Henrywood and Agarwal, Equation (12)

Percentage Accurate: 67.6% → 84.7%

Time: 28.5s

Alternatives: 22

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: 67.6% 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: 84.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := D \cdot \frac{M\_m \cdot 0.5}{d}\\ t_1 := 1 + h \cdot \left(\left(t\_0 \cdot t\_0\right) \cdot \frac{-0.5}{\ell}\right)\\ t_2 := \sqrt{-d}\\ \mathbf{if}\;\ell \leq -7.2 \cdot 10^{+169}:\\ \;\;\;\;\frac{t\_2}{\sqrt{-\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot t\_1\right)\\ \mathbf{elif}\;\ell \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(t\_1 \cdot \frac{t\_2}{\sqrt{-h}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(t\_1 \cdot \left(\sqrt{d} \cdot \sqrt{\frac{1}{h}}\right)\right)\\ \end{array} \end{array} \]

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 (/ (* M_m 0.5) d)))
        (t_1 (+ 1.0 (* h (* (* t_0 t_0) (/ -0.5 l)))))
        (t_2 (sqrt (- d))))
   (if (<= l -7.2e+169)
     (* (/ t_2 (sqrt (- l))) (* (sqrt (/ d h)) t_1))
     (if (<= l -1e-310)
       (* (sqrt (/ d l)) (* t_1 (/ t_2 (sqrt (- h)))))
       (* (/ (sqrt d) (sqrt l)) (* t_1 (* (sqrt d) (sqrt (/ 1.0 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 t_0 = D * ((M_m * 0.5) / d);
	double t_1 = 1.0 + (h * ((t_0 * t_0) * (-0.5 / l)));
	double t_2 = sqrt(-d);
	double tmp;
	if (l <= -7.2e+169) {
		tmp = (t_2 / sqrt(-l)) * (sqrt((d / h)) * t_1);
	} else if (l <= -1e-310) {
		tmp = sqrt((d / l)) * (t_1 * (t_2 / sqrt(-h)));
	} else {
		tmp = (sqrt(d) / sqrt(l)) * (t_1 * (sqrt(d) * sqrt((1.0 / 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) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = d_1 * ((m_m * 0.5d0) / d)
    t_1 = 1.0d0 + (h * ((t_0 * t_0) * ((-0.5d0) / l)))
    t_2 = sqrt(-d)
    if (l <= (-7.2d+169)) then
        tmp = (t_2 / sqrt(-l)) * (sqrt((d / h)) * t_1)
    else if (l <= (-1d-310)) then
        tmp = sqrt((d / l)) * (t_1 * (t_2 / sqrt(-h)))
    else
        tmp = (sqrt(d) / sqrt(l)) * (t_1 * (sqrt(d) * sqrt((1.0d0 / 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 t_0 = D * ((M_m * 0.5) / d);
	double t_1 = 1.0 + (h * ((t_0 * t_0) * (-0.5 / l)));
	double t_2 = Math.sqrt(-d);
	double tmp;
	if (l <= -7.2e+169) {
		tmp = (t_2 / Math.sqrt(-l)) * (Math.sqrt((d / h)) * t_1);
	} else if (l <= -1e-310) {
		tmp = Math.sqrt((d / l)) * (t_1 * (t_2 / Math.sqrt(-h)));
	} else {
		tmp = (Math.sqrt(d) / Math.sqrt(l)) * (t_1 * (Math.sqrt(d) * Math.sqrt((1.0 / 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):
	t_0 = D * ((M_m * 0.5) / d)
	t_1 = 1.0 + (h * ((t_0 * t_0) * (-0.5 / l)))
	t_2 = math.sqrt(-d)
	tmp = 0
	if l <= -7.2e+169:
		tmp = (t_2 / math.sqrt(-l)) * (math.sqrt((d / h)) * t_1)
	elif l <= -1e-310:
		tmp = math.sqrt((d / l)) * (t_1 * (t_2 / math.sqrt(-h)))
	else:
		tmp = (math.sqrt(d) / math.sqrt(l)) * (t_1 * (math.sqrt(d) * math.sqrt((1.0 / 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)
	t_0 = Float64(D * Float64(Float64(M_m * 0.5) / d))
	t_1 = Float64(1.0 + Float64(h * Float64(Float64(t_0 * t_0) * Float64(-0.5 / l))))
	t_2 = sqrt(Float64(-d))
	tmp = 0.0
	if (l <= -7.2e+169)
		tmp = Float64(Float64(t_2 / sqrt(Float64(-l))) * Float64(sqrt(Float64(d / h)) * t_1));
	elseif (l <= -1e-310)
		tmp = Float64(sqrt(Float64(d / l)) * Float64(t_1 * Float64(t_2 / sqrt(Float64(-h)))));
	else
		tmp = Float64(Float64(sqrt(d) / sqrt(l)) * Float64(t_1 * Float64(sqrt(d) * sqrt(Float64(1.0 / 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)
	t_0 = D * ((M_m * 0.5) / d);
	t_1 = 1.0 + (h * ((t_0 * t_0) * (-0.5 / l)));
	t_2 = sqrt(-d);
	tmp = 0.0;
	if (l <= -7.2e+169)
		tmp = (t_2 / sqrt(-l)) * (sqrt((d / h)) * t_1);
	elseif (l <= -1e-310)
		tmp = sqrt((d / l)) * (t_1 * (t_2 / sqrt(-h)));
	else
		tmp = (sqrt(d) / sqrt(l)) * (t_1 * (sqrt(d) * sqrt((1.0 / 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_] := Block[{t$95$0 = N[(D * N[(N[(M$95$m * 0.5), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + N[(h * N[(N[(t$95$0 * t$95$0), $MachinePrecision] * N[(-0.5 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[(-d)], $MachinePrecision]}, If[LessEqual[l, -7.2e+169], N[(N[(t$95$2 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -1e-310], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(t$95$1 * N[(t$95$2 / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * N[(N[Sqrt[d], $MachinePrecision] * N[Sqrt[N[(1.0 / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -7.20000000000000019e169

   1. Initial program 55.3%

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

   3. Simplified 46.1%

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

      1. associate-*l/ 35.2%

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

      2. *-commutative 35.2%

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

      3. associate-*r/ 40.7%

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

      4. *-un-lft-identity 40.7%

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

      5. times-frac 35.2%

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

      6. associate-/l/ 35.2%

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

      7. *-commutative 35.2%

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

      8. times-frac 40.7%

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

      9. *-un-lft-identity 40.7%

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

      10. *-commutative 40.7%

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

      11. associate-/l* 40.7%

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

      12. *-un-lft-identity 40.7%

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

      13. times-frac 40.7%

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

      14. metadata-eval 40.7%

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

   6. Applied egg-rr 40.7%

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

      1. associate-/l* 55.5%

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

      2. *-commutative 55.5%

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

      3. associate-/l* 55.5%

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

      4. *-commutative 55.5%

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

      5. metadata-eval 55.5%

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

      6. times-frac 55.5%

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

      7. *-rgt-identity 55.5%

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

      8. associate-/l* 55.5%

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

      9. *-commutative 55.5%

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

      10. associate-/l* 46.3%

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

      11. *-commutative 46.3%

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

   8. Simplified 46.3%

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

      1. unpow-prod-down 27.5%

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

      2. add-sqr-sqrt 27.5%

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

      3. unpow-prod-down 27.5%

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

      4. sqrt-pow1 27.5%

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

      5. metadata-eval 27.5%

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

      6. pow1 27.5%

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

      7. associate-/r* 27.5%

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

      8. div-inv 27.5%

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

      9. metadata-eval 27.5%

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

      10. unpow-prod-down 42.8%

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

      11. sqrt-pow1 46.3%

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

      12. metadata-eval 46.3%

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

      13. pow1 46.3%

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

      14. associate-/r* 46.3%

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

      15. div-inv 46.3%

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

      16. metadata-eval 46.3%

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

   10. Applied egg-rr 46.3%

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

       1. frac-2neg 46.3%

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

       2. sqrt-div 63.5%

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

   12. Applied egg-rr 63.5%

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

   if -7.20000000000000019e169 < l < -9.999999999999969e-311

   1. Initial program 72.2%

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

   3. Simplified 72.4%

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

      1. associate-*l/ 76.5%

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

      2. *-commutative 76.5%

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

      3. associate-*r/ 77.1%

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

      4. *-un-lft-identity 77.1%

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

      5. times-frac 76.5%

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

      6. associate-/l/ 76.5%

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

      7. *-commutative 76.5%

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

      8. times-frac 77.1%

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

      9. *-un-lft-identity 77.1%

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

      10. *-commutative 77.1%

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

      11. associate-/l* 77.3%

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

      12. *-un-lft-identity 77.3%

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

      13. times-frac 77.3%

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

      14. metadata-eval 77.3%

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

   6. Applied egg-rr 77.3%

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

      1. associate-/l* 77.3%

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

      2. *-commutative 77.3%

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

      3. associate-/l* 77.2%

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

      4. *-commutative 77.2%

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

      5. metadata-eval 77.2%

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

      6. times-frac 77.2%

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

      7. *-rgt-identity 77.2%

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

      8. associate-/l* 76.2%

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

      9. *-commutative 76.2%

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

      10. associate-/l* 77.3%

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

      11. *-commutative 77.3%

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

   8. Simplified 77.3%

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

      1. unpow-prod-down 67.0%

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

      2. add-sqr-sqrt 67.0%

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

      3. unpow-prod-down 67.0%

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

      4. sqrt-pow1 51.2%

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

      5. metadata-eval 51.2%

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

      6. pow1 51.2%

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

      7. associate-/r* 51.2%

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

      8. div-inv 51.2%

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

      9. metadata-eval 51.2%

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

      10. unpow-prod-down 58.7%

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

      11. sqrt-pow1 77.3%

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

      12. metadata-eval 77.3%

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

      13. pow1 77.3%

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

      14. associate-/r* 77.3%

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

      15. div-inv 77.3%

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

      16. metadata-eval 77.3%

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

   10. Applied egg-rr 77.3%

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

       1. frac-2neg 77.3%

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

       2. sqrt-div 88.8%

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

   12. Applied egg-rr 88.8%

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

   if -9.999999999999969e-311 < l

   1. Initial program 72.1%

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

   3. Simplified 70.4%

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

      1. associate-*l/ 73.6%

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

      2. *-commutative 73.6%

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

      3. associate-*r/ 75.2%

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

      4. *-un-lft-identity 75.2%

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

      5. times-frac 73.6%

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

      6. associate-/l/ 73.6%

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

      7. *-commutative 73.6%

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

      8. times-frac 75.2%

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

      9. *-un-lft-identity 75.2%

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

      10. *-commutative 75.2%

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

      11. associate-/l* 74.4%

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

      12. *-un-lft-identity 74.4%

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

      13. times-frac 74.4%

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

      14. metadata-eval 74.4%

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

   6. Applied egg-rr 74.4%

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

      1. associate-/l* 75.3%

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

      2. *-commutative 75.3%

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

      3. associate-/l* 75.3%

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

      4. *-commutative 75.3%

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

      5. metadata-eval 75.3%

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

      6. times-frac 75.3%

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

      7. *-rgt-identity 75.3%

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

      8. associate-/l* 76.1%

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

      9. *-commutative 76.1%

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

      10. associate-/l* 74.5%

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

      11. *-commutative 74.5%

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

   8. Simplified 74.5%

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

      1. unpow-prod-down 64.3%

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

      2. add-sqr-sqrt 64.3%

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

      3. unpow-prod-down 64.3%

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

      4. sqrt-pow1 51.1%

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

      5. metadata-eval 51.1%

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

      6. pow1 51.1%

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

      7. associate-/r* 51.1%

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

      8. div-inv 51.1%

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

      9. metadata-eval 51.1%

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

      10. unpow-prod-down 59.5%

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

      11. sqrt-pow1 74.5%

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

      12. metadata-eval 74.5%

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

      13. pow1 74.5%

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

      14. associate-/r* 74.5%

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

      15. div-inv 74.5%

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

      16. metadata-eval 74.5%

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

   10. Applied egg-rr 74.5%

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

       1. sqrt-div 79.2%

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

   12. Applied egg-rr 79.2%

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

       1. pow1/2 79.2%

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

       2. div-inv 79.1%

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

       3. unpow-prod-down 86.9%

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

       4. pow1/2 86.9%

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

   14. Applied egg-rr 86.9%

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

       1. unpow1/2 86.9%

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

   16. Simplified 86.9%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -7.2 \cdot 10^{+169}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + h \cdot \left(\left(\left(D \cdot \frac{M \cdot 0.5}{d}\right) \cdot \left(D \cdot \frac{M \cdot 0.5}{d}\right)\right) \cdot \frac{-0.5}{\ell}\right)\right)\right)\\ \mathbf{elif}\;\ell \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\left(1 + h \cdot \left(\left(\left(D \cdot \frac{M \cdot 0.5}{d}\right) \cdot \left(D \cdot \frac{M \cdot 0.5}{d}\right)\right) \cdot \frac{-0.5}{\ell}\right)\right) \cdot \frac{\sqrt{-d}}{\sqrt{-h}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(\left(1 + h \cdot \left(\left(\left(D \cdot \frac{M \cdot 0.5}{d}\right) \cdot \left(D \cdot \frac{M \cdot 0.5}{d}\right)\right) \cdot \frac{-0.5}{\ell}\right)\right) \cdot \left(\sqrt{d} \cdot \sqrt{\frac{1}{h}}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 82.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := D \cdot \frac{M\_m \cdot 0.5}{d}\\ t_1 := 1 + h \cdot \left(\left(t\_0 \cdot t\_0\right) \cdot \frac{-0.5}{\ell}\right)\\ t_2 := \sqrt{-d}\\ \mathbf{if}\;\ell \leq -2.4 \cdot 10^{+169}:\\ \;\;\;\;\frac{t\_2}{\sqrt{-\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot t\_1\right)\\ \mathbf{elif}\;\ell \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(t\_1 \cdot \frac{t\_2}{\sqrt{-h}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left({\left(\frac{M\_m \cdot \left(D \cdot 0.5\right)}{d}\right)}^{2}, \frac{h \cdot -0.5}{\ell}, 1\right)}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

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
 (let* ((t_0 (* D (/ (* M_m 0.5) d)))
        (t_1 (+ 1.0 (* h (* (* t_0 t_0) (/ -0.5 l)))))
        (t_2 (sqrt (- d))))
   (if (<= l -2.4e+169)
     (* (/ t_2 (sqrt (- l))) (* (sqrt (/ d h)) t_1))
     (if (<= l -1e-310)
       (* (sqrt (/ d l)) (* t_1 (/ t_2 (sqrt (- h)))))
       (*
        d
        (/
         (fma (pow (/ (* M_m (* D 0.5)) d) 2.0) (/ (* h -0.5) l) 1.0)
         (* (sqrt l) (sqrt 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 t_0 = D * ((M_m * 0.5) / d);
	double t_1 = 1.0 + (h * ((t_0 * t_0) * (-0.5 / l)));
	double t_2 = sqrt(-d);
	double tmp;
	if (l <= -2.4e+169) {
		tmp = (t_2 / sqrt(-l)) * (sqrt((d / h)) * t_1);
	} else if (l <= -1e-310) {
		tmp = sqrt((d / l)) * (t_1 * (t_2 / sqrt(-h)));
	} else {
		tmp = d * (fma(pow(((M_m * (D * 0.5)) / d), 2.0), ((h * -0.5) / l), 1.0) / (sqrt(l) * sqrt(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)
	t_0 = Float64(D * Float64(Float64(M_m * 0.5) / d))
	t_1 = Float64(1.0 + Float64(h * Float64(Float64(t_0 * t_0) * Float64(-0.5 / l))))
	t_2 = sqrt(Float64(-d))
	tmp = 0.0
	if (l <= -2.4e+169)
		tmp = Float64(Float64(t_2 / sqrt(Float64(-l))) * Float64(sqrt(Float64(d / h)) * t_1));
	elseif (l <= -1e-310)
		tmp = Float64(sqrt(Float64(d / l)) * Float64(t_1 * Float64(t_2 / sqrt(Float64(-h)))));
	else
		tmp = Float64(d * Float64(fma((Float64(Float64(M_m * Float64(D * 0.5)) / d) ^ 2.0), Float64(Float64(h * -0.5) / l), 1.0) / Float64(sqrt(l) * sqrt(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_] := Block[{t$95$0 = N[(D * N[(N[(M$95$m * 0.5), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + N[(h * N[(N[(t$95$0 * t$95$0), $MachinePrecision] * N[(-0.5 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[(-d)], $MachinePrecision]}, If[LessEqual[l, -2.4e+169], N[(N[(t$95$2 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -1e-310], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(t$95$1 * N[(t$95$2 / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[(N[Power[N[(N[(M$95$m * N[(D * 0.5), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(h * -0.5), $MachinePrecision] / l), $MachinePrecision] + 1.0), $MachinePrecision] / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -2.3999999999999998e169

   1. Initial program 55.3%

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

   3. Simplified 46.1%

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

      1. associate-*l/ 35.2%

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

      2. *-commutative 35.2%

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

      3. associate-*r/ 40.7%

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

      4. *-un-lft-identity 40.7%

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

      5. times-frac 35.2%

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

      6. associate-/l/ 35.2%

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

      7. *-commutative 35.2%

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

      8. times-frac 40.7%

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

      9. *-un-lft-identity 40.7%

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

      10. *-commutative 40.7%

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

      11. associate-/l* 40.7%

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

      12. *-un-lft-identity 40.7%

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

      13. times-frac 40.7%

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

      14. metadata-eval 40.7%

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

   6. Applied egg-rr 40.7%

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

      1. associate-/l* 55.5%

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

      2. *-commutative 55.5%

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

      3. associate-/l* 55.5%

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

      4. *-commutative 55.5%

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

      5. metadata-eval 55.5%

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

      6. times-frac 55.5%

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

      7. *-rgt-identity 55.5%

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

      8. associate-/l* 55.5%

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

      9. *-commutative 55.5%

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

      10. associate-/l* 46.3%

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

      11. *-commutative 46.3%

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

   8. Simplified 46.3%

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

      1. unpow-prod-down 27.5%

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

      2. add-sqr-sqrt 27.5%

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

      3. unpow-prod-down 27.5%

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

      4. sqrt-pow1 27.5%

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

      5. metadata-eval 27.5%

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

      6. pow1 27.5%

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

      7. associate-/r* 27.5%

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

      8. div-inv 27.5%

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

      9. metadata-eval 27.5%

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

      10. unpow-prod-down 42.8%

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

      11. sqrt-pow1 46.3%

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

      12. metadata-eval 46.3%

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

      13. pow1 46.3%

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

      14. associate-/r* 46.3%

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

      15. div-inv 46.3%

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

      16. metadata-eval 46.3%

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

   10. Applied egg-rr 46.3%

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

       1. frac-2neg 46.3%

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

       2. sqrt-div 63.5%

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

   12. Applied egg-rr 63.5%

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

   if -2.3999999999999998e169 < l < -9.999999999999969e-311

   1. Initial program 72.2%

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

   3. Simplified 72.4%

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

      1. associate-*l/ 76.5%

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

      2. *-commutative 76.5%

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

      3. associate-*r/ 77.1%

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

      4. *-un-lft-identity 77.1%

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

      5. times-frac 76.5%

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

      6. associate-/l/ 76.5%

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

      7. *-commutative 76.5%

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

      8. times-frac 77.1%

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

      9. *-un-lft-identity 77.1%

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

      10. *-commutative 77.1%

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

      11. associate-/l* 77.3%

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

      12. *-un-lft-identity 77.3%

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

      13. times-frac 77.3%

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

      14. metadata-eval 77.3%

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

   6. Applied egg-rr 77.3%

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

      1. associate-/l* 77.3%

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

      2. *-commutative 77.3%

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

      3. associate-/l* 77.2%

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

      4. *-commutative 77.2%

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

      5. metadata-eval 77.2%

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

      6. times-frac 77.2%

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

      7. *-rgt-identity 77.2%

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

      8. associate-/l* 76.2%

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

      9. *-commutative 76.2%

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

      10. associate-/l* 77.3%

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

      11. *-commutative 77.3%

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

   8. Simplified 77.3%

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

      1. unpow-prod-down 67.0%

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

      2. add-sqr-sqrt 67.0%

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

      3. unpow-prod-down 67.0%

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

      4. sqrt-pow1 51.2%

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

      5. metadata-eval 51.2%

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

      6. pow1 51.2%

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

      7. associate-/r* 51.2%

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

      8. div-inv 51.2%

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

      9. metadata-eval 51.2%

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

      10. unpow-prod-down 58.7%

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

      11. sqrt-pow1 77.3%

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

      12. metadata-eval 77.3%

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

      13. pow1 77.3%

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

      14. associate-/r* 77.3%

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

      15. div-inv 77.3%

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

      16. metadata-eval 77.3%

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

   10. Applied egg-rr 77.3%

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

       1. frac-2neg 77.3%

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

       2. sqrt-div 88.8%

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

   12. Applied egg-rr 88.8%

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

   if -9.999999999999969e-311 < l

   1. Initial program 72.1%

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

   3. Simplified 71.3%

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

      1. pow1 71.3%

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

   6. Applied egg-rr 80.9%

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

      1. unpow1 80.9%

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

      2. associate-*l/ 83.3%

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

      3. associate-/l* 83.3%

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

      4. *-commutative 83.3%

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

      5. metadata-eval 83.3%

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

      6. times-frac 83.3%

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

      7. *-rgt-identity 83.3%

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

      8. associate-/l* 84.5%

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

      9. *-commutative 84.5%

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

      10. associate-/l* 82.9%

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

      11. *-commutative 82.9%

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

      12. *-commutative 82.9%

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

   8. Simplified 82.9%

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

   9. Taylor expanded in D around 0 84.5%

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

       1. associate-*r/ 84.5%

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

       2. *-commutative 84.5%

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

       3. associate-*r* 84.5%

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

       4. *-commutative 84.5%

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

       5. associate-*l* 84.5%

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

   11. Simplified 84.5%

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

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.4 \cdot 10^{+169}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + h \cdot \left(\left(\left(D \cdot \frac{M \cdot 0.5}{d}\right) \cdot \left(D \cdot \frac{M \cdot 0.5}{d}\right)\right) \cdot \frac{-0.5}{\ell}\right)\right)\right)\\ \mathbf{elif}\;\ell \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\left(1 + h \cdot \left(\left(\left(D \cdot \frac{M \cdot 0.5}{d}\right) \cdot \left(D \cdot \frac{M \cdot 0.5}{d}\right)\right) \cdot \frac{-0.5}{\ell}\right)\right) \cdot \frac{\sqrt{-d}}{\sqrt{-h}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left({\left(\frac{M \cdot \left(D \cdot 0.5\right)}{d}\right)}^{2}, \frac{h \cdot -0.5}{\ell}, 1\right)}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := D \cdot \frac{M\_m \cdot 0.5}{d}\\ \mathbf{if}\;d \leq -7 \cdot 10^{-309}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\left(1 + h \cdot \left(\left(t\_0 \cdot t\_0\right) \cdot \frac{-0.5}{\ell}\right)\right) \cdot \frac{\sqrt{-d}}{\sqrt{-h}}\right)\\ \mathbf{elif}\;d \leq 4.1 \cdot 10^{+70}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + h \cdot \frac{\left(D \cdot \left(M\_m \cdot 0.5\right)\right) \cdot \left(D \cdot \left(\frac{-0.5}{\ell} \cdot \left(M\_m \cdot \frac{0.5}{d}\right)\right)\right)}{d}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M\_m}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\\ \end{array} \end{array} \]

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 (/ (* M_m 0.5) d))))
   (if (<= d -7e-309)
     (*
      (sqrt (/ d l))
      (*
       (+ 1.0 (* h (* (* t_0 t_0) (/ -0.5 l))))
       (/ (sqrt (- d)) (sqrt (- h)))))
     (if (<= d 4.1e+70)
       (*
        (/ (sqrt d) (sqrt l))
        (*
         (sqrt (/ d h))
         (+
          1.0
          (*
           h
           (/
            (* (* D (* M_m 0.5)) (* D (* (/ -0.5 l) (* M_m (/ 0.5 d)))))
            d)))))
       (*
        (/ d (* (sqrt l) (sqrt h)))
        (- 1.0 (* 0.5 (* (pow (* (/ M_m 2.0) (/ D d)) 2.0) (/ h 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 * ((M_m * 0.5) / d);
	double tmp;
	if (d <= -7e-309) {
		tmp = sqrt((d / l)) * ((1.0 + (h * ((t_0 * t_0) * (-0.5 / l)))) * (sqrt(-d) / sqrt(-h)));
	} else if (d <= 4.1e+70) {
		tmp = (sqrt(d) / sqrt(l)) * (sqrt((d / h)) * (1.0 + (h * (((D * (M_m * 0.5)) * (D * ((-0.5 / l) * (M_m * (0.5 / d))))) / d))));
	} else {
		tmp = (d / (sqrt(l) * sqrt(h))) * (1.0 - (0.5 * (pow(((M_m / 2.0) * (D / d)), 2.0) * (h / 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 = d_1 * ((m_m * 0.5d0) / d)
    if (d <= (-7d-309)) then
        tmp = sqrt((d / l)) * ((1.0d0 + (h * ((t_0 * t_0) * ((-0.5d0) / l)))) * (sqrt(-d) / sqrt(-h)))
    else if (d <= 4.1d+70) then
        tmp = (sqrt(d) / sqrt(l)) * (sqrt((d / h)) * (1.0d0 + (h * (((d_1 * (m_m * 0.5d0)) * (d_1 * (((-0.5d0) / l) * (m_m * (0.5d0 / d))))) / d))))
    else
        tmp = (d / (sqrt(l) * sqrt(h))) * (1.0d0 - (0.5d0 * ((((m_m / 2.0d0) * (d_1 / d)) ** 2.0d0) * (h / 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 = D * ((M_m * 0.5) / d);
	double tmp;
	if (d <= -7e-309) {
		tmp = Math.sqrt((d / l)) * ((1.0 + (h * ((t_0 * t_0) * (-0.5 / l)))) * (Math.sqrt(-d) / Math.sqrt(-h)));
	} else if (d <= 4.1e+70) {
		tmp = (Math.sqrt(d) / Math.sqrt(l)) * (Math.sqrt((d / h)) * (1.0 + (h * (((D * (M_m * 0.5)) * (D * ((-0.5 / l) * (M_m * (0.5 / d))))) / d))));
	} else {
		tmp = (d / (Math.sqrt(l) * Math.sqrt(h))) * (1.0 - (0.5 * (Math.pow(((M_m / 2.0) * (D / d)), 2.0) * (h / 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 = D * ((M_m * 0.5) / d)
	tmp = 0
	if d <= -7e-309:
		tmp = math.sqrt((d / l)) * ((1.0 + (h * ((t_0 * t_0) * (-0.5 / l)))) * (math.sqrt(-d) / math.sqrt(-h)))
	elif d <= 4.1e+70:
		tmp = (math.sqrt(d) / math.sqrt(l)) * (math.sqrt((d / h)) * (1.0 + (h * (((D * (M_m * 0.5)) * (D * ((-0.5 / l) * (M_m * (0.5 / d))))) / d))))
	else:
		tmp = (d / (math.sqrt(l) * math.sqrt(h))) * (1.0 - (0.5 * (math.pow(((M_m / 2.0) * (D / d)), 2.0) * (h / 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 * Float64(Float64(M_m * 0.5) / d))
	tmp = 0.0
	if (d <= -7e-309)
		tmp = Float64(sqrt(Float64(d / l)) * Float64(Float64(1.0 + Float64(h * Float64(Float64(t_0 * t_0) * Float64(-0.5 / l)))) * Float64(sqrt(Float64(-d)) / sqrt(Float64(-h)))));
	elseif (d <= 4.1e+70)
		tmp = Float64(Float64(sqrt(d) / sqrt(l)) * Float64(sqrt(Float64(d / h)) * Float64(1.0 + Float64(h * Float64(Float64(Float64(D * Float64(M_m * 0.5)) * Float64(D * Float64(Float64(-0.5 / l) * Float64(M_m * Float64(0.5 / d))))) / d)))));
	else
		tmp = Float64(Float64(d / Float64(sqrt(l) * sqrt(h))) * Float64(1.0 - Float64(0.5 * Float64((Float64(Float64(M_m / 2.0) * Float64(D / d)) ^ 2.0) * Float64(h / 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 = D * ((M_m * 0.5) / d);
	tmp = 0.0;
	if (d <= -7e-309)
		tmp = sqrt((d / l)) * ((1.0 + (h * ((t_0 * t_0) * (-0.5 / l)))) * (sqrt(-d) / sqrt(-h)));
	elseif (d <= 4.1e+70)
		tmp = (sqrt(d) / sqrt(l)) * (sqrt((d / h)) * (1.0 + (h * (((D * (M_m * 0.5)) * (D * ((-0.5 / l) * (M_m * (0.5 / d))))) / d))));
	else
		tmp = (d / (sqrt(l) * sqrt(h))) * (1.0 - (0.5 * ((((M_m / 2.0) * (D / d)) ^ 2.0) * (h / 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[(D * N[(N[(M$95$m * 0.5), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -7e-309], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(N[(1.0 + N[(h * N[(N[(t$95$0 * t$95$0), $MachinePrecision] * N[(-0.5 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 4.1e+70], N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(h * N[(N[(N[(D * N[(M$95$m * 0.5), $MachinePrecision]), $MachinePrecision] * N[(D * N[(N[(-0.5 / l), $MachinePrecision] * N[(M$95$m * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[Power[N[(N[(M$95$m / 2.0), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if d < -6.9999999999999984e-309

