Henrywood and Agarwal, Equation (12)

Percentage Accurate: 66.9% → 82.5%

Time: 23.5s

Alternatives: 19

Speedup: 1.5×

• Unsound rule application detected in e-graph. Results may not be sound.

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: 66.9% 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: 82.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -100000000:\\ \;\;\;\;\left(d \cdot \left({\left(-\ell\right)}^{-0.5} \cdot {\left(\frac{-1}{h}\right)}^{0.5}\right)\right) \cdot \left(-1 + 0.5 \cdot \left({\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(-1 + 0.5 \cdot \frac{h \cdot {\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2}}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (if (<= l -100000000.0)
   (*
    (* d (* (pow (- l) -0.5) (pow (/ -1.0 h) 0.5)))
    (+ -1.0 (* 0.5 (* (pow (* D (/ (/ M d) 2.0)) 2.0) (/ h l)))))
   (if (<= l -5e-310)
     (*
      (* d (sqrt (/ 1.0 (* l h))))
      (+ -1.0 (* 0.5 (/ (* h (pow (* D (* 0.5 (/ M d))) 2.0)) l))))
     (*
      (/ (sqrt d) (sqrt h))
      (*
       (/ (sqrt d) (sqrt l))
       (- 1.0 (* 0.5 (* (/ h l) (pow (* (/ M 2.0) (/ D d)) 2.0)))))))))

C

assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -100000000.0) {
		tmp = (d * (pow(-l, -0.5) * pow((-1.0 / h), 0.5))) * (-1.0 + (0.5 * (pow((D * ((M / d) / 2.0)), 2.0) * (h / l))));
	} else if (l <= -5e-310) {
		tmp = (d * sqrt((1.0 / (l * h)))) * (-1.0 + (0.5 * ((h * pow((D * (0.5 * (M / d))), 2.0)) / l)));
	} else {
		tmp = (sqrt(d) / sqrt(h)) * ((sqrt(d) / sqrt(l)) * (1.0 - (0.5 * ((h / l) * pow(((M / 2.0) * (D / d)), 2.0)))));
	}
	return tmp;
}

Fortran

NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (l <= (-100000000.0d0)) then
        tmp = (d * ((-l ** (-0.5d0)) * (((-1.0d0) / h) ** 0.5d0))) * ((-1.0d0) + (0.5d0 * (((d_1 * ((m / d) / 2.0d0)) ** 2.0d0) * (h / l))))
    else if (l <= (-5d-310)) then
        tmp = (d * sqrt((1.0d0 / (l * h)))) * ((-1.0d0) + (0.5d0 * ((h * ((d_1 * (0.5d0 * (m / d))) ** 2.0d0)) / l)))
    else
        tmp = (sqrt(d) / sqrt(h)) * ((sqrt(d) / sqrt(l)) * (1.0d0 - (0.5d0 * ((h / l) * (((m / 2.0d0) * (d_1 / d)) ** 2.0d0)))))
    end if
    code = tmp
end function

Java

assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -100000000.0) {
		tmp = (d * (Math.pow(-l, -0.5) * Math.pow((-1.0 / h), 0.5))) * (-1.0 + (0.5 * (Math.pow((D * ((M / d) / 2.0)), 2.0) * (h / l))));
	} else if (l <= -5e-310) {
		tmp = (d * Math.sqrt((1.0 / (l * h)))) * (-1.0 + (0.5 * ((h * Math.pow((D * (0.5 * (M / d))), 2.0)) / l)));
	} else {
		tmp = (Math.sqrt(d) / Math.sqrt(h)) * ((Math.sqrt(d) / Math.sqrt(l)) * (1.0 - (0.5 * ((h / l) * Math.pow(((M / 2.0) * (D / d)), 2.0)))));
	}
	return tmp;
}

Python

[M, D] = sort([M, D])
def code(d, h, l, M, D):
	tmp = 0
	if l <= -100000000.0:
		tmp = (d * (math.pow(-l, -0.5) * math.pow((-1.0 / h), 0.5))) * (-1.0 + (0.5 * (math.pow((D * ((M / d) / 2.0)), 2.0) * (h / l))))
	elif l <= -5e-310:
		tmp = (d * math.sqrt((1.0 / (l * h)))) * (-1.0 + (0.5 * ((h * math.pow((D * (0.5 * (M / d))), 2.0)) / l)))
	else:
		tmp = (math.sqrt(d) / math.sqrt(h)) * ((math.sqrt(d) / math.sqrt(l)) * (1.0 - (0.5 * ((h / l) * math.pow(((M / 2.0) * (D / d)), 2.0)))))
	return tmp

Julia

M, D = sort([M, D])
function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= -100000000.0)
		tmp = Float64(Float64(d * Float64((Float64(-l) ^ -0.5) * (Float64(-1.0 / h) ^ 0.5))) * Float64(-1.0 + Float64(0.5 * Float64((Float64(D * Float64(Float64(M / d) / 2.0)) ^ 2.0) * Float64(h / l)))));
	elseif (l <= -5e-310)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(l * h)))) * Float64(-1.0 + Float64(0.5 * Float64(Float64(h * (Float64(D * Float64(0.5 * Float64(M / d))) ^ 2.0)) / l))));
	else
		tmp = Float64(Float64(sqrt(d) / sqrt(h)) * Float64(Float64(sqrt(d) / sqrt(l)) * Float64(1.0 - Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(M / 2.0) * Float64(D / d)) ^ 2.0))))));
	end
	return tmp
end

MATLAB

M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (l <= -100000000.0)
		tmp = (d * ((-l ^ -0.5) * ((-1.0 / h) ^ 0.5))) * (-1.0 + (0.5 * (((D * ((M / d) / 2.0)) ^ 2.0) * (h / l))));
	elseif (l <= -5e-310)
		tmp = (d * sqrt((1.0 / (l * h)))) * (-1.0 + (0.5 * ((h * ((D * (0.5 * (M / d))) ^ 2.0)) / l)));
	else
		tmp = (sqrt(d) / sqrt(h)) * ((sqrt(d) / sqrt(l)) * (1.0 - (0.5 * ((h / l) * (((M / 2.0) * (D / d)) ^ 2.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := If[LessEqual[l, -100000000.0], N[(N[(d * N[(N[Power[(-l), -0.5], $MachinePrecision] * N[Power[N[(-1.0 / h), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0 + N[(0.5 * N[(N[Power[N[(D * N[(N[(M / d), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -5e-310], N[(N[(d * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-1.0 + N[(0.5 * N[(N[(h * N[Power[N[(D * N[(0.5 * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M / 2.0), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -1e8

   1. Initial program 57.7%

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

      1. metadata-eval 57.7%

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

      2. unpow1/2 57.7%

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

      3. metadata-eval 57.7%

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

      4. unpow1/2 57.7%

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

      5. *-commutative 57.7%

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

      6. associate-*l* 57.7%

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

      7. times-frac 57.9%

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

      8. metadata-eval 57.9%

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

   3. Simplified 57.9%

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

      1. expm1-log1p-u 45.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\right)\right)} \]

      2. expm1-udef 27.7%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\right)} - 1} \]

   5. Applied egg-rr 20.9%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 35.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)\right)} \]

      2. expm1-log1p 46.0%

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

      3. sub-neg 46.0%

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

      4. +-commutative 46.0%

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

      5. associate-*l/ 43.3%

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

      6. associate-/l* 44.7%

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

      7. +-commutative 44.7%

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

      8. sub-neg 44.7%

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

      9. associate-/l* 44.7%

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

      10. associate-*r/ 44.7%

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

      11. associate-*l/ 44.7%

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

      12. associate-/l* 47.4%

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

      13. associate-*r/ 44.7%

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

   7. Simplified 44.6%

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

   8. Taylor expanded in d around -inf 64.1%

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

      1. mul-1-neg 64.1%

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

      2. associate-/r* 65.5%

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

      3. distribute-rgt-neg-in 65.5%

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

      4. associate-/l/ 64.1%

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

      5. unpow-1 64.1%

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

      6. sqr-pow 64.1%

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

      7. rem-sqrt-square 64.1%

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

      8. metadata-eval 64.1%

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

      9. sqr-pow 63.9%

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

      10. fabs-sqr 63.9%

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

      11. sqr-pow 64.1%

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

   10. Simplified 64.1%

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

   11. Taylor expanded in h around -inf 73.0%

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

       1. distribute-lft-in 73.0%

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

       2. exp-sum 73.2%

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

       3. *-commutative 73.2%

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

       4. exp-to-pow 74.1%

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

       5. mul-1-neg 74.1%

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

       6. *-commutative 74.1%

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

       7. *-commutative 74.1%

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

       8. associate-*l* 74.1%

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

       9. metadata-eval 74.1%

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

       10. exp-to-pow 77.7%

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

   13. Simplified 77.7%

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

   if -1e8 < l < -4.999999999999985e-310

   1. Initial program 63.5%

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

      1. metadata-eval 63.5%

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

      2. unpow1/2 63.5%

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

      3. metadata-eval 63.5%

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

      4. unpow1/2 63.5%

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

      5. *-commutative 63.5%

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

      6. associate-*l* 63.5%

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

      7. times-frac 63.3%

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

      8. metadata-eval 63.3%

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

   3. Simplified 63.3%

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

      1. expm1-log1p-u 26.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\right)\right)} \]

      2. expm1-udef 23.1%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\right)} - 1} \]

   5. Applied egg-rr 18.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 21.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)\right)} \]

      2. expm1-log1p 57.8%

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

      3. sub-neg 57.8%

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

      4. +-commutative 57.8%

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

      5. associate-*l/ 57.9%

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

      6. associate-/l* 57.9%

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

      7. +-commutative 57.9%

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

      8. sub-neg 57.9%

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

      9. associate-/l* 57.9%

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

      10. associate-*r/ 58.0%

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

      11. associate-*l/ 58.0%

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

      12. associate-/l* 64.2%

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

      13. associate-*r/ 57.9%

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

   7. Simplified 56.1%

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

   8. Taylor expanded in d around -inf 71.1%

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

      1. associate-*r* 71.1%

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

      2. mul-1-neg 71.1%

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

      3. *-commutative 71.1%

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

   10. Simplified 71.1%

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

       1. associate-*r/ 88.1%

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

       2. div-inv 88.1%

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

       3. metadata-eval 88.1%

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

   12. Applied egg-rr 88.1%

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

   if -4.999999999999985e-310 < l

   1. Initial program 65.1%

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

      1. associate-*l* 65.2%

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

      2. metadata-eval 65.2%

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

      3. unpow1/2 65.2%

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

      4. metadata-eval 65.2%

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

      5. unpow1/2 65.2%

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

      6. associate-*l* 65.2%

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

      7. metadata-eval 65.2%

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

      8. times-frac 65.1%

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

   3. Simplified 65.1%

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

      1. sqrt-div 78.0%

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

   5. Applied egg-rr 78.0%

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

      1. sqrt-div 85.5%

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

   7. Applied egg-rr 85.5%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -100000000:\\ \;\;\;\;\left(d \cdot \left({\left(-\ell\right)}^{-0.5} \cdot {\left(\frac{-1}{h}\right)}^{0.5}\right)\right) \cdot \left(-1 + 0.5 \cdot \left({\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(-1 + 0.5 \cdot \frac{h \cdot {\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2}}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}\right)\right)\right)\\ \end{array} \]

Alternative 2: 79.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -100000000:\\ \;\;\;\;\left(d \cdot \left({\left(-\ell\right)}^{-0.5} \cdot {\left(\frac{-1}{h}\right)}^{0.5}\right)\right) \cdot \left(-1 + 0.5 \cdot \left({\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(-1 + 0.5 \cdot \frac{h \cdot {\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2}}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{\ell}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (if (<= l -100000000.0)
   (*
    (* d (* (pow (- l) -0.5) (pow (/ -1.0 h) 0.5)))
    (+ -1.0 (* 0.5 (* (pow (* D (/ (/ M d) 2.0)) 2.0) (/ h l)))))
   (if (<= l -5e-310)
     (*
      (* d (sqrt (/ 1.0 (* l h))))
      (+ -1.0 (* 0.5 (/ (* h (pow (* D (* 0.5 (/ M d))) 2.0)) l))))
     (*
      (/ (sqrt d) (sqrt h))
      (*
       (- 1.0 (* 0.5 (* (/ h l) (pow (* (/ M 2.0) (/ D d)) 2.0))))
       (sqrt (/ d l)))))))

C

assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -100000000.0) {
		tmp = (d * (pow(-l, -0.5) * pow((-1.0 / h), 0.5))) * (-1.0 + (0.5 * (pow((D * ((M / d) / 2.0)), 2.0) * (h / l))));
	} else if (l <= -5e-310) {
		tmp = (d * sqrt((1.0 / (l * h)))) * (-1.0 + (0.5 * ((h * pow((D * (0.5 * (M / d))), 2.0)) / l)));
	} else {
		tmp = (sqrt(d) / sqrt(h)) * ((1.0 - (0.5 * ((h / l) * pow(((M / 2.0) * (D / d)), 2.0)))) * sqrt((d / l)));
	}
	return tmp;
}

Fortran

NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (l <= (-100000000.0d0)) then
        tmp = (d * ((-l ** (-0.5d0)) * (((-1.0d0) / h) ** 0.5d0))) * ((-1.0d0) + (0.5d0 * (((d_1 * ((m / d) / 2.0d0)) ** 2.0d0) * (h / l))))
    else if (l <= (-5d-310)) then
        tmp = (d * sqrt((1.0d0 / (l * h)))) * ((-1.0d0) + (0.5d0 * ((h * ((d_1 * (0.5d0 * (m / d))) ** 2.0d0)) / l)))
    else
        tmp = (sqrt(d) / sqrt(h)) * ((1.0d0 - (0.5d0 * ((h / l) * (((m / 2.0d0) * (d_1 / d)) ** 2.0d0)))) * sqrt((d / l)))
    end if
    code = tmp
end function

Java

assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -100000000.0) {
		tmp = (d * (Math.pow(-l, -0.5) * Math.pow((-1.0 / h), 0.5))) * (-1.0 + (0.5 * (Math.pow((D * ((M / d) / 2.0)), 2.0) * (h / l))));
	} else if (l <= -5e-310) {
		tmp = (d * Math.sqrt((1.0 / (l * h)))) * (-1.0 + (0.5 * ((h * Math.pow((D * (0.5 * (M / d))), 2.0)) / l)));
	} else {
		tmp = (Math.sqrt(d) / Math.sqrt(h)) * ((1.0 - (0.5 * ((h / l) * Math.pow(((M / 2.0) * (D / d)), 2.0)))) * Math.sqrt((d / l)));
	}
	return tmp;
}

Python

[M, D] = sort([M, D])
def code(d, h, l, M, D):
	tmp = 0
	if l <= -100000000.0:
		tmp = (d * (math.pow(-l, -0.5) * math.pow((-1.0 / h), 0.5))) * (-1.0 + (0.5 * (math.pow((D * ((M / d) / 2.0)), 2.0) * (h / l))))
	elif l <= -5e-310:
		tmp = (d * math.sqrt((1.0 / (l * h)))) * (-1.0 + (0.5 * ((h * math.pow((D * (0.5 * (M / d))), 2.0)) / l)))
	else:
		tmp = (math.sqrt(d) / math.sqrt(h)) * ((1.0 - (0.5 * ((h / l) * math.pow(((M / 2.0) * (D / d)), 2.0)))) * math.sqrt((d / l)))
	return tmp

Julia

M, D = sort([M, D])
function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= -100000000.0)
		tmp = Float64(Float64(d * Float64((Float64(-l) ^ -0.5) * (Float64(-1.0 / h) ^ 0.5))) * Float64(-1.0 + Float64(0.5 * Float64((Float64(D * Float64(Float64(M / d) / 2.0)) ^ 2.0) * Float64(h / l)))));
	elseif (l <= -5e-310)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(l * h)))) * Float64(-1.0 + Float64(0.5 * Float64(Float64(h * (Float64(D * Float64(0.5 * Float64(M / d))) ^ 2.0)) / l))));
	else
		tmp = Float64(Float64(sqrt(d) / sqrt(h)) * Float64(Float64(1.0 - Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(M / 2.0) * Float64(D / d)) ^ 2.0)))) * sqrt(Float64(d / l))));
	end
	return tmp
end

MATLAB

M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (l <= -100000000.0)
		tmp = (d * ((-l ^ -0.5) * ((-1.0 / h) ^ 0.5))) * (-1.0 + (0.5 * (((D * ((M / d) / 2.0)) ^ 2.0) * (h / l))));
	elseif (l <= -5e-310)
		tmp = (d * sqrt((1.0 / (l * h)))) * (-1.0 + (0.5 * ((h * ((D * (0.5 * (M / d))) ^ 2.0)) / l)));
	else
		tmp = (sqrt(d) / sqrt(h)) * ((1.0 - (0.5 * ((h / l) * (((M / 2.0) * (D / d)) ^ 2.0)))) * sqrt((d / l)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := If[LessEqual[l, -100000000.0], N[(N[(d * N[(N[Power[(-l), -0.5], $MachinePrecision] * N[Power[N[(-1.0 / h), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0 + N[(0.5 * N[(N[Power[N[(D * N[(N[(M / d), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -5e-310], N[(N[(d * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-1.0 + N[(0.5 * N[(N[(h * N[Power[N[(D * N[(0.5 * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M / 2.0), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -1e8

   1. Initial program 57.7%

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

      1. metadata-eval 57.7%

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

      2. unpow1/2 57.7%

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

      3. metadata-eval 57.7%

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

      4. unpow1/2 57.7%

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

      5. *-commutative 57.7%

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

      6. associate-*l* 57.7%

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

      7. times-frac 57.9%

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

      8. metadata-eval 57.9%

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

   3. Simplified 57.9%

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

      1. expm1-log1p-u 45.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\right)\right)} \]

      2. expm1-udef 27.7%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\right)} - 1} \]

   5. Applied egg-rr 20.9%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 35.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)\right)} \]

      2. expm1-log1p 46.0%

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

      3. sub-neg 46.0%

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

      4. +-commutative 46.0%

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

      5. associate-*l/ 43.3%

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

      6. associate-/l* 44.7%

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

      7. +-commutative 44.7%

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

      8. sub-neg 44.7%

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

      9. associate-/l* 44.7%

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

      10. associate-*r/ 44.7%

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

      11. associate-*l/ 44.7%

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

      12. associate-/l* 47.4%

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

      13. associate-*r/ 44.7%

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

   7. Simplified 44.6%

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

   8. Taylor expanded in d around -inf 64.1%

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

      1. mul-1-neg 64.1%

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

      2. associate-/r* 65.5%

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

      3. distribute-rgt-neg-in 65.5%

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

      4. associate-/l/ 64.1%

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

      5. unpow-1 64.1%

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

      6. sqr-pow 64.1%

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

      7. rem-sqrt-square 64.1%

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

      8. metadata-eval 64.1%

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

      9. sqr-pow 63.9%

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

      10. fabs-sqr 63.9%

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

      11. sqr-pow 64.1%

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

   10. Simplified 64.1%

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

   11. Taylor expanded in h around -inf 73.0%

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

       1. distribute-lft-in 73.0%

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

       2. exp-sum 73.2%

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

       3. *-commutative 73.2%

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

       4. exp-to-pow 74.1%

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

       5. mul-1-neg 74.1%

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

       6. *-commutative 74.1%

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

       7. *-commutative 74.1%

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

       8. associate-*l* 74.1%

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

       9. metadata-eval 74.1%

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

       10. exp-to-pow 77.7%

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

   13. Simplified 77.7%

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

   if -1e8 < l < -4.999999999999985e-310

   1. Initial program 63.5%

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

      1. metadata-eval 63.5%

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

      2. unpow1/2 63.5%

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

      3. metadata-eval 63.5%

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

      4. unpow1/2 63.5%

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

      5. *-commutative 63.5%

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

      6. associate-*l* 63.5%

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

      7. times-frac 63.3%

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

      8. metadata-eval 63.3%

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

   3. Simplified 63.3%

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

      1. expm1-log1p-u 26.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\right)\right)} \]

      2. expm1-udef 23.1%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\right)} - 1} \]

   5. Applied egg-rr 18.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 21.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)\right)} \]

