0 / (l * h)));
	else
		tmp = d / (sqrt(h) * sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := If[LessEqual[l, 1.52e-229], N[((-d) * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.5199999999999999e-229

   1. Initial program 59.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 61.0%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in l around -inf 0.0%

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. *-commutative 0.0%

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]

      2. *-commutative 0.0%

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot d\right)} \]

      3. unpow2 0.0%

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right) \]

      4. rem-square-sqrt 39.2%

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(\color{blue}{-1} \cdot d\right) \]

      5. neg-mul-1 39.2%

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(-d\right)} \]

   7. Simplified 39.2%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-d\right)} \]

   if 1.5199999999999999e-229 < l

   1. Initial program 70.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 71.1%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 62.9%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. *-un-lft-identity 62.9%

         \[\leadsto d \cdot \color{blue}{\left(1 \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. inv-pow 62.9%

         \[\leadsto d \cdot \left(1 \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}}\right) \]

      3. sqrt-pow1 62.9%

         \[\leadsto d \cdot \left(1 \cdot \color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}}\right) \]

      4. metadata-eval 62.9%

         \[\leadsto d \cdot \left(1 \cdot {\left(h \cdot \ell\right)}^{\color{blue}{-0.5}}\right) \]

   7. Applied egg-rr 62.9%

      \[\leadsto d \cdot \color{blue}{\left(1 \cdot {\left(h \cdot \ell\right)}^{-0.5}\right)} \]
   8. Step-by-step derivation

      1. *-lft-identity 62.9%

         \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

   9. Simplified 62.9%

      \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

   10. Taylor expanded in d around 0 62.9%

       \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   11. Step-by-step derivation

       1. *-commutative 62.9%

          \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

       2. associate-/r* 63.2%

          \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \]

   12. Simplified 63.2%

       \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}} \]
   13. Step-by-step derivation

       1. add-sqr-sqrt 63.0%

          \[\leadsto \color{blue}{\sqrt{d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}} \cdot \sqrt{d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}}} \]

       2. sqrt-unprod 47.2%

          \[\leadsto \color{blue}{\sqrt{\left(d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right) \cdot \left(d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right)}} \]

       3. swap-sqr 40.4%

          \[\leadsto \sqrt{\color{blue}{\left(d \cdot d\right) \cdot \left(\sqrt{\frac{\frac{1}{\ell}}{h}} \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right)}} \]

       4. sqr-neg 40.4%

          \[\leadsto \sqrt{\color{blue}{\left(\left(-d\right) \cdot \left(-d\right)\right)} \cdot \left(\sqrt{\frac{\frac{1}{\ell}}{h}} \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right)} \]

       5. add-sqr-sqrt 40.5%

          \[\leadsto \sqrt{\left(\left(-d\right) \cdot \left(-d\right)\right) \cdot \color{blue}{\frac{\frac{1}{\ell}}{h}}} \]

       6. associate-/l/ 40.4%

          \[\leadsto \sqrt{\left(\left(-d\right) \cdot \left(-d\right)\right) \cdot \color{blue}{\frac{1}{h \cdot \ell}}} \]

       7. div-inv 40.5%

          \[\leadsto \sqrt{\color{blue}{\frac{\left(-d\right) \cdot \left(-d\right)}{h \cdot \ell}}} \]

       8. sqr-neg 40.5%

          \[\leadsto \sqrt{\frac{\color{blue}{d \cdot d}}{h \cdot \ell}} \]

       9. frac-times 45.1%

          \[\leadsto \sqrt{\color{blue}{\frac{d}{h} \cdot \frac{d}{\ell}}} \]

       10. sqrt-prod 51.3%

           \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}} \]

       11. frac-2neg 51.3%

           \[\leadsto \sqrt{\color{blue}{\frac{-d}{-h}}} \cdot \sqrt{\frac{d}{\ell}} \]