   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 67.6%

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

      1. associate-*l/ 68.7%

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

      2. *-commutative 68.7%

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

      3. associate-*r/ 70.3%

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

      4. *-un-lft-identity 70.3%

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

      5. times-frac 68.7%

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

      6. associate-/l/ 68.7%

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

      7. *-commutative 68.7%

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

      8. times-frac 70.3%

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

      9. *-un-lft-identity 70.3%

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

      10. *-commutative 70.3%

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

      11. associate-/l* 70.5%

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

      12. *-un-lft-identity 70.5%

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

      13. times-frac 70.5%

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

      14. metadata-eval 70.5%

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

   6. Applied egg-rr 70.5%

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

      1. associate-/l* 73.4%

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

      2. *-commutative 73.4%

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

      3. associate-/l* 73.4%

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

      4. *-commutative 73.4%

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

      5. metadata-eval 73.4%

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

      6. times-frac 73.4%

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

      7. *-rgt-identity 73.4%

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

      8. associate-/l* 72.6%

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

      9. *-commutative 72.6%

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

      10. associate-/l* 71.6%

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

      11. *-commutative 71.6%

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

   8. Simplified 71.6%

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

      1. unpow-prod-down 59.5%

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

      2. add-sqr-sqrt 59.5%

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

      3. unpow-prod-down 59.5%

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

      4. sqrt-pow1 46.8%

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

      5. metadata-eval 46.8%

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

      6. pow1 46.8%

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

      7. associate-/r* 46.8%

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

      8. div-inv 46.8%

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

      9. metadata-eval 46.8%

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

      10. unpow-prod-down 55.9%

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

      11. sqrt-pow1 71.6%

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

      12. metadata-eval 71.6%

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

      13. pow1 71.6%

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

      14. associate-/r* 71.6%

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

      15. div-inv 71.6%

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

      16. metadata-eval 71.6%

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

   10. Applied egg-rr 71.6%

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

       1. frac-2neg 71.6%

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

       2. sqrt-div 81.5%

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

   12. Applied egg-rr 81.5%

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

   if -6.9999999999999984e-309 < d < 4.1000000000000002e70

   1. Initial program 66.0%

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

   3. Simplified 66.0%

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

      1. associate-*l/ 71.1%

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

      2. *-commutative 71.1%

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

      3. associate-*r/ 71.1%

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

      4. *-un-lft-identity 71.1%

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

      5. times-frac 71.1%

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

      6. associate-/l/ 71.1%

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

      7. *-commutative 71.1%

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

      8. times-frac 71.1%

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

      9. *-un-lft-identity 71.1%

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

      10. *-commutative 71.1%

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

      11. associate-/l* 71.1%

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

      12. *-un-lft-identity 71.1%

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

      13. times-frac 71.1%

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

      14. metadata-eval 71.1%

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

   6. Applied egg-rr 71.1%

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

      1. associate-/l* 71.3%

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

      2. *-commutative 71.3%

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

      3. associate-/l* 71.3%

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

      4. *-commutative 71.3%

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

      5. metadata-eval 71.3%

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

      6. times-frac 71.3%

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

      7. *-rgt-identity 71.3%

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

      8. associate-/l* 71.3%

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

      9. *-commutative 71.3%

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

      10. associate-/l* 71.3%

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

      11. *-commutative 71.3%

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

   8. Simplified 71.3%

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

      1. unpow-prod-down 61.4%

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

      2. add-sqr-sqrt 61.4%

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

      3. unpow-prod-down 61.4%

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

      4. sqrt-pow1 46.6%

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

      5. metadata-eval 46.6%

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

      6. pow1 46.6%

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

      7. associate-/r* 46.6%

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

      8. div-inv 46.6%

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

      9. metadata-eval 46.6%

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

      10. unpow-prod-down 53.7%

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

      11. sqrt-pow1 71.3%

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

      12. metadata-eval 71.3%

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

      13. pow1 71.3%

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

      14. associate-/r* 71.3%

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

      15. div-inv 71.3%

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

      16. metadata-eval 71.3%

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

   10. Applied egg-rr 71.3%

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

       1. sqrt-div 77.8%

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

   12. Applied egg-rr 77.8%

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

       1. associate-*l* 82.9%

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

       2. associate-*r/ 82.9%

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

       3. associate-*l/ 82.9%

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

       4. associate-*l* 82.9%

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

       5. associate-/l* 82.9%

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

   14. Applied egg-rr 82.9%

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

   if 4.1000000000000002e70 < d

   1. Initial program 80.0%

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

   3. Simplified 78.0%

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

      1. *-commutative 78.0%

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

      2. sqrt-div 78.5%

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

      3. sqrt-div 88.0%

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

      4. frac-times 88.0%

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

      5. add-sqr-sqrt 88.2%

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

   6. Applied egg-rr 88.2%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -7 \cdot 10^{-309}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\left(1 + h \cdot \left(\left(\left(D \cdot \frac{M \cdot 0.5}{d}\right) \cdot \left(D \cdot \frac{M \cdot 0.5}{d}\right)\right) \cdot \frac{-0.5}{\ell}\right)\right) \cdot \frac{\sqrt{-d}}{\sqrt{-h}}\right)\\ \mathbf{elif}\;d \leq 4.1 \cdot 10^{+70}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + h \cdot \frac{\left(D \cdot \left(M \cdot 0.5\right)\right) \cdot \left(D \cdot \left(\frac{-0.5}{\ell} \cdot \left(M \cdot \frac{0.5}{d}\right)\right)\right)}{d}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 77.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := 0.5 \cdot \left({\left(\frac{M\_m}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;h \leq -2.6 \cdot 10^{+127}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + h \cdot \left(\frac{-0.5}{\ell} \cdot \left(\left(D \cdot \frac{M\_m \cdot 0.5}{d}\right) \cdot \frac{D}{\frac{d}{M\_m \cdot 0.5}}\right)\right)\right)\right)\\ \mathbf{elif}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(t\_0 + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 - t\_0\right)\\ \end{array} \end{array} \]

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 (* 0.5 (* (pow (* (/ M_m 2.0) (/ D d)) 2.0) (/ h l)))))
   (if (<= h -2.6e+127)
     (*
      (sqrt (/ d l))
      (*
       (sqrt (/ d h))
       (+
        1.0
        (*
         h
         (* (/ -0.5 l) (* (* D (/ (* M_m 0.5) d)) (/ D (/ d (* M_m 0.5)))))))))
     (if (<= h -5e-310)
       (* (* d (sqrt (/ 1.0 (* l h)))) (+ t_0 -1.0))
       (* (/ d (* (sqrt l) (sqrt h))) (- 1.0 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 = 0.5 * (pow(((M_m / 2.0) * (D / d)), 2.0) * (h / l));
	double tmp;
	if (h <= -2.6e+127) {
		tmp = sqrt((d / l)) * (sqrt((d / h)) * (1.0 + (h * ((-0.5 / l) * ((D * ((M_m * 0.5) / d)) * (D / (d / (M_m * 0.5))))))));
	} else if (h <= -5e-310) {
		tmp = (d * sqrt((1.0 / (l * h)))) * (t_0 + -1.0);
	} else {
		tmp = (d / (sqrt(l) * sqrt(h))) * (1.0 - 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) :: tmp
    t_0 = 0.5d0 * ((((m_m / 2.0d0) * (d_1 / d)) ** 2.0d0) * (h / l))
    if (h <= (-2.6d+127)) then
        tmp = sqrt((d / l)) * (sqrt((d / h)) * (1.0d0 + (h * (((-0.5d0) / l) * ((d_1 * ((m_m * 0.5d0) / d)) * (d_1 / (d / (m_m * 0.5d0))))))))
    else if (h <= (-5d-310)) then
        tmp = (d * sqrt((1.0d0 / (l * h)))) * (t_0 + (-1.0d0))
    else
        tmp = (d / (sqrt(l) * sqrt(h))) * (1.0d0 - 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 = 0.5 * (Math.pow(((M_m / 2.0) * (D / d)), 2.0) * (h / l));
	double tmp;
	if (h <= -2.6e+127) {
		tmp = Math.sqrt((d / l)) * (Math.sqrt((d / h)) * (1.0 + (h * ((-0.5 / l) * ((D * ((M_m * 0.5) / d)) * (D / (d / (M_m * 0.5))))))));
	} else if (h <= -5e-310) {
		tmp = (d * Math.sqrt((1.0 / (l * h)))) * (t_0 + -1.0);
	} else {
		tmp = (d / (Math.sqrt(l) * Math.sqrt(h))) * (1.0 - 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 = 0.5 * (math.pow(((M_m / 2.0) * (D / d)), 2.0) * (h / l))
	tmp = 0
	if h <= -2.6e+127:
		tmp = math.sqrt((d / l)) * (math.sqrt((d / h)) * (1.0 + (h * ((-0.5 / l) * ((D * ((M_m * 0.5) / d)) * (D / (d / (M_m * 0.5))))))))
	elif h <= -5e-310:
		tmp = (d * math.sqrt((1.0 / (l * h)))) * (t_0 + -1.0)
	else:
		tmp = (d / (math.sqrt(l) * math.sqrt(h))) * (1.0 - 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 = Float64(0.5 * Float64((Float64(Float64(M_m / 2.0) * Float64(D / d)) ^ 2.0) * Float64(h / l)))
	tmp = 0.0
	if (h <= -2.6e+127)
		tmp = Float64(sqrt(Float64(d / l)) * Float64(sqrt(Float64(d / h)) * Float64(1.0 + Float64(h * Float64(Float64(-0.5 / l) * Float64(Float64(D * Float64(Float64(M_m * 0.5) / d)) * Float64(D / Float64(d / Float64(M_m * 0.5)))))))));
	elseif (h <= -5e-310)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(l * h)))) * Float64(t_0 + -1.0));
	else
		tmp = Float64(Float64(d / Float64(sqrt(l) * sqrt(h))) * Float64(1.0 - 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 = 0.5 * ((((M_m / 2.0) * (D / d)) ^ 2.0) * (h / l));
	tmp = 0.0;
	if (h <= -2.6e+127)
		tmp = sqrt((d / l)) * (sqrt((d / h)) * (1.0 + (h * ((-0.5 / l) * ((D * ((M_m * 0.5) / d)) * (D / (d / (M_m * 0.5))))))));
	elseif (h <= -5e-310)
		tmp = (d * sqrt((1.0 / (l * h)))) * (t_0 + -1.0);
	else
		tmp = (d / (sqrt(l) * sqrt(h))) * (1.0 - 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[(0.5 * N[(N[Power[N[(N[(M$95$m / 2.0), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[h, -2.6e+127], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(h * N[(N[(-0.5 / l), $MachinePrecision] * N[(N[(D * N[(N[(M$95$m * 0.5), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * N[(D / N[(d / N[(M$95$m * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, -5e-310], N[(N[(d * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if h < -2.6000000000000002e127

   1. Initial program 60.2%

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

   3. Simplified 58.1%

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

      1. associate-*l/ 60.8%

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

      2. *-commutative 60.8%

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

      3. associate-*r/ 60.8%

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

      4. *-un-lft-identity 60.8%

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

      5. times-frac 60.8%

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

      6. associate-/l/ 60.8%

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

      7. *-commutative 60.8%

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

      8. times-frac 60.8%

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

      9. *-un-lft-identity 60.8%

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

      10. *-commutative 60.8%

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

      11. associate-/l* 60.8%

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

      12. *-un-lft-identity 60.8%

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

      13. times-frac 60.8%

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

      14. metadata-eval 60.8%

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

   6. Applied egg-rr 60.8%

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

      1. associate-/l* 69.7%

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

      2. *-commutative 69.7%

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

      3. associate-/l* 69.7%

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

      4. *-commutative 69.7%

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

      5. metadata-eval 69.7%

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

      6. times-frac 69.7%

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

      7. *-rgt-identity 69.7%

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

      8. associate-/l* 69.7%

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

      9. *-commutative 69.7%

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

      10. associate-/l* 67.5%

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

      11. *-commutative 67.5%

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

   8. Simplified 67.5%

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

      1. unpow-prod-down 54.4%

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

      2. add-sqr-sqrt 54.4%

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

      3. unpow-prod-down 54.4%

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

      4. sqrt-pow1 47.7%

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

      5. metadata-eval 47.7%

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

      6. pow1 47.7%

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

      7. associate-/r* 47.7%

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

      8. div-inv 47.7%

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

      9. metadata-eval 47.7%

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

      10. unpow-prod-down 54.3%

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

      11. sqrt-pow1 67.5%

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

      12. metadata-eval 67.5%

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

      13. pow1 67.5%

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

      14. associate-/r* 67.5%

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

      15. div-inv 67.5%

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

      16. metadata-eval 67.5%

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

   10. Applied egg-rr 67.5%

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

       1. clear-num 67.5%

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

       2. un-div-inv 67.5%

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

   12. Applied egg-rr 67.5%

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

   if -2.6000000000000002e127 < h < -4.999999999999985e-310

   1. Initial program 73.1%

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

   3. Simplified 74.3%

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

      1. frac-2neg 72.8%

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

      2. sqrt-div 81.2%

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

   6. Applied egg-rr 82.7%

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

   7. Taylor expanded in d around -inf 85.9%

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

   if -4.999999999999985e-310 < h

   1. Initial program 72.1%

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

   3. Simplified 71.3%

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

      1. *-commutative 71.3%

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

      2. sqrt-div 74.5%

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

      3. sqrt-div 80.8%

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

      4. frac-times 80.8%

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

      5. add-sqr-sqrt 80.9%

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

   6. Applied egg-rr 80.9%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -2.6 \cdot 10^{+127}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + h \cdot \left(\frac{-0.5}{\ell} \cdot \left(\left(D \cdot \frac{M \cdot 0.5}{d}\right) \cdot \frac{D}{\frac{d}{M \cdot 0.5}}\right)\right)\right)\right)\\ \mathbf{elif}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right) + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := D \cdot \frac{M\_m \cdot 0.5}{d}\\ \mathbf{if}\;h \leq -8.6 \cdot 10^{+124}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + h \cdot \left(\frac{-0.5}{\ell} \cdot \left(t\_0 \cdot \frac{D}{\frac{d}{M\_m \cdot 0.5}}\right)\right)\right)\right)\\ \mathbf{elif}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(0.5 \cdot \left({\left(\frac{M\_m}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right) + -1\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{1 + {t\_0}^{2} \cdot \left(h \cdot \frac{-0.5}{\ell}\right)}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

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
 (let* ((t_0 (* D (/ (* M_m 0.5) d))))
   (if (<= h -8.6e+124)
     (*
      (sqrt (/ d l))
      (*
       (sqrt (/ d h))
       (+ 1.0 (* h (* (/ -0.5 l) (* t_0 (/ D (/ d (* M_m 0.5)))))))))
     (if (<= h -5e-310)
       (*
        (* d (sqrt (/ 1.0 (* l h))))
        (+ (* 0.5 (* (pow (* (/ M_m 2.0) (/ D d)) 2.0) (/ h l))) -1.0))
       (*
        d
        (/
         (+ 1.0 (* (pow t_0 2.0) (* h (/ -0.5 l))))
         (* (sqrt l) (sqrt 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 t_0 = D * ((M_m * 0.5) / d);
	double tmp;
	if (h <= -8.6e+124) {
		tmp = sqrt((d / l)) * (sqrt((d / h)) * (1.0 + (h * ((-0.5 / l) * (t_0 * (D / (d / (M_m * 0.5))))))));
	} else if (h <= -5e-310) {
		tmp = (d * sqrt((1.0 / (l * h)))) * ((0.5 * (pow(((M_m / 2.0) * (D / d)), 2.0) * (h / l))) + -1.0);
	} else {
		tmp = d * ((1.0 + (pow(t_0, 2.0) * (h * (-0.5 / l)))) / (sqrt(l) * sqrt(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) :: t_0
    real(8) :: tmp
    t_0 = d_1 * ((m_m * 0.5d0) / d)
    if (h <= (-8.6d+124)) then
        tmp = sqrt((d / l)) * (sqrt((d / h)) * (1.0d0 + (h * (((-0.5d0) / l) * (t_0 * (d_1 / (d / (m_m * 0.5d0))))))))
    else if (h <= (-5d-310)) then
        tmp = (d * sqrt((1.0d0 / (l * h)))) * ((0.5d0 * ((((m_m / 2.0d0) * (d_1 / d)) ** 2.0d0) * (h / l))) + (-1.0d0))
    else
        tmp = d * ((1.0d0 + ((t_0 ** 2.0d0) * (h * ((-0.5d0) / l)))) / (sqrt(l) * sqrt(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 t_0 = D * ((M_m * 0.5) / d);
	double tmp;
	if (h <= -8.6e+124) {
		tmp = Math.sqrt((d / l)) * (Math.sqrt((d / h)) * (1.0 + (h * ((-0.5 / l) * (t_0 * (D / (d / (M_m * 0.5))))))));
	} else if (h <= -5e-310) {
		tmp = (d * Math.sqrt((1.0 / (l * h)))) * ((0.5 * (Math.pow(((M_m / 2.0) * (D / d)), 2.0) * (h / l))) + -1.0);
	} else {
		tmp = d * ((1.0 + (Math.pow(t_0, 2.0) * (h * (-0.5 / l)))) / (Math.sqrt(l) * Math.sqrt(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):
	t_0 = D * ((M_m * 0.5) / d)
	tmp = 0
	if h <= -8.6e+124:
		tmp = math.sqrt((d / l)) * (math.sqrt((d / h)) * (1.0 + (h * ((-0.5 / l) * (t_0 * (D / (d / (M_m * 0.5))))))))
	elif h <= -5e-310:
		tmp = (d * math.sqrt((1.0 / (l * h)))) * ((0.5 * (math.pow(((M_m / 2.0) * (D / d)), 2.0) * (h / l))) + -1.0)
	else:
		tmp = d * ((1.0 + (math.pow(t_0, 2.0) * (h * (-0.5 / l)))) / (math.sqrt(l) * math.sqrt(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)
	t_0 = Float64(D * Float64(Float64(M_m * 0.5) / d))
	tmp = 0.0
	if (h <= -8.6e+124)
		tmp = Float64(sqrt(Float64(d / l)) * Float64(sqrt(Float64(d / h)) * Float64(1.0 + Float64(h * Float64(Float64(-0.5 / l) * Float64(t_0 * Float64(D / Float64(d / Float64(M_m * 0.5)))))))));
	elseif (h <= -5e-310)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(l * h)))) * Float64(Float64(0.5 * Float64((Float64(Float64(M_m / 2.0) * Float64(D / d)) ^ 2.0) * Float64(h / l))) + -1.0));
	else
		tmp = Float64(d * Float64(Float64(1.0 + Float64((t_0 ^ 2.0) * Float64(h * Float64(-0.5 / l)))) / Float64(sqrt(l) * sqrt(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)
	t_0 = D * ((M_m * 0.5) / d);
	tmp = 0.0;
	if (h <= -8.6e+124)
		tmp = sqrt((d / l)) * (sqrt((d / h)) * (1.0 + (h * ((-0.5 / l) * (t_0 * (D / (d / (M_m * 0.5))))))));
	elseif (h <= -5e-310)
		tmp = (d * sqrt((1.0 / (l * h)))) * ((0.5 * ((((M_m / 2.0) * (D / d)) ^ 2.0) * (h / l))) + -1.0);
	else
		tmp = d * ((1.0 + ((t_0 ^ 2.0) * (h * (-0.5 / l)))) / (sqrt(l) * sqrt(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_] := Block[{t$95$0 = N[(D * N[(N[(M$95$m * 0.5), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[h, -8.6e+124], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(h * N[(N[(-0.5 / l), $MachinePrecision] * N[(t$95$0 * N[(D / N[(d / N[(M$95$m * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, -5e-310], N[(N[(d * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 * N[(N[Power[N[(N[(M$95$m / 2.0), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[(1.0 + N[(N[Power[t$95$0, 2.0], $MachinePrecision] * N[(h * N[(-0.5 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if h < -8.6e124

   1. Initial program 60.2%

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

   3. Simplified 58.1%

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

      1. associate-*l/ 60.8%

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

      2. *-commutative 60.8%

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

      3. associate-*r/ 60.8%

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

      4. *-un-lft-identity 60.8%

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

      5. times-frac 60.8%

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

      6. associate-/l/ 60.8%

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

      7. *-commutative 60.8%

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

      8. times-frac 60.8%

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

      9. *-un-lft-identity 60.8%

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

      10. *-commutative 60.8%

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

      11. associate-/l* 60.8%

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

      12. *-un-lft-identity 60.8%

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

      13. times-frac 60.8%

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

      14. metadata-eval 60.8%

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

   6. Applied egg-rr 60.8%

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

      1. associate-/l* 69.7%

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

      2. *-commutative 69.7%

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

      3. associate-/l* 69.7%

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

      4. *-commutative 69.7%

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

      5. metadata-eval 69.7%

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

      6. times-frac 69.7%

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

      7. *-rgt-identity 69.7%

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

      8. associate-/l* 69.7%

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

      9. *-commutative 69.7%

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

      10. associate-/l* 67.5%

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

      11. *-commutative 67.5%

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

   8. Simplified 67.5%

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

      1. unpow-prod-down 54.4%

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

      2. add-sqr-sqrt 54.4%

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

      3. unpow-prod-down 54.4%

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

      4. sqrt-pow1 47.7%

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

      5. metadata-eval 47.7%

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

      6. pow1 47.7%

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

      7. associate-/r* 47.7%

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

      8. div-inv 47.7%

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

      9. metadata-eval 47.7%

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

      10. unpow-prod-down 54.3%

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

      11. sqrt-pow1 67.5%

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

      12. metadata-eval 67.5%

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

      13. pow1 67.5%

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

      14. associate-/r* 67.5%

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

      15. div-inv 67.5%

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

      16. metadata-eval 67.5%

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

   10. Applied egg-rr 67.5%

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

       1. clear-num 67.5%

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

       2. un-div-inv 67.5%

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

   12. Applied egg-rr 67.5%

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

   if -8.6e124 < h < -4.999999999999985e-310

   1. Initial program 73.1%

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

   3. Simplified 74.3%

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

      1. frac-2neg 72.8%

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

      2. sqrt-div 81.2%

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

   6. Applied egg-rr 82.7%

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

   7. Taylor expanded in d around -inf 85.9%

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

   if -4.999999999999985e-310 < h

   1. Initial program 72.1%

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

   3. Simplified 71.3%

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

      1. pow1 71.3%

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

   6. Applied egg-rr 80.9%

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

      1. unpow1 80.9%

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

      2. associate-*l/ 83.3%

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

      3. associate-/l* 83.3%

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

      4. *-commutative 83.3%

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

      5. metadata-eval 83.3%

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

      6. times-frac 83.3%

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

      7. *-rgt-identity 83.3%

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

      8. associate-/l* 84.5%

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

      9. *-commutative 84.5%

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

      10. associate-/l* 82.9%

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

      11. *-commutative 82.9%

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

      12. *-commutative 82.9%

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

   8. Simplified 82.9%

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

      1. fma-undefine 82.9%

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

      2. add-sqr-sqrt 82.9%

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

      3. pow2 82.9%

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

      4. sqrt-pow1 82.9%

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

      5. metadata-eval 82.9%

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

      6. pow1 82.9%

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

      7. associate-/r* 82.9%

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

      8. div-inv 82.9%

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

      9. metadata-eval 82.9%

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

      10. associate-/l* 82.9%

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

   10. Applied egg-rr 82.9%

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

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -8.6 \cdot 10^{+124}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + h \cdot \left(\frac{-0.5}{\ell} \cdot \left(\left(D \cdot \frac{M \cdot 0.5}{d}\right) \cdot \frac{D}{\frac{d}{M \cdot 0.5}}\right)\right)\right)\right)\\ \mathbf{elif}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right) + -1\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{1 + {\left(D \cdot \frac{M \cdot 0.5}{d}\right)}^{2} \cdot \left(h \cdot \frac{-0.5}{\ell}\right)}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := \sqrt{\frac{d}{h}}\\ \mathbf{if}\;h \leq -3.5 \cdot 10^{+129}:\\ \;\;\;\;t\_0 \cdot \left(t\_1 \cdot \left(1 + h \cdot \left(\frac{-0.5}{\ell} \cdot \left(\left(D \cdot \frac{M\_m \cdot 0.5}{d}\right) \cdot \frac{D}{\frac{d}{M\_m \cdot 0.5}}\right)\right)\right)\right)\\ \mathbf{elif}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(0.5 \cdot \left({\left(\frac{M\_m}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right) + -1\right)\\ \mathbf{elif}\;h \leq 1.9 \cdot 10^{-46}:\\ \;\;\;\;\left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right) \cdot \left(1 - 0.5 \cdot \frac{h \cdot {\left(M\_m \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(t\_1 \cdot \left(1 + h \cdot \left(\left(0.5 \cdot \left(D \cdot M\_m\right)\right) \cdot \frac{\frac{-0.5}{\ell} \cdot \left(\frac{0.5}{d} \cdot \left(D \cdot M\_m\right)\right)}{d}\right)\right)\right)\\ \end{array} \end{array} \]

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 (sqrt (/ d l))) (t_1 (sqrt (/ d h))))
   (if (<= h -3.5e+129)
     (*
      t_0
      (*
       t_1
       (+
        1.0
        (*
         h
         (* (/ -0.5 l) (* (* D (/ (* M_m 0.5) d)) (/ D (/ d (* M_m 0.5)))))))))
     (if (<= h -5e-310)
       (*
        (* d (sqrt (/ 1.0 (* l h))))
        (+ (* 0.5 (* (pow (* (/ M_m 2.0) (/ D d)) 2.0) (/ h l))) -1.0))
       (if (<= h 1.9e-46)
         (*
          (* d (pow (* l h) -0.5))
          (- 1.0 (* 0.5 (/ (* h (pow (* M_m (* 0.5 (/ D d))) 2.0)) l))))
         (*
          t_0
          (*
           t_1
           (+
            1.0
            (*
             h
             (*
              (* 0.5 (* D M_m))
              (/ (* (/ -0.5 l) (* (/ 0.5 d) (* D M_m))) d)))))))))))

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 / h));
	double tmp;
	if (h <= -3.5e+129) {
		tmp = t_0 * (t_1 * (1.0 + (h * ((-0.5 / l) * ((D * ((M_m * 0.5) / d)) * (D / (d / (M_m * 0.5))))))));
	} else if (h <= -5e-310) {
		tmp = (d * sqrt((1.0 / (l * h)))) * ((0.5 * (pow(((M_m / 2.0) * (D / d)), 2.0) * (h / l))) + -1.0);
	} else if (h <= 1.9e-46) {
		tmp = (d * pow((l * h), -0.5)) * (1.0 - (0.5 * ((h * pow((M_m * (0.5 * (D / d))), 2.0)) / l)));
	} else {
		tmp = t_0 * (t_1 * (1.0 + (h * ((0.5 * (D * M_m)) * (((-0.5 / l) * ((0.5 / d) * (D * M_m))) / d)))));
	}
	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 = sqrt((d / l))
    t_1 = sqrt((d / h))
    if (h <= (-3.5d+129)) then
        tmp = t_0 * (t_1 * (1.0d0 + (h * (((-0.5d0) / l) * ((d_1 * ((m_m * 0.5d0) / d)) * (d_1 / (d / (m_m * 0.5d0))))))))
    else if (h <= (-5d-310)) then
        tmp = (d * sqrt((1.0d0 / (l * h)))) * ((0.5d0 * ((((m_m / 2.0d0) * (d_1 / d)) ** 2.0d0) * (h / l))) + (-1.0d0))
    else if (h <= 1.9d-46) then
        tmp = (d * ((l * h) ** (-0.5d0))) * (1.0d0 - (0.5d0 * ((h * ((m_m * (0.5d0 * (d_1 / d))) ** 2.0d0)) / l)))
    else
        tmp = t_0 * (t_1 * (1.0d0 + (h * ((0.5d0 * (d_1 * m_m)) * ((((-0.5d0) / l) * ((0.5d0 / d) * (d_1 * m_m))) / d)))))
    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 / h));
	double tmp;
	if (h <= -3.5e+129) {
		tmp = t_0 * (t_1 * (1.0 + (h * ((-0.5 / l) * ((D * ((M_m * 0.5) / d)) * (D / (d / (M_m * 0.5))))))));
	} else if (h <= -5e-310) {
		tmp = (d * Math.sqrt((1.0 / (l * h)))) * ((0.5 * (Math.pow(((M_m / 2.0) * (D / d)), 2.0) * (h / l))) + -1.0);
	} else if (h <= 1.9e-46) {
		tmp = (d * Math.pow((l * h), -0.5)) * (1.0 - (0.5 * ((h * Math.pow((M_m * (0.5 * (D / d))), 2.0)) / l)));
	} else {
		tmp = t_0 * (t_1 * (1.0 + (h * ((0.5 * (D * M_m)) * (((-0.5 / l) * ((0.5 / d) * (D * M_m))) / d)))));
	}
	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 / h))
	tmp = 0
	if h <= -3.5e+129:
		tmp = t_0 * (t_1 * (1.0 + (h * ((-0.5 / l) * ((D * ((M_m * 0.5) / d)) * (D / (d / (M_m * 0.5))))))))
	elif h <= -5e-310:
		tmp = (d * math.sqrt((1.0 / (l * h)))) * ((0.5 * (math.pow(((M_m / 2.0) * (D / d)), 2.0) * (h / l))) + -1.0)
	elif h <= 1.9e-46:
		tmp = (d * math.pow((l * h), -0.5)) * (1.0 - (0.5 * ((h * math.pow((M_m * (0.5 * (D / d))), 2.0)) / l)))
	else:
		tmp = t_0 * (t_1 * (1.0 + (h * ((0.5 * (D * M_m)) * (((-0.5 / l) * ((0.5 / d) * (D * M_m))) / d)))))
	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 / h))
	tmp = 0.0
	if (h <= -3.5e+129)
		tmp = Float64(t_0 * Float64(t_1 * Float64(1.0 + Float64(h * Float64(Float64(-0.5 / l) * Float64(Float64(D * Float64(Float64(M_m * 0.5) / d)) * Float64(D / Float64(d / Float64(M_m * 0.5)))))))));
	elseif (h <= -5e-310)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(l * h)))) * Float64(Float64(0.5 * Float64((Float64(Float64(M_m / 2.0) * Float64(D / d)) ^ 2.0) * Float64(h / l))) + -1.0));
	elseif (h <= 1.9e-46)
		tmp = Float64(Float64(d * (Float64(l * h) ^ -0.5)) * Float64(1.0 - Float64(0.5 * Float64(Float64(h * (Float64(M_m * Float64(0.5 * Float64(D / d))) ^ 2.0)) / l))));
	else
		tmp = Float64(t_0 * Float64(t_1 * Float64(1.0 + Float64(h * Float64(Float64(0.5 * Float64(D * M_m)) * Float64(Float64(Float64(-0.5 / l) * Float64(Float64(0.5 / d) * Float64(D * M_m))) / d))))));
	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 / h));
	tmp = 0.0;
	if (h <= -3.5e+129)
		tmp = t_0 * (t_1 * (1.0 + (h * ((-0.5 / l) * ((D * ((M_m * 0.5) / d)) * (D / (d / (M_m * 0.5))))))));
	elseif (h <= -5e-310)
		tmp = (d * sqrt((1.0 / (l * h)))) * ((0.5 * ((((M_m / 2.0) * (D / d)) ^ 2.0) * (h / l))) + -1.0);
	elseif (h <= 1.9e-46)
		tmp = (d * ((l * h) ^ -0.5)) * (1.0 - (0.5 * ((h * ((M_m * (0.5 * (D / d))) ^ 2.0)) / l)));
	else
		tmp = t_0 * (t_1 * (1.0 + (h * ((0.5 * (D * M_m)) * (((-0.5 / l) * ((0.5 / d) * (D * M_m))) / d)))));
	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[N[(d / h), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[h, -3.5e+129], N[(t$95$0 * N[(t$95$1 * N[(1.0 + N[(h * N[(N[(-0.5 / l), $MachinePrecision] * N[(N[(D * N[(N[(M$95$m * 0.5), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * N[(D / N[(d / N[(M$95$m * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, -5e-310], N[(N[(d * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 * N[(N[Power[N[(N[(M$95$m / 2.0), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, 1.9e-46], N[(N[(d * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(h * N[Power[N[(M$95$m * N[(0.5 * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(t$95$1 * N[(1.0 + N[(h * N[(N[(0.5 * N[(D * M$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(-0.5 / l), $MachinePrecision] * N[(N[(0.5 / d), $MachinePrecision] * N[(D * M$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if h < -3.4999999999999998e129