      2. expm1-log1p 57.8%

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

      3. sub-neg 57.8%

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

      4. +-commutative 57.8%

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

      5. associate-*l/ 57.9%

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

      6. associate-/l* 57.9%

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

      7. +-commutative 57.9%

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

      8. sub-neg 57.9%

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

      9. associate-/l* 57.9%

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

      10. associate-*r/ 58.0%

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

      11. associate-*l/ 58.0%

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

      12. associate-/l* 64.2%

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

      13. associate-*r/ 57.9%

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

   7. Simplified 56.1%

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

   8. Taylor expanded in d around -inf 71.1%

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

      1. associate-*r* 71.1%

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

      2. mul-1-neg 71.1%

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

      3. *-commutative 71.1%

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

   10. Simplified 71.1%

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

       1. associate-*r/ 88.1%

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

       2. div-inv 88.1%

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

       3. metadata-eval 88.1%

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

   12. Applied egg-rr 88.1%

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

   if -4.999999999999985e-310 < l

   1. Initial program 65.1%

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

      1. associate-*l* 65.2%

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

      2. metadata-eval 65.2%

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

      3. unpow1/2 65.2%

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

      4. metadata-eval 65.2%

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

      5. unpow1/2 65.2%

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

      6. associate-*l* 65.2%

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

      7. metadata-eval 65.2%

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

      8. times-frac 65.1%

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

   3. Simplified 65.1%

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

      1. sqrt-div 78.0%

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

   5. Applied egg-rr 78.0%

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

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -100000000:\\ \;\;\;\;\left(d \cdot \left({\left(-\ell\right)}^{-0.5} \cdot {\left(\frac{-1}{h}\right)}^{0.5}\right)\right) \cdot \left(-1 + 0.5 \cdot \left({\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(-1 + 0.5 \cdot \frac{h \cdot {\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2}}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{\ell}}\right)\\ \end{array} \]

Alternative 3: 78.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} t_0 := 0.5 \cdot \frac{h \cdot {\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2}}{\ell}\\ \mathbf{if}\;\ell \leq -250000:\\ \;\;\;\;\left(d \cdot \left({\left(-\ell\right)}^{-0.5} \cdot {\left(\frac{-1}{h}\right)}^{0.5}\right)\right) \cdot \left(-1 + 0.5 \cdot \left({\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(-1 + t_0\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - t_0\right) \cdot \left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* 0.5 (/ (* h (pow (* D (* 0.5 (/ M d))) 2.0)) l))))
   (if (<= l -250000.0)
     (*
      (* d (* (pow (- l) -0.5) (pow (/ -1.0 h) 0.5)))
      (+ -1.0 (* 0.5 (* (pow (* D (/ (/ M d) 2.0)) 2.0) (/ h l)))))
     (if (<= l -5e-310)
       (* (* d (sqrt (/ 1.0 (* l h)))) (+ -1.0 t_0))
       (* (- 1.0 t_0) (* d (pow (* l h) -0.5)))))))

C

assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = 0.5 * ((h * pow((D * (0.5 * (M / d))), 2.0)) / l);
	double tmp;
	if (l <= -250000.0) {
		tmp = (d * (pow(-l, -0.5) * pow((-1.0 / h), 0.5))) * (-1.0 + (0.5 * (pow((D * ((M / d) / 2.0)), 2.0) * (h / l))));
	} else if (l <= -5e-310) {
		tmp = (d * sqrt((1.0 / (l * h)))) * (-1.0 + t_0);
	} else {
		tmp = (1.0 - t_0) * (d * pow((l * h), -0.5));
	}
	return tmp;
}

Fortran

NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * ((h * ((d_1 * (0.5d0 * (m / d))) ** 2.0d0)) / l)
    if (l <= (-250000.0d0)) then
        tmp = (d * ((-l ** (-0.5d0)) * (((-1.0d0) / h) ** 0.5d0))) * ((-1.0d0) + (0.5d0 * (((d_1 * ((m / d) / 2.0d0)) ** 2.0d0) * (h / l))))
    else if (l <= (-5d-310)) then
        tmp = (d * sqrt((1.0d0 / (l * h)))) * ((-1.0d0) + t_0)
    else
        tmp = (1.0d0 - t_0) * (d * ((l * h) ** (-0.5d0)))
    end if
    code = tmp
end function

Java

assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double t_0 = 0.5 * ((h * Math.pow((D * (0.5 * (M / d))), 2.0)) / l);
	double tmp;
	if (l <= -250000.0) {
		tmp = (d * (Math.pow(-l, -0.5) * Math.pow((-1.0 / h), 0.5))) * (-1.0 + (0.5 * (Math.pow((D * ((M / d) / 2.0)), 2.0) * (h / l))));
	} else if (l <= -5e-310) {
		tmp = (d * Math.sqrt((1.0 / (l * h)))) * (-1.0 + t_0);
	} else {
		tmp = (1.0 - t_0) * (d * Math.pow((l * h), -0.5));
	}
	return tmp;
}

Python

[M, D] = sort([M, D])
def code(d, h, l, M, D):
	t_0 = 0.5 * ((h * math.pow((D * (0.5 * (M / d))), 2.0)) / l)
	tmp = 0
	if l <= -250000.0:
		tmp = (d * (math.pow(-l, -0.5) * math.pow((-1.0 / h), 0.5))) * (-1.0 + (0.5 * (math.pow((D * ((M / d) / 2.0)), 2.0) * (h / l))))
	elif l <= -5e-310:
		tmp = (d * math.sqrt((1.0 / (l * h)))) * (-1.0 + t_0)
	else:
		tmp = (1.0 - t_0) * (d * math.pow((l * h), -0.5))
	return tmp

Julia

M, D = sort([M, D])
function code(d, h, l, M, D)
	t_0 = Float64(0.5 * Float64(Float64(h * (Float64(D * Float64(0.5 * Float64(M / d))) ^ 2.0)) / l))
	tmp = 0.0
	if (l <= -250000.0)
		tmp = Float64(Float64(d * Float64((Float64(-l) ^ -0.5) * (Float64(-1.0 / h) ^ 0.5))) * Float64(-1.0 + Float64(0.5 * Float64((Float64(D * Float64(Float64(M / d) / 2.0)) ^ 2.0) * Float64(h / l)))));
	elseif (l <= -5e-310)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(l * h)))) * Float64(-1.0 + t_0));
	else
		tmp = Float64(Float64(1.0 - t_0) * Float64(d * (Float64(l * h) ^ -0.5)));
	end
	return tmp
end

MATLAB

M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	t_0 = 0.5 * ((h * ((D * (0.5 * (M / d))) ^ 2.0)) / l);
	tmp = 0.0;
	if (l <= -250000.0)
		tmp = (d * ((-l ^ -0.5) * ((-1.0 / h) ^ 0.5))) * (-1.0 + (0.5 * (((D * ((M / d) / 2.0)) ^ 2.0) * (h / l))));
	elseif (l <= -5e-310)
		tmp = (d * sqrt((1.0 / (l * h)))) * (-1.0 + t_0);
	else
		tmp = (1.0 - t_0) * (d * ((l * h) ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(0.5 * N[(N[(h * N[Power[N[(D * N[(0.5 * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -250000.0], N[(N[(d * N[(N[Power[(-l), -0.5], $MachinePrecision] * N[Power[N[(-1.0 / h), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0 + N[(0.5 * N[(N[Power[N[(D * N[(N[(M / d), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -5e-310], N[(N[(d * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-1.0 + t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - t$95$0), $MachinePrecision] * N[(d * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < -2.5e5

   1. Initial program 57.7%

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

      1. metadata-eval 57.7%

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

      2. unpow1/2 57.7%

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

      3. metadata-eval 57.7%

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

      4. unpow1/2 57.7%

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

      5. *-commutative 57.7%

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

      6. associate-*l* 57.7%

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

      7. times-frac 57.9%

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

      8. metadata-eval 57.9%

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

   3. Simplified 57.9%

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

      1. expm1-log1p-u 45.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\right)\right)} \]

      2. expm1-udef 27.7%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\right)} - 1} \]

   5. Applied egg-rr 20.9%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 35.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)\right)} \]

      2. expm1-log1p 46.0%

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

      3. sub-neg 46.0%

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

      4. +-commutative 46.0%

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

      5. associate-*l/ 43.3%

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

      6. associate-/l* 44.7%

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

      7. +-commutative 44.7%

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

      8. sub-neg 44.7%

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

      9. associate-/l* 44.7%

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

      10. associate-*r/ 44.7%

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

      11. associate-*l/ 44.7%

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

      12. associate-/l* 47.4%

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

      13. associate-*r/ 44.7%

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

   7. Simplified 44.6%

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

   8. Taylor expanded in d around -inf 64.1%

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

      1. mul-1-neg 64.1%

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

      2. associate-/r* 65.5%

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

      3. distribute-rgt-neg-in 65.5%

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

      4. associate-/l/ 64.1%

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

      5. unpow-1 64.1%

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

      6. sqr-pow 64.1%

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

      7. rem-sqrt-square 64.1%

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

      8. metadata-eval 64.1%

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

      9. sqr-pow 63.9%

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

      10. fabs-sqr 63.9%

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

      11. sqr-pow 64.1%

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

   10. Simplified 64.1%

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

   11. Taylor expanded in h around -inf 73.0%

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

       1. distribute-lft-in 73.0%

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

       2. exp-sum 73.2%

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

       3. *-commutative 73.2%

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

       4. exp-to-pow 74.1%

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

       5. mul-1-neg 74.1%

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

       6. *-commutative 74.1%

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

       7. *-commutative 74.1%

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

       8. associate-*l* 74.1%

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

       9. metadata-eval 74.1%

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

       10. exp-to-pow 77.7%

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

   13. Simplified 77.7%

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

   if -2.5e5 < l < -4.999999999999985e-310

   1. Initial program 63.5%

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

      1. metadata-eval 63.5%

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

      2. unpow1/2 63.5%

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

      3. metadata-eval 63.5%

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

      4. unpow1/2 63.5%

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

      5. *-commutative 63.5%

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

      6. associate-*l* 63.5%

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

      7. times-frac 63.3%

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

      8. metadata-eval 63.3%

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

   3. Simplified 63.3%

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

      1. expm1-log1p-u 26.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\right)\right)} \]

      2. expm1-udef 23.1%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\right)} - 1} \]

   5. Applied egg-rr 18.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 21.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)\right)} \]

      2. expm1-log1p 57.8%

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

      3. sub-neg 57.8%

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

      4. +-commutative 57.8%

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

      5. associate-*l/ 57.9%

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

      6. associate-/l* 57.9%

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

      7. +-commutative 57.9%

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

      8. sub-neg 57.9%

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

      9. associate-/l* 57.9%

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

      10. associate-*r/ 58.0%

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

      11. associate-*l/ 58.0%

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

      12. associate-/l* 64.2%

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

      13. associate-*r/ 57.9%

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

   7. Simplified 56.1%

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

   8. Taylor expanded in d around -inf 71.1%

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

      1. associate-*r* 71.1%

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

      2. mul-1-neg 71.1%

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

      3. *-commutative 71.1%

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

   10. Simplified 71.1%

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

       1. associate-*r/ 88.1%

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

       2. div-inv 88.1%

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

       3. metadata-eval 88.1%

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

   12. Applied egg-rr 88.1%

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

   if -4.999999999999985e-310 < l

   1. Initial program 65.1%

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

      1. metadata-eval 65.1%

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

      2. unpow1/2 65.1%

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

      3. metadata-eval 65.1%

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

      4. unpow1/2 65.1%

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

      5. *-commutative 65.1%

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

      6. associate-*l* 65.1%

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

      7. times-frac 65.1%

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

      8. metadata-eval 65.1%

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

   3. Simplified 65.1%

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

      1. expm1-log1p-u 32.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\right)\right)} \]

      2. expm1-udef 28.3%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\right)} - 1} \]

   5. Applied egg-rr 21.9%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 25.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)\right)} \]

      2. expm1-log1p 51.4%

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

      3. sub-neg 51.4%

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

      4. +-commutative 51.4%

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

      5. associate-*l/ 52.9%

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

      6. associate-/l* 52.6%

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

      7. +-commutative 52.6%

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

      8. sub-neg 52.6%

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

      9. associate-/l* 52.6%

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

      10. associate-*r/ 52.6%

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

      11. associate-*l/ 52.6%

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

      12. associate-/l* 55.7%

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

      13. associate-*r/ 52.6%

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

   7. Simplified 52.6%

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

      1. associate-*r/ 4.2%

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

      2. div-inv 4.2%

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

      3. metadata-eval 4.2%

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

   9. Applied egg-rr 55.7%

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

   10. Taylor expanded in d around 0 76.9%

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

       1. *-commutative 71.7%

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

       2. *-commutative 71.7%

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

       3. unpow-1 71.7%

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

       4. sqr-pow 71.7%

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

       5. rem-sqrt-square 71.7%

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

       6. metadata-eval 71.7%

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

       7. sqr-pow 71.5%

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

       8. fabs-sqr 71.5%

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

       9. sqr-pow 71.7%

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

   12. Simplified 76.9%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -250000:\\ \;\;\;\;\left(d \cdot \left({\left(-\ell\right)}^{-0.5} \cdot {\left(\frac{-1}{h}\right)}^{0.5}\right)\right) \cdot \left(-1 + 0.5 \cdot \left({\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(-1 + 0.5 \cdot \frac{h \cdot {\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2}}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - 0.5 \cdot \frac{h \cdot {\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2}}{\ell}\right) \cdot \left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right)\\ \end{array} \]

Alternative 4: 65.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -2.9 \cdot 10^{-273}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(-1 + 0.5 \cdot \left(0.25 \cdot \left(\frac{D}{\frac{\ell}{D}} \cdot \left(\frac{M}{d} \cdot \frac{h \cdot M}{d}\right)\right)\right)\right)\\ \mathbf{elif}\;d \leq 3.1 \cdot 10^{+127}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left({\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \cdot \left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (if (<= d -2.9e-273)
   (*
    (* d (sqrt (/ 1.0 (* l h))))
    (+ -1.0 (* 0.5 (* 0.25 (* (/ D (/ l D)) (* (/ M d) (/ (* h M) d)))))))
   (if (<= d 3.1e+127)
     (*
      (- 1.0 (* 0.5 (* (pow (* D (/ (/ M d) 2.0)) 2.0) (/ h l))))
      (* d (pow (* l h) -0.5)))
     (* d (* (pow h -0.5) (pow l -0.5))))))

C

assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -2.9e-273) {
		tmp = (d * sqrt((1.0 / (l * h)))) * (-1.0 + (0.5 * (0.25 * ((D / (l / D)) * ((M / d) * ((h * M) / d))))));
	} else if (d <= 3.1e+127) {
		tmp = (1.0 - (0.5 * (pow((D * ((M / d) / 2.0)), 2.0) * (h / l)))) * (d * pow((l * h), -0.5));
	} else {
		tmp = d * (pow(h, -0.5) * pow(l, -0.5));
	}
	return tmp;
}

Fortran

NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (d <= (-2.9d-273)) then
        tmp = (d * sqrt((1.0d0 / (l * h)))) * ((-1.0d0) + (0.5d0 * (0.25d0 * ((d_1 / (l / d_1)) * ((m / d) * ((h * m) / d))))))
    else if (d <= 3.1d+127) then
        tmp = (1.0d0 - (0.5d0 * (((d_1 * ((m / d) / 2.0d0)) ** 2.0d0) * (h / l)))) * (d * ((l * h) ** (-0.5d0)))
    else
        tmp = d * ((h ** (-0.5d0)) * (l ** (-0.5d0)))
    end if
    code = tmp
end function

Java

assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -2.9e-273) {
		tmp = (d * Math.sqrt((1.0 / (l * h)))) * (-1.0 + (0.5 * (0.25 * ((D / (l / D)) * ((M / d) * ((h * M) / d))))));
	} else if (d <= 3.1e+127) {
		tmp = (1.0 - (0.5 * (Math.pow((D * ((M / d) / 2.0)), 2.0) * (h / l)))) * (d * Math.pow((l * h), -0.5));
	} else {
		tmp = d * (Math.pow(h, -0.5) * Math.pow(l, -0.5));
	}
	return tmp;
}

Python

[M, D] = sort([M, D])
def code(d, h, l, M, D):
	tmp = 0
	if d <= -2.9e-273:
		tmp = (d * math.sqrt((1.0 / (l * h)))) * (-1.0 + (0.5 * (0.25 * ((D / (l / D)) * ((M / d) * ((h * M) / d))))))
	elif d <= 3.1e+127:
		tmp = (1.0 - (0.5 * (math.pow((D * ((M / d) / 2.0)), 2.0) * (h / l)))) * (d * math.pow((l * h), -0.5))
	else:
		tmp = d * (math.pow(h, -0.5) * math.pow(l, -0.5))
	return tmp

Julia

M, D = sort([M, D])
function code(d, h, l, M, D)
	tmp = 0.0
	if (d <= -2.9e-273)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(l * h)))) * Float64(-1.0 + Float64(0.5 * Float64(0.25 * Float64(Float64(D / Float64(l / D)) * Float64(Float64(M / d) * Float64(Float64(h * M) / d)))))));
	elseif (d <= 3.1e+127)
		tmp = Float64(Float64(1.0 - Float64(0.5 * Float64((Float64(D * Float64(Float64(M / d) / 2.0)) ^ 2.0) * Float64(h / l)))) * Float64(d * (Float64(l * h) ^ -0.5)));
	else
		tmp = Float64(d * Float64((h ^ -0.5) * (l ^ -0.5)));
	end
	return tmp
end

MATLAB

M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (d <= -2.9e-273)
		tmp = (d * sqrt((1.0 / (l * h)))) * (-1.0 + (0.5 * (0.25 * ((D / (l / D)) * ((M / d) * ((h * M) / d))))));
	elseif (d <= 3.1e+127)
		tmp = (1.0 - (0.5 * (((D * ((M / d) / 2.0)) ^ 2.0) * (h / l)))) * (d * ((l * h) ^ -0.5));
	else
		tmp = d * ((h ^ -0.5) * (l ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := If[LessEqual[d, -2.9e-273], N[(N[(d * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-1.0 + N[(0.5 * N[(0.25 * N[(N[(D / N[(l / D), $MachinePrecision]), $MachinePrecision] * N[(N[(M / d), $MachinePrecision] * N[(N[(h * M), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 3.1e+127], N[(N[(1.0 - N[(0.5 * N[(N[Power[N[(D * N[(N[(M / d), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(d * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[h, -0.5], $MachinePrecision] * N[Power[l, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -2.89999999999999986e-273

   1. Initial program 61.1%

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

      1. metadata-eval 61.1%

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

      2. unpow1/2 61.1%

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

      3. metadata-eval 61.1%

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

      4. unpow1/2 61.1%

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

      5. *-commutative 61.1%

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

      6. associate-*l* 61.1%

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

      7. times-frac 61.1%

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

      8. metadata-eval 61.1%

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

   3. Simplified 61.1%

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

      1. expm1-log1p-u 37.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\right)\right)} \]

      2. expm1-udef 26.2%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\right)} - 1} \]

   5. Applied egg-rr 20.1%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 30.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)\right)} \]

      2. expm1-log1p 51.7%

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

      3. sub-neg 51.7%

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

      4. +-commutative 51.7%

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

      5. associate-*l/ 50.2%

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

      6. associate-/l* 51.0%

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

      7. +-commutative 51.0%

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

      8. sub-neg 51.0%

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

      9. associate-/l* 51.0%

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

      10. associate-*r/ 51.0%

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

      11. associate-*l/ 51.0%

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

      12. associate-/l* 55.3%

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

      13. associate-*r/ 51.0%

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

   7. Simplified 50.2%

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

   8. Taylor expanded in d around -inf 68.1%

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

      1. associate-*r* 68.1%

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

      2. mul-1-neg 68.1%

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

      3. *-commutative 68.1%

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

   10. Simplified 68.1%

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

   11. Taylor expanded in D around 0 48.1%

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

       1. times-frac 29.3%

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

       2. *-commutative 29.3%

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

       3. unpow2 29.3%

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

       4. associate-/l* 31.8%

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

       5. *-commutative 31.8%

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

       6. unpow2 31.8%

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

       7. associate-*r* 33.5%

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

       8. unpow2 33.5%

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

       9. times-frac 45.2%

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

   13. Simplified 64.7%

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

   if -2.89999999999999986e-273 < d < 3.1000000000000002e127

   1. Initial program 62.2%

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

      1. metadata-eval 62.2%

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

      2. unpow1/2 62.2%

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

      3. metadata-eval 62.2%

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

      4. unpow1/2 62.2%

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

      5. *-commutative 62.2%

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

      6. associate-*l* 62.2%

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

      7. times-frac 62.2%

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

      8. metadata-eval 62.2%

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

   3. Simplified 62.2%

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

      1. expm1-log1p-u 24.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\right)\right)} \]