       12. sqrt-undiv 0.0%

           \[\leadsto \color{blue}{\frac{\sqrt{-d}}{\sqrt{-h}}} \cdot \sqrt{\frac{d}{\ell}} \]

       13. *-commutative 0.0%

           \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \frac{\sqrt{-d}}{\sqrt{-h}}} \]

       14. sqrt-div 0.0%

           \[\leadsto \color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}} \cdot \frac{\sqrt{-d}}{\sqrt{-h}} \]

       15. frac-times 0.0%

           \[\leadsto \color{blue}{\frac{\sqrt{d} \cdot \sqrt{-d}}{\sqrt{\ell} \cdot \sqrt{-h}}} \]

   14. Applied egg-rr 66.5%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 50.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.52 \cdot 10^{-229}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 42.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;M\_m \leq 5 \cdot 10^{-43}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left(-{\left(\ell \cdot h\right)}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (if (<= M_m 5e-43) (fabs (/ d (sqrt (* l h)))) (* d (- (pow (* l h) -0.5)))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (M_m <= 5e-43) {
		tmp = fabs((d / sqrt((l * h))));
	} else {
		tmp = d * -pow((l * h), -0.5);
	}
	return tmp;
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (m_m <= 5d-43) then
        tmp = abs((d / sqrt((l * h))))
    else
        tmp = d * -((l * h) ** (-0.5d0))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (M_m <= 5e-43) {
		tmp = Math.abs((d / Math.sqrt((l * h))));
	} else {
		tmp = d * -Math.pow((l * h), -0.5);
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	tmp = 0
	if M_m <= 5e-43:
		tmp = math.fabs((d / math.sqrt((l * h))))
	else:
		tmp = d * -math.pow((l * h), -0.5)
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	tmp = 0.0
	if (M_m <= 5e-43)
		tmp = abs(Float64(d / sqrt(Float64(l * h))));
	else
		tmp = Float64(d * Float64(-(Float64(l * h) ^ -0.5)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp_2 = code(d, h, l, M_m, D)
	tmp = 0.0;
	if (M_m <= 5e-43)
		tmp = abs((d / sqrt((l * h))));
	else
		tmp = d * -((l * h) ^ -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := If[LessEqual[M$95$m, 5e-43], N[Abs[N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(d * (-N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 5.00000000000000019e-43

   1. Initial program 61.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 62.8%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 34.6%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. *-un-lft-identity 34.6%

         \[\leadsto d \cdot \color{blue}{\left(1 \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. inv-pow 34.6%

         \[\leadsto d \cdot \left(1 \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}}\right) \]

      3. sqrt-pow1 34.6%

         \[\leadsto d \cdot \left(1 \cdot \color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}}\right) \]

      4. metadata-eval 34.6%

         \[\leadsto d \cdot \left(1 \cdot {\left(h \cdot \ell\right)}^{\color{blue}{-0.5}}\right) \]

   7. Applied egg-rr 34.6%

      \[\leadsto d \cdot \color{blue}{\left(1 \cdot {\left(h \cdot \ell\right)}^{-0.5}\right)} \]
   8. Step-by-step derivation

      1. *-lft-identity 34.6%

         \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

   9. Simplified 34.6%

      \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

   10. Taylor expanded in d around 0 34.6%

       \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   11. Step-by-step derivation

       1. *-commutative 34.6%

          \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

       2. associate-/r* 34.6%

          \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \]

   12. Simplified 34.6%

       \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}} \]
   13. Step-by-step derivation

       1. add-sqr-sqrt 30.9%

          \[\leadsto \color{blue}{\sqrt{d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}} \cdot \sqrt{d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}}} \]

       2. sqrt-unprod 36.7%

          \[\leadsto \color{blue}{\sqrt{\left(d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right) \cdot \left(d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right)}} \]

       3. swap-sqr 33.1%

          \[\leadsto \sqrt{\color{blue}{\left(d \cdot d\right) \cdot \left(\sqrt{\frac{\frac{1}{\ell}}{h}} \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right)}} \]