   1. Initial program 60.2%

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

   3. Simplified 58.1%

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

      1. associate-*l/ 60.8%

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

      2. *-commutative 60.8%

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

      3. associate-*r/ 60.8%

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

      4. *-un-lft-identity 60.8%

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

      5. times-frac 60.8%

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

      6. associate-/l/ 60.8%

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

      7. *-commutative 60.8%

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

      8. times-frac 60.8%

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

      9. *-un-lft-identity 60.8%

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

      10. *-commutative 60.8%

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

      11. associate-/l* 60.8%

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

      12. *-un-lft-identity 60.8%

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

      13. times-frac 60.8%

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

      14. metadata-eval 60.8%

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

   6. Applied egg-rr 60.8%

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

      1. associate-/l* 69.7%

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

      2. *-commutative 69.7%

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

      3. associate-/l* 69.7%

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

      4. *-commutative 69.7%

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

      5. metadata-eval 69.7%

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

      6. times-frac 69.7%

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

      7. *-rgt-identity 69.7%

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

      8. associate-/l* 69.7%

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

      9. *-commutative 69.7%

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

      10. associate-/l* 67.5%

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

      11. *-commutative 67.5%

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

   8. Simplified 67.5%

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

      1. unpow-prod-down 54.4%

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

      2. add-sqr-sqrt 54.4%

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

      3. unpow-prod-down 54.4%

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

      4. sqrt-pow1 47.7%

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

      5. metadata-eval 47.7%

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

      6. pow1 47.7%

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

      7. associate-/r* 47.7%

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

      8. div-inv 47.7%

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

      9. metadata-eval 47.7%

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

      10. unpow-prod-down 54.3%

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

      11. sqrt-pow1 67.5%

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

      12. metadata-eval 67.5%

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

      13. pow1 67.5%

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

      14. associate-/r* 67.5%

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

      15. div-inv 67.5%

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

      16. metadata-eval 67.5%

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

   10. Applied egg-rr 67.5%

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

       1. clear-num 67.5%

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

       2. un-div-inv 67.5%

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

   12. Applied egg-rr 67.5%

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

   if -3.4999999999999998e129 < h < -4.999999999999985e-310

   1. Initial program 73.1%

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

   3. Simplified 74.3%

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

      1. frac-2neg 72.8%

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

      2. sqrt-div 81.2%

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

   6. Applied egg-rr 82.7%

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

   7. Taylor expanded in d around -inf 85.9%

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

   if -4.999999999999985e-310 < h < 1.8999999999999998e-46

   1. Initial program 72.0%

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

   3. Simplified 72.0%

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

      1. *-commutative 72.0%

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

      2. sqrt-div 77.3%

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

      3. sqrt-div 85.6%

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

      4. frac-times 85.7%

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

      5. add-sqr-sqrt 85.8%

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

   6. Applied egg-rr 85.8%

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

      1. div-inv 85.7%

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

      2. *-commutative 85.7%

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

      3. metadata-eval 85.7%

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

      4. sqrt-unprod 80.5%

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

      5. sqrt-div 80.6%

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

      6. *-commutative 80.6%

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

      7. inv-pow 80.6%

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

      8. sqrt-pow1 80.6%

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

      9. metadata-eval 80.6%

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

   8. Applied egg-rr 80.6%

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

      1. associate-*r/ 84.4%

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

      2. frac-times 84.4%

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

      3. associate-/l* 84.4%

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

      4. *-un-lft-identity 84.4%

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

      5. times-frac 84.4%

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

      6. metadata-eval 84.4%

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

   10. Applied egg-rr 84.4%

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

   if 1.8999999999999998e-46 < h

   1. Initial program 72.1%

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

   3. Simplified 69.0%

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

      1. associate-*l/ 71.8%

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

      2. *-commutative 71.8%

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

      3. associate-*r/ 74.9%

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

      4. *-un-lft-identity 74.9%

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

      5. times-frac 71.8%

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

      6. associate-/l/ 71.8%

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

      7. *-commutative 71.8%

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

      8. times-frac 74.9%

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

      9. *-un-lft-identity 74.9%

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

      10. *-commutative 74.9%

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

      11. associate-/l* 73.4%

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

      12. *-un-lft-identity 73.4%

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

      13. times-frac 73.4%

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

      14. metadata-eval 73.4%

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

   6. Applied egg-rr 73.4%

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

      1. associate-/l* 75.1%

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

      2. *-commutative 75.1%

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

      3. associate-/l* 75.1%

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

      4. *-commutative 75.1%

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

      5. metadata-eval 75.1%

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

      6. times-frac 75.1%

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

      7. *-rgt-identity 75.1%

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

      8. associate-/l* 76.6%

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

      9. *-commutative 76.6%

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

      10. associate-/l* 73.5%

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

      11. *-commutative 73.5%

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

   8. Simplified 73.5%

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

      1. unpow-prod-down 61.2%

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

      2. add-sqr-sqrt 61.2%

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

      3. unpow-prod-down 61.2%

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

      4. sqrt-pow1 50.3%

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

      5. metadata-eval 50.3%

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

      6. pow1 50.3%

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

      7. associate-/r* 50.3%

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

      8. div-inv 50.3%

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

      9. metadata-eval 50.3%

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

      10. unpow-prod-down 59.7%

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

      11. sqrt-pow1 73.5%

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

      12. metadata-eval 73.5%

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

      13. pow1 73.5%

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

      14. associate-/r* 73.5%

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

      15. div-inv 73.5%

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

      16. metadata-eval 73.5%

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

   10. Applied egg-rr 73.5%

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

       1. associate-*l* 84.8%

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

       2. associate-*r/ 84.9%

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

       3. associate-*l/ 84.9%

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

       4. associate-*l* 82.0%

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

       5. associate-/l* 82.0%

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

   12. Applied egg-rr 74.8%

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

       1. associate-/l* 73.3%

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

       2. associate-*r* 73.3%

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

       3. associate-*r* 74.8%

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

       4. associate-*r* 77.8%

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

   14. Simplified 77.8%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -3.5 \cdot 10^{+129}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + h \cdot \left(\frac{-0.5}{\ell} \cdot \left(\left(D \cdot \frac{M \cdot 0.5}{d}\right) \cdot \frac{D}{\frac{d}{M \cdot 0.5}}\right)\right)\right)\right)\\ \mathbf{elif}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right) + -1\right)\\ \mathbf{elif}\;h \leq 1.9 \cdot 10^{-46}:\\ \;\;\;\;\left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right) \cdot \left(1 - 0.5 \cdot \frac{h \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + h \cdot \left(\left(0.5 \cdot \left(D \cdot M\right)\right) \cdot \frac{\frac{-0.5}{\ell} \cdot \left(\frac{0.5}{d} \cdot \left(D \cdot M\right)\right)}{d}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 73.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := D \cdot \frac{M\_m \cdot 0.5}{d}\\ t_1 := \sqrt{\frac{d}{h}}\\ t_2 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;h \leq -2.3 \cdot 10^{+119}:\\ \;\;\;\;\left(t\_1 \cdot \left(1 + h \cdot \left(\left(t\_0 \cdot t\_0\right) \cdot \frac{-0.5}{\ell}\right)\right)\right) \cdot t\_2\\ \mathbf{elif}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(0.5 \cdot \left({\left(\frac{M\_m}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right) + -1\right)\\ \mathbf{elif}\;h \leq 1.75 \cdot 10^{-46}:\\ \;\;\;\;\left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right) \cdot \left(1 - 0.5 \cdot \frac{h \cdot {\left(M\_m \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2 \cdot \left(t\_1 \cdot \left(1 + h \cdot \left(\left(0.5 \cdot \left(D \cdot M\_m\right)\right) \cdot \frac{\frac{-0.5}{\ell} \cdot \left(\frac{0.5}{d} \cdot \left(D \cdot M\_m\right)\right)}{d}\right)\right)\right)\\ \end{array} \end{array} \]

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 (/ (* M_m 0.5) d)))
        (t_1 (sqrt (/ d h)))
        (t_2 (sqrt (/ d l))))
   (if (<= h -2.3e+119)
     (* (* t_1 (+ 1.0 (* h (* (* t_0 t_0) (/ -0.5 l))))) t_2)
     (if (<= h -5e-310)
       (*
        (* d (sqrt (/ 1.0 (* l h))))
        (+ (* 0.5 (* (pow (* (/ M_m 2.0) (/ D d)) 2.0) (/ h l))) -1.0))
       (if (<= h 1.75e-46)
         (*
          (* d (pow (* l h) -0.5))
          (- 1.0 (* 0.5 (/ (* h (pow (* M_m (* 0.5 (/ D d))) 2.0)) l))))
         (*
          t_2
          (*
           t_1
           (+
            1.0
            (*
             h
             (*
              (* 0.5 (* D M_m))
              (/ (* (/ -0.5 l) (* (/ 0.5 d) (* D M_m))) d)))))))))))

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 * ((M_m * 0.5) / d);
	double t_1 = sqrt((d / h));
	double t_2 = sqrt((d / l));
	double tmp;
	if (h <= -2.3e+119) {
		tmp = (t_1 * (1.0 + (h * ((t_0 * t_0) * (-0.5 / l))))) * t_2;
	} else if (h <= -5e-310) {
		tmp = (d * sqrt((1.0 / (l * h)))) * ((0.5 * (pow(((M_m / 2.0) * (D / d)), 2.0) * (h / l))) + -1.0);
	} else if (h <= 1.75e-46) {
		tmp = (d * pow((l * h), -0.5)) * (1.0 - (0.5 * ((h * pow((M_m * (0.5 * (D / d))), 2.0)) / l)));
	} else {
		tmp = t_2 * (t_1 * (1.0 + (h * ((0.5 * (D * M_m)) * (((-0.5 / l) * ((0.5 / d) * (D * M_m))) / d)))));
	}
	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 = d_1 * ((m_m * 0.5d0) / d)
    t_1 = sqrt((d / h))
    t_2 = sqrt((d / l))
    if (h <= (-2.3d+119)) then
        tmp = (t_1 * (1.0d0 + (h * ((t_0 * t_0) * ((-0.5d0) / l))))) * t_2
    else if (h <= (-5d-310)) then
        tmp = (d * sqrt((1.0d0 / (l * h)))) * ((0.5d0 * ((((m_m / 2.0d0) * (d_1 / d)) ** 2.0d0) * (h / l))) + (-1.0d0))
    else if (h <= 1.75d-46) then
        tmp = (d * ((l * h) ** (-0.5d0))) * (1.0d0 - (0.5d0 * ((h * ((m_m * (0.5d0 * (d_1 / d))) ** 2.0d0)) / l)))
    else
        tmp = t_2 * (t_1 * (1.0d0 + (h * ((0.5d0 * (d_1 * m_m)) * ((((-0.5d0) / l) * ((0.5d0 / d) * (d_1 * m_m))) / d)))))
    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 * ((M_m * 0.5) / d);
	double t_1 = Math.sqrt((d / h));
	double t_2 = Math.sqrt((d / l));
	double tmp;
	if (h <= -2.3e+119) {
		tmp = (t_1 * (1.0 + (h * ((t_0 * t_0) * (-0.5 / l))))) * t_2;
	} else if (h <= -5e-310) {
		tmp = (d * Math.sqrt((1.0 / (l * h)))) * ((0.5 * (Math.pow(((M_m / 2.0) * (D / d)), 2.0) * (h / l))) + -1.0);
	} else if (h <= 1.75e-46) {
		tmp = (d * Math.pow((l * h), -0.5)) * (1.0 - (0.5 * ((h * Math.pow((M_m * (0.5 * (D / d))), 2.0)) / l)));
	} else {
		tmp = t_2 * (t_1 * (1.0 + (h * ((0.5 * (D * M_m)) * (((-0.5 / l) * ((0.5 / d) * (D * M_m))) / d)))));
	}
	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 * ((M_m * 0.5) / d)
	t_1 = math.sqrt((d / h))
	t_2 = math.sqrt((d / l))
	tmp = 0
	if h <= -2.3e+119:
		tmp = (t_1 * (1.0 + (h * ((t_0 * t_0) * (-0.5 / l))))) * t_2
	elif h <= -5e-310:
		tmp = (d * math.sqrt((1.0 / (l * h)))) * ((0.5 * (math.pow(((M_m / 2.0) * (D / d)), 2.0) * (h / l))) + -1.0)
	elif h <= 1.75e-46:
		tmp = (d * math.pow((l * h), -0.5)) * (1.0 - (0.5 * ((h * math.pow((M_m * (0.5 * (D / d))), 2.0)) / l)))
	else:
		tmp = t_2 * (t_1 * (1.0 + (h * ((0.5 * (D * M_m)) * (((-0.5 / l) * ((0.5 / d) * (D * M_m))) / d)))))
	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 * Float64(Float64(M_m * 0.5) / d))
	t_1 = sqrt(Float64(d / h))
	t_2 = sqrt(Float64(d / l))
	tmp = 0.0
	if (h <= -2.3e+119)
		tmp = Float64(Float64(t_1 * Float64(1.0 + Float64(h * Float64(Float64(t_0 * t_0) * Float64(-0.5 / l))))) * t_2);
	elseif (h <= -5e-310)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(l * h)))) * Float64(Float64(0.5 * Float64((Float64(Float64(M_m / 2.0) * Float64(D / d)) ^ 2.0) * Float64(h / l))) + -1.0));
	elseif (h <= 1.75e-46)
		tmp = Float64(Float64(d * (Float64(l * h) ^ -0.5)) * Float64(1.0 - Float64(0.5 * Float64(Float64(h * (Float64(M_m * Float64(0.5 * Float64(D / d))) ^ 2.0)) / l))));
	else
		tmp = Float64(t_2 * Float64(t_1 * Float64(1.0 + Float64(h * Float64(Float64(0.5 * Float64(D * M_m)) * Float64(Float64(Float64(-0.5 / l) * Float64(Float64(0.5 / d) * Float64(D * M_m))) / d))))));
	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 * ((M_m * 0.5) / d);
	t_1 = sqrt((d / h));
	t_2 = sqrt((d / l));
	tmp = 0.0;
	if (h <= -2.3e+119)
		tmp = (t_1 * (1.0 + (h * ((t_0 * t_0) * (-0.5 / l))))) * t_2;
	elseif (h <= -5e-310)
		tmp = (d * sqrt((1.0 / (l * h)))) * ((0.5 * ((((M_m / 2.0) * (D / d)) ^ 2.0) * (h / l))) + -1.0);
	elseif (h <= 1.75e-46)
		tmp = (d * ((l * h) ^ -0.5)) * (1.0 - (0.5 * ((h * ((M_m * (0.5 * (D / d))) ^ 2.0)) / l)));
	else
		tmp = t_2 * (t_1 * (1.0 + (h * ((0.5 * (D * M_m)) * (((-0.5 / l) * ((0.5 / d) * (D * M_m))) / d)))));
	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[(N[(M$95$m * 0.5), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[h, -2.3e+119], N[(N[(t$95$1 * N[(1.0 + N[(h * N[(N[(t$95$0 * t$95$0), $MachinePrecision] * N[(-0.5 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[h, -5e-310], N[(N[(d * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 * N[(N[Power[N[(N[(M$95$m / 2.0), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, 1.75e-46], N[(N[(d * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(h * N[Power[N[(M$95$m * N[(0.5 * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$2 * N[(t$95$1 * N[(1.0 + N[(h * N[(N[(0.5 * N[(D * M$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(-0.5 / l), $MachinePrecision] * N[(N[(0.5 / d), $MachinePrecision] * N[(D * M$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if h < -2.3000000000000001e119

   1. Initial program 60.2%

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

   3. Simplified 58.1%

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

      1. associate-*l/ 60.8%

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

      2. *-commutative 60.8%

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

      3. associate-*r/ 60.8%

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

      4. *-un-lft-identity 60.8%

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

      5. times-frac 60.8%

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

      6. associate-/l/ 60.8%

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

      7. *-commutative 60.8%

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

      8. times-frac 60.8%

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

      9. *-un-lft-identity 60.8%

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

      10. *-commutative 60.8%

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

      11. associate-/l* 60.8%

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

      12. *-un-lft-identity 60.8%

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

      13. times-frac 60.8%

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

      14. metadata-eval 60.8%

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

   6. Applied egg-rr 60.8%

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

      1. associate-/l* 69.7%

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

      2. *-commutative 69.7%

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

      3. associate-/l* 69.7%

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

      4. *-commutative 69.7%

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

      5. metadata-eval 69.7%

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

      6. times-frac 69.7%

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

      7. *-rgt-identity 69.7%

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

      8. associate-/l* 69.7%

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

      9. *-commutative 69.7%

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

      10. associate-/l* 67.5%

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

      11. *-commutative 67.5%

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

   8. Simplified 67.5%

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

      1. unpow-prod-down 54.4%

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

      2. add-sqr-sqrt 54.4%

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

      3. unpow-prod-down 54.4%

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

      4. sqrt-pow1 47.7%

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

      5. metadata-eval 47.7%

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

      6. pow1 47.7%

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

      7. associate-/r* 47.7%

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

      8. div-inv 47.7%

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

      9. metadata-eval 47.7%

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

      10. unpow-prod-down 54.3%

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

      11. sqrt-pow1 67.5%

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

      12. metadata-eval 67.5%

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

      13. pow1 67.5%

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

      14. associate-/r* 67.5%

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

      15. div-inv 67.5%

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

      16. metadata-eval 67.5%

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

   10. Applied egg-rr 67.5%

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

   if -2.3000000000000001e119 < h < -4.999999999999985e-310

   1. Initial program 73.1%

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

   3. Simplified 74.3%

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

      1. frac-2neg 72.8%

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

      2. sqrt-div 81.2%

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

   6. Applied egg-rr 82.7%

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

   7. Taylor expanded in d around -inf 85.9%

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

   if -4.999999999999985e-310 < h < 1.7500000000000001e-46

   1. Initial program 72.0%

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

   3. Simplified 72.0%

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

      1. *-commutative 72.0%

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

      2. sqrt-div 77.3%

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

      3. sqrt-div 85.6%

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

      4. frac-times 85.7%

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

      5. add-sqr-sqrt 85.8%

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

   6. Applied egg-rr 85.8%

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

      1. div-inv 85.7%

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

      2. *-commutative 85.7%

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

      3. metadata-eval 85.7%

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

      4. sqrt-unprod 80.5%

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

      5. sqrt-div 80.6%

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

      6. *-commutative 80.6%

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

      7. inv-pow 80.6%

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

      8. sqrt-pow1 80.6%

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

      9. metadata-eval 80.6%

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

   8. Applied egg-rr 80.6%

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

      1. associate-*r/ 84.4%

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

      2. frac-times 84.4%

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

      3. associate-/l* 84.4%

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

      4. *-un-lft-identity 84.4%

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

      5. times-frac 84.4%

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

      6. metadata-eval 84.4%

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

   10. Applied egg-rr 84.4%

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

   if 1.7500000000000001e-46 < h

   1. Initial program 72.1%

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

   3. Simplified 69.0%

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

      1. associate-*l/ 71.8%

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

      2. *-commutative 71.8%

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

      3. associate-*r/ 74.9%

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

      4. *-un-lft-identity 74.9%

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

      5. times-frac 71.8%

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

      6. associate-/l/ 71.8%

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

      7. *-commutative 71.8%

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

      8. times-frac 74.9%

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

      9. *-un-lft-identity 74.9%

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

      10. *-commutative 74.9%

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

      11. associate-/l* 73.4%

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

      12. *-un-lft-identity 73.4%

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

      13. times-frac 73.4%

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

      14. metadata-eval 73.4%

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

   6. Applied egg-rr 73.4%

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

      1. associate-/l* 75.1%

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

      2. *-commutative 75.1%

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

      3. associate-/l* 75.1%

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

      4. *-commutative 75.1%

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

      5. metadata-eval 75.1%

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

      6. times-frac 75.1%

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

      7. *-rgt-identity 75.1%

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

      8. associate-/l* 76.6%

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

      9. *-commutative 76.6%

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

      10. associate-/l* 73.5%

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

      11. *-commutative 73.5%

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

   8. Simplified 73.5%

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

      1. unpow-prod-down 61.2%

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

      2. add-sqr-sqrt 61.2%

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

      3. unpow-prod-down 61.2%

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

      4. sqrt-pow1 50.3%

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

      5. metadata-eval 50.3%

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

      6. pow1 50.3%

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

      7. associate-/r* 50.3%

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

      8. div-inv 50.3%

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

      9. metadata-eval 50.3%

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

      10. unpow-prod-down 59.7%

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

      11. sqrt-pow1 73.5%

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

      12. metadata-eval 73.5%

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

      13. pow1 73.5%

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

      14. associate-/r* 73.5%

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

      15. div-inv 73.5%

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

      16. metadata-eval 73.5%

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

   10. Applied egg-rr 73.5%

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

       1. associate-*l* 84.8%

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

       2. associate-*r/ 84.9%

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

       3. associate-*l/ 84.9%

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

       4. associate-*l* 82.0%

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

       5. associate-/l* 82.0%

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

   12. Applied egg-rr 74.8%

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

       1. associate-/l* 73.3%

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

       2. associate-*r* 73.3%

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

       3. associate-*r* 74.8%

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

       4. associate-*r* 77.8%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + h \cdot \left(\left(\left(D \cdot M\right) \cdot 0.5\right) \cdot \frac{\color{blue}{\left(\left(D \cdot M\right) \cdot \frac{0.5}{d}\right)} \cdot \frac{-0.5}{\ell}}{d}\right)\right)\right) \]

   14. Simplified 77.8%

       \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + h \cdot \color{blue}{\left(\left(\left(D \cdot M\right) \cdot 0.5\right) \cdot \frac{\left(\left(D \cdot M\right) \cdot \frac{0.5}{d}\right) \cdot \frac{-0.5}{\ell}}{d}\right)}\right)\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -2.3 \cdot 10^{+119}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \left(1 + h \cdot \left(\left(\left(D \cdot \frac{M \cdot 0.5}{d}\right) \cdot \left(D \cdot \frac{M \cdot 0.5}{d}\right)\right) \cdot \frac{-0.5}{\ell}\right)\right)\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right) + -1\right)\\ \mathbf{elif}\;h \leq 1.75 \cdot 10^{-46}:\\ \;\;\;\;\left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right) \cdot \left(1 - 0.5 \cdot \frac{h \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + h \cdot \left(\left(0.5 \cdot \left(D \cdot M\right)\right) \cdot \frac{\frac{-0.5}{\ell} \cdot \left(\frac{0.5}{d} \cdot \left(D \cdot M\right)\right)}{d}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 73.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + h \cdot \left(\left(0.5 \cdot \left(D \cdot M\_m\right)\right) \cdot \frac{\frac{-0.5}{\ell} \cdot \left(\frac{0.5}{d} \cdot \left(D \cdot M\_m\right)\right)}{d}\right)\right)\right)\\ \mathbf{if}\;h \leq -1 \cdot 10^{+130}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(0.5 \cdot \left({\left(\frac{M\_m}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right) + -1\right)\\ \mathbf{elif}\;h \leq 2 \cdot 10^{-46}:\\ \;\;\;\;\left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right) \cdot \left(1 - 0.5 \cdot \frac{h \cdot {\left(M\_m \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

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))
          (*
           (sqrt (/ d h))
           (+
            1.0
            (*
             h
             (*
              (* 0.5 (* D M_m))
              (/ (* (/ -0.5 l) (* (/ 0.5 d) (* D M_m))) d))))))))
   (if (<= h -1e+130)
     t_0
     (if (<= h -5e-310)
       (*
        (* d (sqrt (/ 1.0 (* l h))))
        (+ (* 0.5 (* (pow (* (/ M_m 2.0) (/ D d)) 2.0) (/ h l))) -1.0))
       (if (<= h 2e-46)
         (*
          (* d (pow (* l h) -0.5))
          (- 1.0 (* 0.5 (/ (* h (pow (* M_m (* 0.5 (/ 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)) * (sqrt((d / h)) * (1.0 + (h * ((0.5 * (D * M_m)) * (((-0.5 / l) * ((0.5 / d) * (D * M_m))) / d)))));
	double tmp;
	if (h <= -1e+130) {
		tmp = t_0;
	} else if (h <= -5e-310) {
		tmp = (d * sqrt((1.0 / (l * h)))) * ((0.5 * (pow(((M_m / 2.0) * (D / d)), 2.0) * (h / l))) + -1.0);
	} else if (h <= 2e-46) {
		tmp = (d * pow((l * h), -0.5)) * (1.0 - (0.5 * ((h * pow((M_m * (0.5 * (D / d))), 2.0)) / l)));
	} else {
		tmp = 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) :: tmp
    t_0 = sqrt((d / l)) * (sqrt((d / h)) * (1.0d0 + (h * ((0.5d0 * (d_1 * m_m)) * ((((-0.5d0) / l) * ((0.5d0 / d) * (d_1 * m_m))) / d)))))
    if (h <= (-1d+130)) then
        tmp = t_0
    else if (h <= (-5d-310)) then
        tmp = (d * sqrt((1.0d0 / (l * h)))) * ((0.5d0 * ((((m_m / 2.0d0) * (d_1 / d)) ** 2.0d0) * (h / l))) + (-1.0d0))
    else if (h <= 2d-46) then
        tmp = (d * ((l * h) ** (-0.5d0))) * (1.0d0 - (0.5d0 * ((h * ((m_m * (0.5d0 * (d_1 / d))) ** 2.0d0)) / l)))
    else
        tmp = 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)) * (Math.sqrt((d / h)) * (1.0 + (h * ((0.5 * (D * M_m)) * (((-0.5 / l) * ((0.5 / d) * (D * M_m))) / d)))));
	double tmp;
	if (h <= -1e+130) {
		tmp = t_0;
	} else if (h <= -5e-310) {
		tmp = (d * Math.sqrt((1.0 / (l * h)))) * ((0.5 * (Math.pow(((M_m / 2.0) * (D / d)), 2.0) * (h / l))) + -1.0);
	} else if (h <= 2e-46) {
		tmp = (d * Math.pow((l * h), -0.5)) * (1.0 - (0.5 * ((h * Math.pow((M_m * (0.5 * (D / d))), 2.0)) / l)));
	} else {
		tmp = 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)) * (math.sqrt((d / h)) * (1.0 + (h * ((0.5 * (D * M_m)) * (((-0.5 / l) * ((0.5 / d) * (D * M_m))) / d)))))
	tmp = 0
	if h <= -1e+130:
		tmp = t_0
	elif h <= -5e-310:
		tmp = (d * math.sqrt((1.0 / (l * h)))) * ((0.5 * (math.pow(((M_m / 2.0) * (D / d)), 2.0) * (h / l))) + -1.0)
	elif h <= 2e-46:
		tmp = (d * math.pow((l * h), -0.5)) * (1.0 - (0.5 * ((h * math.pow((M_m * (0.5 * (D / d))), 2.0)) / l)))
	else:
		tmp = 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 = Float64(sqrt(Float64(d / l)) * Float64(sqrt(Float64(d / h)) * Float64(1.0 + Float64(h * Float64(Float64(0.5 * Float64(D * M_m)) * Float64(Float64(Float64(-0.5 / l) * Float64(Float64(0.5 / d) * Float64(D * M_m))) / d))))))
	tmp = 0.0
	if (h <= -1e+130)
		tmp = t_0;
	elseif (h <= -5e-310)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(l * h)))) * Float64(Float64(0.5 * Float64((Float64(Float64(M_m / 2.0) * Float64(D / d)) ^ 2.0) * Float64(h / l))) + -1.0));
	elseif (h <= 2e-46)
		tmp = Float64(Float64(d * (Float64(l * h) ^ -0.5)) * Float64(1.0 - Float64(0.5 * Float64(Float64(h * (Float64(M_m * Float64(0.5 * Float64(D / d))) ^ 2.0)) / l))));
	else
		tmp = 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)) * (sqrt((d / h)) * (1.0 + (h * ((0.5 * (D * M_m)) * (((-0.5 / l) * ((0.5 / d) * (D * M_m))) / d)))));
	tmp = 0.0;
	if (h <= -1e+130)
		tmp = t_0;
	elseif (h <= -5e-310)
		tmp = (d * sqrt((1.0 / (l * h)))) * ((0.5 * ((((M_m / 2.0) * (D / d)) ^ 2.0) * (h / l))) + -1.0);
	elseif (h <= 2e-46)
		tmp = (d * ((l * h) ^ -0.5)) * (1.0 - (0.5 * ((h * ((M_m * (0.5 * (D / d))) ^ 2.0)) / l)));
	else
		tmp = 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[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(h * N[(N[(0.5 * N[(D * M$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(-0.5 / l), $MachinePrecision] * N[(N[(0.5 / d), $MachinePrecision] * N[(D * M$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[h, -1e+130], t$95$0, If[LessEqual[h, -5e-310], N[(N[(d * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 * N[(N[Power[N[(N[(M$95$m / 2.0), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, 2e-46], N[(N[(d * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(h * N[Power[N[(M$95$m * N[(0.5 * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if h < -1.0000000000000001e130 or 2.00000000000000005e-46 < h