      2. expm1-udef 19.8%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\right)} - 1} \]

   5. Applied egg-rr 16.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 18.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)\right)} \]

      2. expm1-log1p 48.9%

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

      3. sub-neg 48.9%

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

      4. +-commutative 48.9%

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

      5. associate-*l/ 50.0%

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

      6. associate-/l* 48.5%

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

      7. +-commutative 48.5%

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

      8. sub-neg 48.5%

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

      9. associate-/l* 48.5%

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

      10. associate-*r/ 48.5%

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

      11. associate-*l/ 48.5%

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

      12. associate-/l* 50.6%

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

      13. associate-*r/ 48.5%

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

   7. Simplified 48.6%

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

   8. Taylor expanded in d around 0 70.6%

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

      1. *-commutative 70.6%

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

      2. *-commutative 70.6%

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

      3. unpow-1 70.6%

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

      4. sqr-pow 70.6%

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

      5. rem-sqrt-square 70.6%

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

      6. metadata-eval 70.6%

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

      7. sqr-pow 70.5%

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

      8. fabs-sqr 70.5%

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

      9. sqr-pow 70.6%

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

   10. Simplified 70.6%

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

   if 3.1000000000000002e127 < d

   1. Initial program 68.8%

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

      1. metadata-eval 68.8%

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

      2. unpow1/2 68.8%

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

      3. metadata-eval 68.8%

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

      4. unpow1/2 68.8%

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

      5. *-commutative 68.8%

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

      6. associate-*l* 68.8%

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

      7. times-frac 68.8%

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

      8. metadata-eval 68.8%

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

   3. Simplified 68.8%

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

   4. Taylor expanded in d around inf 68.1%

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

      1. *-un-lft-identity 68.1%

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

      2. associate-/r* 69.1%

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

   6. Applied egg-rr 69.1%

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

      1. *-lft-identity 69.1%

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

      2. associate-/l/ 68.1%

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

      3. unpow-1 68.1%

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

      4. sqr-pow 68.1%

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

      5. rem-sqrt-square 68.1%

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

      6. metadata-eval 68.1%

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

      7. sqr-pow 67.8%

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

      8. fabs-sqr 67.8%

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

      9. sqr-pow 68.1%

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

   8. Simplified 68.1%

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

      1. unpow-prod-down 82.1%

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

   10. Applied egg-rr 82.1%

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

4. Final simplification 69.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.9 \cdot 10^{-273}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(-1 + 0.5 \cdot \left(0.25 \cdot \left(\frac{D}{\frac{\ell}{D}} \cdot \left(\frac{M}{d} \cdot \frac{h \cdot M}{d}\right)\right)\right)\right)\\ \mathbf{elif}\;d \leq 3.1 \cdot 10^{+127}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left({\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \cdot \left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \]

Alternative 5: 72.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} t_0 := d \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \mathbf{if}\;h \leq -4.5 \cdot 10^{+205}:\\ \;\;\;\;0.125 \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\left(M \cdot M\right) \cdot \left(D \cdot \frac{D}{d}\right)\right)\right)\\ \mathbf{elif}\;h \leq -2 \cdot 10^{-310}:\\ \;\;\;\;t_0 \cdot \left(-1 + 0.5 \cdot \left({\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - 0.5 \cdot \frac{h \cdot {\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2}}{\ell}\right) \cdot t_0\\ \end{array} \end{array} \]

FPCore

NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* d (pow (* l h) -0.5))))
   (if (<= h -4.5e+205)
     (* 0.125 (* (sqrt (/ h (pow l 3.0))) (* (* M M) (* D (/ D d)))))
     (if (<= h -2e-310)
       (* t_0 (+ -1.0 (* 0.5 (* (pow (* D (/ (/ M d) 2.0)) 2.0) (/ h l)))))
       (* (- 1.0 (* 0.5 (/ (* h (pow (* D (* 0.5 (/ M d))) 2.0)) l))) t_0)))))

C

assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = d * pow((l * h), -0.5);
	double tmp;
	if (h <= -4.5e+205) {
		tmp = 0.125 * (sqrt((h / pow(l, 3.0))) * ((M * M) * (D * (D / d))));
	} else if (h <= -2e-310) {
		tmp = t_0 * (-1.0 + (0.5 * (pow((D * ((M / d) / 2.0)), 2.0) * (h / l))));
	} else {
		tmp = (1.0 - (0.5 * ((h * pow((D * (0.5 * (M / d))), 2.0)) / l))) * t_0;
	}
	return tmp;
}

Fortran

NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = d * ((l * h) ** (-0.5d0))
    if (h <= (-4.5d+205)) then
        tmp = 0.125d0 * (sqrt((h / (l ** 3.0d0))) * ((m * m) * (d_1 * (d_1 / d))))
    else if (h <= (-2d-310)) then
        tmp = t_0 * ((-1.0d0) + (0.5d0 * (((d_1 * ((m / d) / 2.0d0)) ** 2.0d0) * (h / l))))
    else
        tmp = (1.0d0 - (0.5d0 * ((h * ((d_1 * (0.5d0 * (m / d))) ** 2.0d0)) / l))) * t_0
    end if
    code = tmp
end function

Java

assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double t_0 = d * Math.pow((l * h), -0.5);
	double tmp;
	if (h <= -4.5e+205) {
		tmp = 0.125 * (Math.sqrt((h / Math.pow(l, 3.0))) * ((M * M) * (D * (D / d))));
	} else if (h <= -2e-310) {
		tmp = t_0 * (-1.0 + (0.5 * (Math.pow((D * ((M / d) / 2.0)), 2.0) * (h / l))));
	} else {
		tmp = (1.0 - (0.5 * ((h * Math.pow((D * (0.5 * (M / d))), 2.0)) / l))) * t_0;
	}
	return tmp;
}

Python

[M, D] = sort([M, D])
def code(d, h, l, M, D):
	t_0 = d * math.pow((l * h), -0.5)
	tmp = 0
	if h <= -4.5e+205:
		tmp = 0.125 * (math.sqrt((h / math.pow(l, 3.0))) * ((M * M) * (D * (D / d))))
	elif h <= -2e-310:
		tmp = t_0 * (-1.0 + (0.5 * (math.pow((D * ((M / d) / 2.0)), 2.0) * (h / l))))
	else:
		tmp = (1.0 - (0.5 * ((h * math.pow((D * (0.5 * (M / d))), 2.0)) / l))) * t_0
	return tmp

Julia

M, D = sort([M, D])
function code(d, h, l, M, D)
	t_0 = Float64(d * (Float64(l * h) ^ -0.5))
	tmp = 0.0
	if (h <= -4.5e+205)
		tmp = Float64(0.125 * Float64(sqrt(Float64(h / (l ^ 3.0))) * Float64(Float64(M * M) * Float64(D * Float64(D / d)))));
	elseif (h <= -2e-310)
		tmp = Float64(t_0 * Float64(-1.0 + Float64(0.5 * Float64((Float64(D * Float64(Float64(M / d) / 2.0)) ^ 2.0) * Float64(h / l)))));
	else
		tmp = Float64(Float64(1.0 - Float64(0.5 * Float64(Float64(h * (Float64(D * Float64(0.5 * Float64(M / d))) ^ 2.0)) / l))) * t_0);
	end
	return tmp
end

MATLAB

M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	t_0 = d * ((l * h) ^ -0.5);
	tmp = 0.0;
	if (h <= -4.5e+205)
		tmp = 0.125 * (sqrt((h / (l ^ 3.0))) * ((M * M) * (D * (D / d))));
	elseif (h <= -2e-310)
		tmp = t_0 * (-1.0 + (0.5 * (((D * ((M / d) / 2.0)) ^ 2.0) * (h / l))));
	else
		tmp = (1.0 - (0.5 * ((h * ((D * (0.5 * (M / d))) ^ 2.0)) / l))) * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(d * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[h, -4.5e+205], N[(0.125 * N[(N[Sqrt[N[(h / N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(M * M), $MachinePrecision] * N[(D * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, -2e-310], N[(t$95$0 * N[(-1.0 + N[(0.5 * N[(N[Power[N[(D * N[(N[(M / d), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[(0.5 * N[(N[(h * N[Power[N[(D * N[(0.5 * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if h < -4.50000000000000035e205

   1. Initial program 33.2%

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

      1. metadata-eval 33.2%

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

      2. unpow1/2 33.2%

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

      3. metadata-eval 33.2%

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

      4. unpow1/2 33.2%

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

      5. *-commutative 33.2%

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

      6. associate-*l* 33.2%

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

      7. times-frac 32.9%

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

      8. metadata-eval 32.9%

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

   3. Simplified 32.9%

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

      1. expm1-log1p-u 5.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\right)\right)} \]

      2. expm1-udef 1.2%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\right)} - 1} \]

   5. Applied egg-rr 1.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 5.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)\right)} \]

      2. expm1-log1p 32.9%

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

      3. sub-neg 32.9%

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

      4. +-commutative 32.9%

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

      5. associate-*l/ 37.4%

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

      6. associate-/l* 33.0%

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

      7. +-commutative 33.0%

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

      8. sub-neg 33.0%

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

      9. associate-/l* 33.0%

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

      10. associate-*r/ 33.0%

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

      11. associate-*l/ 33.0%

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

      12. associate-/l* 42.1%

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

      13. associate-*r/ 33.0%

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

   7. Simplified 33.0%

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

   8. Taylor expanded in d around -inf 32.7%

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

      1. associate-*r* 32.7%

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

      2. mul-1-neg 32.7%

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

      3. *-commutative 32.7%

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

   10. Simplified 32.7%

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

   11. Taylor expanded in d around 0 51.1%

       \[\leadsto \color{blue}{0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   12. Step-by-step derivation

       1. unpow2 51.1%

          \[\leadsto 0.125 \cdot \left(\frac{\color{blue}{\left(D \cdot D\right)} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \]

       2. associate-*l* 51.5%

          \[\leadsto 0.125 \cdot \left(\frac{\color{blue}{D \cdot \left(D \cdot {M}^{2}\right)}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \]

       3. associate-*r/ 51.5%

          \[\leadsto 0.125 \cdot \left(\color{blue}{\left(D \cdot \frac{D \cdot {M}^{2}}{d}\right)} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \]

       4. associate-*l/ 51.0%

          \[\leadsto 0.125 \cdot \left(\left(D \cdot \color{blue}{\left(\frac{D}{d} \cdot {M}^{2}\right)}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \]

       5. unpow2 51.0%

          \[\leadsto 0.125 \cdot \left(\left(D \cdot \left(\frac{D}{d} \cdot \color{blue}{\left(M \cdot M\right)}\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \]

       6. *-commutative 51.0%

          \[\leadsto 0.125 \cdot \color{blue}{\left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(D \cdot \left(\frac{D}{d} \cdot \left(M \cdot M\right)\right)\right)\right)} \]

       7. *-commutative 51.0%

          \[\leadsto 0.125 \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \color{blue}{\left(\left(\frac{D}{d} \cdot \left(M \cdot M\right)\right) \cdot D\right)}\right) \]

       8. unpow2 51.0%

          \[\leadsto 0.125 \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\left(\frac{D}{d} \cdot \color{blue}{{M}^{2}}\right) \cdot D\right)\right) \]

       9. *-commutative 51.0%

          \[\leadsto 0.125 \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\color{blue}{\left({M}^{2} \cdot \frac{D}{d}\right)} \cdot D\right)\right) \]

       10. associate-*l* 46.5%

           \[\leadsto 0.125 \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \color{blue}{\left({M}^{2} \cdot \left(\frac{D}{d} \cdot D\right)\right)}\right) \]

       11. unpow2 46.5%

           \[\leadsto 0.125 \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\color{blue}{\left(M \cdot M\right)} \cdot \left(\frac{D}{d} \cdot D\right)\right)\right) \]

       12. *-commutative 46.5%

           \[\leadsto 0.125 \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\left(M \cdot M\right) \cdot \color{blue}{\left(D \cdot \frac{D}{d}\right)}\right)\right) \]

   13. Simplified 46.5%

       \[\leadsto \color{blue}{0.125 \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\left(M \cdot M\right) \cdot \left(D \cdot \frac{D}{d}\right)\right)\right)} \]

   if -4.50000000000000035e205 < h < -1.999999999999994e-310

   1. Initial program 66.0%

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

      1. metadata-eval 66.0%

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

      2. unpow1/2 66.0%

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

      3. metadata-eval 66.0%

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

      4. unpow1/2 66.0%

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

      5. *-commutative 66.0%

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

      6. associate-*l* 66.0%

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

      7. times-frac 66.1%

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

      8. metadata-eval 66.1%

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

   3. Simplified 66.1%

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

      1. expm1-log1p-u 44.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\right)\right)} \]

      2. expm1-udef 31.2%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\right)} - 1} \]

   5. Applied egg-rr 23.8%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 34.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)\right)} \]

      2. expm1-log1p 54.8%

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

      3. sub-neg 54.8%

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

      4. +-commutative 54.8%

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

      5. associate-*l/ 52.0%

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

      6. associate-/l* 53.9%

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

      7. +-commutative 53.9%

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

      8. sub-neg 53.9%

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

      9. associate-/l* 53.9%

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

      10. associate-*r/ 53.9%

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

      11. associate-*l/ 53.9%

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

      12. associate-/l* 57.0%

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

      13. associate-*r/ 53.9%

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

   7. Simplified 52.9%

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

   8. Taylor expanded in d around -inf 74.5%

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

      1. mul-1-neg 74.5%

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

      2. associate-/r* 74.5%

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

      3. distribute-rgt-neg-in 74.5%

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

      4. associate-/l/ 74.5%

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

      5. unpow-1 74.5%

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

      6. sqr-pow 74.5%

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

      7. rem-sqrt-square 74.5%

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

      8. metadata-eval 74.5%

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

      9. sqr-pow 74.3%

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

      10. fabs-sqr 74.3%

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

      11. sqr-pow 74.5%

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

   10. Simplified 74.5%

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

   if -1.999999999999994e-310 < h

   1. Initial program 65.1%

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

      1. metadata-eval 65.1%

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

      2. unpow1/2 65.1%

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

      3. metadata-eval 65.1%

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

      4. unpow1/2 65.1%

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

      5. *-commutative 65.1%

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

      6. associate-*l* 65.1%

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

      7. times-frac 65.1%

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

      8. metadata-eval 65.1%

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

   3. Simplified 65.1%

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

      1. expm1-log1p-u 32.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\right)\right)} \]

      2. expm1-udef 28.3%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\right)} - 1} \]

   5. Applied egg-rr 21.9%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 25.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)\right)} \]

      2. expm1-log1p 51.4%

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

      3. sub-neg 51.4%

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

      4. +-commutative 51.4%

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

      5. associate-*l/ 52.9%

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

      6. associate-/l* 52.6%

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

      7. +-commutative 52.6%

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

      8. sub-neg 52.6%

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

      9. associate-/l* 52.6%

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

      10. associate-*r/ 52.6%

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

      11. associate-*l/ 52.6%

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

      12. associate-/l* 55.7%

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

      13. associate-*r/ 52.6%

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

   7. Simplified 52.6%

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

      1. associate-*r/ 4.2%

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

      2. div-inv 4.2%

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

      3. metadata-eval 4.2%

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

   9. Applied egg-rr 55.7%

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

   10. Taylor expanded in d around 0 76.9%

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

       1. *-commutative 71.7%

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

       2. *-commutative 71.7%

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

       3. unpow-1 71.7%

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

       4. sqr-pow 71.7%

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

       5. rem-sqrt-square 71.7%

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

       6. metadata-eval 71.7%

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

       7. sqr-pow 71.5%

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

       8. fabs-sqr 71.5%

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

       9. sqr-pow 71.7%

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

   12. Simplified 76.9%

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

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -4.5 \cdot 10^{+205}:\\ \;\;\;\;0.125 \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\left(M \cdot M\right) \cdot \left(D \cdot \frac{D}{d}\right)\right)\right)\\ \mathbf{elif}\;h \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right) \cdot \left(-1 + 0.5 \cdot \left({\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - 0.5 \cdot \frac{h \cdot {\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2}}{\ell}\right) \cdot \left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right)\\ \end{array} \]

Alternative 6: 70.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(-1 + 0.5 \cdot \left(0.25 \cdot \left(\frac{D}{\frac{\ell}{D}} \cdot \left(\frac{M}{d} \cdot \frac{h \cdot M}{d}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - 0.5 \cdot \frac{h \cdot {\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2}}{\ell}\right) \cdot \left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (if (<= l -5e-310)
   (*
    (* d (sqrt (/ 1.0 (* l h))))
    (+ -1.0 (* 0.5 (* 0.25 (* (/ D (/ l D)) (* (/ M d) (/ (* h M) d)))))))
   (*
    (- 1.0 (* 0.5 (/ (* h (pow (* D (* 0.5 (/ M d))) 2.0)) l)))
    (* d (pow (* l h) -0.5)))))

C

assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -5e-310) {
		tmp = (d * sqrt((1.0 / (l * h)))) * (-1.0 + (0.5 * (0.25 * ((D / (l / D)) * ((M / d) * ((h * M) / d))))));
	} else {
		tmp = (1.0 - (0.5 * ((h * pow((D * (0.5 * (M / d))), 2.0)) / l))) * (d * pow((l * h), -0.5));
	}
	return tmp;
}

Fortran

NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (l <= (-5d-310)) then
        tmp = (d * sqrt((1.0d0 / (l * h)))) * ((-1.0d0) + (0.5d0 * (0.25d0 * ((d_1 / (l / d_1)) * ((m / d) * ((h * m) / d))))))
    else
        tmp = (1.0d0 - (0.5d0 * ((h * ((d_1 * (0.5d0 * (m / d))) ** 2.0d0)) / l))) * (d * ((l * h) ** (-0.5d0)))
    end if
    code = tmp
end function

Java

assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -5e-310) {
		tmp = (d * Math.sqrt((1.0 / (l * h)))) * (-1.0 + (0.5 * (0.25 * ((D / (l / D)) * ((M / d) * ((h * M) / d))))));
	} else {
		tmp = (1.0 - (0.5 * ((h * Math.pow((D * (0.5 * (M / d))), 2.0)) / l))) * (d * Math.pow((l * h), -0.5));
	}
	return tmp;
}

Python

[M, D] = sort([M, D])
def code(d, h, l, M, D):
	tmp = 0
	if l <= -5e-310:
		tmp = (d * math.sqrt((1.0 / (l * h)))) * (-1.0 + (0.5 * (0.25 * ((D / (l / D)) * ((M / d) * ((h * M) / d))))))
	else:
		tmp = (1.0 - (0.5 * ((h * math.pow((D * (0.5 * (M / d))), 2.0)) / l))) * (d * math.pow((l * h), -0.5))
	return tmp

Julia

M, D = sort([M, D])
function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= -5e-310)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(l * h)))) * Float64(-1.0 + Float64(0.5 * Float64(0.25 * Float64(Float64(D / Float64(l / D)) * Float64(Float64(M / d) * Float64(Float64(h * M) / d)))))));
	else
		tmp = Float64(Float64(1.0 - Float64(0.5 * Float64(Float64(h * (Float64(D * Float64(0.5 * Float64(M / d))) ^ 2.0)) / l))) * Float64(d * (Float64(l * h) ^ -0.5)));
	end
	return tmp
end