       4. sqr-neg 33.1%

          \[\leadsto \sqrt{\color{blue}{\left(\left(-d\right) \cdot \left(-d\right)\right)} \cdot \left(\sqrt{\frac{\frac{1}{\ell}}{h}} \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right)} \]

       5. add-sqr-sqrt 33.1%

          \[\leadsto \sqrt{\left(\left(-d\right) \cdot \left(-d\right)\right) \cdot \color{blue}{\frac{\frac{1}{\ell}}{h}}} \]

       6. associate-/l/ 33.1%

          \[\leadsto \sqrt{\left(\left(-d\right) \cdot \left(-d\right)\right) \cdot \color{blue}{\frac{1}{h \cdot \ell}}} \]

       7. div-inv 33.2%

          \[\leadsto \sqrt{\color{blue}{\frac{\left(-d\right) \cdot \left(-d\right)}{h \cdot \ell}}} \]

       8. sqr-neg 33.2%

          \[\leadsto \sqrt{\frac{\color{blue}{d \cdot d}}{h \cdot \ell}} \]

       9. frac-times 35.0%

          \[\leadsto \sqrt{\color{blue}{\frac{d}{h} \cdot \frac{d}{\ell}}} \]

       10. sqrt-prod 41.4%

           \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}} \]

       11. frac-2neg 41.4%

           \[\leadsto \sqrt{\color{blue}{\frac{-d}{-h}}} \cdot \sqrt{\frac{d}{\ell}} \]

       12. sqrt-undiv 20.6%

           \[\leadsto \color{blue}{\frac{\sqrt{-d}}{\sqrt{-h}}} \cdot \sqrt{\frac{d}{\ell}} \]

       13. add-sqr-sqrt 20.6%

           \[\leadsto \color{blue}{\sqrt{\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}} \cdot \sqrt{\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}}} \]

       14. sqrt-prod 14.7%

           \[\leadsto \color{blue}{\sqrt{\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\right)}} \]

       15. rem-sqrt-square 20.6%

           \[\leadsto \color{blue}{\left|\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\right|} \]

   14. Applied egg-rr 50.4%

       \[\leadsto \color{blue}{\left|\frac{d}{\sqrt{h \cdot \ell}}\right|} \]

   if 5.00000000000000019e-43 < M

   1. Initial program 69.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 71.0%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 23.8%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

   6. Taylor expanded in h around -inf 0.0%

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   7. Step-by-step derivation

      1. associate-*l* 0.0%

         \[\leadsto \color{blue}{d \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. *-commutative 0.0%

         \[\leadsto \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot d} \]

      3. unpow2 0.0%

         \[\leadsto \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot d \]

      4. rem-square-sqrt 27.9%

         \[\leadsto \left(\color{blue}{-1} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot d \]

      5. *-commutative 27.9%

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot -1\right)} \cdot d \]

      6. associate-*r* 27.9%

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-1 \cdot d\right)} \]

      7. unpow1/2 27.9%

         \[\leadsto \color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{0.5}} \cdot \left(-1 \cdot d\right) \]

      8. rem-exp-log 26.9%

         \[\leadsto {\left(\frac{1}{\color{blue}{e^{\log \left(h \cdot \ell\right)}}}\right)}^{0.5} \cdot \left(-1 \cdot d\right) \]

      9. exp-neg 26.9%

         \[\leadsto {\color{blue}{\left(e^{-\log \left(h \cdot \ell\right)}\right)}}^{0.5} \cdot \left(-1 \cdot d\right) \]

      10. exp-prod 26.9%

          \[\leadsto \color{blue}{e^{\left(-\log \left(h \cdot \ell\right)\right) \cdot 0.5}} \cdot \left(-1 \cdot d\right) \]

      11. distribute-lft-neg-out 26.9%

          \[\leadsto e^{\color{blue}{-\log \left(h \cdot \ell\right) \cdot 0.5}} \cdot \left(-1 \cdot d\right) \]