   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 64.6%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(D \cdot \frac{\frac{M}{2}}{d}\right)}^{2} \cdot -0.5\right)\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*l/ 67.4%

         \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \color{blue}{\frac{h \cdot \left({\left(D \cdot \frac{\frac{M}{2}}{d}\right)}^{2} \cdot -0.5\right)}{\ell}}\right)\right) \]

      2. *-commutative 67.4%

         \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h \cdot \color{blue}{\left(-0.5 \cdot {\left(D \cdot \frac{\frac{M}{2}}{d}\right)}^{2}\right)}}{\ell}\right)\right) \]

      3. associate-*r/ 69.2%

         \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h \cdot \left(-0.5 \cdot {\color{blue}{\left(\frac{D \cdot \frac{M}{2}}{d}\right)}}^{2}\right)}{\ell}\right)\right) \]

      4. *-un-lft-identity 69.2%

         \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h \cdot \left(-0.5 \cdot {\left(\frac{D \cdot \frac{M}{2}}{\color{blue}{1 \cdot d}}\right)}^{2}\right)}{\ell}\right)\right) \]

      5. times-frac 67.4%

         \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h \cdot \left(-0.5 \cdot {\color{blue}{\left(\frac{D}{1} \cdot \frac{\frac{M}{2}}{d}\right)}}^{2}\right)}{\ell}\right)\right) \]

      6. associate-/l/ 67.4%

         \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h \cdot \left(-0.5 \cdot {\left(\frac{D}{1} \cdot \color{blue}{\frac{M}{d \cdot 2}}\right)}^{2}\right)}{\ell}\right)\right) \]

      7. *-commutative 67.4%

         \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h \cdot \left(-0.5 \cdot {\left(\frac{D}{1} \cdot \frac{M}{\color{blue}{2 \cdot d}}\right)}^{2}\right)}{\ell}\right)\right) \]

      8. times-frac 69.2%

         \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h \cdot \left(-0.5 \cdot {\color{blue}{\left(\frac{D \cdot M}{1 \cdot \left(2 \cdot d\right)}\right)}}^{2}\right)}{\ell}\right)\right) \]

      9. *-un-lft-identity 69.2%

         \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h \cdot \left(-0.5 \cdot {\left(\frac{D \cdot M}{\color{blue}{2 \cdot d}}\right)}^{2}\right)}{\ell}\right)\right) \]

      10. *-commutative 69.2%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h \cdot \left(-0.5 \cdot {\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2}\right)}{\ell}\right)\right) \]

      11. associate-/l* 68.3%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h \cdot \left(-0.5 \cdot {\color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)}}^{2}\right)}{\ell}\right)\right) \]

      12. *-un-lft-identity 68.3%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h \cdot \left(-0.5 \cdot {\left(M \cdot \frac{\color{blue}{1 \cdot D}}{2 \cdot d}\right)}^{2}\right)}{\ell}\right)\right) \]

      13. times-frac 68.3%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h \cdot \left(-0.5 \cdot {\left(M \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{D}{d}\right)}\right)}^{2}\right)}{\ell}\right)\right) \]

      14. metadata-eval 68.3%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h \cdot \left(-0.5 \cdot {\left(M \cdot \left(\color{blue}{0.5} \cdot \frac{D}{d}\right)\right)}^{2}\right)}{\ell}\right)\right) \]

   6. Applied egg-rr 68.3%

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \color{blue}{\frac{h \cdot \left(-0.5 \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right)}{\ell}}\right)\right) \]
   7. Step-by-step derivation

      1. associate-/l* 72.9%

         \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \color{blue}{h \cdot \frac{-0.5 \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell}}\right)\right) \]

      2. *-commutative 72.9%

         \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + h \cdot \frac{\color{blue}{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot -0.5}}{\ell}\right)\right) \]

      3. associate-/l* 72.9%

         \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + h \cdot \color{blue}{\left({\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \frac{-0.5}{\ell}\right)}\right)\right) \]

      4. *-commutative 72.9%

         \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + h \cdot \left({\left(M \cdot \color{blue}{\left(\frac{D}{d} \cdot 0.5\right)}\right)}^{2} \cdot \frac{-0.5}{\ell}\right)\right)\right) \]

      5. metadata-eval 72.9%

         \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + h \cdot \left({\left(M \cdot \left(\frac{D}{d} \cdot \color{blue}{\frac{1}{2}}\right)\right)}^{2} \cdot \frac{-0.5}{\ell}\right)\right)\right) \]

      6. times-frac 72.9%

         \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + h \cdot \left({\left(M \cdot \color{blue}{\frac{D \cdot 1}{d \cdot 2}}\right)}^{2} \cdot \frac{-0.5}{\ell}\right)\right)\right) \]

      7. *-rgt-identity 72.9%

         \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + h \cdot \left({\left(M \cdot \frac{\color{blue}{D}}{d \cdot 2}\right)}^{2} \cdot \frac{-0.5}{\ell}\right)\right)\right) \]

      8. associate-/l* 73.8%

         \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + h \cdot \left({\color{blue}{\left(\frac{M \cdot D}{d \cdot 2}\right)}}^{2} \cdot \frac{-0.5}{\ell}\right)\right)\right) \]

      9. *-commutative 73.8%

         \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + h \cdot \left({\left(\frac{\color{blue}{D \cdot M}}{d \cdot 2}\right)}^{2} \cdot \frac{-0.5}{\ell}\right)\right)\right) \]

      10. associate-/l* 71.1%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + h \cdot \left({\color{blue}{\left(D \cdot \frac{M}{d \cdot 2}\right)}}^{2} \cdot \frac{-0.5}{\ell}\right)\right)\right) \]

      11. *-commutative 71.1%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + h \cdot \left({\left(D \cdot \frac{M}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot \frac{-0.5}{\ell}\right)\right)\right) \]

   8. Simplified 71.1%

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \color{blue}{h \cdot \left({\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2} \cdot \frac{-0.5}{\ell}\right)}\right)\right) \]
   9. Step-by-step derivation

      1. unpow-prod-down 58.4%

         \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + h \cdot \left(\color{blue}{\left({D}^{2} \cdot {\left(\frac{M}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{-0.5}{\ell}\right)\right)\right) \]

      2. add-sqr-sqrt 58.4%

         \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + h \cdot \left(\color{blue}{\left(\sqrt{{D}^{2} \cdot {\left(\frac{M}{2 \cdot d}\right)}^{2}} \cdot \sqrt{{D}^{2} \cdot {\left(\frac{M}{2 \cdot d}\right)}^{2}}\right)} \cdot \frac{-0.5}{\ell}\right)\right)\right) \]

      3. unpow-prod-down 58.4%

         \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + h \cdot \left(\left(\sqrt{\color{blue}{{\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}}} \cdot \sqrt{{D}^{2} \cdot {\left(\frac{M}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{-0.5}{\ell}\right)\right)\right) \]

      4. sqrt-pow1 49.3%

         \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + h \cdot \left(\left(\color{blue}{{\left(D \cdot \frac{M}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}} \cdot \sqrt{{D}^{2} \cdot {\left(\frac{M}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{-0.5}{\ell}\right)\right)\right) \]

      5. metadata-eval 49.3%

         \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + h \cdot \left(\left({\left(D \cdot \frac{M}{2 \cdot d}\right)}^{\color{blue}{1}} \cdot \sqrt{{D}^{2} \cdot {\left(\frac{M}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{-0.5}{\ell}\right)\right)\right) \]

      6. pow1 49.3%

         \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + h \cdot \left(\left(\color{blue}{\left(D \cdot \frac{M}{2 \cdot d}\right)} \cdot \sqrt{{D}^{2} \cdot {\left(\frac{M}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{-0.5}{\ell}\right)\right)\right) \]

      7. associate-/r* 49.3%

         \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + h \cdot \left(\left(\left(D \cdot \color{blue}{\frac{\frac{M}{2}}{d}}\right) \cdot \sqrt{{D}^{2} \cdot {\left(\frac{M}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{-0.5}{\ell}\right)\right)\right) \]

      8. div-inv 49.3%

         \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + h \cdot \left(\left(\left(D \cdot \frac{\color{blue}{M \cdot \frac{1}{2}}}{d}\right) \cdot \sqrt{{D}^{2} \cdot {\left(\frac{M}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{-0.5}{\ell}\right)\right)\right) \]

      9. metadata-eval 49.3%

         \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + h \cdot \left(\left(\left(D \cdot \frac{M \cdot \color{blue}{0.5}}{d}\right) \cdot \sqrt{{D}^{2} \cdot {\left(\frac{M}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{-0.5}{\ell}\right)\right)\right) \]

      10. unpow-prod-down 57.5%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + h \cdot \left(\left(\left(D \cdot \frac{M \cdot 0.5}{d}\right) \cdot \sqrt{\color{blue}{{\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}}}\right) \cdot \frac{-0.5}{\ell}\right)\right)\right) \]

      11. sqrt-pow1 71.1%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + h \cdot \left(\left(\left(D \cdot \frac{M \cdot 0.5}{d}\right) \cdot \color{blue}{{\left(D \cdot \frac{M}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}}\right) \cdot \frac{-0.5}{\ell}\right)\right)\right) \]

      12. metadata-eval 71.1%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + h \cdot \left(\left(\left(D \cdot \frac{M \cdot 0.5}{d}\right) \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{\color{blue}{1}}\right) \cdot \frac{-0.5}{\ell}\right)\right)\right) \]

      13. pow1 71.1%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + h \cdot \left(\left(\left(D \cdot \frac{M \cdot 0.5}{d}\right) \cdot \color{blue}{\left(D \cdot \frac{M}{2 \cdot d}\right)}\right) \cdot \frac{-0.5}{\ell}\right)\right)\right) \]

      14. associate-/r* 71.1%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + h \cdot \left(\left(\left(D \cdot \frac{M \cdot 0.5}{d}\right) \cdot \left(D \cdot \color{blue}{\frac{\frac{M}{2}}{d}}\right)\right) \cdot \frac{-0.5}{\ell}\right)\right)\right) \]

      15. div-inv 71.1%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + h \cdot \left(\left(\left(D \cdot \frac{M \cdot 0.5}{d}\right) \cdot \left(D \cdot \frac{\color{blue}{M \cdot \frac{1}{2}}}{d}\right)\right) \cdot \frac{-0.5}{\ell}\right)\right)\right) \]

      16. metadata-eval 71.1%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + h \cdot \left(\left(\left(D \cdot \frac{M \cdot 0.5}{d}\right) \cdot \left(D \cdot \frac{M \cdot \color{blue}{0.5}}{d}\right)\right) \cdot \frac{-0.5}{\ell}\right)\right)\right) \]

   10. Applied egg-rr 71.1%

       \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + h \cdot \left(\color{blue}{\left(\left(D \cdot \frac{M \cdot 0.5}{d}\right) \cdot \left(D \cdot \frac{M \cdot 0.5}{d}\right)\right)} \cdot \frac{-0.5}{\ell}\right)\right)\right) \]
   11. Step-by-step derivation

       1. associate-*l* 50.4%

          \[\leadsto \frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + h \cdot \color{blue}{\left(\left(D \cdot \frac{M \cdot 0.5}{d}\right) \cdot \left(\left(D \cdot \frac{M \cdot 0.5}{d}\right) \cdot \frac{-0.5}{\ell}\right)\right)}\right)\right) \]

       2. associate-*r/ 50.5%

          \[\leadsto \frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + h \cdot \left(\color{blue}{\frac{D \cdot \left(M \cdot 0.5\right)}{d}} \cdot \left(\left(D \cdot \frac{M \cdot 0.5}{d}\right) \cdot \frac{-0.5}{\ell}\right)\right)\right)\right) \]

       3. associate-*l/ 50.5%

          \[\leadsto \frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + h \cdot \color{blue}{\frac{\left(D \cdot \left(M \cdot 0.5\right)\right) \cdot \left(\left(D \cdot \frac{M \cdot 0.5}{d}\right) \cdot \frac{-0.5}{\ell}\right)}{d}}\right)\right) \]

       4. associate-*l* 48.7%

          \[\leadsto \frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + h \cdot \frac{\left(D \cdot \left(M \cdot 0.5\right)\right) \cdot \color{blue}{\left(D \cdot \left(\frac{M \cdot 0.5}{d} \cdot \frac{-0.5}{\ell}\right)\right)}}{d}\right)\right) \]

       5. associate-/l* 48.7%

          \[\leadsto \frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + h \cdot \frac{\left(D \cdot \left(M \cdot 0.5\right)\right) \cdot \left(D \cdot \left(\color{blue}{\left(M \cdot \frac{0.5}{d}\right)} \cdot \frac{-0.5}{\ell}\right)\right)}{d}\right)\right) \]

   12. Applied egg-rr 70.1%

       \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + h \cdot \color{blue}{\frac{\left(D \cdot \left(M \cdot 0.5\right)\right) \cdot \left(D \cdot \left(\left(M \cdot \frac{0.5}{d}\right) \cdot \frac{-0.5}{\ell}\right)\right)}{d}}\right)\right) \]
   13. Step-by-step derivation

       1. associate-/l* 69.2%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + h \cdot \color{blue}{\left(\left(D \cdot \left(M \cdot 0.5\right)\right) \cdot \frac{D \cdot \left(\left(M \cdot \frac{0.5}{d}\right) \cdot \frac{-0.5}{\ell}\right)}{d}\right)}\right)\right) \]

       2. associate-*r* 69.2%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + h \cdot \left(\color{blue}{\left(\left(D \cdot M\right) \cdot 0.5\right)} \cdot \frac{D \cdot \left(\left(M \cdot \frac{0.5}{d}\right) \cdot \frac{-0.5}{\ell}\right)}{d}\right)\right)\right) \]

       3. associate-*r* 71.8%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + h \cdot \left(\left(\left(D \cdot M\right) \cdot 0.5\right) \cdot \frac{\color{blue}{\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right) \cdot \frac{-0.5}{\ell}}}{d}\right)\right)\right) \]

       4. associate-*r* 74.5%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + h \cdot \left(\left(\left(D \cdot M\right) \cdot 0.5\right) \cdot \frac{\color{blue}{\left(\left(D \cdot M\right) \cdot \frac{0.5}{d}\right)} \cdot \frac{-0.5}{\ell}}{d}\right)\right)\right) \]

   14. Simplified 74.5%

       \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + h \cdot \color{blue}{\left(\left(\left(D \cdot M\right) \cdot 0.5\right) \cdot \frac{\left(\left(D \cdot M\right) \cdot \frac{0.5}{d}\right) \cdot \frac{-0.5}{\ell}}{d}\right)}\right)\right) \]

   if -1.0000000000000001e130 < h < -4.999999999999985e-310

   1. Initial program 73.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 74.3%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. frac-2neg 72.8%

         \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\color{blue}{\frac{-d}{-h}}} \cdot \left(1 + h \cdot \left(\left(\left(D \cdot \frac{M \cdot 0.5}{d}\right) \cdot \left(D \cdot \frac{M \cdot 0.5}{d}\right)\right) \cdot \frac{-0.5}{\ell}\right)\right)\right) \]

      2. sqrt-div 81.2%

         \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\color{blue}{\frac{\sqrt{-d}}{\sqrt{-h}}} \cdot \left(1 + h \cdot \left(\left(\left(D \cdot \frac{M \cdot 0.5}{d}\right) \cdot \left(D \cdot \frac{M \cdot 0.5}{d}\right)\right) \cdot \frac{-0.5}{\ell}\right)\right)\right) \]

   6. Applied egg-rr 82.7%

      \[\leadsto \left(\color{blue}{\frac{\sqrt{-d}}{\sqrt{-h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   7. Taylor expanded in d around -inf 85.9%

      \[\leadsto \color{blue}{\left(-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)\right)} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   if -4.999999999999985e-310 < h < 2.00000000000000005e-46

   1. Initial program 72.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 72.0%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 72.0%

         \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      2. sqrt-div 77.3%

         \[\leadsto \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      3. sqrt-div 85.6%

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{h}}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      4. frac-times 85.7%

         \[\leadsto \color{blue}{\frac{\sqrt{d} \cdot \sqrt{d}}{\sqrt{\ell} \cdot \sqrt{h}}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      5. add-sqr-sqrt 85.8%

         \[\leadsto \frac{\color{blue}{d}}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   6. Applied egg-rr 85.8%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
   7. Step-by-step derivation

      1. div-inv 85.7%

         \[\leadsto \color{blue}{\left(d \cdot \frac{1}{\sqrt{\ell} \cdot \sqrt{h}}\right)} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      2. *-commutative 85.7%

         \[\leadsto \left(d \cdot \frac{1}{\color{blue}{\sqrt{h} \cdot \sqrt{\ell}}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      3. metadata-eval 85.7%

         \[\leadsto \left(d \cdot \frac{\color{blue}{\sqrt{1}}}{\sqrt{h} \cdot \sqrt{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      4. sqrt-unprod 80.5%

         \[\leadsto \left(d \cdot \frac{\sqrt{1}}{\color{blue}{\sqrt{h \cdot \ell}}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      5. sqrt-div 80.6%

         \[\leadsto \left(d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      6. *-commutative 80.6%

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot d\right)} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      7. inv-pow 80.6%

         \[\leadsto \left(\sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \cdot d\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. sqrt-pow1 80.6%

         \[\leadsto \left(\color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}} \cdot d\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      9. metadata-eval 80.6%

         \[\leadsto \left({\left(h \cdot \ell\right)}^{\color{blue}{-0.5}} \cdot d\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   8. Applied egg-rr 80.6%

      \[\leadsto \color{blue}{\left({\left(h \cdot \ell\right)}^{-0.5} \cdot d\right)} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
   9. Step-by-step derivation

      1. associate-*r/ 84.4%

         \[\leadsto \left({\left(h \cdot \ell\right)}^{-0.5} \cdot d\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot h}{\ell}}\right) \]

      2. frac-times 84.4%

         \[\leadsto \left({\left(h \cdot \ell\right)}^{-0.5} \cdot d\right) \cdot \left(1 - 0.5 \cdot \frac{{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot h}{\ell}\right) \]

      3. associate-/l* 84.4%

         \[\leadsto \left({\left(h \cdot \ell\right)}^{-0.5} \cdot d\right) \cdot \left(1 - 0.5 \cdot \frac{{\color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)}}^{2} \cdot h}{\ell}\right) \]

      4. *-un-lft-identity 84.4%

         \[\leadsto \left({\left(h \cdot \ell\right)}^{-0.5} \cdot d\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(M \cdot \frac{\color{blue}{1 \cdot D}}{2 \cdot d}\right)}^{2} \cdot h}{\ell}\right) \]

      5. times-frac 84.4%

         \[\leadsto \left({\left(h \cdot \ell\right)}^{-0.5} \cdot d\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(M \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{D}{d}\right)}\right)}^{2} \cdot h}{\ell}\right) \]

      6. metadata-eval 84.4%

         \[\leadsto \left({\left(h \cdot \ell\right)}^{-0.5} \cdot d\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(M \cdot \left(\color{blue}{0.5} \cdot \frac{D}{d}\right)\right)}^{2} \cdot h}{\ell}\right) \]

   10. Applied egg-rr 84.4%

       \[\leadsto \left({\left(h \cdot \ell\right)}^{-0.5} \cdot d\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot h}{\ell}}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -1 \cdot 10^{+130}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + h \cdot \left(\left(0.5 \cdot \left(D \cdot M\right)\right) \cdot \frac{\frac{-0.5}{\ell} \cdot \left(\frac{0.5}{d} \cdot \left(D \cdot M\right)\right)}{d}\right)\right)\right)\\ \mathbf{elif}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right) + -1\right)\\ \mathbf{elif}\;h \leq 2 \cdot 10^{-46}:\\ \;\;\;\;\left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right) \cdot \left(1 - 0.5 \cdot \frac{h \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + h \cdot \left(\left(0.5 \cdot \left(D \cdot M\right)\right) \cdot \frac{\frac{-0.5}{\ell} \cdot \left(\frac{0.5}{d} \cdot \left(D \cdot M\right)\right)}{d}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 73.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -5.9 \cdot 10^{+259}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;\ell \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(0.5 \cdot \left({\left(\frac{M\_m}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right) + -1\right)\\ \mathbf{elif}\;\ell \leq 2.45 \cdot 10^{+203}:\\ \;\;\;\;\left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right) \cdot \left(1 - 0.5 \cdot \frac{h \cdot {\left(M\_m \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

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 -5.9e+259)
   (* (sqrt (/ d h)) (sqrt (/ d l)))
   (if (<= l -1e-310)
     (*
      (* d (sqrt (/ 1.0 (* l h))))
      (+ (* 0.5 (* (pow (* (/ M_m 2.0) (/ D d)) 2.0) (/ h l))) -1.0))
     (if (<= l 2.45e+203)
       (*
        (* d (pow (* l h) -0.5))
        (- 1.0 (* 0.5 (/ (* h (pow (* M_m (* 0.5 (/ D d))) 2.0)) l))))
       (/ d (* (sqrt l) (sqrt 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 <= -5.9e+259) {
		tmp = sqrt((d / h)) * sqrt((d / l));
	} else if (l <= -1e-310) {
		tmp = (d * sqrt((1.0 / (l * h)))) * ((0.5 * (pow(((M_m / 2.0) * (D / d)), 2.0) * (h / l))) + -1.0);
	} else if (l <= 2.45e+203) {
		tmp = (d * pow((l * h), -0.5)) * (1.0 - (0.5 * ((h * pow((M_m * (0.5 * (D / d))), 2.0)) / l)));
	} else {
		tmp = d / (sqrt(l) * sqrt(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 <= (-5.9d+259)) then
        tmp = sqrt((d / h)) * sqrt((d / l))
    else if (l <= (-1d-310)) then
        tmp = (d * sqrt((1.0d0 / (l * h)))) * ((0.5d0 * ((((m_m / 2.0d0) * (d_1 / d)) ** 2.0d0) * (h / l))) + (-1.0d0))
    else if (l <= 2.45d+203) then
        tmp = (d * ((l * h) ** (-0.5d0))) * (1.0d0 - (0.5d0 * ((h * ((m_m * (0.5d0 * (d_1 / d))) ** 2.0d0)) / l)))
    else
        tmp = d / (sqrt(l) * sqrt(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 <= -5.9e+259) {
		tmp = Math.sqrt((d / h)) * Math.sqrt((d / l));
	} else if (l <= -1e-310) {
		tmp = (d * Math.sqrt((1.0 / (l * h)))) * ((0.5 * (Math.pow(((M_m / 2.0) * (D / d)), 2.0) * (h / l))) + -1.0);
	} else if (l <= 2.45e+203) {
		tmp = (d * Math.pow((l * h), -0.5)) * (1.0 - (0.5 * ((h * Math.pow((M_m * (0.5 * (D / d))), 2.0)) / l)));
	} else {
		tmp = d / (Math.sqrt(l) * Math.sqrt(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 <= -5.9e+259:
		tmp = math.sqrt((d / h)) * math.sqrt((d / l))
	elif l <= -1e-310:
		tmp = (d * math.sqrt((1.0 / (l * h)))) * ((0.5 * (math.pow(((M_m / 2.0) * (D / d)), 2.0) * (h / l))) + -1.0)
	elif l <= 2.45e+203:
		tmp = (d * math.pow((l * h), -0.5)) * (1.0 - (0.5 * ((h * math.pow((M_m * (0.5 * (D / d))), 2.0)) / l)))
	else:
		tmp = d / (math.sqrt(l) * math.sqrt(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 <= -5.9e+259)
		tmp = Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l)));
	elseif (l <= -1e-310)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(l * h)))) * Float64(Float64(0.5 * Float64((Float64(Float64(M_m / 2.0) * Float64(D / d)) ^ 2.0) * Float64(h / l))) + -1.0));
	elseif (l <= 2.45e+203)
		tmp = Float64(Float64(d * (Float64(l * h) ^ -0.5)) * Float64(1.0 - Float64(0.5 * Float64(Float64(h * (Float64(M_m * Float64(0.5 * Float64(D / d))) ^ 2.0)) / l))));
	else
		tmp = Float64(d / Float64(sqrt(l) * sqrt(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 <= -5.9e+259)
		tmp = sqrt((d / h)) * sqrt((d / l));
	elseif (l <= -1e-310)
		tmp = (d * sqrt((1.0 / (l * h)))) * ((0.5 * ((((M_m / 2.0) * (D / d)) ^ 2.0) * (h / l))) + -1.0);
	elseif (l <= 2.45e+203)
		tmp = (d * ((l * h) ^ -0.5)) * (1.0 - (0.5 * ((h * ((M_m * (0.5 * (D / d))) ^ 2.0)) / l)));
	else
		tmp = d / (sqrt(l) * sqrt(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, -5.9e+259], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -1e-310], N[(N[(d * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 * N[(N[Power[N[(N[(M$95$m / 2.0), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 2.45e+203], N[(N[(d * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(h * N[Power[N[(M$95$m * N[(0.5 * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if l < -5.89999999999999972e259

   1. Initial program 66.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 66.5%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 67.7%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]

   if -5.89999999999999972e259 < l < -9.999999999999969e-311

   1. Initial program 69.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 69.8%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. frac-2neg 73.0%

         \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\color{blue}{\frac{-d}{-h}}} \cdot \left(1 + h \cdot \left(\left(\left(D \cdot \frac{M \cdot 0.5}{d}\right) \cdot \left(D \cdot \frac{M \cdot 0.5}{d}\right)\right) \cdot \frac{-0.5}{\ell}\right)\right)\right) \]

      2. sqrt-div 83.6%

         \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\color{blue}{\frac{\sqrt{-d}}{\sqrt{-h}}} \cdot \left(1 + h \cdot \left(\left(\left(D \cdot \frac{M \cdot 0.5}{d}\right) \cdot \left(D \cdot \frac{M \cdot 0.5}{d}\right)\right) \cdot \frac{-0.5}{\ell}\right)\right)\right) \]

   6. Applied egg-rr 78.2%

      \[\leadsto \left(\color{blue}{\frac{\sqrt{-d}}{\sqrt{-h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   7. Taylor expanded in d around -inf 76.4%

      \[\leadsto \color{blue}{\left(-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)\right)} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   if -9.999999999999969e-311 < l < 2.4499999999999999e203

   1. Initial program 77.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 76.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 76.7%

         \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      2. sqrt-div 77.0%

         \[\leadsto \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      3. sqrt-div 82.9%

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{h}}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      4. frac-times 82.9%

         \[\leadsto \color{blue}{\frac{\sqrt{d} \cdot \sqrt{d}}{\sqrt{\ell} \cdot \sqrt{h}}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      5. add-sqr-sqrt 83.1%

         \[\leadsto \frac{\color{blue}{d}}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   6. Applied egg-rr 83.1%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
   7. Step-by-step derivation

      1. div-inv 83.0%

         \[\leadsto \color{blue}{\left(d \cdot \frac{1}{\sqrt{\ell} \cdot \sqrt{h}}\right)} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      2. *-commutative 83.0%

         \[\leadsto \left(d \cdot \frac{1}{\color{blue}{\sqrt{h} \cdot \sqrt{\ell}}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      3. metadata-eval 83.0%

         \[\leadsto \left(d \cdot \frac{\color{blue}{\sqrt{1}}}{\sqrt{h} \cdot \sqrt{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      4. sqrt-unprod 78.0%

         \[\leadsto \left(d \cdot \frac{\sqrt{1}}{\color{blue}{\sqrt{h \cdot \ell}}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      5. sqrt-div 78.0%

         \[\leadsto \left(d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      6. *-commutative 78.0%

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot d\right)} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      7. inv-pow 78.0%

         \[\leadsto \left(\sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \cdot d\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. sqrt-pow1 78.1%

         \[\leadsto \left(\color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}} \cdot d\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      9. metadata-eval 78.1%

         \[\leadsto \left({\left(h \cdot \ell\right)}^{\color{blue}{-0.5}} \cdot d\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   8. Applied egg-rr 78.1%

      \[\leadsto \color{blue}{\left({\left(h \cdot \ell\right)}^{-0.5} \cdot d\right)} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
   9. Step-by-step derivation

      1. associate-*r/ 83.2%

         \[\leadsto \left({\left(h \cdot \ell\right)}^{-0.5} \cdot d\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot h}{\ell}}\right) \]

      2. frac-times 84.8%

         \[\leadsto \left({\left(h \cdot \ell\right)}^{-0.5} \cdot d\right) \cdot \left(1 - 0.5 \cdot \frac{{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot h}{\ell}\right) \]

      3. associate-/l* 83.2%

         \[\leadsto \left({\left(h \cdot \ell\right)}^{-0.5} \cdot d\right) \cdot \left(1 - 0.5 \cdot \frac{{\color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)}}^{2} \cdot h}{\ell}\right) \]

      4. *-un-lft-identity 83.2%

         \[\leadsto \left({\left(h \cdot \ell\right)}^{-0.5} \cdot d\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(M \cdot \frac{\color{blue}{1 \cdot D}}{2 \cdot d}\right)}^{2} \cdot h}{\ell}\right) \]

      5. times-frac 83.2%

         \[\leadsto \left({\left(h \cdot \ell\right)}^{-0.5} \cdot d\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(M \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{D}{d}\right)}\right)}^{2} \cdot h}{\ell}\right) \]

      6. metadata-eval 83.2%

         \[\leadsto \left({\left(h \cdot \ell\right)}^{-0.5} \cdot d\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(M \cdot \left(\color{blue}{0.5} \cdot \frac{D}{d}\right)\right)}^{2} \cdot h}{\ell}\right) \]