MATLAB

M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (l <= -5e-310)
		tmp = (d * sqrt((1.0 / (l * h)))) * (-1.0 + (0.5 * (0.25 * ((D / (l / D)) * ((M / d) * ((h * M) / d))))));
	else
		tmp = (1.0 - (0.5 * ((h * ((D * (0.5 * (M / d))) ^ 2.0)) / l))) * (d * ((l * h) ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := If[LessEqual[l, -5e-310], N[(N[(d * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-1.0 + N[(0.5 * N[(0.25 * N[(N[(D / N[(l / D), $MachinePrecision]), $MachinePrecision] * N[(N[(M / d), $MachinePrecision] * N[(N[(h * M), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[(0.5 * N[(N[(h * N[Power[N[(D * N[(0.5 * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(d * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -4.999999999999985e-310

   1. Initial program 60.1%

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

      1. metadata-eval 60.1%

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

      2. unpow1/2 60.1%

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

      3. metadata-eval 60.1%

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

      4. unpow1/2 60.1%

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

      5. *-commutative 60.1%

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

      6. associate-*l* 60.1%

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

      7. times-frac 60.1%

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

      8. metadata-eval 60.1%

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

   3. Simplified 60.1%

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

      1. expm1-log1p-u 37.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\right)\right)} \]

      2. expm1-udef 25.8%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\right)} - 1} \]

   5. Applied egg-rr 19.8%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 29.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)\right)} \]

      2. expm1-log1p 50.9%

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

      3. sub-neg 50.9%

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

      4. +-commutative 50.9%

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

      5. associate-*l/ 49.4%

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

      6. associate-/l* 50.2%

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

      7. +-commutative 50.2%

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

      8. sub-neg 50.2%

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

      9. associate-/l* 50.2%

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

      10. associate-*r/ 50.2%

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

      11. associate-*l/ 50.2%

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

      12. associate-/l* 54.4%

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

      13. associate-*r/ 50.2%

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

   7. Simplified 49.3%

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

   8. Taylor expanded in d around -inf 67.0%

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

      1. associate-*r* 67.0%

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

      2. mul-1-neg 67.0%

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

      3. *-commutative 67.0%

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

   10. Simplified 67.0%

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

   11. Taylor expanded in D around 0 47.3%

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

       1. times-frac 28.8%

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

       2. *-commutative 28.8%

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

       3. unpow2 28.8%

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

       4. associate-/l* 31.2%

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

       5. *-commutative 31.2%

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

       6. unpow2 31.2%

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

       7. associate-*r* 32.9%

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

       8. unpow2 32.9%

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

       9. times-frac 44.5%

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

   13. Simplified 63.7%

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

   if -4.999999999999985e-310 < l

   1. Initial program 65.1%

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

      1. metadata-eval 65.1%

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

      2. unpow1/2 65.1%

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

      3. metadata-eval 65.1%

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

      4. unpow1/2 65.1%

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

      5. *-commutative 65.1%

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

      6. associate-*l* 65.1%

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

      7. times-frac 65.1%

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

      8. metadata-eval 65.1%

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

   3. Simplified 65.1%

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

      1. expm1-log1p-u 32.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\right)\right)} \]

      2. expm1-udef 28.3%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\right)} - 1} \]

   5. Applied egg-rr 21.9%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 25.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)\right)} \]

      2. expm1-log1p 51.4%

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

      3. sub-neg 51.4%

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

      4. +-commutative 51.4%

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

      5. associate-*l/ 52.9%

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

      6. associate-/l* 52.6%

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

      7. +-commutative 52.6%

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

      8. sub-neg 52.6%

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

      9. associate-/l* 52.6%

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

      10. associate-*r/ 52.6%

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

      11. associate-*l/ 52.6%

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

      12. associate-/l* 55.7%

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

      13. associate-*r/ 52.6%

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

   7. Simplified 52.6%

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

      1. associate-*r/ 4.2%

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

      2. div-inv 4.2%

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

      3. metadata-eval 4.2%

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

   9. Applied egg-rr 55.7%

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

   10. Taylor expanded in d around 0 76.9%

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

       1. *-commutative 71.7%

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

       2. *-commutative 71.7%

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

       3. unpow-1 71.7%

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

       4. sqr-pow 71.7%

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

       5. rem-sqrt-square 71.7%

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

       6. metadata-eval 71.7%

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

       7. sqr-pow 71.5%

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

       8. fabs-sqr 71.5%

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

       9. sqr-pow 71.7%

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

   12. Simplified 76.9%

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

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(-1 + 0.5 \cdot \left(0.25 \cdot \left(\frac{D}{\frac{\ell}{D}} \cdot \left(\frac{M}{d} \cdot \frac{h \cdot M}{d}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - 0.5 \cdot \frac{h \cdot {\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2}}{\ell}\right) \cdot \left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right)\\ \end{array} \]

Alternative 7: 74.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} t_0 := 0.5 \cdot \frac{h \cdot {\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2}}{\ell}\\ t_1 := d \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \mathbf{if}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;t_1 \cdot \left(-1 + t_0\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - t_0\right) \cdot t_1\\ \end{array} \end{array} \]

FPCore

NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* 0.5 (/ (* h (pow (* D (* 0.5 (/ M d))) 2.0)) l)))
        (t_1 (* d (pow (* l h) -0.5))))
   (if (<= l -5e-310) (* t_1 (+ -1.0 t_0)) (* (- 1.0 t_0) t_1))))

C

assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = 0.5 * ((h * pow((D * (0.5 * (M / d))), 2.0)) / l);
	double t_1 = d * pow((l * h), -0.5);
	double tmp;
	if (l <= -5e-310) {
		tmp = t_1 * (-1.0 + t_0);
	} else {
		tmp = (1.0 - t_0) * t_1;
	}
	return tmp;
}

Fortran

NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 0.5d0 * ((h * ((d_1 * (0.5d0 * (m / d))) ** 2.0d0)) / l)
    t_1 = d * ((l * h) ** (-0.5d0))
    if (l <= (-5d-310)) then
        tmp = t_1 * ((-1.0d0) + t_0)
    else
        tmp = (1.0d0 - t_0) * t_1
    end if
    code = tmp
end function

Java

assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double t_0 = 0.5 * ((h * Math.pow((D * (0.5 * (M / d))), 2.0)) / l);
	double t_1 = d * Math.pow((l * h), -0.5);
	double tmp;
	if (l <= -5e-310) {
		tmp = t_1 * (-1.0 + t_0);
	} else {
		tmp = (1.0 - t_0) * t_1;
	}
	return tmp;
}

Python

[M, D] = sort([M, D])
def code(d, h, l, M, D):
	t_0 = 0.5 * ((h * math.pow((D * (0.5 * (M / d))), 2.0)) / l)
	t_1 = d * math.pow((l * h), -0.5)
	tmp = 0
	if l <= -5e-310:
		tmp = t_1 * (-1.0 + t_0)
	else:
		tmp = (1.0 - t_0) * t_1
	return tmp

Julia

M, D = sort([M, D])
function code(d, h, l, M, D)
	t_0 = Float64(0.5 * Float64(Float64(h * (Float64(D * Float64(0.5 * Float64(M / d))) ^ 2.0)) / l))
	t_1 = Float64(d * (Float64(l * h) ^ -0.5))
	tmp = 0.0
	if (l <= -5e-310)
		tmp = Float64(t_1 * Float64(-1.0 + t_0));
	else
		tmp = Float64(Float64(1.0 - t_0) * t_1);
	end
	return tmp
end

MATLAB

M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	t_0 = 0.5 * ((h * ((D * (0.5 * (M / d))) ^ 2.0)) / l);
	t_1 = d * ((l * h) ^ -0.5);
	tmp = 0.0;
	if (l <= -5e-310)
		tmp = t_1 * (-1.0 + t_0);
	else
		tmp = (1.0 - t_0) * t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(0.5 * N[(N[(h * N[Power[N[(D * N[(0.5 * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(d * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -5e-310], N[(t$95$1 * N[(-1.0 + t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - t$95$0), $MachinePrecision] * t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if l < -4.999999999999985e-310

   1. Initial program 60.1%

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

      1. metadata-eval 60.1%

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

      2. unpow1/2 60.1%

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

      3. metadata-eval 60.1%

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

      4. unpow1/2 60.1%

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

      5. *-commutative 60.1%

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

      6. associate-*l* 60.1%

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

      7. times-frac 60.1%

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

      8. metadata-eval 60.1%

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

   3. Simplified 60.1%

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

      1. expm1-log1p-u 37.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\right)\right)} \]

      2. expm1-udef 25.8%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\right)} - 1} \]

   5. Applied egg-rr 19.8%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 29.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)\right)} \]

      2. expm1-log1p 50.9%

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

      3. sub-neg 50.9%

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

      4. +-commutative 50.9%

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

      5. associate-*l/ 49.4%

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

      6. associate-/l* 50.2%

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

      7. +-commutative 50.2%

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

      8. sub-neg 50.2%

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

      9. associate-/l* 50.2%

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

      10. associate-*r/ 50.2%

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

      11. associate-*l/ 50.2%

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

      12. associate-/l* 54.4%

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

      13. associate-*r/ 50.2%

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

   7. Simplified 49.3%

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

      1. associate-*r/ 75.8%

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

      2. div-inv 75.8%

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

      3. metadata-eval 75.8%

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

   9. Applied egg-rr 53.6%

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

   10. Taylor expanded in d around -inf 75.8%

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

       1. mul-1-neg 67.0%

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

       2. associate-/r* 67.8%

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

       3. distribute-rgt-neg-in 67.8%

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

       4. associate-/l/ 67.0%

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

       5. unpow-1 67.0%

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

       6. sqr-pow 67.0%

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

       7. rem-sqrt-square 67.0%

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

       8. metadata-eval 67.0%

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

       9. sqr-pow 66.9%

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

       10. fabs-sqr 66.9%

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

       11. sqr-pow 67.0%

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

   12. Simplified 75.8%

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

   if -4.999999999999985e-310 < l

   1. Initial program 65.1%

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

      1. metadata-eval 65.1%

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

      2. unpow1/2 65.1%

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

      3. metadata-eval 65.1%

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

      4. unpow1/2 65.1%

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

      5. *-commutative 65.1%

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

      6. associate-*l* 65.1%

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

      7. times-frac 65.1%

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

      8. metadata-eval 65.1%

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

   3. Simplified 65.1%

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

      1. expm1-log1p-u 32.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\right)\right)} \]

      2. expm1-udef 28.3%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\right)} - 1} \]

   5. Applied egg-rr 21.9%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 25.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)\right)} \]

      2. expm1-log1p 51.4%

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

      3. sub-neg 51.4%

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

      4. +-commutative 51.4%

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

      5. associate-*l/ 52.9%

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

      6. associate-/l* 52.6%

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

      7. +-commutative 52.6%

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

      8. sub-neg 52.6%

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

      9. associate-/l* 52.6%

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

      10. associate-*r/ 52.6%

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

      11. associate-*l/ 52.6%

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

      12. associate-/l* 55.7%

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

      13. associate-*r/ 52.6%

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

   7. Simplified 52.6%

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

      1. associate-*r/ 4.2%

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

      2. div-inv 4.2%

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

      3. metadata-eval 4.2%

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

   9. Applied egg-rr 55.7%

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

   10. Taylor expanded in d around 0 76.9%

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

       1. *-commutative 71.7%

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

       2. *-commutative 71.7%

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

       3. unpow-1 71.7%

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

       4. sqr-pow 71.7%

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

       5. rem-sqrt-square 71.7%

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

       6. metadata-eval 71.7%

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

       7. sqr-pow 71.5%

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

       8. fabs-sqr 71.5%

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

       9. sqr-pow 71.7%

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

   12. Simplified 76.9%

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

4. Final simplification 76.4%

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

Alternative 8: 61.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} t_0 := 0.5 \cdot \left(0.25 \cdot \left(\frac{D}{\frac{\ell}{D}} \cdot \left(\frac{M}{d} \cdot \frac{h \cdot M}{d}\right)\right)\right)\\ \mathbf{if}\;d \leq -2.9 \cdot 10^{-273}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(-1 + t_0\right)\\ \mathbf{elif}\;d \leq 1.82 \cdot 10^{-134}:\\ \;\;\;\;-0.125 \cdot \left(D \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(M \cdot \left(M \cdot \frac{D}{d}\right)\right)\right)\right)\\ \mathbf{elif}\;d \leq 5.6 \cdot 10^{+34}:\\ \;\;\;\;\left(1 - t_0\right) \cdot \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* 0.5 (* 0.25 (* (/ D (/ l D)) (* (/ M d) (/ (* h M) d)))))))
   (if (<= d -2.9e-273)
     (* (* d (sqrt (/ 1.0 (* l h)))) (+ -1.0 t_0))
     (if (<= d 1.82e-134)
       (* -0.125 (* D (* (sqrt (/ h (pow l 3.0))) (* M (* M (/ D d))))))
       (if (<= d 5.6e+34)
         (* (- 1.0 t_0) (sqrt (/ d (/ h (/ d l)))))
         (* d (* (pow h -0.5) (pow l -0.5))))))))

C

assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = 0.5 * (0.25 * ((D / (l / D)) * ((M / d) * ((h * M) / d))));
	double tmp;
	if (d <= -2.9e-273) {
		tmp = (d * sqrt((1.0 / (l * h)))) * (-1.0 + t_0);
	} else if (d <= 1.82e-134) {
		tmp = -0.125 * (D * (sqrt((h / pow(l, 3.0))) * (M * (M * (D / d)))));
	} else if (d <= 5.6e+34) {
		tmp = (1.0 - t_0) * sqrt((d / (h / (d / l))));
	} else {
		tmp = d * (pow(h, -0.5) * pow(l, -0.5));
	}
	return tmp;
}

Fortran

NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * (0.25d0 * ((d_1 / (l / d_1)) * ((m / d) * ((h * m) / d))))
    if (d <= (-2.9d-273)) then
        tmp = (d * sqrt((1.0d0 / (l * h)))) * ((-1.0d0) + t_0)
    else if (d <= 1.82d-134) then
        tmp = (-0.125d0) * (d_1 * (sqrt((h / (l ** 3.0d0))) * (m * (m * (d_1 / d)))))
    else if (d <= 5.6d+34) then
        tmp = (1.0d0 - t_0) * sqrt((d / (h / (d / l))))
    else
        tmp = d * ((h ** (-0.5d0)) * (l ** (-0.5d0)))
    end if
    code = tmp
end function

Java

assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double t_0 = 0.5 * (0.25 * ((D / (l / D)) * ((M / d) * ((h * M) / d))));
	double tmp;
	if (d <= -2.9e-273) {
		tmp = (d * Math.sqrt((1.0 / (l * h)))) * (-1.0 + t_0);
	} else if (d <= 1.82e-134) {
		tmp = -0.125 * (D * (Math.sqrt((h / Math.pow(l, 3.0))) * (M * (M * (D / d)))));
	} else if (d <= 5.6e+34) {
		tmp = (1.0 - t_0) * Math.sqrt((d / (h / (d / l))));
	} else {
		tmp = d * (Math.pow(h, -0.5) * Math.pow(l, -0.5));
	}
	return tmp;
}

Python

[M, D] = sort([M, D])
def code(d, h, l, M, D):
	t_0 = 0.5 * (0.25 * ((D / (l / D)) * ((M / d) * ((h * M) / d))))
	tmp = 0
	if d <= -2.9e-273:
		tmp = (d * math.sqrt((1.0 / (l * h)))) * (-1.0 + t_0)
	elif d <= 1.82e-134:
		tmp = -0.125 * (D * (math.sqrt((h / math.pow(l, 3.0))) * (M * (M * (D / d)))))
	elif d <= 5.6e+34:
		tmp = (1.0 - t_0) * math.sqrt((d / (h / (d / l))))
	else:
		tmp = d * (math.pow(h, -0.5) * math.pow(l, -0.5))
	return tmp

Julia

M, D = sort([M, D])
function code(d, h, l, M, D)
	t_0 = Float64(0.5 * Float64(0.25 * Float64(Float64(D / Float64(l / D)) * Float64(Float64(M / d) * Float64(Float64(h * M) / d)))))
	tmp = 0.0
	if (d <= -2.9e-273)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(l * h)))) * Float64(-1.0 + t_0));
	elseif (d <= 1.82e-134)
		tmp = Float64(-0.125 * Float64(D * Float64(sqrt(Float64(h / (l ^ 3.0))) * Float64(M * Float64(M * Float64(D / d))))));
	elseif (d <= 5.6e+34)
		tmp = Float64(Float64(1.0 - t_0) * sqrt(Float64(d / Float64(h / Float64(d / l)))));
	else
		tmp = Float64(d * Float64((h ^ -0.5) * (l ^ -0.5)));
	end
	return tmp
end

MATLAB

M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	t_0 = 0.5 * (0.25 * ((D / (l / D)) * ((M / d) * ((h * M) / d))));
	tmp = 0.0;
	if (d <= -2.9e-273)
		tmp = (d * sqrt((1.0 / (l * h)))) * (-1.0 + t_0);
	elseif (d <= 1.82e-134)
		tmp = -0.125 * (D * (sqrt((h / (l ^ 3.0))) * (M * (M * (D / d)))));
	elseif (d <= 5.6e+34)
		tmp = (1.0 - t_0) * sqrt((d / (h / (d / l))));
	else
		tmp = d * ((h ^ -0.5) * (l ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(0.5 * N[(0.25 * N[(N[(D / N[(l / D), $MachinePrecision]), $MachinePrecision] * N[(N[(M / d), $MachinePrecision] * N[(N[(h * M), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -2.9e-273], N[(N[(d * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-1.0 + t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.82e-134], N[(-0.125 * N[(D * N[(N[Sqrt[N[(h / N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(M * N[(M * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 5.6e+34], N[(N[(1.0 - t$95$0), $MachinePrecision] * N[Sqrt[N[(d / N[(h / N[(d / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[h, -0.5], $MachinePrecision] * N[Power[l, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -2.89999999999999986e-273

   1. Initial program 61.1%

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

      1. metadata-eval 61.1%

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

      2. unpow1/2 61.1%

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

      3. metadata-eval 61.1%

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

      4. unpow1/2 61.1%

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

      5. *-commutative 61.1%

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

      6. associate-*l* 61.1%

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

      7. times-frac 61.1%

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

      8. metadata-eval 61.1%

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

   3. Simplified 61.1%

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

      1. expm1-log1p-u 37.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\right)\right)} \]

      2. expm1-udef 26.2%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\right)} - 1} \]

   5. Applied egg-rr 20.1%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 30.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)\right)} \]

      2. expm1-log1p 51.7%

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

      3. sub-neg 51.7%

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

      4. +-commutative 51.7%

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

      5. associate-*l/ 50.2%

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

      6. associate-/l* 51.0%

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

      7. +-commutative 51.0%

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

      8. sub-neg 51.0%

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

      9. associate-/l* 51.0%

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

      10. associate-*r/ 51.0%

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

      11. associate-*l/ 51.0%

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

      12. associate-/l* 55.3%

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

      13. associate-*r/ 51.0%

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

   7. Simplified 50.2%

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

   8. Taylor expanded in d around -inf 68.1%

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

      1. associate-*r* 68.1%

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

      2. mul-1-neg 68.1%

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

      3. *-commutative 68.1%

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

   10. Simplified 68.1%

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

   11. Taylor expanded in D around 0 48.1%

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

       1. times-frac 29.3%

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

       2. *-commutative 29.3%

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

       3. unpow2 29.3%

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

       4. associate-/l* 31.8%

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

       5. *-commutative 31.8%

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

       6. unpow2 31.8%

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

       7. associate-*r* 33.5%

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

       8. unpow2 33.5%

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

       9. times-frac 45.2%

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

   13. Simplified 64.7%

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

   if -2.89999999999999986e-273 < d < 1.82000000000000006e-134

   1. Initial program 40.8%

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

      1. metadata-eval 40.8%

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

      2. unpow1/2 40.8%

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

      3. metadata-eval 40.8%

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

      4. unpow1/2 40.8%

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

      5. *-commutative 40.8%

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

      6. associate-*l* 40.8%

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

      7. times-frac 40.7%

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

      8. metadata-eval 40.7%

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

   3. Simplified 40.7%

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

      1. expm1-log1p-u 18.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\right)\right)} \]

      2. expm1-udef 14.4%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\right)} - 1} \]

   5. Applied egg-rr 14.4%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 14.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)\right)} \]