      12. distribute-rgt-neg-in 26.9%

          \[\leadsto e^{\color{blue}{\log \left(h \cdot \ell\right) \cdot \left(-0.5\right)}} \cdot \left(-1 \cdot d\right) \]

      13. metadata-eval 26.9%

          \[\leadsto e^{\log \left(h \cdot \ell\right) \cdot \color{blue}{-0.5}} \cdot \left(-1 \cdot d\right) \]

      14. exp-to-pow 27.9%

          \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \left(-1 \cdot d\right) \]

      15. neg-mul-1 27.9%

          \[\leadsto {\left(h \cdot \ell\right)}^{-0.5} \cdot \color{blue}{\left(-d\right)} \]

   8. Simplified 27.9%

      \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot \left(-d\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 44.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 5 \cdot 10^{-43}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left(-{\left(\ell \cdot h\right)}^{-0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 42.1% accurate, 2.9× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 6 \cdot 10^{-220}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (if (<= l 6e-220)
   (* (- d) (sqrt (/ 1.0 (* l h))))
   (* d (sqrt (/ (/ 1.0 l) h)))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (l <= 6e-220) {
		tmp = -d * sqrt((1.0 / (l * h)));
	} else {
		tmp = d * sqrt(((1.0 / l) / h));
	}
	return tmp;
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (l <= 6d-220) then
        tmp = -d * sqrt((1.0d0 / (l * h)))
    else
        tmp = d * sqrt(((1.0d0 / l) / h))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (l <= 6e-220) {
		tmp = -d * Math.sqrt((1.0 / (l * h)));
	} else {
		tmp = d * Math.sqrt(((1.0 / l) / h));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	tmp = 0
	if l <= 6e-220:
		tmp = -d * math.sqrt((1.0 / (l * h)))
	else:
		tmp = d * math.sqrt(((1.0 / l) / h))
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	tmp = 0.0
	if (l <= 6e-220)
		tmp = Float64(Float64(-d) * sqrt(Float64(1.0 / Float64(l * h))));
	else
		tmp = Float64(d * sqrt(Float64(Float64(1.0 / l) / h)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp_2 = code(d, h, l, M_m, D)
	tmp = 0.0;
	if (l <= 6e-220)
		tmp = -d * sqrt((1.0 / (l * h)));
	else
		tmp = d * sqrt(((1.0 / l) / h));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := If[LessEqual[l, 6e-220], N[((-d) * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d * N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 6.00000000000000035e-220

   1. Initial program 59.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 61.0%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in l around -inf 0.0%

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. *-commutative 0.0%

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]

      2. *-commutative 0.0%

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot d\right)} \]

      3. unpow2 0.0%

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right) \]

      4. rem-square-sqrt 39.2%

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(\color{blue}{-1} \cdot d\right) \]

      5. neg-mul-1 39.2%

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(-d\right)} \]

   7. Simplified 39.2%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-d\right)} \]

   if 6.00000000000000035e-220 < l

   1. Initial program 70.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 71.1%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 62.9%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. *-un-lft-identity 62.9%

         \[\leadsto d \cdot \color{blue}{\left(1 \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. inv-pow 62.9%

         \[\leadsto d \cdot \left(1 \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}}\right) \]

      3. sqrt-pow1 62.9%

         \[\leadsto d \cdot \left(1 \cdot \color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}}\right) \]

      4. metadata-eval 62.9%

         \[\leadsto d \cdot \left(1 \cdot {\left(h \cdot \ell\right)}^{\color{blue}{-0.5}}\right) \]

   7. Applied egg-rr 62.9%

      \[\leadsto d \cdot \color{blue}{\left(1 \cdot {\left(h \cdot \ell\right)}^{-0.5}\right)} \]
   8. Step-by-step derivation

      1. *-lft-identity 62.9%

         \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

   9. Simplified 62.9%

      \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

   10. Taylor expanded in d around 0 62.9%

       \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   11. Step-by-step derivation