   10. Applied egg-rr 83.2%

       \[\leadsto \left({\left(h \cdot \ell\right)}^{-0.5} \cdot d\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot h}{\ell}}\right) \]

   if 2.4499999999999999e203 < l

   1. Initial program 50.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 50.1%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 53.2%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. sqrt-div 53.1%

         \[\leadsto d \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{h \cdot \ell}}} \]

      2. metadata-eval 53.1%

         \[\leadsto d \cdot \frac{\color{blue}{1}}{\sqrt{h \cdot \ell}} \]

      3. sqrt-unprod 71.4%

         \[\leadsto d \cdot \frac{1}{\color{blue}{\sqrt{h} \cdot \sqrt{\ell}}} \]

      4. *-commutative 71.4%

         \[\leadsto d \cdot \frac{1}{\color{blue}{\sqrt{\ell} \cdot \sqrt{h}}} \]

      5. div-inv 71.5%

         \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}} \]

      6. associate-/r* 68.0%

         \[\leadsto \color{blue}{\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}} \]

   7. Applied egg-rr 68.0%

      \[\leadsto \color{blue}{\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}} \]
   8. Step-by-step derivation

      1. associate-/l/ 71.5%

         \[\leadsto \color{blue}{\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}} \]

      2. *-commutative 71.5%

         \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell} \cdot \sqrt{h}}} \]

   9. Simplified 71.5%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5.9 \cdot 10^{+259}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;\ell \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right) + -1\right)\\ \mathbf{elif}\;\ell \leq 2.45 \cdot 10^{+203}:\\ \;\;\;\;\left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right) \cdot \left(1 - 0.5 \cdot \frac{h \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 67.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := {\left(\ell \cdot h\right)}^{-0.5}\\ \mathbf{if}\;\ell \leq -1.7 \cdot 10^{+166}:\\ \;\;\;\;\left(-d\right) \cdot t\_0\\ \mathbf{elif}\;\ell \leq 5.8 \cdot 10^{-306}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left({\left(\frac{M\_m}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{elif}\;\ell \leq 1.32 \cdot 10^{+203}:\\ \;\;\;\;\left(d \cdot t\_0\right) \cdot \left(1 - 0.5 \cdot \frac{h \cdot {\left(M\_m \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

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
 (let* ((t_0 (pow (* l h) -0.5)))
   (if (<= l -1.7e+166)
     (* (- d) t_0)
     (if (<= l 5.8e-306)
       (*
        (- 1.0 (* 0.5 (* (pow (* (/ M_m 2.0) (/ D d)) 2.0) (/ h l))))
        (sqrt (* (/ d h) (/ d l))))
       (if (<= l 1.32e+203)
         (*
          (* d t_0)
          (- 1.0 (* 0.5 (/ (* h (pow (* M_m (* 0.5 (/ D d))) 2.0)) l))))
         (/ d (* (sqrt l) (sqrt 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 t_0 = pow((l * h), -0.5);
	double tmp;
	if (l <= -1.7e+166) {
		tmp = -d * t_0;
	} else if (l <= 5.8e-306) {
		tmp = (1.0 - (0.5 * (pow(((M_m / 2.0) * (D / d)), 2.0) * (h / l)))) * sqrt(((d / h) * (d / l)));
	} else if (l <= 1.32e+203) {
		tmp = (d * t_0) * (1.0 - (0.5 * ((h * pow((M_m * (0.5 * (D / d))), 2.0)) / l)));
	} else {
		tmp = d / (sqrt(l) * sqrt(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) :: t_0
    real(8) :: tmp
    t_0 = (l * h) ** (-0.5d0)
    if (l <= (-1.7d+166)) then
        tmp = -d * t_0
    else if (l <= 5.8d-306) then
        tmp = (1.0d0 - (0.5d0 * ((((m_m / 2.0d0) * (d_1 / d)) ** 2.0d0) * (h / l)))) * sqrt(((d / h) * (d / l)))
    else if (l <= 1.32d+203) then
        tmp = (d * t_0) * (1.0d0 - (0.5d0 * ((h * ((m_m * (0.5d0 * (d_1 / d))) ** 2.0d0)) / l)))
    else
        tmp = d / (sqrt(l) * sqrt(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 t_0 = Math.pow((l * h), -0.5);
	double tmp;
	if (l <= -1.7e+166) {
		tmp = -d * t_0;
	} else if (l <= 5.8e-306) {
		tmp = (1.0 - (0.5 * (Math.pow(((M_m / 2.0) * (D / d)), 2.0) * (h / l)))) * Math.sqrt(((d / h) * (d / l)));
	} else if (l <= 1.32e+203) {
		tmp = (d * t_0) * (1.0 - (0.5 * ((h * Math.pow((M_m * (0.5 * (D / d))), 2.0)) / l)));
	} else {
		tmp = d / (Math.sqrt(l) * Math.sqrt(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):
	t_0 = math.pow((l * h), -0.5)
	tmp = 0
	if l <= -1.7e+166:
		tmp = -d * t_0
	elif l <= 5.8e-306:
		tmp = (1.0 - (0.5 * (math.pow(((M_m / 2.0) * (D / d)), 2.0) * (h / l)))) * math.sqrt(((d / h) * (d / l)))
	elif l <= 1.32e+203:
		tmp = (d * t_0) * (1.0 - (0.5 * ((h * math.pow((M_m * (0.5 * (D / d))), 2.0)) / l)))
	else:
		tmp = d / (math.sqrt(l) * math.sqrt(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)
	t_0 = Float64(l * h) ^ -0.5
	tmp = 0.0
	if (l <= -1.7e+166)
		tmp = Float64(Float64(-d) * t_0);
	elseif (l <= 5.8e-306)
		tmp = Float64(Float64(1.0 - Float64(0.5 * Float64((Float64(Float64(M_m / 2.0) * Float64(D / d)) ^ 2.0) * Float64(h / l)))) * sqrt(Float64(Float64(d / h) * Float64(d / l))));
	elseif (l <= 1.32e+203)
		tmp = Float64(Float64(d * t_0) * Float64(1.0 - Float64(0.5 * Float64(Float64(h * (Float64(M_m * Float64(0.5 * Float64(D / d))) ^ 2.0)) / l))));
	else
		tmp = Float64(d / Float64(sqrt(l) * sqrt(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)
	t_0 = (l * h) ^ -0.5;
	tmp = 0.0;
	if (l <= -1.7e+166)
		tmp = -d * t_0;
	elseif (l <= 5.8e-306)
		tmp = (1.0 - (0.5 * ((((M_m / 2.0) * (D / d)) ^ 2.0) * (h / l)))) * sqrt(((d / h) * (d / l)));
	elseif (l <= 1.32e+203)
		tmp = (d * t_0) * (1.0 - (0.5 * ((h * ((M_m * (0.5 * (D / d))) ^ 2.0)) / l)));
	else
		tmp = d / (sqrt(l) * sqrt(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_] := Block[{t$95$0 = N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]}, If[LessEqual[l, -1.7e+166], N[((-d) * t$95$0), $MachinePrecision], If[LessEqual[l, 5.8e-306], N[(N[(1.0 - N[(0.5 * N[(N[Power[N[(N[(M$95$m / 2.0), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(d / h), $MachinePrecision] * N[(d / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.32e+203], N[(N[(d * t$95$0), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(h * N[Power[N[(M$95$m * N[(0.5 * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -1.7e166

   1. Initial program 55.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 55.2%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 19.5%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

   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. *-commutative 0.0%

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]

      2. unpow-1 0.0%

         \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      3. metadata-eval 0.0%

         \[\leadsto \sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      4. pow-sqr 0.0%

         \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      5. rem-sqrt-square 0.0%

         \[\leadsto \color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      6. rem-square-sqrt 0.0%

         \[\leadsto \left|\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right| \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      7. fabs-sqr 0.0%

         \[\leadsto \color{blue}{\left(\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}\right)} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      8. rem-square-sqrt 0.0%

         \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      9. *-commutative 0.0%

         \[\leadsto {\left(h \cdot \ell\right)}^{-0.5} \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot d\right)} \]

      10. unpow2 0.0%

          \[\leadsto {\left(h \cdot \ell\right)}^{-0.5} \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right) \]

      11. rem-square-sqrt 58.4%

          \[\leadsto {\left(h \cdot \ell\right)}^{-0.5} \cdot \left(\color{blue}{-1} \cdot d\right) \]

      12. mul-1-neg 58.4%

          \[\leadsto {\left(h \cdot \ell\right)}^{-0.5} \cdot \color{blue}{\left(-d\right)} \]

   8. Simplified 58.4%

      \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot \left(-d\right)} \]

   if -1.7e166 < l < 5.7999999999999998e-306

   1. Initial program 72.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 73.2%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 73.2%

         \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      2. sqrt-unprod 62.6%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   6. Applied egg-rr 62.6%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   if 5.7999999999999998e-306 < l < 1.32e203

   1. Initial program 77.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 76.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 76.7%

         \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      2. sqrt-div 77.0%

         \[\leadsto \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      3. sqrt-div 82.9%

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{h}}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      4. frac-times 82.9%

         \[\leadsto \color{blue}{\frac{\sqrt{d} \cdot \sqrt{d}}{\sqrt{\ell} \cdot \sqrt{h}}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      5. add-sqr-sqrt 83.1%

         \[\leadsto \frac{\color{blue}{d}}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   6. Applied egg-rr 83.1%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
   7. Step-by-step derivation

      1. div-inv 83.0%

         \[\leadsto \color{blue}{\left(d \cdot \frac{1}{\sqrt{\ell} \cdot \sqrt{h}}\right)} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      2. *-commutative 83.0%

         \[\leadsto \left(d \cdot \frac{1}{\color{blue}{\sqrt{h} \cdot \sqrt{\ell}}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      3. metadata-eval 83.0%

         \[\leadsto \left(d \cdot \frac{\color{blue}{\sqrt{1}}}{\sqrt{h} \cdot \sqrt{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      4. sqrt-unprod 78.0%

         \[\leadsto \left(d \cdot \frac{\sqrt{1}}{\color{blue}{\sqrt{h \cdot \ell}}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      5. sqrt-div 78.0%

         \[\leadsto \left(d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      6. *-commutative 78.0%

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot d\right)} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      7. inv-pow 78.0%

         \[\leadsto \left(\sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \cdot d\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. sqrt-pow1 78.1%

         \[\leadsto \left(\color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}} \cdot d\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      9. metadata-eval 78.1%

         \[\leadsto \left({\left(h \cdot \ell\right)}^{\color{blue}{-0.5}} \cdot d\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   8. Applied egg-rr 78.1%

      \[\leadsto \color{blue}{\left({\left(h \cdot \ell\right)}^{-0.5} \cdot d\right)} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
   9. Step-by-step derivation

      1. associate-*r/ 83.2%

         \[\leadsto \left({\left(h \cdot \ell\right)}^{-0.5} \cdot d\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot h}{\ell}}\right) \]

      2. frac-times 84.8%

         \[\leadsto \left({\left(h \cdot \ell\right)}^{-0.5} \cdot d\right) \cdot \left(1 - 0.5 \cdot \frac{{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot h}{\ell}\right) \]

      3. associate-/l* 83.2%

         \[\leadsto \left({\left(h \cdot \ell\right)}^{-0.5} \cdot d\right) \cdot \left(1 - 0.5 \cdot \frac{{\color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)}}^{2} \cdot h}{\ell}\right) \]

      4. *-un-lft-identity 83.2%

         \[\leadsto \left({\left(h \cdot \ell\right)}^{-0.5} \cdot d\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(M \cdot \frac{\color{blue}{1 \cdot D}}{2 \cdot d}\right)}^{2} \cdot h}{\ell}\right) \]

      5. times-frac 83.2%

         \[\leadsto \left({\left(h \cdot \ell\right)}^{-0.5} \cdot d\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(M \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{D}{d}\right)}\right)}^{2} \cdot h}{\ell}\right) \]

      6. metadata-eval 83.2%

         \[\leadsto \left({\left(h \cdot \ell\right)}^{-0.5} \cdot d\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(M \cdot \left(\color{blue}{0.5} \cdot \frac{D}{d}\right)\right)}^{2} \cdot h}{\ell}\right) \]

   10. Applied egg-rr 83.2%

       \[\leadsto \left({\left(h \cdot \ell\right)}^{-0.5} \cdot d\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot h}{\ell}}\right) \]

   if 1.32e203 < l

   1. Initial program 50.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 50.1%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 53.2%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. sqrt-div 53.1%

         \[\leadsto d \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{h \cdot \ell}}} \]

      2. metadata-eval 53.1%

         \[\leadsto d \cdot \frac{\color{blue}{1}}{\sqrt{h \cdot \ell}} \]

      3. sqrt-unprod 71.4%

         \[\leadsto d \cdot \frac{1}{\color{blue}{\sqrt{h} \cdot \sqrt{\ell}}} \]

      4. *-commutative 71.4%

         \[\leadsto d \cdot \frac{1}{\color{blue}{\sqrt{\ell} \cdot \sqrt{h}}} \]

      5. div-inv 71.5%

         \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}} \]

      6. associate-/r* 68.0%

         \[\leadsto \color{blue}{\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}} \]

   7. Applied egg-rr 68.0%

      \[\leadsto \color{blue}{\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}} \]
   8. Step-by-step derivation

      1. associate-/l/ 71.5%

         \[\leadsto \color{blue}{\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}} \]

      2. *-commutative 71.5%

         \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell} \cdot \sqrt{h}}} \]

   9. Simplified 71.5%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.7 \cdot 10^{+166}:\\ \;\;\;\;\left(-d\right) \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \mathbf{elif}\;\ell \leq 5.8 \cdot 10^{-306}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{elif}\;\ell \leq 1.32 \cdot 10^{+203}:\\ \;\;\;\;\left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right) \cdot \left(1 - 0.5 \cdot \frac{h \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 63.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := {\left(\ell \cdot h\right)}^{-0.5}\\ \mathbf{if}\;\ell \leq -6 \cdot 10^{-25}:\\ \;\;\;\;\left(-d\right) \cdot t\_0\\ \mathbf{elif}\;\ell \leq -1 \cdot 10^{-310}:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{0}}\\ \mathbf{elif}\;\ell \leq 1.35 \cdot 10^{+203}:\\ \;\;\;\;\left(d \cdot t\_0\right) \cdot \left(1 - 0.5 \cdot \frac{h \cdot {\left(M\_m \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

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
 (let* ((t_0 (pow (* l h) -0.5)))
   (if (<= l -6e-25)
     (* (- d) t_0)
     (if (<= l -1e-310)
       (* d (sqrt (/ 1.0 0.0)))
       (if (<= l 1.35e+203)
         (*
          (* d t_0)
          (- 1.0 (* 0.5 (/ (* h (pow (* M_m (* 0.5 (/ D d))) 2.0)) l))))
         (/ d (* (sqrt l) (sqrt 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 t_0 = pow((l * h), -0.5);
	double tmp;
	if (l <= -6e-25) {
		tmp = -d * t_0;
	} else if (l <= -1e-310) {
		tmp = d * sqrt((1.0 / 0.0));
	} else if (l <= 1.35e+203) {
		tmp = (d * t_0) * (1.0 - (0.5 * ((h * pow((M_m * (0.5 * (D / d))), 2.0)) / l)));
	} else {
		tmp = d / (sqrt(l) * sqrt(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) :: t_0
    real(8) :: tmp
    t_0 = (l * h) ** (-0.5d0)
    if (l <= (-6d-25)) then
        tmp = -d * t_0
    else if (l <= (-1d-310)) then
        tmp = d * sqrt((1.0d0 / 0.0d0))
    else if (l <= 1.35d+203) then
        tmp = (d * t_0) * (1.0d0 - (0.5d0 * ((h * ((m_m * (0.5d0 * (d_1 / d))) ** 2.0d0)) / l)))
    else
        tmp = d / (sqrt(l) * sqrt(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 t_0 = Math.pow((l * h), -0.5);
	double tmp;
	if (l <= -6e-25) {
		tmp = -d * t_0;
	} else if (l <= -1e-310) {
		tmp = d * Math.sqrt((1.0 / 0.0));
	} else if (l <= 1.35e+203) {
		tmp = (d * t_0) * (1.0 - (0.5 * ((h * Math.pow((M_m * (0.5 * (D / d))), 2.0)) / l)));
	} else {
		tmp = d / (Math.sqrt(l) * Math.sqrt(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):
	t_0 = math.pow((l * h), -0.5)
	tmp = 0
	if l <= -6e-25:
		tmp = -d * t_0
	elif l <= -1e-310:
		tmp = d * math.sqrt((1.0 / 0.0))
	elif l <= 1.35e+203:
		tmp = (d * t_0) * (1.0 - (0.5 * ((h * math.pow((M_m * (0.5 * (D / d))), 2.0)) / l)))
	else:
		tmp = d / (math.sqrt(l) * math.sqrt(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)
	t_0 = Float64(l * h) ^ -0.5
	tmp = 0.0
	if (l <= -6e-25)
		tmp = Float64(Float64(-d) * t_0);
	elseif (l <= -1e-310)
		tmp = Float64(d * sqrt(Float64(1.0 / 0.0)));
	elseif (l <= 1.35e+203)
		tmp = Float64(Float64(d * t_0) * Float64(1.0 - Float64(0.5 * Float64(Float64(h * (Float64(M_m * Float64(0.5 * Float64(D / d))) ^ 2.0)) / l))));
	else
		tmp = Float64(d / Float64(sqrt(l) * sqrt(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)
	t_0 = (l * h) ^ -0.5;
	tmp = 0.0;
	if (l <= -6e-25)
		tmp = -d * t_0;
	elseif (l <= -1e-310)
		tmp = d * sqrt((1.0 / 0.0));
	elseif (l <= 1.35e+203)
		tmp = (d * t_0) * (1.0 - (0.5 * ((h * ((M_m * (0.5 * (D / d))) ^ 2.0)) / l)));
	else
		tmp = d / (sqrt(l) * sqrt(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_] := Block[{t$95$0 = N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]}, If[LessEqual[l, -6e-25], N[((-d) * t$95$0), $MachinePrecision], If[LessEqual[l, -1e-310], N[(d * N[Sqrt[N[(1.0 / 0.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.35e+203], N[(N[(d * t$95$0), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(h * N[Power[N[(M$95$m * N[(0.5 * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -5.9999999999999995e-25

   1. Initial program 64.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 65.5%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 11.6%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

   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. *-commutative 0.0%

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]

      2. unpow-1 0.0%

         \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      3. metadata-eval 0.0%

         \[\leadsto \sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      4. pow-sqr 0.0%

         \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      5. rem-sqrt-square 0.0%

         \[\leadsto \color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      6. rem-square-sqrt 0.0%

         \[\leadsto \left|\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right| \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      7. fabs-sqr 0.0%

         \[\leadsto \color{blue}{\left(\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}\right)} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      8. rem-square-sqrt 0.0%

         \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      9. *-commutative 0.0%

         \[\leadsto {\left(h \cdot \ell\right)}^{-0.5} \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot d\right)} \]

      10. unpow2 0.0%

          \[\leadsto {\left(h \cdot \ell\right)}^{-0.5} \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right) \]

      11. rem-square-sqrt 53.0%

          \[\leadsto {\left(h \cdot \ell\right)}^{-0.5} \cdot \left(\color{blue}{-1} \cdot d\right) \]

      12. mul-1-neg 53.0%

          \[\leadsto {\left(h \cdot \ell\right)}^{-0.5} \cdot \color{blue}{\left(-d\right)} \]

   8. Simplified 53.0%

      \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot \left(-d\right)} \]

   if -5.9999999999999995e-25 < l < -9.999999999999969e-311

   1. Initial program 74.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 74.4%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 24.4%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. add-log-exp 38.2%

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\log \left(e^{h \cdot \ell}\right)}}} \]

      2. *-commutative 38.2%

         \[\leadsto d \cdot \sqrt{\frac{1}{\log \left(e^{\color{blue}{\ell \cdot h}}\right)}} \]

      3. exp-prod 52.1%

         \[\leadsto d \cdot \sqrt{\frac{1}{\log \color{blue}{\left({\left(e^{\ell}\right)}^{h}\right)}}} \]

   7. Applied egg-rr 52.1%

      \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\log \left({\left(e^{\ell}\right)}^{h}\right)}}} \]

   8. Taylor expanded in l around 0 52.1%

      \[\leadsto d \cdot \sqrt{\frac{1}{\log \color{blue}{1}}} \]

   if -9.999999999999969e-311 < l < 1.35e203

   1. Initial program 77.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 76.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 76.7%

         \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      2. sqrt-div 77.0%

         \[\leadsto \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      3. sqrt-div 82.9%

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{h}}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      4. frac-times 82.9%

         \[\leadsto \color{blue}{\frac{\sqrt{d} \cdot \sqrt{d}}{\sqrt{\ell} \cdot \sqrt{h}}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      5. add-sqr-sqrt 83.1%

         \[\leadsto \frac{\color{blue}{d}}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   6. Applied egg-rr 83.1%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
   7. Step-by-step derivation

      1. div-inv 83.0%

         \[\leadsto \color{blue}{\left(d \cdot \frac{1}{\sqrt{\ell} \cdot \sqrt{h}}\right)} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      2. *-commutative 83.0%

         \[\leadsto \left(d \cdot \frac{1}{\color{blue}{\sqrt{h} \cdot \sqrt{\ell}}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      3. metadata-eval 83.0%

         \[\leadsto \left(d \cdot \frac{\color{blue}{\sqrt{1}}}{\sqrt{h} \cdot \sqrt{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      4. sqrt-unprod 78.0%

         \[\leadsto \left(d \cdot \frac{\sqrt{1}}{\color{blue}{\sqrt{h \cdot \ell}}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      5. sqrt-div 78.0%

         \[\leadsto \left(d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      6. *-commutative 78.0%

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot d\right)} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      7. inv-pow 78.0%

         \[\leadsto \left(\sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \cdot d\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. sqrt-pow1 78.1%

         \[\leadsto \left(\color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}} \cdot d\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      9. metadata-eval 78.1%

         \[\leadsto \left({\left(h \cdot \ell\right)}^{\color{blue}{-0.5}} \cdot d\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   8. Applied egg-rr 78.1%

      \[\leadsto \color{blue}{\left({\left(h \cdot \ell\right)}^{-0.5} \cdot d\right)} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
   9. Step-by-step derivation

      1. associate-*r/ 83.2%

         \[\leadsto \left({\left(h \cdot \ell\right)}^{-0.5} \cdot d\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot h}{\ell}}\right) \]

      2. frac-times 84.8%

         \[\leadsto \left({\left(h \cdot \ell\right)}^{-0.5} \cdot d\right) \cdot \left(1 - 0.5 \cdot \frac{{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot h}{\ell}\right) \]

      3. associate-/l* 83.2%

         \[\leadsto \left({\left(h \cdot \ell\right)}^{-0.5} \cdot d\right) \cdot \left(1 - 0.5 \cdot \frac{{\color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)}}^{2} \cdot h}{\ell}\right) \]

      4. *-un-lft-identity 83.2%

         \[\leadsto \left({\left(h \cdot \ell\right)}^{-0.5} \cdot d\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(M \cdot \frac{\color{blue}{1 \cdot D}}{2 \cdot d}\right)}^{2} \cdot h}{\ell}\right) \]

      5. times-frac 83.2%

         \[\leadsto \left({\left(h \cdot \ell\right)}^{-0.5} \cdot d\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(M \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{D}{d}\right)}\right)}^{2} \cdot h}{\ell}\right) \]

      6. metadata-eval 83.2%

         \[\leadsto \left({\left(h \cdot \ell\right)}^{-0.5} \cdot d\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(M \cdot \left(\color{blue}{0.5} \cdot \frac{D}{d}\right)\right)}^{2} \cdot h}{\ell}\right) \]

   10. Applied egg-rr 83.2%

       \[\leadsto \left({\left(h \cdot \ell\right)}^{-0.5} \cdot d\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot h}{\ell}}\right) \]

   if 1.35e203 < l

   1. Initial program 50.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 50.1%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 53.2%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. sqrt-div 53.1%

         \[\leadsto d \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{h \cdot \ell}}} \]

      2. metadata-eval 53.1%

         \[\leadsto d \cdot \frac{\color{blue}{1}}{\sqrt{h \cdot \ell}} \]

      3. sqrt-unprod 71.4%

         \[\leadsto d \cdot \frac{1}{\color{blue}{\sqrt{h} \cdot \sqrt{\ell}}} \]

      4. *-commutative 71.4%

         \[\leadsto d \cdot \frac{1}{\color{blue}{\sqrt{\ell} \cdot \sqrt{h}}} \]

      5. div-inv 71.5%

         \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}} \]

      6. associate-/r* 68.0%

         \[\leadsto \color{blue}{\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}} \]

   7. Applied egg-rr 68.0%

      \[\leadsto \color{blue}{\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}} \]
   8. Step-by-step derivation

      1. associate-/l/ 71.5%

         \[\leadsto \color{blue}{\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}} \]

      2. *-commutative 71.5%

         \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell} \cdot \sqrt{h}}} \]

   9. Simplified 71.5%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 66.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -6 \cdot 10^{-25}:\\ \;\;\;\;\left(-d\right) \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \mathbf{elif}\;\ell \leq -1 \cdot 10^{-310}:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{0}}\\ \mathbf{elif}\;\ell \leq 1.35 \cdot 10^{+203}:\\ \;\;\;\;\left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right) \cdot \left(1 - 0.5 \cdot \frac{h \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 62.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := {\left(\ell \cdot h\right)}^{-0.5}\\ \mathbf{if}\;\ell \leq -1.2 \cdot 10^{-24}:\\ \;\;\;\;\left(-d\right) \cdot t\_0\\ \mathbf{elif}\;\ell \leq -1 \cdot 10^{-310}:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{0}}\\ \mathbf{elif}\;\ell \leq 1.45 \cdot 10^{+157}:\\ \;\;\;\;d \cdot \left(t\_0 \cdot \left(1 + -0.5 \cdot \left({\left(\frac{M\_m \cdot \left(D \cdot 0.5\right)}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

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
 (let* ((t_0 (pow (* l h) -0.5)))
   (if (<= l -1.2e-24)
     (* (- d) t_0)
     (if (<= l -1e-310)
       (* d (sqrt (/ 1.0 0.0)))
       (if (<= l 1.45e+157)
         (*
          d
          (*
           t_0
           (+ 1.0 (* -0.5 (* (pow (/ (* M_m (* D 0.5)) d) 2.0) (/ h l))))))
         (/ d (* (sqrt l) (sqrt 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 t_0 = pow((l * h), -0.5);
	double tmp;
	if (l <= -1.2e-24) {
		tmp = -d * t_0;
	} else if (l <= -1e-310) {
		tmp = d * sqrt((1.0 / 0.0));
	} else if (l <= 1.45e+157) {
		tmp = d * (t_0 * (1.0 + (-0.5 * (pow(((M_m * (D * 0.5)) / d), 2.0) * (h / l)))));
	} else {
		tmp = d / (sqrt(l) * sqrt(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) :: t_0
    real(8) :: tmp
    t_0 = (l * h) ** (-0.5d0)
    if (l <= (-1.2d-24)) then
        tmp = -d * t_0
    else if (l <= (-1d-310)) then
        tmp = d * sqrt((1.0d0 / 0.0d0))
    else if (l <= 1.45d+157) then
        tmp = d * (t_0 * (1.0d0 + ((-0.5d0) * ((((m_m * (d_1 * 0.5d0)) / d) ** 2.0d0) * (h / l)))))
    else
        tmp = d / (sqrt(l) * sqrt(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 t_0 = Math.pow((l * h), -0.5);
	double tmp;
	if (l <= -1.2e-24) {
		tmp = -d * t_0;
	} else if (l <= -1e-310) {
		tmp = d * Math.sqrt((1.0 / 0.0));
	} else if (l <= 1.45e+157) {
		tmp = d * (t_0 * (1.0 + (-0.5 * (Math.pow(((M_m * (D * 0.5)) / d), 2.0) * (h / l)))));
	} else {
		tmp = d / (Math.sqrt(l) * Math.sqrt(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):
	t_0 = math.pow((l * h), -0.5)
	tmp = 0
	if l <= -1.2e-24:
		tmp = -d * t_0
	elif l <= -1e-310:
		tmp = d * math.sqrt((1.0 / 0.0))
	elif l <= 1.45e+157:
		tmp = d * (t_0 * (1.0 + (-0.5 * (math.pow(((M_m * (D * 0.5)) / d), 2.0) * (h / l)))))
	else:
		tmp = d / (math.sqrt(l) * math.sqrt(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)
	t_0 = Float64(l * h) ^ -0.5
	tmp = 0.0
	if (l <= -1.2e-24)
		tmp = Float64(Float64(-d) * t_0);
	elseif (l <= -1e-310)
		tmp = Float64(d * sqrt(Float64(1.0 / 0.0)));
	elseif (l <= 1.45e+157)
		tmp = Float64(d * Float64(t_0 * Float64(1.0 + Float64(-0.5 * Float64((Float64(Float64(M_m * Float64(D * 0.5)) / d) ^ 2.0) * Float64(h / l))))));
	else
		tmp = Float64(d / Float64(sqrt(l) * sqrt(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)
	t_0 = (l * h) ^ -0.5;
	tmp = 0.0;
	if (l <= -1.2e-24)
		tmp = -d * t_0;
	elseif (l <= -1e-310)
		tmp = d * sqrt((1.0 / 0.0));
	elseif (l <= 1.45e+157)
		tmp = d * (t_0 * (1.0 + (-0.5 * ((((M_m * (D * 0.5)) / d) ^ 2.0) * (h / l)))));
	else
		tmp = d / (sqrt(l) * sqrt(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_] := Block[{t$95$0 = N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]}, If[LessEqual[l, -1.2e-24], N[((-d) * t$95$0), $MachinePrecision], If[LessEqual[l, -1e-310], N[(d * N[Sqrt[N[(1.0 / 0.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.45e+157], N[(d * N[(t$95$0 * N[(1.0 + N[(-0.5 * N[(N[Power[N[(N[(M$95$m * N[(D * 0.5), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -1.1999999999999999e-24

   1. Initial program 64.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 65.5%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 11.6%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

   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. *-commutative 0.0%

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]

      2. unpow-1 0.0%

         \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      3. metadata-eval 0.0%

         \[\leadsto \sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      4. pow-sqr 0.0%