      2. expm1-log1p 29.0%

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

      3. sub-neg 29.0%

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

      4. +-commutative 29.0%

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

      5. associate-*l/ 26.6%

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

      6. associate-/l* 29.0%

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

      7. +-commutative 29.0%

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

      8. sub-neg 29.0%

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

      9. associate-/l* 29.0%

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

      10. associate-*r/ 29.0%

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

      11. associate-*l/ 29.0%

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

      12. associate-/l* 28.9%

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

      13. associate-*r/ 29.0%

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

   7. Simplified 29.2%

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

   8. Taylor expanded in d around 0 46.0%

      \[\leadsto \color{blue}{-0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   9. Step-by-step derivation

      1. associate-/l* 45.8%

         \[\leadsto -0.125 \cdot \left(\color{blue}{\frac{{D}^{2}}{\frac{d}{{M}^{2}}}} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \]

      2. associate-/r/ 45.2%

         \[\leadsto -0.125 \cdot \left(\color{blue}{\left(\frac{{D}^{2}}{d} \cdot {M}^{2}\right)} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \]

      3. unpow2 45.2%

         \[\leadsto -0.125 \cdot \left(\left(\frac{\color{blue}{D \cdot D}}{d} \cdot {M}^{2}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \]

      4. associate-*r/ 49.9%

         \[\leadsto -0.125 \cdot \left(\left(\color{blue}{\left(D \cdot \frac{D}{d}\right)} \cdot {M}^{2}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \]

      5. unpow2 49.9%

         \[\leadsto -0.125 \cdot \left(\left(\left(D \cdot \frac{D}{d}\right) \cdot \color{blue}{\left(M \cdot M\right)}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \]

   10. Simplified 49.9%

       \[\leadsto \color{blue}{-0.125 \cdot \left(\left(\left(D \cdot \frac{D}{d}\right) \cdot \left(M \cdot M\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]

   11. Taylor expanded in D around 0 46.0%

       \[\leadsto -0.125 \cdot \color{blue}{\left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   12. Step-by-step derivation

       1. unpow2 46.0%

          \[\leadsto -0.125 \cdot \left(\frac{\color{blue}{\left(D \cdot D\right)} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \]

       2. associate-*l* 50.8%

          \[\leadsto -0.125 \cdot \left(\frac{\color{blue}{D \cdot \left(D \cdot {M}^{2}\right)}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \]

       3. associate-*r/ 50.8%

          \[\leadsto -0.125 \cdot \left(\color{blue}{\left(D \cdot \frac{D \cdot {M}^{2}}{d}\right)} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \]

       4. associate-*l/ 50.4%

          \[\leadsto -0.125 \cdot \left(\left(D \cdot \color{blue}{\left(\frac{D}{d} \cdot {M}^{2}\right)}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \]

       5. unpow2 50.4%

          \[\leadsto -0.125 \cdot \left(\left(D \cdot \left(\frac{D}{d} \cdot \color{blue}{\left(M \cdot M\right)}\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \]

       6. associate-*l* 50.4%

          \[\leadsto -0.125 \cdot \color{blue}{\left(D \cdot \left(\left(\frac{D}{d} \cdot \left(M \cdot M\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)\right)} \]

       7. unpow2 50.4%

          \[\leadsto -0.125 \cdot \left(D \cdot \left(\left(\frac{D}{d} \cdot \color{blue}{{M}^{2}}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)\right) \]

       8. *-commutative 50.4%

          \[\leadsto -0.125 \cdot \left(D \cdot \left(\color{blue}{\left({M}^{2} \cdot \frac{D}{d}\right)} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)\right) \]

       9. unpow2 50.4%

          \[\leadsto -0.125 \cdot \left(D \cdot \left(\left(\color{blue}{\left(M \cdot M\right)} \cdot \frac{D}{d}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)\right) \]

       10. associate-*l* 52.8%

           \[\leadsto -0.125 \cdot \left(D \cdot \left(\color{blue}{\left(M \cdot \left(M \cdot \frac{D}{d}\right)\right)} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)\right) \]

   13. Simplified 52.8%

       \[\leadsto -0.125 \cdot \color{blue}{\left(D \cdot \left(\left(M \cdot \left(M \cdot \frac{D}{d}\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)\right)} \]

   if 1.82000000000000006e-134 < d < 5.60000000000000016e34

   1. Initial program 82.4%

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

      1. metadata-eval 82.4%

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

      2. unpow1/2 82.4%

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

      3. metadata-eval 82.4%

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

      4. unpow1/2 82.4%

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

      5. *-commutative 82.4%

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

      6. associate-*l* 82.4%

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

      7. times-frac 82.4%

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

      8. metadata-eval 82.4%

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

   3. Simplified 82.4%

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

      1. expm1-log1p-u 23.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\right)\right)} \]

      2. expm1-udef 17.7%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\right)} - 1} \]

   5. Applied egg-rr 17.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 23.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)\right)} \]

      2. expm1-log1p 74.8%

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

      3. sub-neg 74.8%

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

      4. +-commutative 74.8%

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

      5. associate-*l/ 75.1%

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

      6. associate-/l* 69.0%

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

      7. +-commutative 69.0%

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

      8. sub-neg 69.0%

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

      9. associate-/l* 69.0%

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

      10. associate-*r/ 69.0%

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

      11. associate-*l/ 69.0%

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

      12. associate-/l* 74.1%

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

      13. associate-*r/ 69.0%

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

   7. Simplified 68.9%

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

   8. Taylor expanded in D around 0 52.1%

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

      1. times-frac 52.3%

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

      2. *-commutative 52.3%

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

      3. unpow2 52.3%

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

      4. associate-/l* 61.8%

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

      5. *-commutative 61.8%

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

      6. unpow2 61.8%

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

      7. associate-*r* 66.6%

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

      8. unpow2 66.6%

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

      9. times-frac 66.6%

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

   10. Simplified 66.6%

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

   if 5.60000000000000016e34 < d

   1. Initial program 69.0%

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

      1. metadata-eval 69.0%

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

      2. unpow1/2 69.0%

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

      3. metadata-eval 69.0%

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

      4. unpow1/2 69.0%

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

      5. *-commutative 69.0%

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

      6. associate-*l* 69.0%

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

      7. times-frac 69.0%

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

      8. metadata-eval 69.0%

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

   3. Simplified 69.0%

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

   4. Taylor expanded in d around inf 63.0%

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

      1. *-un-lft-identity 63.0%

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

      2. associate-/r* 63.7%

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

   6. Applied egg-rr 63.7%

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

      1. *-lft-identity 63.7%

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

      2. associate-/l/ 63.0%

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

      3. unpow-1 63.0%

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

      4. sqr-pow 63.0%

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

      5. rem-sqrt-square 63.0%

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

      6. metadata-eval 63.0%

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

      7. sqr-pow 62.7%

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

      8. fabs-sqr 62.7%

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

      9. sqr-pow 63.0%

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

   8. Simplified 63.0%

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

      1. unpow-prod-down 77.0%

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

   10. Applied egg-rr 77.0%

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

4. Final simplification 65.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.9 \cdot 10^{-273}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(-1 + 0.5 \cdot \left(0.25 \cdot \left(\frac{D}{\frac{\ell}{D}} \cdot \left(\frac{M}{d} \cdot \frac{h \cdot M}{d}\right)\right)\right)\right)\\ \mathbf{elif}\;d \leq 1.82 \cdot 10^{-134}:\\ \;\;\;\;-0.125 \cdot \left(D \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(M \cdot \left(M \cdot \frac{D}{d}\right)\right)\right)\right)\\ \mathbf{elif}\;d \leq 5.6 \cdot 10^{+34}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{D}{\frac{\ell}{D}} \cdot \left(\frac{M}{d} \cdot \frac{h \cdot M}{d}\right)\right)\right)\right) \cdot \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \]

Alternative 9: 61.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} t_0 := 0.5 \cdot \left(0.25 \cdot \left(\frac{D}{\frac{\ell}{D}} \cdot \left(\frac{M}{d} \cdot \frac{h \cdot M}{d}\right)\right)\right)\\ \mathbf{if}\;d \leq -2.9 \cdot 10^{-273}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(-1 + t_0\right)\\ \mathbf{elif}\;d \leq 4.2 \cdot 10^{-138}:\\ \;\;\;\;-0.125 \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(D \cdot \left(M \cdot \left(M \cdot \frac{D}{d}\right)\right)\right)\right)\\ \mathbf{elif}\;d \leq 6.2 \cdot 10^{+33}:\\ \;\;\;\;\left(1 - t_0\right) \cdot \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* 0.5 (* 0.25 (* (/ D (/ l D)) (* (/ M d) (/ (* h M) d)))))))
   (if (<= d -2.9e-273)
     (* (* d (sqrt (/ 1.0 (* l h)))) (+ -1.0 t_0))
     (if (<= d 4.2e-138)
       (* -0.125 (* (sqrt (/ h (pow l 3.0))) (* D (* M (* M (/ D d))))))
       (if (<= d 6.2e+33)
         (* (- 1.0 t_0) (sqrt (/ d (/ h (/ d l)))))
         (* d (* (pow h -0.5) (pow l -0.5))))))))

C

assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = 0.5 * (0.25 * ((D / (l / D)) * ((M / d) * ((h * M) / d))));
	double tmp;
	if (d <= -2.9e-273) {
		tmp = (d * sqrt((1.0 / (l * h)))) * (-1.0 + t_0);
	} else if (d <= 4.2e-138) {
		tmp = -0.125 * (sqrt((h / pow(l, 3.0))) * (D * (M * (M * (D / d)))));
	} else if (d <= 6.2e+33) {
		tmp = (1.0 - t_0) * sqrt((d / (h / (d / l))));
	} else {
		tmp = d * (pow(h, -0.5) * pow(l, -0.5));
	}
	return tmp;
}

Fortran

NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * (0.25d0 * ((d_1 / (l / d_1)) * ((m / d) * ((h * m) / d))))
    if (d <= (-2.9d-273)) then
        tmp = (d * sqrt((1.0d0 / (l * h)))) * ((-1.0d0) + t_0)
    else if (d <= 4.2d-138) then
        tmp = (-0.125d0) * (sqrt((h / (l ** 3.0d0))) * (d_1 * (m * (m * (d_1 / d)))))
    else if (d <= 6.2d+33) then
        tmp = (1.0d0 - t_0) * sqrt((d / (h / (d / l))))
    else
        tmp = d * ((h ** (-0.5d0)) * (l ** (-0.5d0)))
    end if
    code = tmp
end function

Java

assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double t_0 = 0.5 * (0.25 * ((D / (l / D)) * ((M / d) * ((h * M) / d))));
	double tmp;
	if (d <= -2.9e-273) {
		tmp = (d * Math.sqrt((1.0 / (l * h)))) * (-1.0 + t_0);
	} else if (d <= 4.2e-138) {
		tmp = -0.125 * (Math.sqrt((h / Math.pow(l, 3.0))) * (D * (M * (M * (D / d)))));
	} else if (d <= 6.2e+33) {
		tmp = (1.0 - t_0) * Math.sqrt((d / (h / (d / l))));
	} else {
		tmp = d * (Math.pow(h, -0.5) * Math.pow(l, -0.5));
	}
	return tmp;
}

Python

[M, D] = sort([M, D])
def code(d, h, l, M, D):
	t_0 = 0.5 * (0.25 * ((D / (l / D)) * ((M / d) * ((h * M) / d))))
	tmp = 0
	if d <= -2.9e-273:
		tmp = (d * math.sqrt((1.0 / (l * h)))) * (-1.0 + t_0)
	elif d <= 4.2e-138:
		tmp = -0.125 * (math.sqrt((h / math.pow(l, 3.0))) * (D * (M * (M * (D / d)))))
	elif d <= 6.2e+33:
		tmp = (1.0 - t_0) * math.sqrt((d / (h / (d / l))))
	else:
		tmp = d * (math.pow(h, -0.5) * math.pow(l, -0.5))
	return tmp

Julia

M, D = sort([M, D])
function code(d, h, l, M, D)
	t_0 = Float64(0.5 * Float64(0.25 * Float64(Float64(D / Float64(l / D)) * Float64(Float64(M / d) * Float64(Float64(h * M) / d)))))
	tmp = 0.0
	if (d <= -2.9e-273)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(l * h)))) * Float64(-1.0 + t_0));
	elseif (d <= 4.2e-138)
		tmp = Float64(-0.125 * Float64(sqrt(Float64(h / (l ^ 3.0))) * Float64(D * Float64(M * Float64(M * Float64(D / d))))));
	elseif (d <= 6.2e+33)
		tmp = Float64(Float64(1.0 - t_0) * sqrt(Float64(d / Float64(h / Float64(d / l)))));
	else
		tmp = Float64(d * Float64((h ^ -0.5) * (l ^ -0.5)));
	end
	return tmp
end

MATLAB

M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	t_0 = 0.5 * (0.25 * ((D / (l / D)) * ((M / d) * ((h * M) / d))));
	tmp = 0.0;
	if (d <= -2.9e-273)
		tmp = (d * sqrt((1.0 / (l * h)))) * (-1.0 + t_0);
	elseif (d <= 4.2e-138)
		tmp = -0.125 * (sqrt((h / (l ^ 3.0))) * (D * (M * (M * (D / d)))));
	elseif (d <= 6.2e+33)
		tmp = (1.0 - t_0) * sqrt((d / (h / (d / l))));
	else
		tmp = d * ((h ^ -0.5) * (l ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(0.5 * N[(0.25 * N[(N[(D / N[(l / D), $MachinePrecision]), $MachinePrecision] * N[(N[(M / d), $MachinePrecision] * N[(N[(h * M), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -2.9e-273], N[(N[(d * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-1.0 + t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 4.2e-138], N[(-0.125 * N[(N[Sqrt[N[(h / N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(D * N[(M * N[(M * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 6.2e+33], N[(N[(1.0 - t$95$0), $MachinePrecision] * N[Sqrt[N[(d / N[(h / N[(d / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[h, -0.5], $MachinePrecision] * N[Power[l, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -2.89999999999999986e-273

   1. Initial program 61.1%

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

      1. metadata-eval 61.1%

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

      2. unpow1/2 61.1%

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

      3. metadata-eval 61.1%

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

      4. unpow1/2 61.1%

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

      5. *-commutative 61.1%

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

      6. associate-*l* 61.1%

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

      7. times-frac 61.1%

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

      8. metadata-eval 61.1%

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

   3. Simplified 61.1%

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

      1. expm1-log1p-u 37.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\right)\right)} \]

      2. expm1-udef 26.2%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\right)} - 1} \]

   5. Applied egg-rr 20.1%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 30.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)\right)} \]

      2. expm1-log1p 51.7%

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

      3. sub-neg 51.7%

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

      4. +-commutative 51.7%

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

      5. associate-*l/ 50.2%

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

      6. associate-/l* 51.0%

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

      7. +-commutative 51.0%

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

      8. sub-neg 51.0%

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

      9. associate-/l* 51.0%

         \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \color{blue}{\frac{0.5}{\frac{\ell}{h}}}\right) \]

      10. associate-*r/ 51.0%

          \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - \color{blue}{\frac{{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot 0.5}{\frac{\ell}{h}}}\right) \]

      11. associate-*l/ 51.0%

          \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - \color{blue}{\frac{{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}}{\frac{\ell}{h}} \cdot 0.5}\right) \]

      12. associate-/l* 55.3%

          \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - \color{blue}{\frac{{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot h}{\ell}} \cdot 0.5\right) \]

      13. associate-*r/ 51.0%

          \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - \color{blue}{\left({\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\ell}\right)} \cdot 0.5\right) \]

   7. Simplified 50.2%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - 0.5 \cdot \left({\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]

   8. Taylor expanded in d around -inf 68.1%

      \[\leadsto \color{blue}{\left(-1 \cdot \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)\right)} \cdot \left(1 - 0.5 \cdot \left({\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
   9. Step-by-step derivation

      1. associate-*r* 68.1%

         \[\leadsto \color{blue}{\left(\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot \left(1 - 0.5 \cdot \left({\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      2. mul-1-neg 68.1%

         \[\leadsto \left(\color{blue}{\left(-d\right)} \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      3. *-commutative 68.1%

         \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   10. Simplified 68.1%

       \[\leadsto \color{blue}{\left(\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot \left(1 - 0.5 \cdot \left({\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   11. Taylor expanded in D around 0 48.1%

       \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}\right)}\right) \]
   12. Step-by-step derivation

       1. times-frac 29.3%

          \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \color{blue}{\left(\frac{{D}^{2}}{\ell} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right)}\right)\right) \]

       2. *-commutative 29.3%

          \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{{D}^{2}}{\ell} \cdot \frac{\color{blue}{h \cdot {M}^{2}}}{{d}^{2}}\right)\right)\right) \]

       3. unpow2 29.3%

          \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{\color{blue}{D \cdot D}}{\ell} \cdot \frac{h \cdot {M}^{2}}{{d}^{2}}\right)\right)\right) \]

       4. associate-/l* 31.8%

          \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\color{blue}{\frac{D}{\frac{\ell}{D}}} \cdot \frac{h \cdot {M}^{2}}{{d}^{2}}\right)\right)\right) \]

       5. *-commutative 31.8%

          \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{D}{\frac{\ell}{D}} \cdot \frac{\color{blue}{{M}^{2} \cdot h}}{{d}^{2}}\right)\right)\right) \]

       6. unpow2 31.8%

          \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{D}{\frac{\ell}{D}} \cdot \frac{\color{blue}{\left(M \cdot M\right)} \cdot h}{{d}^{2}}\right)\right)\right) \]

       7. associate-*r* 33.5%

          \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{D}{\frac{\ell}{D}} \cdot \frac{\color{blue}{M \cdot \left(M \cdot h\right)}}{{d}^{2}}\right)\right)\right) \]

       8. unpow2 33.5%

          \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{D}{\frac{\ell}{D}} \cdot \frac{M \cdot \left(M \cdot h\right)}{\color{blue}{d \cdot d}}\right)\right)\right) \]

       9. times-frac 45.2%

          \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{D}{\frac{\ell}{D}} \cdot \color{blue}{\left(\frac{M}{d} \cdot \frac{M \cdot h}{d}\right)}\right)\right)\right) \]

   13. Simplified 64.7%

       \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(0.25 \cdot \left(\frac{D}{\frac{\ell}{D}} \cdot \left(\frac{M}{d} \cdot \frac{M \cdot h}{d}\right)\right)\right)}\right) \]

   if -2.89999999999999986e-273 < d < 4.19999999999999972e-138

   1. Initial program 40.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 40.8%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 40.8%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 40.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 40.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 40.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 40.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 40.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 40.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 40.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]
   4. Step-by-step derivation

      1. expm1-log1p-u 18.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\right)\right)} \]

      2. expm1-udef 14.4%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\right)} - 1} \]

   5. Applied egg-rr 14.4%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 14.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)\right)} \]

      2. expm1-log1p 29.0%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)} \]

      3. sub-neg 29.0%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \color{blue}{\left(1 + \left(-{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)} \]

      4. +-commutative 29.0%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \color{blue}{\left(\left(-{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) + 1\right)} \]

      5. associate-*l/ 26.6%

         \[\leadsto \sqrt{\color{blue}{\frac{d \cdot \frac{d}{\ell}}{h}}} \cdot \left(\left(-{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) + 1\right) \]

      6. associate-/l* 29.0%

         \[\leadsto \sqrt{\color{blue}{\frac{d}{\frac{h}{\frac{d}{\ell}}}}} \cdot \left(\left(-{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) + 1\right) \]

      7. +-commutative 29.0%

         \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \color{blue}{\left(1 + \left(-{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)} \]

      8. sub-neg 29.0%

         \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \color{blue}{\left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)} \]

      9. associate-/l* 29.0%

         \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \color{blue}{\frac{0.5}{\frac{\ell}{h}}}\right) \]

      10. associate-*r/ 29.0%

          \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - \color{blue}{\frac{{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot 0.5}{\frac{\ell}{h}}}\right) \]

      11. associate-*l/ 29.0%

          \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - \color{blue}{\frac{{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}}{\frac{\ell}{h}} \cdot 0.5}\right) \]

      12. associate-/l* 28.9%

          \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - \color{blue}{\frac{{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot h}{\ell}} \cdot 0.5\right) \]