       1. *-commutative 62.9%

          \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

       2. associate-/r* 63.2%

          \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \]

   12. Simplified 63.2%

       \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 49.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 6 \cdot 10^{-220}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 42.2% accurate, 3.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.95 \cdot 10^{-229}:\\ \;\;\;\;d \cdot \left(-{\left(\ell \cdot h\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (if (<= l 1.95e-229)
   (* d (- (pow (* l h) -0.5)))
   (* d (sqrt (/ (/ 1.0 l) h)))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (l <= 1.95e-229) {
		tmp = d * -pow((l * h), -0.5);
	} else {
		tmp = d * sqrt(((1.0 / l) / h));
	}
	return tmp;
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (l <= 1.95d-229) then
        tmp = d * -((l * h) ** (-0.5d0))
    else
        tmp = d * sqrt(((1.0d0 / l) / h))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (l <= 1.95e-229) {
		tmp = d * -Math.pow((l * h), -0.5);
	} else {
		tmp = d * Math.sqrt(((1.0 / l) / h));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	tmp = 0
	if l <= 1.95e-229:
		tmp = d * -math.pow((l * h), -0.5)
	else:
		tmp = d * math.sqrt(((1.0 / l) / h))
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	tmp = 0.0
	if (l <= 1.95e-229)
		tmp = Float64(d * Float64(-(Float64(l * h) ^ -0.5)));
	else
		tmp = Float64(d * sqrt(Float64(Float64(1.0 / l) / h)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp_2 = code(d, h, l, M_m, D)
	tmp = 0.0;
	if (l <= 1.95e-229)
		tmp = d * -((l * h) ^ -0.5);
	else
		tmp = d * sqrt(((1.0 / l) / h));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := If[LessEqual[l, 1.95e-229], N[(d * (-N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision])), $MachinePrecision], N[(d * N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.94999999999999992e-229

   1. Initial program 59.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 61.0%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 10.1%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

   6. Taylor expanded in h around -inf 0.0%

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   7. Step-by-step derivation

      1. associate-*l* 0.0%

         \[\leadsto \color{blue}{d \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. *-commutative 0.0%

         \[\leadsto \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot d} \]

      3. unpow2 0.0%

         \[\leadsto \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot d \]

      4. rem-square-sqrt 39.2%

         \[\leadsto \left(\color{blue}{-1} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot d \]

      5. *-commutative 39.2%

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot -1\right)} \cdot d \]

      6. associate-*r* 39.2%

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-1 \cdot d\right)} \]

      7. unpow1/2 39.2%

         \[\leadsto \color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{0.5}} \cdot \left(-1 \cdot d\right) \]

      8. rem-exp-log 37.3%

         \[\leadsto {\left(\frac{1}{\color{blue}{e^{\log \left(h \cdot \ell\right)}}}\right)}^{0.5} \cdot \left(-1 \cdot d\right) \]

      9. exp-neg 37.3%

         \[\leadsto {\color{blue}{\left(e^{-\log \left(h \cdot \ell\right)}\right)}}^{0.5} \cdot \left(-1 \cdot d\right) \]

      10. exp-prod 37.3%

          \[\leadsto \color{blue}{e^{\left(-\log \left(h \cdot \ell\right)\right) \cdot 0.5}} \cdot \left(-1 \cdot d\right) \]

      11. distribute-lft-neg-out 37.3%

          \[\leadsto e^{\color{blue}{-\log \left(h \cdot \ell\right) \cdot 0.5}} \cdot \left(-1 \cdot d\right) \]

      12. distribute-rgt-neg-in 37.3%

          \[\leadsto e^{\color{blue}{\log \left(h \cdot \ell\right) \cdot \left(-0.5\right)}} \cdot \left(-1 \cdot d\right) \]