         \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      5. rem-sqrt-square 0.0%

         \[\leadsto \color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      6. rem-square-sqrt 0.0%

         \[\leadsto \left|\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right| \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      7. fabs-sqr 0.0%

         \[\leadsto \color{blue}{\left(\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}\right)} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      8. rem-square-sqrt 0.0%

         \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      9. *-commutative 0.0%

         \[\leadsto {\left(h \cdot \ell\right)}^{-0.5} \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot d\right)} \]

      10. unpow2 0.0%

          \[\leadsto {\left(h \cdot \ell\right)}^{-0.5} \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right) \]

      11. rem-square-sqrt 53.0%

          \[\leadsto {\left(h \cdot \ell\right)}^{-0.5} \cdot \left(\color{blue}{-1} \cdot d\right) \]

      12. mul-1-neg 53.0%

          \[\leadsto {\left(h \cdot \ell\right)}^{-0.5} \cdot \color{blue}{\left(-d\right)} \]

   8. Simplified 53.0%

      \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot \left(-d\right)} \]

   if -1.1999999999999999e-24 < l < -9.999999999999969e-311

   1. Initial program 74.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 74.4%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 24.4%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. add-log-exp 38.2%

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\log \left(e^{h \cdot \ell}\right)}}} \]

      2. *-commutative 38.2%

         \[\leadsto d \cdot \sqrt{\frac{1}{\log \left(e^{\color{blue}{\ell \cdot h}}\right)}} \]

      3. exp-prod 52.1%

         \[\leadsto d \cdot \sqrt{\frac{1}{\log \color{blue}{\left({\left(e^{\ell}\right)}^{h}\right)}}} \]

   7. Applied egg-rr 52.1%

      \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\log \left({\left(e^{\ell}\right)}^{h}\right)}}} \]

   8. Taylor expanded in l around 0 52.1%

      \[\leadsto d \cdot \sqrt{\frac{1}{\log \color{blue}{1}}} \]

   if -9.999999999999969e-311 < l < 1.44999999999999994e157

   1. Initial program 78.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 77.4%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 77.4%

         \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      2. sqrt-div 77.7%

         \[\leadsto \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      3. sqrt-div 84.0%

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{h}}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      4. frac-times 84.0%

         \[\leadsto \color{blue}{\frac{\sqrt{d} \cdot \sqrt{d}}{\sqrt{\ell} \cdot \sqrt{h}}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      5. add-sqr-sqrt 84.2%

         \[\leadsto \frac{\color{blue}{d}}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   6. Applied egg-rr 84.2%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
   7. Step-by-step derivation

      1. div-inv 84.1%

         \[\leadsto \color{blue}{\left(d \cdot \frac{1}{\sqrt{\ell} \cdot \sqrt{h}}\right)} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      2. *-commutative 84.1%

         \[\leadsto \left(d \cdot \frac{1}{\color{blue}{\sqrt{h} \cdot \sqrt{\ell}}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      3. metadata-eval 84.1%

         \[\leadsto \left(d \cdot \frac{\color{blue}{\sqrt{1}}}{\sqrt{h} \cdot \sqrt{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      4. sqrt-unprod 78.8%

         \[\leadsto \left(d \cdot \frac{\sqrt{1}}{\color{blue}{\sqrt{h \cdot \ell}}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      5. sqrt-div 78.8%

         \[\leadsto \left(d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      6. *-commutative 78.8%

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot d\right)} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      7. inv-pow 78.8%

         \[\leadsto \left(\sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \cdot d\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. sqrt-pow1 78.9%

         \[\leadsto \left(\color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}} \cdot d\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      9. metadata-eval 78.9%

         \[\leadsto \left({\left(h \cdot \ell\right)}^{\color{blue}{-0.5}} \cdot d\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   8. Applied egg-rr 78.9%

      \[\leadsto \color{blue}{\left({\left(h \cdot \ell\right)}^{-0.5} \cdot d\right)} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
   9. Step-by-step derivation

      1. pow1 78.9%

         \[\leadsto \color{blue}{{\left(\left({\left(h \cdot \ell\right)}^{-0.5} \cdot d\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)}^{1}} \]

   10. Applied egg-rr 78.9%

       \[\leadsto \color{blue}{{\left(\left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right) \cdot \left(1 + -0.5 \cdot \left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)}^{1}} \]
   11. Step-by-step derivation

       1. unpow1 78.9%

          \[\leadsto \color{blue}{\left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right) \cdot \left(1 + -0.5 \cdot \left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]

       2. associate-*l* 81.1%

          \[\leadsto \color{blue}{d \cdot \left({\left(h \cdot \ell\right)}^{-0.5} \cdot \left(1 + -0.5 \cdot \left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)} \]

       3. *-commutative 81.1%

          \[\leadsto d \cdot \left({\left(h \cdot \ell\right)}^{-0.5} \cdot \left(1 + -0.5 \cdot \color{blue}{\left(\frac{h}{\ell} \cdot {\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2}\right)}\right)\right) \]

       4. associate-*r/ 82.8%

          \[\leadsto d \cdot \left({\left(h \cdot \ell\right)}^{-0.5} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\color{blue}{\left(\frac{\left(M \cdot 0.5\right) \cdot D}{d}\right)}}^{2}\right)\right)\right) \]

       5. associate-*l* 82.8%

          \[\leadsto d \cdot \left({\left(h \cdot \ell\right)}^{-0.5} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{\color{blue}{M \cdot \left(0.5 \cdot D\right)}}{d}\right)}^{2}\right)\right)\right) \]

   12. Simplified 82.8%

       \[\leadsto \color{blue}{d \cdot \left({\left(h \cdot \ell\right)}^{-0.5} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M \cdot \left(0.5 \cdot D\right)}{d}\right)}^{2}\right)\right)\right)} \]

   if 1.44999999999999994e157 < l

   1. Initial program 53.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 53.3%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 55.8%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. sqrt-div 55.7%

         \[\leadsto d \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{h \cdot \ell}}} \]

      2. metadata-eval 55.7%

         \[\leadsto d \cdot \frac{\color{blue}{1}}{\sqrt{h \cdot \ell}} \]

      3. sqrt-unprod 70.4%

         \[\leadsto d \cdot \frac{1}{\color{blue}{\sqrt{h} \cdot \sqrt{\ell}}} \]

      4. *-commutative 70.4%

         \[\leadsto d \cdot \frac{1}{\color{blue}{\sqrt{\ell} \cdot \sqrt{h}}} \]

      5. div-inv 70.6%

         \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}} \]

      6. associate-/r* 67.7%

         \[\leadsto \color{blue}{\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}} \]

   7. Applied egg-rr 67.7%

      \[\leadsto \color{blue}{\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}} \]
   8. Step-by-step derivation

      1. associate-/l/ 70.6%

         \[\leadsto \color{blue}{\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}} \]

      2. *-commutative 70.6%

         \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell} \cdot \sqrt{h}}} \]

   9. Simplified 70.6%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 65.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.2 \cdot 10^{-24}:\\ \;\;\;\;\left(-d\right) \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \mathbf{elif}\;\ell \leq -1 \cdot 10^{-310}:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{0}}\\ \mathbf{elif}\;\ell \leq 1.45 \cdot 10^{+157}:\\ \;\;\;\;d \cdot \left({\left(\ell \cdot h\right)}^{-0.5} \cdot \left(1 + -0.5 \cdot \left({\left(\frac{M \cdot \left(D \cdot 0.5\right)}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 61.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -2.8 \cdot 10^{-25}:\\ \;\;\;\;\left(-d\right) \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \mathbf{elif}\;\ell \leq -1 \cdot 10^{-310}:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{0}}\\ \mathbf{elif}\;\ell \leq 6.6 \cdot 10^{+144}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left({\left(\frac{M\_m}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \cdot \frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

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 -2.8e-25)
   (* (- d) (pow (* l h) -0.5))
   (if (<= l -1e-310)
     (* d (sqrt (/ 1.0 0.0)))
     (if (<= l 6.6e+144)
       (*
        (- 1.0 (* 0.5 (* (pow (* (/ M_m 2.0) (/ D d)) 2.0) (/ h l))))
        (/ d (sqrt (* l h))))
       (/ d (* (sqrt l) (sqrt 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 <= -2.8e-25) {
		tmp = -d * pow((l * h), -0.5);
	} else if (l <= -1e-310) {
		tmp = d * sqrt((1.0 / 0.0));
	} else if (l <= 6.6e+144) {
		tmp = (1.0 - (0.5 * (pow(((M_m / 2.0) * (D / d)), 2.0) * (h / l)))) * (d / sqrt((l * h)));
	} else {
		tmp = d / (sqrt(l) * sqrt(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 <= (-2.8d-25)) then
        tmp = -d * ((l * h) ** (-0.5d0))
    else if (l <= (-1d-310)) then
        tmp = d * sqrt((1.0d0 / 0.0d0))
    else if (l <= 6.6d+144) then
        tmp = (1.0d0 - (0.5d0 * ((((m_m / 2.0d0) * (d_1 / d)) ** 2.0d0) * (h / l)))) * (d / sqrt((l * h)))
    else
        tmp = d / (sqrt(l) * sqrt(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 <= -2.8e-25) {
		tmp = -d * Math.pow((l * h), -0.5);
	} else if (l <= -1e-310) {
		tmp = d * Math.sqrt((1.0 / 0.0));
	} else if (l <= 6.6e+144) {
		tmp = (1.0 - (0.5 * (Math.pow(((M_m / 2.0) * (D / d)), 2.0) * (h / l)))) * (d / Math.sqrt((l * h)));
	} else {
		tmp = d / (Math.sqrt(l) * Math.sqrt(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 <= -2.8e-25:
		tmp = -d * math.pow((l * h), -0.5)
	elif l <= -1e-310:
		tmp = d * math.sqrt((1.0 / 0.0))
	elif l <= 6.6e+144:
		tmp = (1.0 - (0.5 * (math.pow(((M_m / 2.0) * (D / d)), 2.0) * (h / l)))) * (d / math.sqrt((l * h)))
	else:
		tmp = d / (math.sqrt(l) * math.sqrt(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 <= -2.8e-25)
		tmp = Float64(Float64(-d) * (Float64(l * h) ^ -0.5));
	elseif (l <= -1e-310)
		tmp = Float64(d * sqrt(Float64(1.0 / 0.0)));
	elseif (l <= 6.6e+144)
		tmp = Float64(Float64(1.0 - Float64(0.5 * Float64((Float64(Float64(M_m / 2.0) * Float64(D / d)) ^ 2.0) * Float64(h / l)))) * Float64(d / sqrt(Float64(l * h))));
	else
		tmp = Float64(d / Float64(sqrt(l) * sqrt(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 <= -2.8e-25)
		tmp = -d * ((l * h) ^ -0.5);
	elseif (l <= -1e-310)
		tmp = d * sqrt((1.0 / 0.0));
	elseif (l <= 6.6e+144)
		tmp = (1.0 - (0.5 * ((((M_m / 2.0) * (D / d)) ^ 2.0) * (h / l)))) * (d / sqrt((l * h)));
	else
		tmp = d / (sqrt(l) * sqrt(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, -2.8e-25], N[((-d) * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -1e-310], N[(d * N[Sqrt[N[(1.0 / 0.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 6.6e+144], N[(N[(1.0 - N[(0.5 * N[(N[Power[N[(N[(M$95$m / 2.0), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if l < -2.79999999999999988e-25

   1. Initial program 64.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 65.5%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 11.6%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

   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. *-commutative 0.0%

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]

      2. unpow-1 0.0%

         \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      3. metadata-eval 0.0%

         \[\leadsto \sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      4. pow-sqr 0.0%

         \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      5. rem-sqrt-square 0.0%

         \[\leadsto \color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      6. rem-square-sqrt 0.0%

         \[\leadsto \left|\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right| \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      7. fabs-sqr 0.0%

         \[\leadsto \color{blue}{\left(\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}\right)} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      8. rem-square-sqrt 0.0%

         \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      9. *-commutative 0.0%

         \[\leadsto {\left(h \cdot \ell\right)}^{-0.5} \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot d\right)} \]

      10. unpow2 0.0%

          \[\leadsto {\left(h \cdot \ell\right)}^{-0.5} \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right) \]

      11. rem-square-sqrt 53.0%

          \[\leadsto {\left(h \cdot \ell\right)}^{-0.5} \cdot \left(\color{blue}{-1} \cdot d\right) \]

      12. mul-1-neg 53.0%

          \[\leadsto {\left(h \cdot \ell\right)}^{-0.5} \cdot \color{blue}{\left(-d\right)} \]

   8. Simplified 53.0%

      \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot \left(-d\right)} \]

   if -2.79999999999999988e-25 < l < -9.999999999999969e-311

   1. Initial program 74.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 74.4%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 24.4%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. add-log-exp 38.2%

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\log \left(e^{h \cdot \ell}\right)}}} \]

      2. *-commutative 38.2%

         \[\leadsto d \cdot \sqrt{\frac{1}{\log \left(e^{\color{blue}{\ell \cdot h}}\right)}} \]

      3. exp-prod 52.1%

         \[\leadsto d \cdot \sqrt{\frac{1}{\log \color{blue}{\left({\left(e^{\ell}\right)}^{h}\right)}}} \]

   7. Applied egg-rr 52.1%

      \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\log \left({\left(e^{\ell}\right)}^{h}\right)}}} \]

   8. Taylor expanded in l around 0 52.1%

      \[\leadsto d \cdot \sqrt{\frac{1}{\log \color{blue}{1}}} \]

   if -9.999999999999969e-311 < l < 6.6e144

   1. Initial program 78.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 77.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 77.7%

         \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      2. sqrt-div 78.0%

         \[\leadsto \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      3. sqrt-div 83.5%

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{h}}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      4. frac-times 83.5%

         \[\leadsto \color{blue}{\frac{\sqrt{d} \cdot \sqrt{d}}{\sqrt{\ell} \cdot \sqrt{h}}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      5. add-sqr-sqrt 83.7%

         \[\leadsto \frac{\color{blue}{d}}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   6. Applied egg-rr 83.7%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   7. Taylor expanded in d around 0 78.1%

      \[\leadsto \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
   8. Step-by-step derivation

      1. associate-/r* 78.1%

         \[\leadsto \left(d \cdot \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      2. unpow1/2 78.1%

         \[\leadsto \left(d \cdot \color{blue}{{\left(\frac{\frac{1}{h}}{\ell}\right)}^{0.5}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      3. associate-/r* 78.1%

         \[\leadsto \left(d \cdot {\color{blue}{\left(\frac{1}{h \cdot \ell}\right)}}^{0.5}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      4. rem-exp-log 76.3%

         \[\leadsto \left(d \cdot {\left(\frac{1}{\color{blue}{e^{\log \left(h \cdot \ell\right)}}}\right)}^{0.5}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      5. exp-neg 76.3%

         \[\leadsto \left(d \cdot {\color{blue}{\left(e^{-\log \left(h \cdot \ell\right)}\right)}}^{0.5}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      6. exp-prod 76.3%

         \[\leadsto \left(d \cdot \color{blue}{e^{\left(-\log \left(h \cdot \ell\right)\right) \cdot 0.5}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      7. distribute-lft-neg-out 76.3%

         \[\leadsto \left(d \cdot e^{\color{blue}{-\log \left(h \cdot \ell\right) \cdot 0.5}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. exp-neg 76.3%

         \[\leadsto \left(d \cdot \color{blue}{\frac{1}{e^{\log \left(h \cdot \ell\right) \cdot 0.5}}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      9. exp-to-pow 78.1%

         \[\leadsto \left(d \cdot \frac{1}{\color{blue}{{\left(h \cdot \ell\right)}^{0.5}}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      10. unpow1/2 78.1%

          \[\leadsto \left(d \cdot \frac{1}{\color{blue}{\sqrt{h \cdot \ell}}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      11. associate-/l* 78.2%

          \[\leadsto \color{blue}{\frac{d \cdot 1}{\sqrt{h \cdot \ell}}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      12. associate-*l/ 78.2%

          \[\leadsto \color{blue}{\left(\frac{d}{\sqrt{h \cdot \ell}} \cdot 1\right)} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      13. *-rgt-identity 78.2%

          \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   9. Simplified 78.2%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   if 6.6e144 < l

   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 d around inf 59.6%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. sqrt-div 59.6%

         \[\leadsto d \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{h \cdot \ell}}} \]

      2. metadata-eval 59.6%

         \[\leadsto d \cdot \frac{\color{blue}{1}}{\sqrt{h \cdot \ell}} \]

      3. sqrt-unprod 73.0%

         \[\leadsto d \cdot \frac{1}{\color{blue}{\sqrt{h} \cdot \sqrt{\ell}}} \]

      4. *-commutative 73.0%

         \[\leadsto d \cdot \frac{1}{\color{blue}{\sqrt{\ell} \cdot \sqrt{h}}} \]

      5. div-inv 73.1%

         \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}} \]

      6. associate-/r* 70.4%

         \[\leadsto \color{blue}{\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}} \]

   7. Applied egg-rr 70.4%

      \[\leadsto \color{blue}{\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}} \]
   8. Step-by-step derivation

      1. associate-/l/ 73.1%

         \[\leadsto \color{blue}{\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}} \]

      2. *-commutative 73.1%

         \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell} \cdot \sqrt{h}}} \]

   9. Simplified 73.1%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 64.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.8 \cdot 10^{-25}:\\ \;\;\;\;\left(-d\right) \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \mathbf{elif}\;\ell \leq -1 \cdot 10^{-310}:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{0}}\\ \mathbf{elif}\;\ell \leq 6.6 \cdot 10^{+144}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \cdot \frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 73.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;h \leq -8.2 \cdot 10^{+126}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + h \cdot \left(\left(0.5 \cdot \left(D \cdot M\_m\right)\right) \cdot \frac{-0.25 \cdot \frac{D \cdot M\_m}{\ell \cdot d}}{d}\right)\right)\right)\\ \mathbf{elif}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(0.5 \cdot \left({\left(\frac{M\_m}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right) + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right) \cdot \left(1 - 0.5 \cdot \frac{h \cdot {\left(M\_m \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell}\right)\\ \end{array} \end{array} \]

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
 (if (<= h -8.2e+126)
   (*
    (sqrt (/ d l))
    (*
     (sqrt (/ d h))
     (+
      1.0
      (* h (* (* 0.5 (* D M_m)) (/ (* -0.25 (/ (* D M_m) (* l d))) d))))))
   (if (<= h -5e-310)
     (*
      (* d (sqrt (/ 1.0 (* l h))))
      (+ (* 0.5 (* (pow (* (/ M_m 2.0) (/ D d)) 2.0) (/ h l))) -1.0))
     (*
      (* d (pow (* l h) -0.5))
      (- 1.0 (* 0.5 (/ (* h (pow (* M_m (* 0.5 (/ D d))) 2.0)) 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 (h <= -8.2e+126) {
		tmp = sqrt((d / l)) * (sqrt((d / h)) * (1.0 + (h * ((0.5 * (D * M_m)) * ((-0.25 * ((D * M_m) / (l * d))) / d)))));
	} else if (h <= -5e-310) {
		tmp = (d * sqrt((1.0 / (l * h)))) * ((0.5 * (pow(((M_m / 2.0) * (D / d)), 2.0) * (h / l))) + -1.0);
	} else {
		tmp = (d * pow((l * h), -0.5)) * (1.0 - (0.5 * ((h * pow((M_m * (0.5 * (D / d))), 2.0)) / 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 (h <= (-8.2d+126)) then
        tmp = sqrt((d / l)) * (sqrt((d / h)) * (1.0d0 + (h * ((0.5d0 * (d_1 * m_m)) * (((-0.25d0) * ((d_1 * m_m) / (l * d))) / d)))))
    else if (h <= (-5d-310)) then
        tmp = (d * sqrt((1.0d0 / (l * h)))) * ((0.5d0 * ((((m_m / 2.0d0) * (d_1 / d)) ** 2.0d0) * (h / l))) + (-1.0d0))
    else
        tmp = (d * ((l * h) ** (-0.5d0))) * (1.0d0 - (0.5d0 * ((h * ((m_m * (0.5d0 * (d_1 / d))) ** 2.0d0)) / 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 (h <= -8.2e+126) {
		tmp = Math.sqrt((d / l)) * (Math.sqrt((d / h)) * (1.0 + (h * ((0.5 * (D * M_m)) * ((-0.25 * ((D * M_m) / (l * d))) / d)))));
	} else if (h <= -5e-310) {
		tmp = (d * Math.sqrt((1.0 / (l * h)))) * ((0.5 * (Math.pow(((M_m / 2.0) * (D / d)), 2.0) * (h / l))) + -1.0);
	} else {
		tmp = (d * Math.pow((l * h), -0.5)) * (1.0 - (0.5 * ((h * Math.pow((M_m * (0.5 * (D / d))), 2.0)) / 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 h <= -8.2e+126:
		tmp = math.sqrt((d / l)) * (math.sqrt((d / h)) * (1.0 + (h * ((0.5 * (D * M_m)) * ((-0.25 * ((D * M_m) / (l * d))) / d)))))
	elif h <= -5e-310:
		tmp = (d * math.sqrt((1.0 / (l * h)))) * ((0.5 * (math.pow(((M_m / 2.0) * (D / d)), 2.0) * (h / l))) + -1.0)
	else:
		tmp = (d * math.pow((l * h), -0.5)) * (1.0 - (0.5 * ((h * math.pow((M_m * (0.5 * (D / d))), 2.0)) / 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 (h <= -8.2e+126)
		tmp = Float64(sqrt(Float64(d / l)) * Float64(sqrt(Float64(d / h)) * Float64(1.0 + Float64(h * Float64(Float64(0.5 * Float64(D * M_m)) * Float64(Float64(-0.25 * Float64(Float64(D * M_m) / Float64(l * d))) / d))))));
	elseif (h <= -5e-310)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(l * h)))) * Float64(Float64(0.5 * Float64((Float64(Float64(M_m / 2.0) * Float64(D / d)) ^ 2.0) * Float64(h / l))) + -1.0));
	else
		tmp = Float64(Float64(d * (Float64(l * h) ^ -0.5)) * Float64(1.0 - Float64(0.5 * Float64(Float64(h * (Float64(M_m * Float64(0.5 * Float64(D / d))) ^ 2.0)) / 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 (h <= -8.2e+126)
		tmp = sqrt((d / l)) * (sqrt((d / h)) * (1.0 + (h * ((0.5 * (D * M_m)) * ((-0.25 * ((D * M_m) / (l * d))) / d)))));
	elseif (h <= -5e-310)
		tmp = (d * sqrt((1.0 / (l * h)))) * ((0.5 * ((((M_m / 2.0) * (D / d)) ^ 2.0) * (h / l))) + -1.0);
	else
		tmp = (d * ((l * h) ^ -0.5)) * (1.0 - (0.5 * ((h * ((M_m * (0.5 * (D / d))) ^ 2.0)) / 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[h, -8.2e+126], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(h * N[(N[(0.5 * N[(D * M$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(-0.25 * N[(N[(D * M$95$m), $MachinePrecision] / N[(l * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, -5e-310], N[(N[(d * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 * N[(N[Power[N[(N[(M$95$m / 2.0), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(d * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(h * N[Power[N[(M$95$m * N[(0.5 * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if h < -8.2000000000000001e126

   1. Initial program 60.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 58.1%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(D \cdot \frac{\frac{M}{2}}{d}\right)}^{2} \cdot -0.5\right)\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*l/ 60.8%

         \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \color{blue}{\frac{h \cdot \left({\left(D \cdot \frac{\frac{M}{2}}{d}\right)}^{2} \cdot -0.5\right)}{\ell}}\right)\right) \]

      2. *-commutative 60.8%

         \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h \cdot \color{blue}{\left(-0.5 \cdot {\left(D \cdot \frac{\frac{M}{2}}{d}\right)}^{2}\right)}}{\ell}\right)\right) \]

      3. associate-*r/ 60.8%

         \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h \cdot \left(-0.5 \cdot {\color{blue}{\left(\frac{D \cdot \frac{M}{2}}{d}\right)}}^{2}\right)}{\ell}\right)\right) \]

      4. *-un-lft-identity 60.8%

         \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h \cdot \left(-0.5 \cdot {\left(\frac{D \cdot \frac{M}{2}}{\color{blue}{1 \cdot d}}\right)}^{2}\right)}{\ell}\right)\right) \]

      5. times-frac 60.8%

         \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h \cdot \left(-0.5 \cdot {\color{blue}{\left(\frac{D}{1} \cdot \frac{\frac{M}{2}}{d}\right)}}^{2}\right)}{\ell}\right)\right) \]

      6. associate-/l/ 60.8%

         \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h \cdot \left(-0.5 \cdot {\left(\frac{D}{1} \cdot \color{blue}{\frac{M}{d \cdot 2}}\right)}^{2}\right)}{\ell}\right)\right) \]

      7. *-commutative 60.8%

         \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h \cdot \left(-0.5 \cdot {\left(\frac{D}{1} \cdot \frac{M}{\color{blue}{2 \cdot d}}\right)}^{2}\right)}{\ell}\right)\right) \]

      8. times-frac 60.8%

         \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h \cdot \left(-0.5 \cdot {\color{blue}{\left(\frac{D \cdot M}{1 \cdot \left(2 \cdot d\right)}\right)}}^{2}\right)}{\ell}\right)\right) \]

      9. *-un-lft-identity 60.8%

         \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h \cdot \left(-0.5 \cdot {\left(\frac{D \cdot M}{\color{blue}{2 \cdot d}}\right)}^{2}\right)}{\ell}\right)\right) \]

      10. *-commutative 60.8%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h \cdot \left(-0.5 \cdot {\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2}\right)}{\ell}\right)\right) \]

      11. associate-/l* 60.8%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h \cdot \left(-0.5 \cdot {\color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)}}^{2}\right)}{\ell}\right)\right) \]

      12. *-un-lft-identity 60.8%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h \cdot \left(-0.5 \cdot {\left(M \cdot \frac{\color{blue}{1 \cdot D}}{2 \cdot d}\right)}^{2}\right)}{\ell}\right)\right) \]

      13. times-frac 60.8%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h \cdot \left(-0.5 \cdot {\left(M \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{D}{d}\right)}\right)}^{2}\right)}{\ell}\right)\right) \]

      14. metadata-eval 60.8%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h \cdot \left(-0.5 \cdot {\left(M \cdot \left(\color{blue}{0.5} \cdot \frac{D}{d}\right)\right)}^{2}\right)}{\ell}\right)\right) \]

   6. Applied egg-rr 60.8%

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \color{blue}{\frac{h \cdot \left(-0.5 \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right)}{\ell}}\right)\right) \]
   7. Step-by-step derivation

      1. associate-/l* 69.7%

         \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \color{blue}{h \cdot \frac{-0.5 \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell}}\right)\right) \]

      2. *-commutative 69.7%

         \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + h \cdot \frac{\color{blue}{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot -0.5}}{\ell}\right)\right) \]

      3. associate-/l* 69.7%

         \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + h \cdot \color{blue}{\left({\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \frac{-0.5}{\ell}\right)}\right)\right) \]

      4. *-commutative 69.7%

         \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + h \cdot \left({\left(M \cdot \color{blue}{\left(\frac{D}{d} \cdot 0.5\right)}\right)}^{2} \cdot \frac{-0.5}{\ell}\right)\right)\right) \]

      5. metadata-eval 69.7%

         \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + h \cdot \left({\left(M \cdot \left(\frac{D}{d} \cdot \color{blue}{\frac{1}{2}}\right)\right)}^{2} \cdot \frac{-0.5}{\ell}\right)\right)\right) \]

      6. times-frac 69.7%

         \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + h \cdot \left({\left(M \cdot \color{blue}{\frac{D \cdot 1}{d \cdot 2}}\right)}^{2} \cdot \frac{-0.5}{\ell}\right)\right)\right) \]

      7. *-rgt-identity 69.7%

         \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + h \cdot \left({\left(M \cdot \frac{\color{blue}{D}}{d \cdot 2}\right)}^{2} \cdot \frac{-0.5}{\ell}\right)\right)\right) \]

      8. associate-/l* 69.7%

         \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + h \cdot \left({\color{blue}{\left(\frac{M \cdot D}{d \cdot 2}\right)}}^{2} \cdot \frac{-0.5}{\ell}\right)\right)\right) \]

      9. *-commutative 69.7%

         \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + h \cdot \left({\left(\frac{\color{blue}{D \cdot M}}{d \cdot 2}\right)}^{2} \cdot \frac{-0.5}{\ell}\right)\right)\right) \]

      10. associate-/l* 67.5%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + h \cdot \left({\color{blue}{\left(D \cdot \frac{M}{d \cdot 2}\right)}}^{2} \cdot \frac{-0.5}{\ell}\right)\right)\right) \]

      11. *-commutative 67.5%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + h \cdot \left({\left(D \cdot \frac{M}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot \frac{-0.5}{\ell}\right)\right)\right) \]

   8. Simplified 67.5%

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \color{blue}{h \cdot \left({\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2} \cdot \frac{-0.5}{\ell}\right)}\right)\right) \]
   9. Step-by-step derivation

      1. unpow-prod-down 54.4%

         \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + h \cdot \left(\color{blue}{\left({D}^{2} \cdot {\left(\frac{M}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{-0.5}{\ell}\right)\right)\right) \]

      2. add-sqr-sqrt 54.4%

         \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + h \cdot \left(\color{blue}{\left(\sqrt{{D}^{2} \cdot {\left(\frac{M}{2 \cdot d}\right)}^{2}} \cdot \sqrt{{D}^{2} \cdot {\left(\frac{M}{2 \cdot d}\right)}^{2}}\right)} \cdot \frac{-0.5}{\ell}\right)\right)\right) \]

      3. unpow-prod-down 54.4%

         \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + h \cdot \left(\left(\sqrt{\color{blue}{{\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}}} \cdot \sqrt{{D}^{2} \cdot {\left(\frac{M}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{-0.5}{\ell}\right)\right)\right) \]

      4. sqrt-pow1 47.7%

         \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + h \cdot \left(\left(\color{blue}{{\left(D \cdot \frac{M}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}} \cdot \sqrt{{D}^{2} \cdot {\left(\frac{M}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{-0.5}{\ell}\right)\right)\right) \]