      13. associate-*r/ 29.0%

          \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - \color{blue}{\left({\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\ell}\right)} \cdot 0.5\right) \]

   7. Simplified 29.2%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - 0.5 \cdot \left({\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]

   8. Taylor expanded in d around 0 46.0%

      \[\leadsto \color{blue}{-0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   9. Step-by-step derivation

      1. associate-/l* 45.8%

         \[\leadsto -0.125 \cdot \left(\color{blue}{\frac{{D}^{2}}{\frac{d}{{M}^{2}}}} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \]

      2. associate-/r/ 45.2%

         \[\leadsto -0.125 \cdot \left(\color{blue}{\left(\frac{{D}^{2}}{d} \cdot {M}^{2}\right)} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \]

      3. unpow2 45.2%

         \[\leadsto -0.125 \cdot \left(\left(\frac{\color{blue}{D \cdot D}}{d} \cdot {M}^{2}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \]

      4. associate-*r/ 49.9%

         \[\leadsto -0.125 \cdot \left(\left(\color{blue}{\left(D \cdot \frac{D}{d}\right)} \cdot {M}^{2}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \]

      5. unpow2 49.9%

         \[\leadsto -0.125 \cdot \left(\left(\left(D \cdot \frac{D}{d}\right) \cdot \color{blue}{\left(M \cdot M\right)}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \]

   10. Simplified 49.9%

       \[\leadsto \color{blue}{-0.125 \cdot \left(\left(\left(D \cdot \frac{D}{d}\right) \cdot \left(M \cdot M\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]

   11. Taylor expanded in D around 0 46.0%

       \[\leadsto -0.125 \cdot \left(\color{blue}{\frac{{D}^{2} \cdot {M}^{2}}{d}} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \]
   12. Step-by-step derivation

       1. unpow2 46.0%

          \[\leadsto -0.125 \cdot \left(\frac{\color{blue}{\left(D \cdot D\right)} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \]

       2. associate-*l* 50.8%

          \[\leadsto -0.125 \cdot \left(\frac{\color{blue}{D \cdot \left(D \cdot {M}^{2}\right)}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \]

       3. associate-*r/ 50.8%

          \[\leadsto -0.125 \cdot \left(\color{blue}{\left(D \cdot \frac{D \cdot {M}^{2}}{d}\right)} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \]

       4. associate-*l/ 50.4%

          \[\leadsto -0.125 \cdot \left(\left(D \cdot \color{blue}{\left(\frac{D}{d} \cdot {M}^{2}\right)}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \]

       5. *-commutative 50.4%

          \[\leadsto -0.125 \cdot \left(\left(D \cdot \color{blue}{\left({M}^{2} \cdot \frac{D}{d}\right)}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \]

       6. unpow2 50.4%

          \[\leadsto -0.125 \cdot \left(\left(D \cdot \left(\color{blue}{\left(M \cdot M\right)} \cdot \frac{D}{d}\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \]

       7. associate-*l* 52.9%

          \[\leadsto -0.125 \cdot \left(\left(D \cdot \color{blue}{\left(M \cdot \left(M \cdot \frac{D}{d}\right)\right)}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \]

   13. Simplified 52.9%

       \[\leadsto -0.125 \cdot \left(\color{blue}{\left(D \cdot \left(M \cdot \left(M \cdot \frac{D}{d}\right)\right)\right)} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \]

   if 4.19999999999999972e-138 < d < 6.2e33

   1. Initial program 82.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 82.4%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 82.4%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 82.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 82.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 82.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 82.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 82.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 82.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 82.4%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]
   4. Step-by-step derivation

      1. expm1-log1p-u 23.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\right)\right)} \]

      2. expm1-udef 17.7%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\right)} - 1} \]

   5. Applied egg-rr 17.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 23.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)\right)} \]

      2. expm1-log1p 74.8%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)} \]

      3. sub-neg 74.8%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \color{blue}{\left(1 + \left(-{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)} \]

      4. +-commutative 74.8%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \color{blue}{\left(\left(-{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) + 1\right)} \]

      5. associate-*l/ 75.1%

         \[\leadsto \sqrt{\color{blue}{\frac{d \cdot \frac{d}{\ell}}{h}}} \cdot \left(\left(-{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) + 1\right) \]

      6. associate-/l* 69.0%

         \[\leadsto \sqrt{\color{blue}{\frac{d}{\frac{h}{\frac{d}{\ell}}}}} \cdot \left(\left(-{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) + 1\right) \]

      7. +-commutative 69.0%

         \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \color{blue}{\left(1 + \left(-{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)} \]

      8. sub-neg 69.0%

         \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \color{blue}{\left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)} \]

      9. associate-/l* 69.0%

         \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \color{blue}{\frac{0.5}{\frac{\ell}{h}}}\right) \]

      10. associate-*r/ 69.0%

          \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - \color{blue}{\frac{{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot 0.5}{\frac{\ell}{h}}}\right) \]

      11. associate-*l/ 69.0%

          \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - \color{blue}{\frac{{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}}{\frac{\ell}{h}} \cdot 0.5}\right) \]

      12. associate-/l* 74.1%

          \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - \color{blue}{\frac{{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot h}{\ell}} \cdot 0.5\right) \]

      13. associate-*r/ 69.0%

          \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - \color{blue}{\left({\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\ell}\right)} \cdot 0.5\right) \]

   7. Simplified 68.9%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - 0.5 \cdot \left({\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]

   8. Taylor expanded in D around 0 52.1%

      \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - 0.5 \cdot \color{blue}{\left(0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}\right)}\right) \]
   9. Step-by-step derivation

      1. times-frac 52.3%

         \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \color{blue}{\left(\frac{{D}^{2}}{\ell} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right)}\right)\right) \]

      2. *-commutative 52.3%

         \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{{D}^{2}}{\ell} \cdot \frac{\color{blue}{h \cdot {M}^{2}}}{{d}^{2}}\right)\right)\right) \]

      3. unpow2 52.3%

         \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{\color{blue}{D \cdot D}}{\ell} \cdot \frac{h \cdot {M}^{2}}{{d}^{2}}\right)\right)\right) \]

      4. associate-/l* 61.8%

         \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\color{blue}{\frac{D}{\frac{\ell}{D}}} \cdot \frac{h \cdot {M}^{2}}{{d}^{2}}\right)\right)\right) \]

      5. *-commutative 61.8%

         \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{D}{\frac{\ell}{D}} \cdot \frac{\color{blue}{{M}^{2} \cdot h}}{{d}^{2}}\right)\right)\right) \]

      6. unpow2 61.8%

         \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{D}{\frac{\ell}{D}} \cdot \frac{\color{blue}{\left(M \cdot M\right)} \cdot h}{{d}^{2}}\right)\right)\right) \]

      7. associate-*r* 66.6%

         \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{D}{\frac{\ell}{D}} \cdot \frac{\color{blue}{M \cdot \left(M \cdot h\right)}}{{d}^{2}}\right)\right)\right) \]

      8. unpow2 66.6%

         \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{D}{\frac{\ell}{D}} \cdot \frac{M \cdot \left(M \cdot h\right)}{\color{blue}{d \cdot d}}\right)\right)\right) \]

      9. times-frac 66.6%

         \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{D}{\frac{\ell}{D}} \cdot \color{blue}{\left(\frac{M}{d} \cdot \frac{M \cdot h}{d}\right)}\right)\right)\right) \]

   10. Simplified 66.6%

       \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - 0.5 \cdot \color{blue}{\left(0.25 \cdot \left(\frac{D}{\frac{\ell}{D}} \cdot \left(\frac{M}{d} \cdot \frac{M \cdot h}{d}\right)\right)\right)}\right) \]

   if 6.2e33 < d

   1. Initial program 69.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 69.0%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 69.0%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 69.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 69.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 69.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 69.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 69.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 69.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 69.0%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]

   4. Taylor expanded in d around inf 63.0%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   5. Step-by-step derivation

      1. *-un-lft-identity 63.0%

         \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot d \]

      2. associate-/r* 63.7%

         \[\leadsto \left(1 \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}}\right) \cdot d \]

   6. Applied egg-rr 63.7%

      \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right)} \cdot d \]
   7. Step-by-step derivation

      1. *-lft-identity 63.7%

         \[\leadsto \color{blue}{\sqrt{\frac{\frac{1}{\ell}}{h}}} \cdot d \]

      2. associate-/l/ 63.0%

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      3. unpow-1 63.0%

         \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \cdot d \]

      4. sqr-pow 63.0%

         \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}}} \cdot d \]

      5. rem-sqrt-square 63.0%

         \[\leadsto \color{blue}{\left|{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}\right|} \cdot d \]

      6. metadata-eval 63.0%

         \[\leadsto \left|{\left(h \cdot \ell\right)}^{\color{blue}{-0.5}}\right| \cdot d \]

      7. sqr-pow 62.7%

         \[\leadsto \left|\color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-0.5}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{-0.5}{2}\right)}}\right| \cdot d \]

      8. fabs-sqr 62.7%

         \[\leadsto \color{blue}{\left({\left(h \cdot \ell\right)}^{\left(\frac{-0.5}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{-0.5}{2}\right)}\right)} \cdot d \]

      9. sqr-pow 63.0%

         \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \cdot d \]

   8. Simplified 63.0%

      \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \cdot d \]
   9. Step-by-step derivation

      1. unpow-prod-down 77.0%

         \[\leadsto \color{blue}{\left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)} \cdot d \]

   10. Applied egg-rr 77.0%

       \[\leadsto \color{blue}{\left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)} \cdot d \]
3. Recombined 4 regimes into one program.

4. Final simplification 65.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.9 \cdot 10^{-273}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(-1 + 0.5 \cdot \left(0.25 \cdot \left(\frac{D}{\frac{\ell}{D}} \cdot \left(\frac{M}{d} \cdot \frac{h \cdot M}{d}\right)\right)\right)\right)\\ \mathbf{elif}\;d \leq 4.2 \cdot 10^{-138}:\\ \;\;\;\;-0.125 \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(D \cdot \left(M \cdot \left(M \cdot \frac{D}{d}\right)\right)\right)\right)\\ \mathbf{elif}\;d \leq 6.2 \cdot 10^{+33}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{D}{\frac{\ell}{D}} \cdot \left(\frac{M}{d} \cdot \frac{h \cdot M}{d}\right)\right)\right)\right) \cdot \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \]

Alternative 10: 62.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} t_0 := 0.5 \cdot \left(0.25 \cdot \left(\frac{D}{\frac{\ell}{D}} \cdot \left(\frac{M}{d} \cdot \frac{h \cdot M}{d}\right)\right)\right)\\ \mathbf{if}\;d \leq -2.9 \cdot 10^{-273}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(-1 + t_0\right)\\ \mathbf{elif}\;d \leq 2.05 \cdot 10^{-81}:\\ \;\;\;\;-0.125 \cdot \left(\left(\left(M \cdot M\right) \cdot \left(D \cdot \frac{D}{d}\right)\right) \cdot \frac{\sqrt{h}}{{\ell}^{1.5}}\right)\\ \mathbf{elif}\;d \leq 5.2 \cdot 10^{+34}:\\ \;\;\;\;\left(1 - t_0\right) \cdot \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* 0.5 (* 0.25 (* (/ D (/ l D)) (* (/ M d) (/ (* h M) d)))))))
   (if (<= d -2.9e-273)
     (* (* d (sqrt (/ 1.0 (* l h)))) (+ -1.0 t_0))
     (if (<= d 2.05e-81)
       (* -0.125 (* (* (* M M) (* D (/ D d))) (/ (sqrt h) (pow l 1.5))))
       (if (<= d 5.2e+34)
         (* (- 1.0 t_0) (sqrt (/ d (/ h (/ d l)))))
         (* d (* (pow h -0.5) (pow l -0.5))))))))

C

assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = 0.5 * (0.25 * ((D / (l / D)) * ((M / d) * ((h * M) / d))));
	double tmp;
	if (d <= -2.9e-273) {
		tmp = (d * sqrt((1.0 / (l * h)))) * (-1.0 + t_0);
	} else if (d <= 2.05e-81) {
		tmp = -0.125 * (((M * M) * (D * (D / d))) * (sqrt(h) / pow(l, 1.5)));
	} else if (d <= 5.2e+34) {
		tmp = (1.0 - t_0) * sqrt((d / (h / (d / l))));
	} else {
		tmp = d * (pow(h, -0.5) * pow(l, -0.5));
	}
	return tmp;
}

Fortran

NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * (0.25d0 * ((d_1 / (l / d_1)) * ((m / d) * ((h * m) / d))))
    if (d <= (-2.9d-273)) then
        tmp = (d * sqrt((1.0d0 / (l * h)))) * ((-1.0d0) + t_0)
    else if (d <= 2.05d-81) then
        tmp = (-0.125d0) * (((m * m) * (d_1 * (d_1 / d))) * (sqrt(h) / (l ** 1.5d0)))
    else if (d <= 5.2d+34) then
        tmp = (1.0d0 - t_0) * sqrt((d / (h / (d / l))))
    else
        tmp = d * ((h ** (-0.5d0)) * (l ** (-0.5d0)))
    end if
    code = tmp
end function

Java

assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double t_0 = 0.5 * (0.25 * ((D / (l / D)) * ((M / d) * ((h * M) / d))));
	double tmp;
	if (d <= -2.9e-273) {
		tmp = (d * Math.sqrt((1.0 / (l * h)))) * (-1.0 + t_0);
	} else if (d <= 2.05e-81) {
		tmp = -0.125 * (((M * M) * (D * (D / d))) * (Math.sqrt(h) / Math.pow(l, 1.5)));
	} else if (d <= 5.2e+34) {
		tmp = (1.0 - t_0) * Math.sqrt((d / (h / (d / l))));
	} else {
		tmp = d * (Math.pow(h, -0.5) * Math.pow(l, -0.5));
	}
	return tmp;
}

Python

[M, D] = sort([M, D])
def code(d, h, l, M, D):
	t_0 = 0.5 * (0.25 * ((D / (l / D)) * ((M / d) * ((h * M) / d))))
	tmp = 0
	if d <= -2.9e-273:
		tmp = (d * math.sqrt((1.0 / (l * h)))) * (-1.0 + t_0)
	elif d <= 2.05e-81:
		tmp = -0.125 * (((M * M) * (D * (D / d))) * (math.sqrt(h) / math.pow(l, 1.5)))
	elif d <= 5.2e+34:
		tmp = (1.0 - t_0) * math.sqrt((d / (h / (d / l))))
	else:
		tmp = d * (math.pow(h, -0.5) * math.pow(l, -0.5))
	return tmp

Julia

M, D = sort([M, D])
function code(d, h, l, M, D)
	t_0 = Float64(0.5 * Float64(0.25 * Float64(Float64(D / Float64(l / D)) * Float64(Float64(M / d) * Float64(Float64(h * M) / d)))))
	tmp = 0.0
	if (d <= -2.9e-273)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(l * h)))) * Float64(-1.0 + t_0));
	elseif (d <= 2.05e-81)
		tmp = Float64(-0.125 * Float64(Float64(Float64(M * M) * Float64(D * Float64(D / d))) * Float64(sqrt(h) / (l ^ 1.5))));
	elseif (d <= 5.2e+34)
		tmp = Float64(Float64(1.0 - t_0) * sqrt(Float64(d / Float64(h / Float64(d / l)))));
	else
		tmp = Float64(d * Float64((h ^ -0.5) * (l ^ -0.5)));
	end
	return tmp
end

MATLAB

M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	t_0 = 0.5 * (0.25 * ((D / (l / D)) * ((M / d) * ((h * M) / d))));
	tmp = 0.0;
	if (d <= -2.9e-273)
		tmp = (d * sqrt((1.0 / (l * h)))) * (-1.0 + t_0);
	elseif (d <= 2.05e-81)
		tmp = -0.125 * (((M * M) * (D * (D / d))) * (sqrt(h) / (l ^ 1.5)));
	elseif (d <= 5.2e+34)
		tmp = (1.0 - t_0) * sqrt((d / (h / (d / l))));
	else
		tmp = d * ((h ^ -0.5) * (l ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(0.5 * N[(0.25 * N[(N[(D / N[(l / D), $MachinePrecision]), $MachinePrecision] * N[(N[(M / d), $MachinePrecision] * N[(N[(h * M), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -2.9e-273], N[(N[(d * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-1.0 + t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 2.05e-81], N[(-0.125 * N[(N[(N[(M * M), $MachinePrecision] * N[(D * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[h], $MachinePrecision] / N[Power[l, 1.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 5.2e+34], N[(N[(1.0 - t$95$0), $MachinePrecision] * N[Sqrt[N[(d / N[(h / N[(d / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[h, -0.5], $MachinePrecision] * N[Power[l, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -2.89999999999999986e-273

   1. Initial program 61.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 61.1%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 61.1%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 61.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 61.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 61.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 61.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 61.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 61.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 61.1%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]
   4. Step-by-step derivation

      1. expm1-log1p-u 37.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\right)\right)} \]

      2. expm1-udef 26.2%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\right)} - 1} \]

   5. Applied egg-rr 20.1%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 30.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)\right)} \]

      2. expm1-log1p 51.7%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)} \]

      3. sub-neg 51.7%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \color{blue}{\left(1 + \left(-{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)} \]

      4. +-commutative 51.7%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \color{blue}{\left(\left(-{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) + 1\right)} \]

      5. associate-*l/ 50.2%

         \[\leadsto \sqrt{\color{blue}{\frac{d \cdot \frac{d}{\ell}}{h}}} \cdot \left(\left(-{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) + 1\right) \]

      6. associate-/l* 51.0%

         \[\leadsto \sqrt{\color{blue}{\frac{d}{\frac{h}{\frac{d}{\ell}}}}} \cdot \left(\left(-{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) + 1\right) \]

      7. +-commutative 51.0%

         \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \color{blue}{\left(1 + \left(-{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)} \]

      8. sub-neg 51.0%

         \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \color{blue}{\left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)} \]

      9. associate-/l* 51.0%

         \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \color{blue}{\frac{0.5}{\frac{\ell}{h}}}\right) \]

      10. associate-*r/ 51.0%

          \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - \color{blue}{\frac{{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot 0.5}{\frac{\ell}{h}}}\right) \]

      11. associate-*l/ 51.0%

          \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - \color{blue}{\frac{{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}}{\frac{\ell}{h}} \cdot 0.5}\right) \]

      12. associate-/l* 55.3%

          \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - \color{blue}{\frac{{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot h}{\ell}} \cdot 0.5\right) \]

      13. associate-*r/ 51.0%

          \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - \color{blue}{\left({\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\ell}\right)} \cdot 0.5\right) \]

   7. Simplified 50.2%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - 0.5 \cdot \left({\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]

   8. Taylor expanded in d around -inf 68.1%

      \[\leadsto \color{blue}{\left(-1 \cdot \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)\right)} \cdot \left(1 - 0.5 \cdot \left({\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
   9. Step-by-step derivation

      1. associate-*r* 68.1%

         \[\leadsto \color{blue}{\left(\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot \left(1 - 0.5 \cdot \left({\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      2. mul-1-neg 68.1%

         \[\leadsto \left(\color{blue}{\left(-d\right)} \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      3. *-commutative 68.1%

         \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   10. Simplified 68.1%

       \[\leadsto \color{blue}{\left(\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot \left(1 - 0.5 \cdot \left({\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   11. Taylor expanded in D around 0 48.1%

       \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}\right)}\right) \]
   12. Step-by-step derivation

       1. times-frac 29.3%

          \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \color{blue}{\left(\frac{{D}^{2}}{\ell} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right)}\right)\right) \]

       2. *-commutative 29.3%

          \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{{D}^{2}}{\ell} \cdot \frac{\color{blue}{h \cdot {M}^{2}}}{{d}^{2}}\right)\right)\right) \]

       3. unpow2 29.3%

          \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{\color{blue}{D \cdot D}}{\ell} \cdot \frac{h \cdot {M}^{2}}{{d}^{2}}\right)\right)\right) \]

       4. associate-/l* 31.8%

          \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\color{blue}{\frac{D}{\frac{\ell}{D}}} \cdot \frac{h \cdot {M}^{2}}{{d}^{2}}\right)\right)\right) \]