      13. metadata-eval 37.3%

          \[\leadsto e^{\log \left(h \cdot \ell\right) \cdot \color{blue}{-0.5}} \cdot \left(-1 \cdot d\right) \]

      14. exp-to-pow 39.2%

          \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \left(-1 \cdot d\right) \]

      15. neg-mul-1 39.2%

          \[\leadsto {\left(h \cdot \ell\right)}^{-0.5} \cdot \color{blue}{\left(-d\right)} \]

   8. Simplified 39.2%

      \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot \left(-d\right)} \]

   if 1.94999999999999992e-229 < l

   1. Initial program 70.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 71.1%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 62.9%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. *-un-lft-identity 62.9%

         \[\leadsto d \cdot \color{blue}{\left(1 \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. inv-pow 62.9%

         \[\leadsto d \cdot \left(1 \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}}\right) \]

      3. sqrt-pow1 62.9%

         \[\leadsto d \cdot \left(1 \cdot \color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}}\right) \]

      4. metadata-eval 62.9%

         \[\leadsto d \cdot \left(1 \cdot {\left(h \cdot \ell\right)}^{\color{blue}{-0.5}}\right) \]

   7. Applied egg-rr 62.9%

      \[\leadsto d \cdot \color{blue}{\left(1 \cdot {\left(h \cdot \ell\right)}^{-0.5}\right)} \]
   8. Step-by-step derivation

      1. *-lft-identity 62.9%

         \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

   9. Simplified 62.9%

      \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

   10. Taylor expanded in d around 0 62.9%

       \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   11. Step-by-step derivation

       1. *-commutative 62.9%

          \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

       2. associate-/r* 63.2%

          \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \]

   12. Simplified 63.2%

       \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 49.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.95 \cdot 10^{-229}:\\ \;\;\;\;d \cdot \left(-{\left(\ell \cdot h\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 25: 26.9% accurate, 3.1× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D) :precision binary64 (* d (sqrt (/ (/ 1.0 l) h))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	return d * sqrt(((1.0 / l) / h));
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_1
    code = d * sqrt(((1.0d0 / l) / h))
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	return d * Math.sqrt(((1.0 / l) / h));
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	return d * math.sqrt(((1.0 / l) / h))

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	return Float64(d * sqrt(Float64(Float64(1.0 / l) / h)))
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp = code(d, h, l, M_m, D)
	tmp = d * sqrt(((1.0 / l) / h));
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := N[(d * N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 64.0%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

3. Simplified 65.1%

   \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in d around inf 31.6%

   \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
6. Step-by-step derivation

   1. *-un-lft-identity 31.6%

      \[\leadsto d \cdot \color{blue}{\left(1 \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

   2. inv-pow 31.6%

      \[\leadsto d \cdot \left(1 \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}}\right) \]

   3. sqrt-pow1 31.6%

      \[\leadsto d \cdot \left(1 \cdot \color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}}\right) \]

   4. metadata-eval 31.6%

      \[\leadsto d \cdot \left(1 \cdot {\left(h \cdot \ell\right)}^{\color{blue}{-0.5}}\right) \]

7. Applied egg-rr 31.6%

   \[\leadsto d \cdot \color{blue}{\left(1 \cdot {\left(h \cdot \ell\right)}^{-0.5}\right)} \]
8. Step-by-step derivation

   1. *-lft-identity 31.6%

      \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

9. Simplified 31.6%

   \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

10. Taylor expanded in d around 0 31.6%

    \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
11. Step-by-step derivation

    1. *-commutative 31.6%

       \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

    2. associate-/r* 31.7%

       \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \]

12. Simplified 31.7%

    \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}} \]
13. Add Preprocessing

Alternative 26: 26.7% accurate, 3.2× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \frac{d}{\sqrt{\ell \cdot h}} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D) :precision binary64 (/ d (sqrt (* l h))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	return d / sqrt((l * h));
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_1
    code = d / sqrt((l * h))
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	return d / Math.sqrt((l * h));
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	return d / math.sqrt((l * h))