      5. metadata-eval 47.7%

         \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + h \cdot \left(\left({\left(D \cdot \frac{M}{2 \cdot d}\right)}^{\color{blue}{1}} \cdot \sqrt{{D}^{2} \cdot {\left(\frac{M}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{-0.5}{\ell}\right)\right)\right) \]

      6. pow1 47.7%

         \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + h \cdot \left(\left(\color{blue}{\left(D \cdot \frac{M}{2 \cdot d}\right)} \cdot \sqrt{{D}^{2} \cdot {\left(\frac{M}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{-0.5}{\ell}\right)\right)\right) \]

      7. associate-/r* 47.7%

         \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + h \cdot \left(\left(\left(D \cdot \color{blue}{\frac{\frac{M}{2}}{d}}\right) \cdot \sqrt{{D}^{2} \cdot {\left(\frac{M}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{-0.5}{\ell}\right)\right)\right) \]

      8. div-inv 47.7%

         \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + h \cdot \left(\left(\left(D \cdot \frac{\color{blue}{M \cdot \frac{1}{2}}}{d}\right) \cdot \sqrt{{D}^{2} \cdot {\left(\frac{M}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{-0.5}{\ell}\right)\right)\right) \]

      9. metadata-eval 47.7%

         \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + h \cdot \left(\left(\left(D \cdot \frac{M \cdot \color{blue}{0.5}}{d}\right) \cdot \sqrt{{D}^{2} \cdot {\left(\frac{M}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{-0.5}{\ell}\right)\right)\right) \]

      10. unpow-prod-down 54.3%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + h \cdot \left(\left(\left(D \cdot \frac{M \cdot 0.5}{d}\right) \cdot \sqrt{\color{blue}{{\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}}}\right) \cdot \frac{-0.5}{\ell}\right)\right)\right) \]

      11. sqrt-pow1 67.5%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + h \cdot \left(\left(\left(D \cdot \frac{M \cdot 0.5}{d}\right) \cdot \color{blue}{{\left(D \cdot \frac{M}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}}\right) \cdot \frac{-0.5}{\ell}\right)\right)\right) \]

      12. metadata-eval 67.5%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + h \cdot \left(\left(\left(D \cdot \frac{M \cdot 0.5}{d}\right) \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{\color{blue}{1}}\right) \cdot \frac{-0.5}{\ell}\right)\right)\right) \]

      13. pow1 67.5%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + h \cdot \left(\left(\left(D \cdot \frac{M \cdot 0.5}{d}\right) \cdot \color{blue}{\left(D \cdot \frac{M}{2 \cdot d}\right)}\right) \cdot \frac{-0.5}{\ell}\right)\right)\right) \]

      14. associate-/r* 67.5%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + h \cdot \left(\left(\left(D \cdot \frac{M \cdot 0.5}{d}\right) \cdot \left(D \cdot \color{blue}{\frac{\frac{M}{2}}{d}}\right)\right) \cdot \frac{-0.5}{\ell}\right)\right)\right) \]

      15. div-inv 67.5%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + h \cdot \left(\left(\left(D \cdot \frac{M \cdot 0.5}{d}\right) \cdot \left(D \cdot \frac{\color{blue}{M \cdot \frac{1}{2}}}{d}\right)\right) \cdot \frac{-0.5}{\ell}\right)\right)\right) \]

      16. metadata-eval 67.5%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + h \cdot \left(\left(\left(D \cdot \frac{M \cdot 0.5}{d}\right) \cdot \left(D \cdot \frac{M \cdot \color{blue}{0.5}}{d}\right)\right) \cdot \frac{-0.5}{\ell}\right)\right)\right) \]

   10. Applied egg-rr 67.5%

       \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + h \cdot \left(\color{blue}{\left(\left(D \cdot \frac{M \cdot 0.5}{d}\right) \cdot \left(D \cdot \frac{M \cdot 0.5}{d}\right)\right)} \cdot \frac{-0.5}{\ell}\right)\right)\right) \]
   11. Step-by-step derivation

       1. associate-*l* 0.0%

          \[\leadsto \frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + h \cdot \color{blue}{\left(\left(D \cdot \frac{M \cdot 0.5}{d}\right) \cdot \left(\left(D \cdot \frac{M \cdot 0.5}{d}\right) \cdot \frac{-0.5}{\ell}\right)\right)}\right)\right) \]

       2. associate-*r/ 0.0%

          \[\leadsto \frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + h \cdot \left(\color{blue}{\frac{D \cdot \left(M \cdot 0.5\right)}{d}} \cdot \left(\left(D \cdot \frac{M \cdot 0.5}{d}\right) \cdot \frac{-0.5}{\ell}\right)\right)\right)\right) \]

       3. associate-*l/ 0.0%

          \[\leadsto \frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + h \cdot \color{blue}{\frac{\left(D \cdot \left(M \cdot 0.5\right)\right) \cdot \left(\left(D \cdot \frac{M \cdot 0.5}{d}\right) \cdot \frac{-0.5}{\ell}\right)}{d}}\right)\right) \]

       4. associate-*l* 0.0%

          \[\leadsto \frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + h \cdot \frac{\left(D \cdot \left(M \cdot 0.5\right)\right) \cdot \color{blue}{\left(D \cdot \left(\frac{M \cdot 0.5}{d} \cdot \frac{-0.5}{\ell}\right)\right)}}{d}\right)\right) \]

       5. associate-/l* 0.0%

          \[\leadsto \frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + h \cdot \frac{\left(D \cdot \left(M \cdot 0.5\right)\right) \cdot \left(D \cdot \left(\color{blue}{\left(M \cdot \frac{0.5}{d}\right)} \cdot \frac{-0.5}{\ell}\right)\right)}{d}\right)\right) \]

   12. Applied egg-rr 63.3%

       \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + h \cdot \color{blue}{\frac{\left(D \cdot \left(M \cdot 0.5\right)\right) \cdot \left(D \cdot \left(\left(M \cdot \frac{0.5}{d}\right) \cdot \frac{-0.5}{\ell}\right)\right)}{d}}\right)\right) \]
   13. Step-by-step derivation

       1. associate-/l* 63.2%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + h \cdot \color{blue}{\left(\left(D \cdot \left(M \cdot 0.5\right)\right) \cdot \frac{D \cdot \left(\left(M \cdot \frac{0.5}{d}\right) \cdot \frac{-0.5}{\ell}\right)}{d}\right)}\right)\right) \]

       2. associate-*r* 63.2%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + h \cdot \left(\color{blue}{\left(\left(D \cdot M\right) \cdot 0.5\right)} \cdot \frac{D \cdot \left(\left(M \cdot \frac{0.5}{d}\right) \cdot \frac{-0.5}{\ell}\right)}{d}\right)\right)\right) \]

       3. associate-*r* 67.4%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + h \cdot \left(\left(\left(D \cdot M\right) \cdot 0.5\right) \cdot \frac{\color{blue}{\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right) \cdot \frac{-0.5}{\ell}}}{d}\right)\right)\right) \]

       4. associate-*r* 69.6%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + h \cdot \left(\left(\left(D \cdot M\right) \cdot 0.5\right) \cdot \frac{\color{blue}{\left(\left(D \cdot M\right) \cdot \frac{0.5}{d}\right)} \cdot \frac{-0.5}{\ell}}{d}\right)\right)\right) \]

   14. Simplified 69.6%

       \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + h \cdot \color{blue}{\left(\left(\left(D \cdot M\right) \cdot 0.5\right) \cdot \frac{\left(\left(D \cdot M\right) \cdot \frac{0.5}{d}\right) \cdot \frac{-0.5}{\ell}}{d}\right)}\right)\right) \]

   15. Taylor expanded in D around 0 69.6%

       \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + h \cdot \left(\left(\left(D \cdot M\right) \cdot 0.5\right) \cdot \frac{\color{blue}{-0.25 \cdot \frac{D \cdot M}{d \cdot \ell}}}{d}\right)\right)\right) \]

   if -8.2000000000000001e126 < h < -4.999999999999985e-310

   1. Initial program 73.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 74.3%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. frac-2neg 72.8%

         \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\color{blue}{\frac{-d}{-h}}} \cdot \left(1 + h \cdot \left(\left(\left(D \cdot \frac{M \cdot 0.5}{d}\right) \cdot \left(D \cdot \frac{M \cdot 0.5}{d}\right)\right) \cdot \frac{-0.5}{\ell}\right)\right)\right) \]

      2. sqrt-div 81.2%

         \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\color{blue}{\frac{\sqrt{-d}}{\sqrt{-h}}} \cdot \left(1 + h \cdot \left(\left(\left(D \cdot \frac{M \cdot 0.5}{d}\right) \cdot \left(D \cdot \frac{M \cdot 0.5}{d}\right)\right) \cdot \frac{-0.5}{\ell}\right)\right)\right) \]

   6. Applied egg-rr 82.7%

      \[\leadsto \left(\color{blue}{\frac{\sqrt{-d}}{\sqrt{-h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   7. Taylor expanded in d around -inf 85.9%

      \[\leadsto \color{blue}{\left(-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)\right)} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   if -4.999999999999985e-310 < h

   1. Initial program 72.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 71.3%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 71.3%

         \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      2. sqrt-div 74.5%

         \[\leadsto \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      3. sqrt-div 80.8%

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{h}}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      4. frac-times 80.8%

         \[\leadsto \color{blue}{\frac{\sqrt{d} \cdot \sqrt{d}}{\sqrt{\ell} \cdot \sqrt{h}}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      5. add-sqr-sqrt 80.9%

         \[\leadsto \frac{\color{blue}{d}}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   6. Applied egg-rr 80.9%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
   7. Step-by-step derivation

      1. div-inv 80.9%

         \[\leadsto \color{blue}{\left(d \cdot \frac{1}{\sqrt{\ell} \cdot \sqrt{h}}\right)} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      2. *-commutative 80.9%

         \[\leadsto \left(d \cdot \frac{1}{\color{blue}{\sqrt{h} \cdot \sqrt{\ell}}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      3. metadata-eval 80.9%

         \[\leadsto \left(d \cdot \frac{\color{blue}{\sqrt{1}}}{\sqrt{h} \cdot \sqrt{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      4. sqrt-unprod 71.4%

         \[\leadsto \left(d \cdot \frac{\sqrt{1}}{\color{blue}{\sqrt{h \cdot \ell}}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      5. sqrt-div 71.5%

         \[\leadsto \left(d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      6. *-commutative 71.5%

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot d\right)} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      7. inv-pow 71.5%

         \[\leadsto \left(\sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \cdot d\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. sqrt-pow1 71.5%

         \[\leadsto \left(\color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}} \cdot d\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      9. metadata-eval 71.5%

         \[\leadsto \left({\left(h \cdot \ell\right)}^{\color{blue}{-0.5}} \cdot d\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   8. Applied egg-rr 71.5%

      \[\leadsto \color{blue}{\left({\left(h \cdot \ell\right)}^{-0.5} \cdot d\right)} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
   9. Step-by-step derivation

      1. associate-*r/ 75.5%

         \[\leadsto \left({\left(h \cdot \ell\right)}^{-0.5} \cdot d\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot h}{\ell}}\right) \]

      2. frac-times 76.8%

         \[\leadsto \left({\left(h \cdot \ell\right)}^{-0.5} \cdot d\right) \cdot \left(1 - 0.5 \cdot \frac{{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot h}{\ell}\right) \]

      3. associate-/l* 75.5%

         \[\leadsto \left({\left(h \cdot \ell\right)}^{-0.5} \cdot d\right) \cdot \left(1 - 0.5 \cdot \frac{{\color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)}}^{2} \cdot h}{\ell}\right) \]

      4. *-un-lft-identity 75.5%

         \[\leadsto \left({\left(h \cdot \ell\right)}^{-0.5} \cdot d\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(M \cdot \frac{\color{blue}{1 \cdot D}}{2 \cdot d}\right)}^{2} \cdot h}{\ell}\right) \]

      5. times-frac 75.5%

         \[\leadsto \left({\left(h \cdot \ell\right)}^{-0.5} \cdot d\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(M \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{D}{d}\right)}\right)}^{2} \cdot h}{\ell}\right) \]

      6. metadata-eval 75.5%

         \[\leadsto \left({\left(h \cdot \ell\right)}^{-0.5} \cdot d\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(M \cdot \left(\color{blue}{0.5} \cdot \frac{D}{d}\right)\right)}^{2} \cdot h}{\ell}\right) \]

   10. Applied egg-rr 75.5%

       \[\leadsto \left({\left(h \cdot \ell\right)}^{-0.5} \cdot d\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot h}{\ell}}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -8.2 \cdot 10^{+126}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + h \cdot \left(\left(0.5 \cdot \left(D \cdot M\right)\right) \cdot \frac{-0.25 \cdot \frac{D \cdot M}{\ell \cdot d}}{d}\right)\right)\right)\\ \mathbf{elif}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right) + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right) \cdot \left(1 - 0.5 \cdot \frac{h \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 49.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -3.25 \cdot 10^{-24}:\\ \;\;\;\;\left(-d\right) \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \mathbf{elif}\;\ell \leq -1 \cdot 10^{-310}:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

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 -3.25e-24)
   (* (- d) (pow (* l h) -0.5))
   (if (<= l -1e-310) (* d (sqrt (/ 1.0 0.0))) (/ d (* (sqrt l) (sqrt 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 <= -3.25e-24) {
		tmp = -d * pow((l * h), -0.5);
	} else if (l <= -1e-310) {
		tmp = d * sqrt((1.0 / 0.0));
	} else {
		tmp = d / (sqrt(l) * sqrt(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 <= (-3.25d-24)) then
        tmp = -d * ((l * h) ** (-0.5d0))
    else if (l <= (-1d-310)) then
        tmp = d * sqrt((1.0d0 / 0.0d0))
    else
        tmp = d / (sqrt(l) * sqrt(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 <= -3.25e-24) {
		tmp = -d * Math.pow((l * h), -0.5);
	} else if (l <= -1e-310) {
		tmp = d * Math.sqrt((1.0 / 0.0));
	} else {
		tmp = d / (Math.sqrt(l) * Math.sqrt(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 <= -3.25e-24:
		tmp = -d * math.pow((l * h), -0.5)
	elif l <= -1e-310:
		tmp = d * math.sqrt((1.0 / 0.0))
	else:
		tmp = d / (math.sqrt(l) * math.sqrt(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 <= -3.25e-24)
		tmp = Float64(Float64(-d) * (Float64(l * h) ^ -0.5));
	elseif (l <= -1e-310)
		tmp = Float64(d * sqrt(Float64(1.0 / 0.0)));
	else
		tmp = Float64(d / Float64(sqrt(l) * sqrt(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 <= -3.25e-24)
		tmp = -d * ((l * h) ^ -0.5);
	elseif (l <= -1e-310)
		tmp = d * sqrt((1.0 / 0.0));
	else
		tmp = d / (sqrt(l) * sqrt(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, -3.25e-24], N[((-d) * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -1e-310], N[(d * N[Sqrt[N[(1.0 / 0.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -3.25e-24

   1. Initial program 64.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 65.5%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 11.6%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

   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. *-commutative 0.0%

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]

      2. unpow-1 0.0%

         \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      3. metadata-eval 0.0%

         \[\leadsto \sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      4. pow-sqr 0.0%

         \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      5. rem-sqrt-square 0.0%

         \[\leadsto \color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      6. rem-square-sqrt 0.0%

         \[\leadsto \left|\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right| \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      7. fabs-sqr 0.0%

         \[\leadsto \color{blue}{\left(\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}\right)} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      8. rem-square-sqrt 0.0%

         \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      9. *-commutative 0.0%

         \[\leadsto {\left(h \cdot \ell\right)}^{-0.5} \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot d\right)} \]

      10. unpow2 0.0%

          \[\leadsto {\left(h \cdot \ell\right)}^{-0.5} \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right) \]

      11. rem-square-sqrt 53.0%

          \[\leadsto {\left(h \cdot \ell\right)}^{-0.5} \cdot \left(\color{blue}{-1} \cdot d\right) \]

      12. mul-1-neg 53.0%

          \[\leadsto {\left(h \cdot \ell\right)}^{-0.5} \cdot \color{blue}{\left(-d\right)} \]

   8. Simplified 53.0%

      \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot \left(-d\right)} \]

   if -3.25e-24 < l < -9.999999999999969e-311

   1. Initial program 74.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 74.4%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 24.4%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. add-log-exp 38.2%

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\log \left(e^{h \cdot \ell}\right)}}} \]

      2. *-commutative 38.2%

         \[\leadsto d \cdot \sqrt{\frac{1}{\log \left(e^{\color{blue}{\ell \cdot h}}\right)}} \]

      3. exp-prod 52.1%

         \[\leadsto d \cdot \sqrt{\frac{1}{\log \color{blue}{\left({\left(e^{\ell}\right)}^{h}\right)}}} \]

   7. Applied egg-rr 52.1%

      \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\log \left({\left(e^{\ell}\right)}^{h}\right)}}} \]

   8. Taylor expanded in l around 0 52.1%

      \[\leadsto d \cdot \sqrt{\frac{1}{\log \color{blue}{1}}} \]

   if -9.999999999999969e-311 < l

   1. Initial program 72.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 71.3%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 47.2%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. sqrt-div 47.2%

         \[\leadsto d \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{h \cdot \ell}}} \]

      2. metadata-eval 47.2%

         \[\leadsto d \cdot \frac{\color{blue}{1}}{\sqrt{h \cdot \ell}} \]

      3. sqrt-unprod 53.3%

         \[\leadsto d \cdot \frac{1}{\color{blue}{\sqrt{h} \cdot \sqrt{\ell}}} \]

      4. *-commutative 53.3%

         \[\leadsto d \cdot \frac{1}{\color{blue}{\sqrt{\ell} \cdot \sqrt{h}}} \]

      5. div-inv 53.4%

         \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}} \]

      6. associate-/r* 52.7%

         \[\leadsto \color{blue}{\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}} \]

   7. Applied egg-rr 52.7%

      \[\leadsto \color{blue}{\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}} \]
   8. Step-by-step derivation

      1. associate-/l/ 53.4%

         \[\leadsto \color{blue}{\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}} \]

      2. *-commutative 53.4%

         \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell} \cdot \sqrt{h}}} \]

   9. Simplified 53.4%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 53.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -3.25 \cdot 10^{-24}:\\ \;\;\;\;\left(-d\right) \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \mathbf{elif}\;\ell \leq -1 \cdot 10^{-310}:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 43.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;M\_m \leq 1.4 \cdot 10^{-26}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{0}}\\ \end{array} \end{array} \]

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 1.4e-26)
   (* (sqrt (/ d h)) (sqrt (/ d l)))
   (* d (sqrt (/ 1.0 0.0)))))

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 tmp;
	if (M_m <= 1.4e-26) {
		tmp = sqrt((d / h)) * sqrt((d / l));
	} else {
		tmp = d * sqrt((1.0 / 0.0));
	}
	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) :: tmp
    if (m_m <= 1.4d-26) then
        tmp = sqrt((d / h)) * sqrt((d / l))
    else
        tmp = d * sqrt((1.0d0 / 0.0d0))
    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 <= 1.4e-26) {
		tmp = Math.sqrt((d / h)) * Math.sqrt((d / l));
	} else {
		tmp = d * Math.sqrt((1.0 / 0.0));
	}
	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):
	tmp = 0
	if M_m <= 1.4e-26:
		tmp = math.sqrt((d / h)) * math.sqrt((d / l))
	else:
		tmp = d * math.sqrt((1.0 / 0.0))
	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)
	tmp = 0.0
	if (M_m <= 1.4e-26)
		tmp = Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l)));
	else
		tmp = Float64(d * sqrt(Float64(1.0 / 0.0)));
	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)
	tmp = 0.0;
	if (M_m <= 1.4e-26)
		tmp = sqrt((d / h)) * sqrt((d / l));
	else
		tmp = d * sqrt((1.0 / 0.0));
	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_] := If[LessEqual[M$95$m, 1.4e-26], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d * N[Sqrt[N[(1.0 / 0.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 1.4000000000000001e-26

   1. Initial program 70.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 70.6%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 50.3%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]

   if 1.4000000000000001e-26 < M

   1. Initial program 70.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 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 20.2%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. add-log-exp 21.1%

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\log \left(e^{h \cdot \ell}\right)}}} \]

      2. *-commutative 21.1%

         \[\leadsto d \cdot \sqrt{\frac{1}{\log \left(e^{\color{blue}{\ell \cdot h}}\right)}} \]

      3. exp-prod 28.1%

         \[\leadsto d \cdot \sqrt{\frac{1}{\log \color{blue}{\left({\left(e^{\ell}\right)}^{h}\right)}}} \]

   7. Applied egg-rr 28.1%

      \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\log \left({\left(e^{\ell}\right)}^{h}\right)}}} \]

   8. Taylor expanded in l around 0 33.0%

      \[\leadsto d \cdot \sqrt{\frac{1}{\log \color{blue}{1}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 45.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 1.4 \cdot 10^{-26}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{0}}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 45.6% accurate, 2.9× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -4.8 \cdot 10^{-25}:\\ \;\;\;\;\left(-d\right) \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \mathbf{elif}\;\ell \leq -1 \cdot 10^{-310}:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}}\\ \end{array} \end{array} \]

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 (<= l -4.8e-25)
   (* (- d) (pow (* l h) -0.5))
   (if (<= l -1e-310) (* d (sqrt (/ 1.0 0.0))) (/ 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 (l <= -4.8e-25) {
		tmp = -d * pow((l * h), -0.5);
	} else if (l <= -1e-310) {
		tmp = d * sqrt((1.0 / 0.0));
	} else {
		tmp = 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 (l <= (-4.8d-25)) then
        tmp = -d * ((l * h) ** (-0.5d0))
    else if (l <= (-1d-310)) then
        tmp = d * sqrt((1.0d0 / 0.0d0))
    else
        tmp = 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 (l <= -4.8e-25) {
		tmp = -d * Math.pow((l * h), -0.5);
	} else if (l <= -1e-310) {
		tmp = d * Math.sqrt((1.0 / 0.0));
	} else {
		tmp = 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 l <= -4.8e-25:
		tmp = -d * math.pow((l * h), -0.5)
	elif l <= -1e-310:
		tmp = d * math.sqrt((1.0 / 0.0))
	else:
		tmp = 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 (l <= -4.8e-25)
		tmp = Float64(Float64(-d) * (Float64(l * h) ^ -0.5));
	elseif (l <= -1e-310)
		tmp = Float64(d * sqrt(Float64(1.0 / 0.0)));
	else
		tmp = 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 (l <= -4.8e-25)
		tmp = -d * ((l * h) ^ -0.5);
	elseif (l <= -1e-310)
		tmp = d * sqrt((1.0 / 0.0));
	else
		tmp = 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[l, -4.8e-25], N[((-d) * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -1e-310], N[(d * N[Sqrt[N[(1.0 / 0.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -4.80000000000000018e-25

   1. Initial program 64.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 65.5%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 11.6%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

   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. *-commutative 0.0%

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]

      2. unpow-1 0.0%

         \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      3. metadata-eval 0.0%

         \[\leadsto \sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      4. pow-sqr 0.0%

         \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      5. rem-sqrt-square 0.0%

         \[\leadsto \color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      6. rem-square-sqrt 0.0%

         \[\leadsto \left|\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right| \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      7. fabs-sqr 0.0%

         \[\leadsto \color{blue}{\left(\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}\right)} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      8. rem-square-sqrt 0.0%

         \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      9. *-commutative 0.0%

         \[\leadsto {\left(h \cdot \ell\right)}^{-0.5} \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot d\right)} \]

      10. unpow2 0.0%

          \[\leadsto {\left(h \cdot \ell\right)}^{-0.5} \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right) \]

      11. rem-square-sqrt 53.0%

          \[\leadsto {\left(h \cdot \ell\right)}^{-0.5} \cdot \left(\color{blue}{-1} \cdot d\right) \]

      12. mul-1-neg 53.0%

          \[\leadsto {\left(h \cdot \ell\right)}^{-0.5} \cdot \color{blue}{\left(-d\right)} \]

   8. Simplified 53.0%

      \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot \left(-d\right)} \]

   if -4.80000000000000018e-25 < l < -9.999999999999969e-311

   1. Initial program 74.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 74.4%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 24.4%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. add-log-exp 38.2%

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\log \left(e^{h \cdot \ell}\right)}}} \]

      2. *-commutative 38.2%

         \[\leadsto d \cdot \sqrt{\frac{1}{\log \left(e^{\color{blue}{\ell \cdot h}}\right)}} \]

      3. exp-prod 52.1%

         \[\leadsto d \cdot \sqrt{\frac{1}{\log \color{blue}{\left({\left(e^{\ell}\right)}^{h}\right)}}} \]

   7. Applied egg-rr 52.1%

      \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\log \left({\left(e^{\ell}\right)}^{h}\right)}}} \]

   8. Taylor expanded in l around 0 52.1%

      \[\leadsto d \cdot \sqrt{\frac{1}{\log \color{blue}{1}}} \]

   if -9.999999999999969e-311 < l

   1. Initial program 72.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 71.3%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. pow1 71.3%

         \[\leadsto \color{blue}{{\left(\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)\right)}^{1}} \]

   6. Applied egg-rr 80.9%

      \[\leadsto \color{blue}{{\left(\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \mathsf{fma}\left({\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}, \frac{-0.5 \cdot h}{\ell}, 1\right)\right)}^{1}} \]
   7. Step-by-step derivation

      1. unpow1 80.9%

         \[\leadsto \color{blue}{\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \mathsf{fma}\left({\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}, \frac{-0.5 \cdot h}{\ell}, 1\right)} \]

      2. associate-*l/ 83.3%

         \[\leadsto \color{blue}{\frac{d \cdot \mathsf{fma}\left({\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}, \frac{-0.5 \cdot h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}}} \]

      3. associate-/l* 83.3%

         \[\leadsto \color{blue}{d \cdot \frac{\mathsf{fma}\left({\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}, \frac{-0.5 \cdot h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}}} \]

      4. *-commutative 83.3%

         \[\leadsto d \cdot \frac{\mathsf{fma}\left({\left(M \cdot \color{blue}{\left(\frac{D}{d} \cdot 0.5\right)}\right)}^{2}, \frac{-0.5 \cdot h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      5. metadata-eval 83.3%

         \[\leadsto d \cdot \frac{\mathsf{fma}\left({\left(M \cdot \left(\frac{D}{d} \cdot \color{blue}{\frac{1}{2}}\right)\right)}^{2}, \frac{-0.5 \cdot h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      6. times-frac 83.3%

         \[\leadsto d \cdot \frac{\mathsf{fma}\left({\left(M \cdot \color{blue}{\frac{D \cdot 1}{d \cdot 2}}\right)}^{2}, \frac{-0.5 \cdot h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      7. *-rgt-identity 83.3%

         \[\leadsto d \cdot \frac{\mathsf{fma}\left({\left(M \cdot \frac{\color{blue}{D}}{d \cdot 2}\right)}^{2}, \frac{-0.5 \cdot h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      8. associate-/l* 84.5%

         \[\leadsto d \cdot \frac{\mathsf{fma}\left({\color{blue}{\left(\frac{M \cdot D}{d \cdot 2}\right)}}^{2}, \frac{-0.5 \cdot h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      9. *-commutative 84.5%

         \[\leadsto d \cdot \frac{\mathsf{fma}\left({\left(\frac{\color{blue}{D \cdot M}}{d \cdot 2}\right)}^{2}, \frac{-0.5 \cdot h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      10. associate-/l* 82.9%

          \[\leadsto d \cdot \frac{\mathsf{fma}\left({\color{blue}{\left(D \cdot \frac{M}{d \cdot 2}\right)}}^{2}, \frac{-0.5 \cdot h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      11. *-commutative 82.9%

          \[\leadsto d \cdot \frac{\mathsf{fma}\left({\left(D \cdot \frac{M}{\color{blue}{2 \cdot d}}\right)}^{2}, \frac{-0.5 \cdot h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      12. *-commutative 82.9%

          \[\leadsto d \cdot \frac{\mathsf{fma}\left({\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}, \frac{\color{blue}{h \cdot -0.5}}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

   8. Simplified 82.9%

      \[\leadsto \color{blue}{d \cdot \frac{\mathsf{fma}\left({\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}, \frac{h \cdot -0.5}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}}} \]

   9. Taylor expanded in D around 0 47.2%

      \[\leadsto d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]
   10. Step-by-step derivation

       1. unpow-1 47.2%

          \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \]

       2. metadata-eval 47.2%

          \[\leadsto d \cdot \sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \]

       3. pow-sqr 47.3%

          \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}} \]

       4. rem-sqrt-square 47.3%

          \[\leadsto d \cdot \color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|} \]

       5. rem-square-sqrt 47.2%

          \[\leadsto d \cdot \left|\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right| \]

       6. fabs-sqr 47.2%

          \[\leadsto d \cdot \color{blue}{\left(\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}\right)} \]

       7. rem-square-sqrt 47.3%

          \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

   11. Simplified 47.3%

       \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]
   12. Step-by-step derivation

       1. metadata-eval 47.3%

          \[\leadsto d \cdot {\left(h \cdot \ell\right)}^{\color{blue}{\left(\frac{-1}{2}\right)}} \]

       2. sqrt-pow1 47.2%

          \[\leadsto d \cdot \color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-1}}} \]

       3. add-log-exp 16.5%

          \[\leadsto d \cdot \sqrt{{\left(h \cdot \color{blue}{\log \left(e^{\ell}\right)}\right)}^{-1}} \]

       4. log-pow 15.1%

          \[\leadsto d \cdot \sqrt{{\color{blue}{\log \left({\left(e^{\ell}\right)}^{h}\right)}}^{-1}} \]

       5. inv-pow 15.1%

          \[\leadsto d \cdot \sqrt{\color{blue}{\frac{1}{\log \left({\left(e^{\ell}\right)}^{h}\right)}}} \]

       6. sqrt-div 15.1%

          \[\leadsto d \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\log \left({\left(e^{\ell}\right)}^{h}\right)}}} \]