       5. *-commutative 31.8%

          \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{D}{\frac{\ell}{D}} \cdot \frac{\color{blue}{{M}^{2} \cdot h}}{{d}^{2}}\right)\right)\right) \]

       6. unpow2 31.8%

          \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{D}{\frac{\ell}{D}} \cdot \frac{\color{blue}{\left(M \cdot M\right)} \cdot h}{{d}^{2}}\right)\right)\right) \]

       7. associate-*r* 33.5%

          \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{D}{\frac{\ell}{D}} \cdot \frac{\color{blue}{M \cdot \left(M \cdot h\right)}}{{d}^{2}}\right)\right)\right) \]

       8. unpow2 33.5%

          \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{D}{\frac{\ell}{D}} \cdot \frac{M \cdot \left(M \cdot h\right)}{\color{blue}{d \cdot d}}\right)\right)\right) \]

       9. times-frac 45.2%

          \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{D}{\frac{\ell}{D}} \cdot \color{blue}{\left(\frac{M}{d} \cdot \frac{M \cdot h}{d}\right)}\right)\right)\right) \]

   13. Simplified 64.7%

       \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(0.25 \cdot \left(\frac{D}{\frac{\ell}{D}} \cdot \left(\frac{M}{d} \cdot \frac{M \cdot h}{d}\right)\right)\right)}\right) \]

   if -2.89999999999999986e-273 < d < 2.04999999999999992e-81

   1. Initial program 45.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 45.2%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 45.2%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 45.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 45.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 45.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 45.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 45.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 45.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 45.0%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]
   4. Step-by-step derivation

      1. expm1-log1p-u 18.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\right)\right)} \]

      2. expm1-udef 14.3%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\right)} - 1} \]

   5. Applied egg-rr 14.3%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 14.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)\right)} \]

      2. expm1-log1p 33.0%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)} \]

      3. sub-neg 33.0%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \color{blue}{\left(1 + \left(-{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)} \]

      4. +-commutative 33.0%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \color{blue}{\left(\left(-{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) + 1\right)} \]

      5. associate-*l/ 31.0%

         \[\leadsto \sqrt{\color{blue}{\frac{d \cdot \frac{d}{\ell}}{h}}} \cdot \left(\left(-{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) + 1\right) \]

      6. associate-/l* 33.0%

         \[\leadsto \sqrt{\color{blue}{\frac{d}{\frac{h}{\frac{d}{\ell}}}}} \cdot \left(\left(-{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) + 1\right) \]

      7. +-commutative 33.0%

         \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \color{blue}{\left(1 + \left(-{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)} \]

      8. sub-neg 33.0%

         \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \color{blue}{\left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)} \]

      9. associate-/l* 33.0%

         \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \color{blue}{\frac{0.5}{\frac{\ell}{h}}}\right) \]

      10. associate-*r/ 33.0%

          \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - \color{blue}{\frac{{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot 0.5}{\frac{\ell}{h}}}\right) \]

      11. associate-*l/ 33.0%

          \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - \color{blue}{\frac{{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}}{\frac{\ell}{h}} \cdot 0.5}\right) \]

      12. associate-/l* 32.9%

          \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - \color{blue}{\frac{{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot h}{\ell}} \cdot 0.5\right) \]

      13. associate-*r/ 33.0%

          \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - \color{blue}{\left({\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\ell}\right)} \cdot 0.5\right) \]

   7. Simplified 33.1%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - 0.5 \cdot \left({\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]

   8. Taylor expanded in d around 0 43.7%

      \[\leadsto \color{blue}{-0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   9. Step-by-step derivation

      1. associate-/l* 43.4%

         \[\leadsto -0.125 \cdot \left(\color{blue}{\frac{{D}^{2}}{\frac{d}{{M}^{2}}}} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \]

      2. associate-/r/ 43.0%

         \[\leadsto -0.125 \cdot \left(\color{blue}{\left(\frac{{D}^{2}}{d} \cdot {M}^{2}\right)} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \]

      3. unpow2 43.0%

         \[\leadsto -0.125 \cdot \left(\left(\frac{\color{blue}{D \cdot D}}{d} \cdot {M}^{2}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \]

      4. associate-*r/ 47.1%

         \[\leadsto -0.125 \cdot \left(\left(\color{blue}{\left(D \cdot \frac{D}{d}\right)} \cdot {M}^{2}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \]

      5. unpow2 47.1%

         \[\leadsto -0.125 \cdot \left(\left(\left(D \cdot \frac{D}{d}\right) \cdot \color{blue}{\left(M \cdot M\right)}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \]

   10. Simplified 47.1%

       \[\leadsto \color{blue}{-0.125 \cdot \left(\left(\left(D \cdot \frac{D}{d}\right) \cdot \left(M \cdot M\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   11. Step-by-step derivation

       1. sqrt-div 53.1%

          \[\leadsto -0.125 \cdot \left(\left(\left(D \cdot \frac{D}{d}\right) \cdot \left(M \cdot M\right)\right) \cdot \color{blue}{\frac{\sqrt{h}}{\sqrt{{\ell}^{3}}}}\right) \]

   12. Applied egg-rr 53.1%

       \[\leadsto -0.125 \cdot \left(\left(\left(D \cdot \frac{D}{d}\right) \cdot \left(M \cdot M\right)\right) \cdot \color{blue}{\frac{\sqrt{h}}{\sqrt{{\ell}^{3}}}}\right) \]
   13. Step-by-step derivation

       1. sqr-pow 53.1%

          \[\leadsto -0.125 \cdot \left(\left(\left(D \cdot \frac{D}{d}\right) \cdot \left(M \cdot M\right)\right) \cdot \frac{\sqrt{h}}{\sqrt{\color{blue}{{\ell}^{\left(\frac{3}{2}\right)} \cdot {\ell}^{\left(\frac{3}{2}\right)}}}}\right) \]

       2. rem-sqrt-square 57.3%

          \[\leadsto -0.125 \cdot \left(\left(\left(D \cdot \frac{D}{d}\right) \cdot \left(M \cdot M\right)\right) \cdot \frac{\sqrt{h}}{\color{blue}{\left|{\ell}^{\left(\frac{3}{2}\right)}\right|}}\right) \]

       3. sqr-pow 57.3%

          \[\leadsto -0.125 \cdot \left(\left(\left(D \cdot \frac{D}{d}\right) \cdot \left(M \cdot M\right)\right) \cdot \frac{\sqrt{h}}{\left|\color{blue}{{\ell}^{\left(\frac{\frac{3}{2}}{2}\right)} \cdot {\ell}^{\left(\frac{\frac{3}{2}}{2}\right)}}\right|}\right) \]

       4. fabs-sqr 57.3%

          \[\leadsto -0.125 \cdot \left(\left(\left(D \cdot \frac{D}{d}\right) \cdot \left(M \cdot M\right)\right) \cdot \frac{\sqrt{h}}{\color{blue}{{\ell}^{\left(\frac{\frac{3}{2}}{2}\right)} \cdot {\ell}^{\left(\frac{\frac{3}{2}}{2}\right)}}}\right) \]

       5. sqr-pow 57.3%

          \[\leadsto -0.125 \cdot \left(\left(\left(D \cdot \frac{D}{d}\right) \cdot \left(M \cdot M\right)\right) \cdot \frac{\sqrt{h}}{\color{blue}{{\ell}^{\left(\frac{3}{2}\right)}}}\right) \]

       6. metadata-eval 57.3%

          \[\leadsto -0.125 \cdot \left(\left(\left(D \cdot \frac{D}{d}\right) \cdot \left(M \cdot M\right)\right) \cdot \frac{\sqrt{h}}{{\ell}^{\color{blue}{1.5}}}\right) \]

   14. Simplified 57.3%

       \[\leadsto -0.125 \cdot \left(\left(\left(D \cdot \frac{D}{d}\right) \cdot \left(M \cdot M\right)\right) \cdot \color{blue}{\frac{\sqrt{h}}{{\ell}^{1.5}}}\right) \]

   if 2.04999999999999992e-81 < d < 5.19999999999999995e34

   1. Initial program 84.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 84.7%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 84.7%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 84.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 84.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 84.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 84.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 84.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 84.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 84.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]
   4. Step-by-step derivation

      1. expm1-log1p-u 25.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\right)\right)} \]

      2. expm1-udef 18.7%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\right)} - 1} \]

   5. Applied egg-rr 18.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 25.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)\right)} \]

      2. expm1-log1p 78.6%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)} \]

      3. sub-neg 78.6%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \color{blue}{\left(1 + \left(-{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)} \]

      4. +-commutative 78.6%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \color{blue}{\left(\left(-{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) + 1\right)} \]

      5. associate-*l/ 78.9%

         \[\leadsto \sqrt{\color{blue}{\frac{d \cdot \frac{d}{\ell}}{h}}} \cdot \left(\left(-{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) + 1\right) \]

      6. associate-/l* 71.5%

         \[\leadsto \sqrt{\color{blue}{\frac{d}{\frac{h}{\frac{d}{\ell}}}}} \cdot \left(\left(-{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) + 1\right) \]

      7. +-commutative 71.5%

         \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \color{blue}{\left(1 + \left(-{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)} \]

      8. sub-neg 71.5%

         \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \color{blue}{\left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)} \]

      9. associate-/l* 71.6%

         \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \color{blue}{\frac{0.5}{\frac{\ell}{h}}}\right) \]

      10. associate-*r/ 71.6%

          \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - \color{blue}{\frac{{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot 0.5}{\frac{\ell}{h}}}\right) \]

      11. associate-*l/ 71.6%

          \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - \color{blue}{\frac{{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}}{\frac{\ell}{h}} \cdot 0.5}\right) \]

      12. associate-/l* 77.7%

          \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - \color{blue}{\frac{{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot h}{\ell}} \cdot 0.5\right) \]

      13. associate-*r/ 71.5%

          \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - \color{blue}{\left({\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\ell}\right)} \cdot 0.5\right) \]

   7. Simplified 71.5%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - 0.5 \cdot \left({\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]

   8. Taylor expanded in D around 0 56.8%

      \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - 0.5 \cdot \color{blue}{\left(0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}\right)}\right) \]
   9. Step-by-step derivation

      1. times-frac 57.1%

         \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \color{blue}{\left(\frac{{D}^{2}}{\ell} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right)}\right)\right) \]

      2. *-commutative 57.1%

         \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{{D}^{2}}{\ell} \cdot \frac{\color{blue}{h \cdot {M}^{2}}}{{d}^{2}}\right)\right)\right) \]

      3. unpow2 57.1%

         \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{\color{blue}{D \cdot D}}{\ell} \cdot \frac{h \cdot {M}^{2}}{{d}^{2}}\right)\right)\right) \]

      4. associate-/l* 68.6%

         \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\color{blue}{\frac{D}{\frac{\ell}{D}}} \cdot \frac{h \cdot {M}^{2}}{{d}^{2}}\right)\right)\right) \]

      5. *-commutative 68.6%

         \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{D}{\frac{\ell}{D}} \cdot \frac{\color{blue}{{M}^{2} \cdot h}}{{d}^{2}}\right)\right)\right) \]

      6. unpow2 68.6%

         \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{D}{\frac{\ell}{D}} \cdot \frac{\color{blue}{\left(M \cdot M\right)} \cdot h}{{d}^{2}}\right)\right)\right) \]

      7. associate-*r* 71.5%

         \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{D}{\frac{\ell}{D}} \cdot \frac{\color{blue}{M \cdot \left(M \cdot h\right)}}{{d}^{2}}\right)\right)\right) \]

      8. unpow2 71.5%

         \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{D}{\frac{\ell}{D}} \cdot \frac{M \cdot \left(M \cdot h\right)}{\color{blue}{d \cdot d}}\right)\right)\right) \]

      9. times-frac 71.5%

         \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{D}{\frac{\ell}{D}} \cdot \color{blue}{\left(\frac{M}{d} \cdot \frac{M \cdot h}{d}\right)}\right)\right)\right) \]

   10. Simplified 71.5%

       \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - 0.5 \cdot \color{blue}{\left(0.25 \cdot \left(\frac{D}{\frac{\ell}{D}} \cdot \left(\frac{M}{d} \cdot \frac{M \cdot h}{d}\right)\right)\right)}\right) \]

   if 5.19999999999999995e34 < d

   1. Initial program 69.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 69.0%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 69.0%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 69.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 69.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 69.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 69.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 69.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 69.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 69.0%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]

   4. Taylor expanded in d around inf 63.0%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   5. Step-by-step derivation

      1. *-un-lft-identity 63.0%

         \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot d \]

      2. associate-/r* 63.7%

         \[\leadsto \left(1 \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}}\right) \cdot d \]

   6. Applied egg-rr 63.7%

      \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right)} \cdot d \]
   7. Step-by-step derivation

      1. *-lft-identity 63.7%

         \[\leadsto \color{blue}{\sqrt{\frac{\frac{1}{\ell}}{h}}} \cdot d \]

      2. associate-/l/ 63.0%

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      3. unpow-1 63.0%

         \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \cdot d \]

      4. sqr-pow 63.0%

         \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}}} \cdot d \]

      5. rem-sqrt-square 63.0%

         \[\leadsto \color{blue}{\left|{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}\right|} \cdot d \]

      6. metadata-eval 63.0%

         \[\leadsto \left|{\left(h \cdot \ell\right)}^{\color{blue}{-0.5}}\right| \cdot d \]

      7. sqr-pow 62.7%

         \[\leadsto \left|\color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-0.5}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{-0.5}{2}\right)}}\right| \cdot d \]

      8. fabs-sqr 62.7%

         \[\leadsto \color{blue}{\left({\left(h \cdot \ell\right)}^{\left(\frac{-0.5}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{-0.5}{2}\right)}\right)} \cdot d \]

      9. sqr-pow 63.0%

         \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \cdot d \]

   8. Simplified 63.0%

      \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \cdot d \]
   9. Step-by-step derivation

      1. unpow-prod-down 77.0%

         \[\leadsto \color{blue}{\left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)} \cdot d \]

   10. Applied egg-rr 77.0%

       \[\leadsto \color{blue}{\left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)} \cdot d \]
3. Recombined 4 regimes into one program.

4. Final simplification 66.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.9 \cdot 10^{-273}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(-1 + 0.5 \cdot \left(0.25 \cdot \left(\frac{D}{\frac{\ell}{D}} \cdot \left(\frac{M}{d} \cdot \frac{h \cdot M}{d}\right)\right)\right)\right)\\ \mathbf{elif}\;d \leq 2.05 \cdot 10^{-81}:\\ \;\;\;\;-0.125 \cdot \left(\left(\left(M \cdot M\right) \cdot \left(D \cdot \frac{D}{d}\right)\right) \cdot \frac{\sqrt{h}}{{\ell}^{1.5}}\right)\\ \mathbf{elif}\;d \leq 5.2 \cdot 10^{+34}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{D}{\frac{\ell}{D}} \cdot \left(\frac{M}{d} \cdot \frac{h \cdot M}{d}\right)\right)\right)\right) \cdot \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \]

Alternative 11: 58.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} t_0 := 0.5 \cdot \left(0.25 \cdot \left(\frac{D}{\frac{\ell}{D}} \cdot \left(\frac{M}{d} \cdot \frac{h \cdot M}{d}\right)\right)\right)\\ \mathbf{if}\;d \leq -2.9 \cdot 10^{-273}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(-1 + t_0\right)\\ \mathbf{elif}\;d \leq 1.05 \cdot 10^{-119} \lor \neg \left(d \leq 1.9 \cdot 10^{+34}\right):\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - t_0\right) \cdot \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}}\\ \end{array} \end{array} \]

FPCore

NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* 0.5 (* 0.25 (* (/ D (/ l D)) (* (/ M d) (/ (* h M) d)))))))
   (if (<= d -2.9e-273)
     (* (* d (sqrt (/ 1.0 (* l h)))) (+ -1.0 t_0))
     (if (or (<= d 1.05e-119) (not (<= d 1.9e+34)))
       (* d (* (pow h -0.5) (pow l -0.5)))
       (* (- 1.0 t_0) (sqrt (/ d (/ h (/ d l)))))))))

C

assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = 0.5 * (0.25 * ((D / (l / D)) * ((M / d) * ((h * M) / d))));
	double tmp;
	if (d <= -2.9e-273) {
		tmp = (d * sqrt((1.0 / (l * h)))) * (-1.0 + t_0);
	} else if ((d <= 1.05e-119) || !(d <= 1.9e+34)) {
		tmp = d * (pow(h, -0.5) * pow(l, -0.5));
	} else {
		tmp = (1.0 - t_0) * sqrt((d / (h / (d / l))));
	}
	return tmp;
}

Fortran

NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * (0.25d0 * ((d_1 / (l / d_1)) * ((m / d) * ((h * m) / d))))
    if (d <= (-2.9d-273)) then
        tmp = (d * sqrt((1.0d0 / (l * h)))) * ((-1.0d0) + t_0)
    else if ((d <= 1.05d-119) .or. (.not. (d <= 1.9d+34))) then
        tmp = d * ((h ** (-0.5d0)) * (l ** (-0.5d0)))
    else
        tmp = (1.0d0 - t_0) * sqrt((d / (h / (d / l))))
    end if
    code = tmp
end function

Java

assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double t_0 = 0.5 * (0.25 * ((D / (l / D)) * ((M / d) * ((h * M) / d))));
	double tmp;
	if (d <= -2.9e-273) {
		tmp = (d * Math.sqrt((1.0 / (l * h)))) * (-1.0 + t_0);
	} else if ((d <= 1.05e-119) || !(d <= 1.9e+34)) {
		tmp = d * (Math.pow(h, -0.5) * Math.pow(l, -0.5));
	} else {
		tmp = (1.0 - t_0) * Math.sqrt((d / (h / (d / l))));
	}
	return tmp;
}

Python

[M, D] = sort([M, D])
def code(d, h, l, M, D):
	t_0 = 0.5 * (0.25 * ((D / (l / D)) * ((M / d) * ((h * M) / d))))
	tmp = 0
	if d <= -2.9e-273:
		tmp = (d * math.sqrt((1.0 / (l * h)))) * (-1.0 + t_0)
	elif (d <= 1.05e-119) or not (d <= 1.9e+34):
		tmp = d * (math.pow(h, -0.5) * math.pow(l, -0.5))
	else:
		tmp = (1.0 - t_0) * math.sqrt((d / (h / (d / l))))
	return tmp

Julia

M, D = sort([M, D])
function code(d, h, l, M, D)
	t_0 = Float64(0.5 * Float64(0.25 * Float64(Float64(D / Float64(l / D)) * Float64(Float64(M / d) * Float64(Float64(h * M) / d)))))
	tmp = 0.0
	if (d <= -2.9e-273)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(l * h)))) * Float64(-1.0 + t_0));
	elseif ((d <= 1.05e-119) || !(d <= 1.9e+34))
		tmp = Float64(d * Float64((h ^ -0.5) * (l ^ -0.5)));
	else
		tmp = Float64(Float64(1.0 - t_0) * sqrt(Float64(d / Float64(h / Float64(d / l)))));
	end
	return tmp
end

MATLAB

M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	t_0 = 0.5 * (0.25 * ((D / (l / D)) * ((M / d) * ((h * M) / d))));
	tmp = 0.0;
	if (d <= -2.9e-273)
		tmp = (d * sqrt((1.0 / (l * h)))) * (-1.0 + t_0);
	elseif ((d <= 1.05e-119) || ~((d <= 1.9e+34)))
		tmp = d * ((h ^ -0.5) * (l ^ -0.5));
	else
		tmp = (1.0 - t_0) * sqrt((d / (h / (d / l))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(0.5 * N[(0.25 * N[(N[(D / N[(l / D), $MachinePrecision]), $MachinePrecision] * N[(N[(M / d), $MachinePrecision] * N[(N[(h * M), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -2.9e-273], N[(N[(d * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-1.0 + t$95$0), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[d, 1.05e-119], N[Not[LessEqual[d, 1.9e+34]], $MachinePrecision]], N[(d * N[(N[Power[h, -0.5], $MachinePrecision] * N[Power[l, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - t$95$0), $MachinePrecision] * N[Sqrt[N[(d / N[(h / N[(d / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if d < -2.89999999999999986e-273