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	return Float64(d / sqrt(Float64(l * h)))
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp = code(d, h, l, M_m, D)
	tmp = d / sqrt((l * h));
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 64.0%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

3. Simplified 65.1%

   \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in d around inf 31.6%

   \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
6. Step-by-step derivation

   1. *-un-lft-identity 31.6%

      \[\leadsto d \cdot \color{blue}{\left(1 \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

   2. inv-pow 31.6%

      \[\leadsto d \cdot \left(1 \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}}\right) \]

   3. sqrt-pow1 31.6%

      \[\leadsto d \cdot \left(1 \cdot \color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}}\right) \]

   4. metadata-eval 31.6%

      \[\leadsto d \cdot \left(1 \cdot {\left(h \cdot \ell\right)}^{\color{blue}{-0.5}}\right) \]

7. Applied egg-rr 31.6%

   \[\leadsto d \cdot \color{blue}{\left(1 \cdot {\left(h \cdot \ell\right)}^{-0.5}\right)} \]
8. Step-by-step derivation

   1. *-lft-identity 31.6%

      \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

9. Simplified 31.6%

   \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

10. Taylor expanded in d around 0 31.6%

    \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
11. Step-by-step derivation

    1. *-commutative 31.6%

       \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

    2. associate-/r* 31.7%

       \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \]

12. Simplified 31.7%

    \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}} \]
13. Step-by-step derivation

    1. add-sqr-sqrt 28.1%

       \[\leadsto \color{blue}{\sqrt{d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}} \cdot \sqrt{d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}}} \]

    2. sqrt-unprod 34.1%

       \[\leadsto \color{blue}{\sqrt{\left(d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right) \cdot \left(d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right)}} \]

    3. swap-sqr 29.5%

       \[\leadsto \sqrt{\color{blue}{\left(d \cdot d\right) \cdot \left(\sqrt{\frac{\frac{1}{\ell}}{h}} \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right)}} \]

    4. sqr-neg 29.5%

       \[\leadsto \sqrt{\color{blue}{\left(\left(-d\right) \cdot \left(-d\right)\right)} \cdot \left(\sqrt{\frac{\frac{1}{\ell}}{h}} \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right)} \]

    5. add-sqr-sqrt 29.5%

       \[\leadsto \sqrt{\left(\left(-d\right) \cdot \left(-d\right)\right) \cdot \color{blue}{\frac{\frac{1}{\ell}}{h}}} \]

    6. associate-/l/ 29.5%

       \[\leadsto \sqrt{\left(\left(-d\right) \cdot \left(-d\right)\right) \cdot \color{blue}{\frac{1}{h \cdot \ell}}} \]

    7. div-inv 29.6%

       \[\leadsto \sqrt{\color{blue}{\frac{\left(-d\right) \cdot \left(-d\right)}{h \cdot \ell}}} \]

    8. sqr-neg 29.6%

       \[\leadsto \sqrt{\frac{\color{blue}{d \cdot d}}{h \cdot \ell}} \]

    9. frac-times 33.9%

       \[\leadsto \sqrt{\color{blue}{\frac{d}{h} \cdot \frac{d}{\ell}}} \]

    10. frac-times 29.6%

        \[\leadsto \sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}} \]

    11. sqrt-div 33.4%

        \[\leadsto \color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}} \]

    12. sqrt-unprod 27.5%

        \[\leadsto \frac{\color{blue}{\sqrt{d} \cdot \sqrt{d}}}{\sqrt{h \cdot \ell}} \]

    13. add-sqr-sqrt 31.6%

        \[\leadsto \frac{\color{blue}{d}}{\sqrt{h \cdot \ell}} \]

14. Applied egg-rr 31.6%

    \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]

15. Final simplification 31.6%

    \[\leadsto \frac{d}{\sqrt{\ell \cdot h}} \]
16. Add Preprocessing

Reproduce

herbie shell --seed 2024149 
(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)))))