       7. metadata-eval 15.1%

          \[\leadsto d \cdot \frac{\color{blue}{1}}{\sqrt{\log \left({\left(e^{\ell}\right)}^{h}\right)}} \]

       8. un-div-inv 15.1%

          \[\leadsto \color{blue}{\frac{d}{\sqrt{\log \left({\left(e^{\ell}\right)}^{h}\right)}}} \]

       9. log-pow 16.5%

          \[\leadsto \frac{d}{\sqrt{\color{blue}{h \cdot \log \left(e^{\ell}\right)}}} \]

       10. add-log-exp 47.3%

           \[\leadsto \frac{d}{\sqrt{h \cdot \color{blue}{\ell}}} \]

   13. Applied egg-rr 47.3%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 50.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4.8 \cdot 10^{-25}:\\ \;\;\;\;\left(-d\right) \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \mathbf{elif}\;\ell \leq -1 \cdot 10^{-310}:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 42.1% accurate, 3.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.55 \cdot 10^{-192}:\\ \;\;\;\;\left(-d\right) \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \end{array} \end{array} \]

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 -1.55e-192)
   (* (- 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 <= -1.55e-192) {
		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 <= (-1.55d-192)) 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 <= -1.55e-192) {
		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 <= -1.55e-192:
		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 <= -1.55e-192)
		tmp = Float64(Float64(-d) * (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 <= -1.55e-192)
		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, -1.55e-192], 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 < -1.55e-192

   1. Initial program 68.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 69.0%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 11.9%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

   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. *-commutative 0.0%

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]

      2. unpow-1 0.0%

         \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      3. metadata-eval 0.0%

         \[\leadsto \sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      4. pow-sqr 0.0%

         \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      5. rem-sqrt-square 0.0%

         \[\leadsto \color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      6. rem-square-sqrt 0.0%

         \[\leadsto \left|\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right| \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      7. fabs-sqr 0.0%

         \[\leadsto \color{blue}{\left(\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}\right)} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      8. rem-square-sqrt 0.0%

         \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      9. *-commutative 0.0%

         \[\leadsto {\left(h \cdot \ell\right)}^{-0.5} \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot d\right)} \]

      10. unpow2 0.0%

          \[\leadsto {\left(h \cdot \ell\right)}^{-0.5} \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right) \]

      11. rem-square-sqrt 49.2%

          \[\leadsto {\left(h \cdot \ell\right)}^{-0.5} \cdot \left(\color{blue}{-1} \cdot d\right) \]

      12. mul-1-neg 49.2%

          \[\leadsto {\left(h \cdot \ell\right)}^{-0.5} \cdot \color{blue}{\left(-d\right)} \]

   8. Simplified 49.2%

      \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot \left(-d\right)} \]

   if -1.55e-192 < l

   1. Initial program 72.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 71.5%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 47.1%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. add-exp-log 45.5%

         \[\leadsto d \cdot \sqrt{\color{blue}{e^{\log \left(\frac{1}{h \cdot \ell}\right)}}} \]

      2. log-rec 45.5%

         \[\leadsto d \cdot \sqrt{e^{\color{blue}{-\log \left(h \cdot \ell\right)}}} \]

   7. Applied egg-rr 45.5%

      \[\leadsto d \cdot \sqrt{\color{blue}{e^{-\log \left(h \cdot \ell\right)}}} \]
   8. Step-by-step derivation

      1. exp-neg 45.5%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{1}{e^{\log \left(h \cdot \ell\right)}}}} \]

      2. add-exp-log 47.1%

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \]

      3. *-commutative 47.1%

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      4. associate-/r* 47.2%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \]

   9. Applied egg-rr 47.2%

      \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 48.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.55 \cdot 10^{-192}:\\ \;\;\;\;\left(-d\right) \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 42.1% accurate, 3.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.32 \cdot 10^{-192}:\\ \;\;\;\;\frac{-d}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \end{array} \end{array} \]

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 -1.32e-192) (/ (- d) (sqrt (* 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 <= -1.32e-192) {
		tmp = -d / sqrt((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 <= (-1.32d-192)) then
        tmp = -d / sqrt((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 <= -1.32e-192) {
		tmp = -d / Math.sqrt((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 <= -1.32e-192:
		tmp = -d / math.sqrt((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 <= -1.32e-192)
		tmp = Float64(Float64(-d) / sqrt(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 <= -1.32e-192)
		tmp = -d / sqrt((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, -1.32e-192], N[((-d) / N[Sqrt[N[(l * h), $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 < -1.32e-192

   1. Initial program 68.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 69.0%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 11.9%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

   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. unpow2 0.0%

         \[\leadsto d \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      3. rem-square-sqrt 49.2%

         \[\leadsto d \cdot \left(\color{blue}{-1} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      4. mul-1-neg 49.2%

         \[\leadsto d \cdot \color{blue}{\left(-\sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      5. distribute-rgt-neg-in 49.2%

         \[\leadsto \color{blue}{-d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      6. *-commutative 49.2%

         \[\leadsto -\color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      7. associate-/r* 49.1%

         \[\leadsto -\sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \cdot d \]

      8. unpow1/2 49.1%

         \[\leadsto -\color{blue}{{\left(\frac{\frac{1}{h}}{\ell}\right)}^{0.5}} \cdot d \]

      9. associate-/r* 49.2%

         \[\leadsto -{\color{blue}{\left(\frac{1}{h \cdot \ell}\right)}}^{0.5} \cdot d \]

      10. rem-exp-log 46.4%

          \[\leadsto -{\left(\frac{1}{\color{blue}{e^{\log \left(h \cdot \ell\right)}}}\right)}^{0.5} \cdot d \]

      11. exp-neg 46.4%

          \[\leadsto -{\color{blue}{\left(e^{-\log \left(h \cdot \ell\right)}\right)}}^{0.5} \cdot d \]

      12. exp-prod 46.4%

          \[\leadsto -\color{blue}{e^{\left(-\log \left(h \cdot \ell\right)\right) \cdot 0.5}} \cdot d \]

      13. distribute-lft-neg-out 46.4%

          \[\leadsto -e^{\color{blue}{-\log \left(h \cdot \ell\right) \cdot 0.5}} \cdot d \]

      14. exp-neg 46.4%

          \[\leadsto -\color{blue}{\frac{1}{e^{\log \left(h \cdot \ell\right) \cdot 0.5}}} \cdot d \]

      15. exp-to-pow 49.1%

          \[\leadsto -\frac{1}{\color{blue}{{\left(h \cdot \ell\right)}^{0.5}}} \cdot d \]

      16. unpow1/2 49.1%

          \[\leadsto -\frac{1}{\color{blue}{\sqrt{h \cdot \ell}}} \cdot d \]

      17. unpow-1 49.1%

          \[\leadsto -\color{blue}{{\left(\sqrt{h \cdot \ell}\right)}^{-1}} \cdot d \]

      18. unpow-1 49.1%

          \[\leadsto -\color{blue}{\frac{1}{\sqrt{h \cdot \ell}}} \cdot d \]

      19. associate-*l/ 49.1%

          \[\leadsto -\color{blue}{\frac{1 \cdot d}{\sqrt{h \cdot \ell}}} \]

   8. Simplified 49.1%

      \[\leadsto \color{blue}{\frac{d}{-\sqrt{h \cdot \ell}}} \]

   if -1.32e-192 < l

   1. Initial program 72.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 71.5%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 47.1%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. add-exp-log 45.5%

         \[\leadsto d \cdot \sqrt{\color{blue}{e^{\log \left(\frac{1}{h \cdot \ell}\right)}}} \]

      2. log-rec 45.5%

         \[\leadsto d \cdot \sqrt{e^{\color{blue}{-\log \left(h \cdot \ell\right)}}} \]

   7. Applied egg-rr 45.5%

      \[\leadsto d \cdot \sqrt{\color{blue}{e^{-\log \left(h \cdot \ell\right)}}} \]
   8. Step-by-step derivation

      1. exp-neg 45.5%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{1}{e^{\log \left(h \cdot \ell\right)}}}} \]

      2. add-exp-log 47.1%

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \]

      3. *-commutative 47.1%

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      4. associate-/r* 47.2%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \]

   9. Applied egg-rr 47.2%

      \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 48.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.32 \cdot 10^{-192}:\\ \;\;\;\;\frac{-d}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 42.3% accurate, 3.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := \sqrt{\ell \cdot h}\\ \mathbf{if}\;\ell \leq -3.4 \cdot 10^{-224}:\\ \;\;\;\;\frac{-d}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{t\_0}\\ \end{array} \end{array} \]

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 (* l h)))) (if (<= l -3.4e-224) (/ (- d) t_0) (/ d 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((l * h));
	double tmp;
	if (l <= -3.4e-224) {
		tmp = -d / t_0;
	} else {
		tmp = d / 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) :: tmp
    t_0 = sqrt((l * h))
    if (l <= (-3.4d-224)) then
        tmp = -d / t_0
    else
        tmp = d / 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((l * h));
	double tmp;
	if (l <= -3.4e-224) {
		tmp = -d / t_0;
	} else {
		tmp = d / 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((l * h))
	tmp = 0
	if l <= -3.4e-224:
		tmp = -d / t_0
	else:
		tmp = d / 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(l * h))
	tmp = 0.0
	if (l <= -3.4e-224)
		tmp = Float64(Float64(-d) / t_0);
	else
		tmp = Float64(d / 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((l * h));
	tmp = 0.0;
	if (l <= -3.4e-224)
		tmp = -d / t_0;
	else
		tmp = d / 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[(l * h), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -3.4e-224], N[((-d) / t$95$0), $MachinePrecision], N[(d / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < -3.39999999999999992e-224

   1. Initial program 68.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 68.8%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 13.0%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

   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. unpow2 0.0%

         \[\leadsto d \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      3. rem-square-sqrt 48.4%

         \[\leadsto d \cdot \left(\color{blue}{-1} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      4. mul-1-neg 48.4%

         \[\leadsto d \cdot \color{blue}{\left(-\sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      5. distribute-rgt-neg-in 48.4%

         \[\leadsto \color{blue}{-d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      6. *-commutative 48.4%

         \[\leadsto -\color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      7. associate-/r* 48.3%

         \[\leadsto -\sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \cdot d \]

      8. unpow1/2 48.3%

         \[\leadsto -\color{blue}{{\left(\frac{\frac{1}{h}}{\ell}\right)}^{0.5}} \cdot d \]

      9. associate-/r* 48.4%

         \[\leadsto -{\color{blue}{\left(\frac{1}{h \cdot \ell}\right)}}^{0.5} \cdot d \]

      10. rem-exp-log 45.7%

          \[\leadsto -{\left(\frac{1}{\color{blue}{e^{\log \left(h \cdot \ell\right)}}}\right)}^{0.5} \cdot d \]

      11. exp-neg 45.7%

          \[\leadsto -{\color{blue}{\left(e^{-\log \left(h \cdot \ell\right)}\right)}}^{0.5} \cdot d \]

      12. exp-prod 45.7%

          \[\leadsto -\color{blue}{e^{\left(-\log \left(h \cdot \ell\right)\right) \cdot 0.5}} \cdot d \]

      13. distribute-lft-neg-out 45.7%

          \[\leadsto -e^{\color{blue}{-\log \left(h \cdot \ell\right) \cdot 0.5}} \cdot d \]

      14. exp-neg 45.7%

          \[\leadsto -\color{blue}{\frac{1}{e^{\log \left(h \cdot \ell\right) \cdot 0.5}}} \cdot d \]

      15. exp-to-pow 48.3%

          \[\leadsto -\frac{1}{\color{blue}{{\left(h \cdot \ell\right)}^{0.5}}} \cdot d \]

      16. unpow1/2 48.3%

          \[\leadsto -\frac{1}{\color{blue}{\sqrt{h \cdot \ell}}} \cdot d \]

      17. unpow-1 48.3%

          \[\leadsto -\color{blue}{{\left(\sqrt{h \cdot \ell}\right)}^{-1}} \cdot d \]

      18. unpow-1 48.3%

          \[\leadsto -\color{blue}{\frac{1}{\sqrt{h \cdot \ell}}} \cdot d \]

      19. associate-*l/ 48.3%

          \[\leadsto -\color{blue}{\frac{1 \cdot d}{\sqrt{h \cdot \ell}}} \]

   8. Simplified 48.3%

      \[\leadsto \color{blue}{\frac{d}{-\sqrt{h \cdot \ell}}} \]

   if -3.39999999999999992e-224 < l

   1. Initial program 72.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 71.7%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. pow1 71.7%

         \[\leadsto \color{blue}{{\left(\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)\right)}^{1}} \]

   6. Applied egg-rr 71.6%

      \[\leadsto \color{blue}{{\left(\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \mathsf{fma}\left({\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}, \frac{-0.5 \cdot h}{\ell}, 1\right)\right)}^{1}} \]
   7. Step-by-step derivation

      1. unpow1 71.6%

         \[\leadsto \color{blue}{\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \mathsf{fma}\left({\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}, \frac{-0.5 \cdot h}{\ell}, 1\right)} \]

      2. associate-*l/ 73.7%

         \[\leadsto \color{blue}{\frac{d \cdot \mathsf{fma}\left({\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}, \frac{-0.5 \cdot h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}}} \]

      3. associate-/l* 73.6%

         \[\leadsto \color{blue}{d \cdot \frac{\mathsf{fma}\left({\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}, \frac{-0.5 \cdot h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}}} \]

      4. *-commutative 73.6%

         \[\leadsto d \cdot \frac{\mathsf{fma}\left({\left(M \cdot \color{blue}{\left(\frac{D}{d} \cdot 0.5\right)}\right)}^{2}, \frac{-0.5 \cdot h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      5. metadata-eval 73.6%

         \[\leadsto d \cdot \frac{\mathsf{fma}\left({\left(M \cdot \left(\frac{D}{d} \cdot \color{blue}{\frac{1}{2}}\right)\right)}^{2}, \frac{-0.5 \cdot h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      6. times-frac 73.6%

         \[\leadsto d \cdot \frac{\mathsf{fma}\left({\left(M \cdot \color{blue}{\frac{D \cdot 1}{d \cdot 2}}\right)}^{2}, \frac{-0.5 \cdot h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      7. *-rgt-identity 73.6%

         \[\leadsto d \cdot \frac{\mathsf{fma}\left({\left(M \cdot \frac{\color{blue}{D}}{d \cdot 2}\right)}^{2}, \frac{-0.5 \cdot h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      8. associate-/l* 74.7%

         \[\leadsto d \cdot \frac{\mathsf{fma}\left({\color{blue}{\left(\frac{M \cdot D}{d \cdot 2}\right)}}^{2}, \frac{-0.5 \cdot h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      9. *-commutative 74.7%

         \[\leadsto d \cdot \frac{\mathsf{fma}\left({\left(\frac{\color{blue}{D \cdot M}}{d \cdot 2}\right)}^{2}, \frac{-0.5 \cdot h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      10. associate-/l* 73.3%

          \[\leadsto d \cdot \frac{\mathsf{fma}\left({\color{blue}{\left(D \cdot \frac{M}{d \cdot 2}\right)}}^{2}, \frac{-0.5 \cdot h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      11. *-commutative 73.3%

          \[\leadsto d \cdot \frac{\mathsf{fma}\left({\left(D \cdot \frac{M}{\color{blue}{2 \cdot d}}\right)}^{2}, \frac{-0.5 \cdot h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

      12. *-commutative 73.3%

          \[\leadsto d \cdot \frac{\mathsf{fma}\left({\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}, \frac{\color{blue}{h \cdot -0.5}}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

   8. Simplified 73.3%

      \[\leadsto \color{blue}{d \cdot \frac{\mathsf{fma}\left({\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}, \frac{h \cdot -0.5}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}}} \]

   9. Taylor expanded in D around 0 47.7%

      \[\leadsto d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]
   10. Step-by-step derivation

       1. unpow-1 47.7%

          \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \]

       2. metadata-eval 47.7%

          \[\leadsto d \cdot \sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \]

       3. pow-sqr 47.7%

          \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}} \]

       4. rem-sqrt-square 47.7%

          \[\leadsto d \cdot \color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|} \]

       5. rem-square-sqrt 47.6%

          \[\leadsto d \cdot \left|\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right| \]

       6. fabs-sqr 47.6%

          \[\leadsto d \cdot \color{blue}{\left(\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}\right)} \]

       7. rem-square-sqrt 47.7%

          \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

   11. Simplified 47.7%

       \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]
   12. Step-by-step derivation

       1. metadata-eval 47.7%

          \[\leadsto d \cdot {\left(h \cdot \ell\right)}^{\color{blue}{\left(\frac{-1}{2}\right)}} \]

       2. sqrt-pow1 47.7%

          \[\leadsto d \cdot \color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-1}}} \]

       3. add-log-exp 14.6%

          \[\leadsto d \cdot \sqrt{{\left(h \cdot \color{blue}{\log \left(e^{\ell}\right)}\right)}^{-1}} \]

       4. log-pow 22.1%

          \[\leadsto d \cdot \sqrt{{\color{blue}{\log \left({\left(e^{\ell}\right)}^{h}\right)}}^{-1}} \]

       5. inv-pow 22.1%

          \[\leadsto d \cdot \sqrt{\color{blue}{\frac{1}{\log \left({\left(e^{\ell}\right)}^{h}\right)}}} \]

       6. sqrt-div 22.1%

          \[\leadsto d \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\log \left({\left(e^{\ell}\right)}^{h}\right)}}} \]

       7. metadata-eval 22.1%

          \[\leadsto d \cdot \frac{\color{blue}{1}}{\sqrt{\log \left({\left(e^{\ell}\right)}^{h}\right)}} \]

       8. un-div-inv 22.1%

          \[\leadsto \color{blue}{\frac{d}{\sqrt{\log \left({\left(e^{\ell}\right)}^{h}\right)}}} \]

       9. log-pow 16.1%

          \[\leadsto \frac{d}{\sqrt{\color{blue}{h \cdot \log \left(e^{\ell}\right)}}} \]

       10. add-log-exp 47.8%

           \[\leadsto \frac{d}{\sqrt{h \cdot \color{blue}{\ell}}} \]

   13. Applied egg-rr 47.8%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 48.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -3.4 \cdot 10^{-224}:\\ \;\;\;\;\frac{-d}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 26.6% 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 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 70.4%

   \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. pow1 70.4%

      \[\leadsto \color{blue}{{\left(\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)\right)}^{1}} \]

6. Applied egg-rr 38.6%

   \[\leadsto \color{blue}{{\left(\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \mathsf{fma}\left({\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}, \frac{-0.5 \cdot h}{\ell}, 1\right)\right)}^{1}} \]
7. Step-by-step derivation

   1. unpow1 38.6%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \mathsf{fma}\left({\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}, \frac{-0.5 \cdot h}{\ell}, 1\right)} \]

   2. associate-*l/ 39.7%

      \[\leadsto \color{blue}{\frac{d \cdot \mathsf{fma}\left({\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}, \frac{-0.5 \cdot h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}}} \]

   3. associate-/l* 39.7%

      \[\leadsto \color{blue}{d \cdot \frac{\mathsf{fma}\left({\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}, \frac{-0.5 \cdot h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}}} \]

   4. *-commutative 39.7%

      \[\leadsto d \cdot \frac{\mathsf{fma}\left({\left(M \cdot \color{blue}{\left(\frac{D}{d} \cdot 0.5\right)}\right)}^{2}, \frac{-0.5 \cdot h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

   5. metadata-eval 39.7%

      \[\leadsto d \cdot \frac{\mathsf{fma}\left({\left(M \cdot \left(\frac{D}{d} \cdot \color{blue}{\frac{1}{2}}\right)\right)}^{2}, \frac{-0.5 \cdot h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

   6. times-frac 39.7%

      \[\leadsto d \cdot \frac{\mathsf{fma}\left({\left(M \cdot \color{blue}{\frac{D \cdot 1}{d \cdot 2}}\right)}^{2}, \frac{-0.5 \cdot h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

   7. *-rgt-identity 39.7%

      \[\leadsto d \cdot \frac{\mathsf{fma}\left({\left(M \cdot \frac{\color{blue}{D}}{d \cdot 2}\right)}^{2}, \frac{-0.5 \cdot h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

   8. associate-/l* 40.3%

      \[\leadsto d \cdot \frac{\mathsf{fma}\left({\color{blue}{\left(\frac{M \cdot D}{d \cdot 2}\right)}}^{2}, \frac{-0.5 \cdot h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

   9. *-commutative 40.3%

      \[\leadsto d \cdot \frac{\mathsf{fma}\left({\left(\frac{\color{blue}{D \cdot M}}{d \cdot 2}\right)}^{2}, \frac{-0.5 \cdot h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

   10. associate-/l* 39.5%

       \[\leadsto d \cdot \frac{\mathsf{fma}\left({\color{blue}{\left(D \cdot \frac{M}{d \cdot 2}\right)}}^{2}, \frac{-0.5 \cdot h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

   11. *-commutative 39.5%

       \[\leadsto d \cdot \frac{\mathsf{fma}\left({\left(D \cdot \frac{M}{\color{blue}{2 \cdot d}}\right)}^{2}, \frac{-0.5 \cdot h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

   12. *-commutative 39.5%

       \[\leadsto d \cdot \frac{\mathsf{fma}\left({\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}, \frac{\color{blue}{h \cdot -0.5}}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

8. Simplified 39.5%

   \[\leadsto \color{blue}{d \cdot \frac{\mathsf{fma}\left({\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}, \frac{h \cdot -0.5}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}}} \]

9. Taylor expanded in D around 0 31.7%

   \[\leadsto d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]
10. Step-by-step derivation

    1. unpow-1 31.7%

       \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \]

    2. metadata-eval 31.7%

       \[\leadsto d \cdot \sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \]

    3. pow-sqr 31.7%

       \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}} \]

    4. rem-sqrt-square 31.4%

       \[\leadsto d \cdot \color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|} \]

    5. rem-square-sqrt 31.3%

       \[\leadsto d \cdot \left|\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right| \]

    6. fabs-sqr 31.3%

       \[\leadsto d \cdot \color{blue}{\left(\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}\right)} \]

    7. rem-square-sqrt 31.4%

       \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

11. Simplified 31.4%

    \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]
12. Step-by-step derivation

    1. metadata-eval 31.4%

       \[\leadsto d \cdot {\left(h \cdot \ell\right)}^{\color{blue}{\left(\frac{-1}{2}\right)}} \]

    2. sqrt-pow1 31.7%

       \[\leadsto d \cdot \color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-1}}} \]

    3. add-log-exp 11.1%

       \[\leadsto d \cdot \sqrt{{\left(h \cdot \color{blue}{\log \left(e^{\ell}\right)}\right)}^{-1}} \]

    4. log-pow 23.4%

       \[\leadsto d \cdot \sqrt{{\color{blue}{\log \left({\left(e^{\ell}\right)}^{h}\right)}}^{-1}} \]

    5. inv-pow 23.4%

       \[\leadsto d \cdot \sqrt{\color{blue}{\frac{1}{\log \left({\left(e^{\ell}\right)}^{h}\right)}}} \]

    6. sqrt-div 23.4%

       \[\leadsto d \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\log \left({\left(e^{\ell}\right)}^{h}\right)}}} \]

    7. metadata-eval 23.4%

       \[\leadsto d \cdot \frac{\color{blue}{1}}{\sqrt{\log \left({\left(e^{\ell}\right)}^{h}\right)}} \]

    8. un-div-inv 23.4%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{\log \left({\left(e^{\ell}\right)}^{h}\right)}}} \]

    9. log-pow 13.8%

       \[\leadsto \frac{d}{\sqrt{\color{blue}{h \cdot \log \left(e^{\ell}\right)}}} \]

    10. add-log-exp 31.4%

        \[\leadsto \frac{d}{\sqrt{h \cdot \color{blue}{\ell}}} \]

13. Applied egg-rr 31.4%

    \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]

14. Final simplification 31.4%

    \[\leadsto \frac{d}{\sqrt{\ell \cdot h}} \]
15. Add Preprocessing

Alternative 22: 4.0% accurate, 3.2× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ d \cdot \sqrt{\ell \cdot h} \end{array} \]

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 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 70.4%

   \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. pow1 70.4%

      \[\leadsto \color{blue}{{\left(\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)\right)}^{1}} \]

6. Applied egg-rr 38.6%

   \[\leadsto \color{blue}{{\left(\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \mathsf{fma}\left({\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}, \frac{-0.5 \cdot h}{\ell}, 1\right)\right)}^{1}} \]
7. Step-by-step derivation

   1. unpow1 38.6%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \mathsf{fma}\left({\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}, \frac{-0.5 \cdot h}{\ell}, 1\right)} \]

   2. associate-*l/ 39.7%

      \[\leadsto \color{blue}{\frac{d \cdot \mathsf{fma}\left({\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}, \frac{-0.5 \cdot h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}}} \]

   3. associate-/l* 39.7%

      \[\leadsto \color{blue}{d \cdot \frac{\mathsf{fma}\left({\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}, \frac{-0.5 \cdot h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}}} \]

   4. *-commutative 39.7%

      \[\leadsto d \cdot \frac{\mathsf{fma}\left({\left(M \cdot \color{blue}{\left(\frac{D}{d} \cdot 0.5\right)}\right)}^{2}, \frac{-0.5 \cdot h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

   5. metadata-eval 39.7%

      \[\leadsto d \cdot \frac{\mathsf{fma}\left({\left(M \cdot \left(\frac{D}{d} \cdot \color{blue}{\frac{1}{2}}\right)\right)}^{2}, \frac{-0.5 \cdot h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

   6. times-frac 39.7%

      \[\leadsto d \cdot \frac{\mathsf{fma}\left({\left(M \cdot \color{blue}{\frac{D \cdot 1}{d \cdot 2}}\right)}^{2}, \frac{-0.5 \cdot h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

   7. *-rgt-identity 39.7%

      \[\leadsto d \cdot \frac{\mathsf{fma}\left({\left(M \cdot \frac{\color{blue}{D}}{d \cdot 2}\right)}^{2}, \frac{-0.5 \cdot h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

   8. associate-/l* 40.3%

      \[\leadsto d \cdot \frac{\mathsf{fma}\left({\color{blue}{\left(\frac{M \cdot D}{d \cdot 2}\right)}}^{2}, \frac{-0.5 \cdot h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

   9. *-commutative 40.3%

      \[\leadsto d \cdot \frac{\mathsf{fma}\left({\left(\frac{\color{blue}{D \cdot M}}{d \cdot 2}\right)}^{2}, \frac{-0.5 \cdot h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

   10. associate-/l* 39.5%

       \[\leadsto d \cdot \frac{\mathsf{fma}\left({\color{blue}{\left(D \cdot \frac{M}{d \cdot 2}\right)}}^{2}, \frac{-0.5 \cdot h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

   11. *-commutative 39.5%

       \[\leadsto d \cdot \frac{\mathsf{fma}\left({\left(D \cdot \frac{M}{\color{blue}{2 \cdot d}}\right)}^{2}, \frac{-0.5 \cdot h}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

   12. *-commutative 39.5%

       \[\leadsto d \cdot \frac{\mathsf{fma}\left({\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}, \frac{\color{blue}{h \cdot -0.5}}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}} \]

8. Simplified 39.5%

   \[\leadsto \color{blue}{d \cdot \frac{\mathsf{fma}\left({\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}, \frac{h \cdot -0.5}{\ell}, 1\right)}{\sqrt{h} \cdot \sqrt{\ell}}} \]

9. Taylor expanded in D around 0 31.7%

   \[\leadsto d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]
10. Step-by-step derivation

    1. unpow-1 31.7%

       \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \]

    2. metadata-eval 31.7%

       \[\leadsto d \cdot \sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \]

    3. pow-sqr 31.7%

       \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}} \]

    4. rem-sqrt-square 31.4%

       \[\leadsto d \cdot \color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|} \]

    5. rem-square-sqrt 31.3%

       \[\leadsto d \cdot \left|\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right| \]

    6. fabs-sqr 31.3%

       \[\leadsto d \cdot \color{blue}{\left(\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}\right)} \]

    7. rem-square-sqrt 31.4%

       \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

11. Simplified 31.4%

    \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]
12. Step-by-step derivation

    1. metadata-eval 31.4%

       \[\leadsto d \cdot {\left(h \cdot \ell\right)}^{\color{blue}{\left(-1 \cdot 0.5\right)}} \]

    2. pow-pow 31.7%

       \[\leadsto d \cdot \color{blue}{{\left({\left(h \cdot \ell\right)}^{-1}\right)}^{0.5}} \]

    3. inv-pow 31.7%

       \[\leadsto d \cdot {\color{blue}{\left(\frac{1}{h \cdot \ell}\right)}}^{0.5} \]

    4. associate-/r* 31.7%

       \[\leadsto d \cdot {\color{blue}{\left(\frac{\frac{1}{h}}{\ell}\right)}}^{0.5} \]

    5. metadata-eval 31.7%

       \[\leadsto d \cdot {\left(\frac{\frac{1}{h}}{\ell}\right)}^{\color{blue}{\left(1.5 \cdot 0.3333333333333333\right)}} \]

    6. pow-pow 29.4%

       \[\leadsto d \cdot \color{blue}{{\left({\left(\frac{\frac{1}{h}}{\ell}\right)}^{1.5}\right)}^{0.3333333333333333}} \]

    7. pow1/3 30.3%

       \[\leadsto d \cdot \color{blue}{\sqrt[3]{{\left(\frac{\frac{1}{h}}{\ell}\right)}^{1.5}}} \]

    8. pow1 30.3%

       \[\leadsto \color{blue}{{\left(d \cdot \sqrt[3]{{\left(\frac{\frac{1}{h}}{\ell}\right)}^{1.5}}\right)}^{1}} \]

13. Applied egg-rr 3.1%

    \[\leadsto \color{blue}{{\left(d \cdot \sqrt{h \cdot \ell}\right)}^{1}} \]
14. Step-by-step derivation

    1. unpow1 3.1%

       \[\leadsto \color{blue}{d \cdot \sqrt{h \cdot \ell}} \]

15. Simplified 3.1%

    \[\leadsto \color{blue}{d \cdot \sqrt{h \cdot \ell}} \]

16. Final simplification 3.1%

    \[\leadsto d \cdot \sqrt{\ell \cdot h} \]
17. Add Preprocessing

Reproduce

herbie shell --seed 2024136 
(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)))))