   1. Initial program 61.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 61.1%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 61.1%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 61.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 61.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 61.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 61.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 61.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 61.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 61.1%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]
   4. Step-by-step derivation

      1. expm1-log1p-u 37.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\right)\right)} \]

      2. expm1-udef 26.2%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\right)} - 1} \]

   5. Applied egg-rr 20.1%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 30.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)\right)} \]

      2. expm1-log1p 51.7%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)} \]

      3. sub-neg 51.7%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \color{blue}{\left(1 + \left(-{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)} \]

      4. +-commutative 51.7%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \color{blue}{\left(\left(-{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) + 1\right)} \]

      5. associate-*l/ 50.2%

         \[\leadsto \sqrt{\color{blue}{\frac{d \cdot \frac{d}{\ell}}{h}}} \cdot \left(\left(-{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) + 1\right) \]

      6. associate-/l* 51.0%

         \[\leadsto \sqrt{\color{blue}{\frac{d}{\frac{h}{\frac{d}{\ell}}}}} \cdot \left(\left(-{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) + 1\right) \]

      7. +-commutative 51.0%

         \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \color{blue}{\left(1 + \left(-{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)} \]

      8. sub-neg 51.0%

         \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \color{blue}{\left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)} \]

      9. associate-/l* 51.0%

         \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \color{blue}{\frac{0.5}{\frac{\ell}{h}}}\right) \]

      10. associate-*r/ 51.0%

          \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - \color{blue}{\frac{{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot 0.5}{\frac{\ell}{h}}}\right) \]

      11. associate-*l/ 51.0%

          \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - \color{blue}{\frac{{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}}{\frac{\ell}{h}} \cdot 0.5}\right) \]

      12. associate-/l* 55.3%

          \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - \color{blue}{\frac{{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot h}{\ell}} \cdot 0.5\right) \]

      13. associate-*r/ 51.0%

          \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - \color{blue}{\left({\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\ell}\right)} \cdot 0.5\right) \]

   7. Simplified 50.2%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - 0.5 \cdot \left({\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]

   8. Taylor expanded in d around -inf 68.1%

      \[\leadsto \color{blue}{\left(-1 \cdot \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)\right)} \cdot \left(1 - 0.5 \cdot \left({\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
   9. Step-by-step derivation

      1. associate-*r* 68.1%

         \[\leadsto \color{blue}{\left(\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot \left(1 - 0.5 \cdot \left({\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      2. mul-1-neg 68.1%

         \[\leadsto \left(\color{blue}{\left(-d\right)} \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      3. *-commutative 68.1%

         \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   10. Simplified 68.1%

       \[\leadsto \color{blue}{\left(\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot \left(1 - 0.5 \cdot \left({\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   11. Taylor expanded in D around 0 48.1%

       \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}\right)}\right) \]
   12. Step-by-step derivation

       1. times-frac 29.3%

          \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \color{blue}{\left(\frac{{D}^{2}}{\ell} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right)}\right)\right) \]

       2. *-commutative 29.3%

          \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{{D}^{2}}{\ell} \cdot \frac{\color{blue}{h \cdot {M}^{2}}}{{d}^{2}}\right)\right)\right) \]

       3. unpow2 29.3%

          \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{\color{blue}{D \cdot D}}{\ell} \cdot \frac{h \cdot {M}^{2}}{{d}^{2}}\right)\right)\right) \]

       4. associate-/l* 31.8%

          \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\color{blue}{\frac{D}{\frac{\ell}{D}}} \cdot \frac{h \cdot {M}^{2}}{{d}^{2}}\right)\right)\right) \]

       5. *-commutative 31.8%

          \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{D}{\frac{\ell}{D}} \cdot \frac{\color{blue}{{M}^{2} \cdot h}}{{d}^{2}}\right)\right)\right) \]

       6. unpow2 31.8%

          \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{D}{\frac{\ell}{D}} \cdot \frac{\color{blue}{\left(M \cdot M\right)} \cdot h}{{d}^{2}}\right)\right)\right) \]

       7. associate-*r* 33.5%

          \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{D}{\frac{\ell}{D}} \cdot \frac{\color{blue}{M \cdot \left(M \cdot h\right)}}{{d}^{2}}\right)\right)\right) \]

       8. unpow2 33.5%

          \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{D}{\frac{\ell}{D}} \cdot \frac{M \cdot \left(M \cdot h\right)}{\color{blue}{d \cdot d}}\right)\right)\right) \]

       9. times-frac 45.2%

          \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{D}{\frac{\ell}{D}} \cdot \color{blue}{\left(\frac{M}{d} \cdot \frac{M \cdot h}{d}\right)}\right)\right)\right) \]

   13. Simplified 64.7%

       \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(0.25 \cdot \left(\frac{D}{\frac{\ell}{D}} \cdot \left(\frac{M}{d} \cdot \frac{M \cdot h}{d}\right)\right)\right)}\right) \]

   if -2.89999999999999986e-273 < d < 1.05e-119 or 1.9000000000000001e34 < d

   1. Initial program 55.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 55.4%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 55.4%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 55.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 55.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 55.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 55.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 55.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 55.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 55.3%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]

   4. Taylor expanded in d around inf 48.0%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   5. Step-by-step derivation

      1. *-un-lft-identity 48.0%

         \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot d \]

      2. associate-/r* 48.4%

         \[\leadsto \left(1 \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}}\right) \cdot d \]

   6. Applied egg-rr 48.4%

      \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right)} \cdot d \]
   7. Step-by-step derivation

      1. *-lft-identity 48.4%

         \[\leadsto \color{blue}{\sqrt{\frac{\frac{1}{\ell}}{h}}} \cdot d \]

      2. associate-/l/ 48.0%

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      3. unpow-1 48.0%

         \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \cdot d \]

      4. sqr-pow 48.0%

         \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}}} \cdot d \]

      5. rem-sqrt-square 48.0%

         \[\leadsto \color{blue}{\left|{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}\right|} \cdot d \]

      6. metadata-eval 48.0%

         \[\leadsto \left|{\left(h \cdot \ell\right)}^{\color{blue}{-0.5}}\right| \cdot d \]

      7. sqr-pow 47.8%

         \[\leadsto \left|\color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-0.5}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{-0.5}{2}\right)}}\right| \cdot d \]

      8. fabs-sqr 47.8%

         \[\leadsto \color{blue}{\left({\left(h \cdot \ell\right)}^{\left(\frac{-0.5}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{-0.5}{2}\right)}\right)} \cdot d \]

      9. sqr-pow 48.0%

         \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \cdot d \]

   8. Simplified 48.0%

      \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \cdot d \]
   9. Step-by-step derivation

      1. unpow-prod-down 57.1%

         \[\leadsto \color{blue}{\left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)} \cdot d \]

   10. Applied egg-rr 57.1%

       \[\leadsto \color{blue}{\left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)} \cdot d \]

   if 1.05e-119 < d < 1.9000000000000001e34

   1. Initial program 86.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 86.7%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 86.7%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 86.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 86.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 86.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 86.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 86.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 86.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 86.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]
   4. Step-by-step derivation

      1. expm1-log1p-u 24.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\right)\right)} \]

      2. expm1-udef 18.7%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\right)} - 1} \]

   5. Applied egg-rr 18.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 24.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)\right)} \]

      2. expm1-log1p 78.7%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)} \]

      3. sub-neg 78.7%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \color{blue}{\left(1 + \left(-{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)} \]

      4. +-commutative 78.7%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \color{blue}{\left(\left(-{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) + 1\right)} \]

      5. associate-*l/ 79.0%

         \[\leadsto \sqrt{\color{blue}{\frac{d \cdot \frac{d}{\ell}}{h}}} \cdot \left(\left(-{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) + 1\right) \]

      6. associate-/l* 72.6%

         \[\leadsto \sqrt{\color{blue}{\frac{d}{\frac{h}{\frac{d}{\ell}}}}} \cdot \left(\left(-{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) + 1\right) \]

      7. +-commutative 72.6%

         \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \color{blue}{\left(1 + \left(-{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)} \]

      8. sub-neg 72.6%

         \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \color{blue}{\left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)} \]

      9. associate-/l* 72.7%

         \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \color{blue}{\frac{0.5}{\frac{\ell}{h}}}\right) \]

      10. associate-*r/ 72.7%

          \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - \color{blue}{\frac{{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot 0.5}{\frac{\ell}{h}}}\right) \]

      11. associate-*l/ 72.7%

          \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - \color{blue}{\frac{{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}}{\frac{\ell}{h}} \cdot 0.5}\right) \]

      12. associate-/l* 78.0%

          \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - \color{blue}{\frac{{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot h}{\ell}} \cdot 0.5\right) \]

      13. associate-*r/ 72.6%

          \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - \color{blue}{\left({\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\ell}\right)} \cdot 0.5\right) \]

   7. Simplified 72.6%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - 0.5 \cdot \left({\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]

   8. Taylor expanded in D around 0 54.7%

      \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - 0.5 \cdot \color{blue}{\left(0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}\right)}\right) \]
   9. Step-by-step derivation

      1. times-frac 55.0%

         \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \color{blue}{\left(\frac{{D}^{2}}{\ell} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right)}\right)\right) \]

      2. *-commutative 55.0%

         \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{{D}^{2}}{\ell} \cdot \frac{\color{blue}{h \cdot {M}^{2}}}{{d}^{2}}\right)\right)\right) \]

      3. unpow2 55.0%

         \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{\color{blue}{D \cdot D}}{\ell} \cdot \frac{h \cdot {M}^{2}}{{d}^{2}}\right)\right)\right) \]

      4. associate-/l* 64.9%

         \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\color{blue}{\frac{D}{\frac{\ell}{D}}} \cdot \frac{h \cdot {M}^{2}}{{d}^{2}}\right)\right)\right) \]

      5. *-commutative 64.9%

         \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{D}{\frac{\ell}{D}} \cdot \frac{\color{blue}{{M}^{2} \cdot h}}{{d}^{2}}\right)\right)\right) \]

      6. unpow2 64.9%

         \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{D}{\frac{\ell}{D}} \cdot \frac{\color{blue}{\left(M \cdot M\right)} \cdot h}{{d}^{2}}\right)\right)\right) \]

      7. associate-*r* 70.0%

         \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{D}{\frac{\ell}{D}} \cdot \frac{\color{blue}{M \cdot \left(M \cdot h\right)}}{{d}^{2}}\right)\right)\right) \]

      8. unpow2 70.0%

         \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{D}{\frac{\ell}{D}} \cdot \frac{M \cdot \left(M \cdot h\right)}{\color{blue}{d \cdot d}}\right)\right)\right) \]

      9. times-frac 69.9%

         \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{D}{\frac{\ell}{D}} \cdot \color{blue}{\left(\frac{M}{d} \cdot \frac{M \cdot h}{d}\right)}\right)\right)\right) \]

   10. Simplified 69.9%

       \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - 0.5 \cdot \color{blue}{\left(0.25 \cdot \left(\frac{D}{\frac{\ell}{D}} \cdot \left(\frac{M}{d} \cdot \frac{M \cdot h}{d}\right)\right)\right)}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 62.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.9 \cdot 10^{-273}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(-1 + 0.5 \cdot \left(0.25 \cdot \left(\frac{D}{\frac{\ell}{D}} \cdot \left(\frac{M}{d} \cdot \frac{h \cdot M}{d}\right)\right)\right)\right)\\ \mathbf{elif}\;d \leq 1.05 \cdot 10^{-119} \lor \neg \left(d \leq 1.9 \cdot 10^{+34}\right):\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{D}{\frac{\ell}{D}} \cdot \left(\frac{M}{d} \cdot \frac{h \cdot M}{d}\right)\right)\right)\right) \cdot \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}}\\ \end{array} \]

Alternative 12: 51.7% accurate, 2.4× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} t_0 := \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{D}{\frac{\ell}{D}} \cdot \left(\frac{M}{d} \cdot \frac{h \cdot M}{d}\right)\right)\right)\right) \cdot \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}}\\ t_1 := d \cdot \left(-\sqrt{\frac{\frac{1}{h}}{\ell}}\right)\\ \mathbf{if}\;d \leq -1.4 \cdot 10^{+115}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq -3 \cdot 10^{-138}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 1.35 \cdot 10^{-278}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq 1.05 \cdot 10^{-119} \lor \neg \left(d \leq 3.2 \cdot 10^{+151}\right):\\ \;\;\;\;d \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0
         (*
          (- 1.0 (* 0.5 (* 0.25 (* (/ D (/ l D)) (* (/ M d) (/ (* h M) d))))))
          (sqrt (/ d (/ h (/ d l))))))
        (t_1 (* d (- (sqrt (/ (/ 1.0 h) l))))))
   (if (<= d -1.4e+115)
     t_1
     (if (<= d -3e-138)
       t_0
       (if (<= d 1.35e-278)
         t_1
         (if (or (<= d 1.05e-119) (not (<= d 3.2e+151)))
           (* d (pow (* l h) -0.5))
           t_0))))))

C

assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = (1.0 - (0.5 * (0.25 * ((D / (l / D)) * ((M / d) * ((h * M) / d)))))) * sqrt((d / (h / (d / l))));
	double t_1 = d * -sqrt(((1.0 / h) / l));
	double tmp;
	if (d <= -1.4e+115) {
		tmp = t_1;
	} else if (d <= -3e-138) {
		tmp = t_0;
	} else if (d <= 1.35e-278) {
		tmp = t_1;
	} else if ((d <= 1.05e-119) || !(d <= 3.2e+151)) {
		tmp = d * pow((l * h), -0.5);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (1.0d0 - (0.5d0 * (0.25d0 * ((d_1 / (l / d_1)) * ((m / d) * ((h * m) / d)))))) * sqrt((d / (h / (d / l))))
    t_1 = d * -sqrt(((1.0d0 / h) / l))
    if (d <= (-1.4d+115)) then
        tmp = t_1
    else if (d <= (-3d-138)) then
        tmp = t_0
    else if (d <= 1.35d-278) then
        tmp = t_1
    else if ((d <= 1.05d-119) .or. (.not. (d <= 3.2d+151))) then
        tmp = d * ((l * h) ** (-0.5d0))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double t_0 = (1.0 - (0.5 * (0.25 * ((D / (l / D)) * ((M / d) * ((h * M) / d)))))) * Math.sqrt((d / (h / (d / l))));
	double t_1 = d * -Math.sqrt(((1.0 / h) / l));
	double tmp;
	if (d <= -1.4e+115) {
		tmp = t_1;
	} else if (d <= -3e-138) {
		tmp = t_0;
	} else if (d <= 1.35e-278) {
		tmp = t_1;
	} else if ((d <= 1.05e-119) || !(d <= 3.2e+151)) {
		tmp = d * Math.pow((l * h), -0.5);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

[M, D] = sort([M, D])
def code(d, h, l, M, D):
	t_0 = (1.0 - (0.5 * (0.25 * ((D / (l / D)) * ((M / d) * ((h * M) / d)))))) * math.sqrt((d / (h / (d / l))))
	t_1 = d * -math.sqrt(((1.0 / h) / l))
	tmp = 0
	if d <= -1.4e+115:
		tmp = t_1
	elif d <= -3e-138:
		tmp = t_0
	elif d <= 1.35e-278:
		tmp = t_1
	elif (d <= 1.05e-119) or not (d <= 3.2e+151):
		tmp = d * math.pow((l * h), -0.5)
	else:
		tmp = t_0
	return tmp

Julia

M, D = sort([M, D])
function code(d, h, l, M, D)
	t_0 = Float64(Float64(1.0 - Float64(0.5 * Float64(0.25 * Float64(Float64(D / Float64(l / D)) * Float64(Float64(M / d) * Float64(Float64(h * M) / d)))))) * sqrt(Float64(d / Float64(h / Float64(d / l)))))
	t_1 = Float64(d * Float64(-sqrt(Float64(Float64(1.0 / h) / l))))
	tmp = 0.0
	if (d <= -1.4e+115)
		tmp = t_1;
	elseif (d <= -3e-138)
		tmp = t_0;
	elseif (d <= 1.35e-278)
		tmp = t_1;
	elseif ((d <= 1.05e-119) || !(d <= 3.2e+151))
		tmp = Float64(d * (Float64(l * h) ^ -0.5));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	t_0 = (1.0 - (0.5 * (0.25 * ((D / (l / D)) * ((M / d) * ((h * M) / d)))))) * sqrt((d / (h / (d / l))));
	t_1 = d * -sqrt(((1.0 / h) / l));
	tmp = 0.0;
	if (d <= -1.4e+115)
		tmp = t_1;
	elseif (d <= -3e-138)
		tmp = t_0;
	elseif (d <= 1.35e-278)
		tmp = t_1;
	elseif ((d <= 1.05e-119) || ~((d <= 3.2e+151)))
		tmp = d * ((l * h) ^ -0.5);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(1.0 - N[(0.5 * N[(0.25 * N[(N[(D / N[(l / D), $MachinePrecision]), $MachinePrecision] * N[(N[(M / d), $MachinePrecision] * N[(N[(h * M), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / N[(h / N[(d / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(d * (-N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]}, If[LessEqual[d, -1.4e+115], t$95$1, If[LessEqual[d, -3e-138], t$95$0, If[LessEqual[d, 1.35e-278], t$95$1, If[Or[LessEqual[d, 1.05e-119], N[Not[LessEqual[d, 3.2e+151]], $MachinePrecision]], N[(d * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.4e115 or -3.0000000000000001e-138 < d < 1.3500000000000001e-278

   1. Initial program 48.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 48.2%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 48.2%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 48.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 48.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 48.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 48.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 48.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 48.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 48.4%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]
   4. Step-by-step derivation

      1. expm1-log1p-u 29.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\right)\right)} \]

      2. expm1-udef 25.4%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\right)} - 1} \]

   5. Applied egg-rr 17.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 21.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)\right)} \]

      2. expm1-log1p 37.6%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)} \]

      3. sub-neg 37.6%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \color{blue}{\left(1 + \left(-{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)} \]

      4. +-commutative 37.6%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \color{blue}{\left(\left(-{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) + 1\right)} \]

      5. associate-*l/ 34.8%

         \[\leadsto \sqrt{\color{blue}{\frac{d \cdot \frac{d}{\ell}}{h}}} \cdot \left(\left(-{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) + 1\right) \]

      6. associate-/l* 36.2%

         \[\leadsto \sqrt{\color{blue}{\frac{d}{\frac{h}{\frac{d}{\ell}}}}} \cdot \left(\left(-{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) + 1\right) \]

      7. +-commutative 36.2%

         \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \color{blue}{\left(1 + \left(-{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)} \]

      8. sub-neg 36.2%

         \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \color{blue}{\left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)} \]

      9. associate-/l* 36.2%

         \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \color{blue}{\frac{0.5}{\frac{\ell}{h}}}\right) \]

      10. associate-*r/ 36.2%

          \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - \color{blue}{\frac{{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot 0.5}{\frac{\ell}{h}}}\right) \]

      11. associate-*l/ 36.2%

          \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - \color{blue}{\frac{{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}}{\frac{\ell}{h}} \cdot 0.5}\right) \]

      12. associate-/l* 35.0%

          \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - \color{blue}{\frac{{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot h}{\ell}} \cdot 0.5\right) \]

      13. associate-*r/ 36.2%

          \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - \color{blue}{\left({\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\ell}\right)} \cdot 0.5\right) \]

   7. Simplified 36.1%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - 0.5 \cdot \left({\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   8. Step-by-step derivation

      1. associate-*r/ 66.0%

         \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{{\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2} \cdot h}{\ell}}\right) \]

      2. div-inv 66.0%

         \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(D \cdot \color{blue}{\left(\frac{M}{d} \cdot \frac{1}{2}\right)}\right)}^{2} \cdot h}{\ell}\right) \]

      3. metadata-eval 66.0%

         \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(D \cdot \left(\frac{M}{d} \cdot \color{blue}{0.5}\right)\right)}^{2} \cdot h}{\ell}\right) \]

   9. Applied egg-rr 34.9%

      \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{{\left(D \cdot \left(\frac{M}{d} \cdot 0.5\right)\right)}^{2} \cdot h}{\ell}}\right) \]

   10. Taylor expanded in d around -inf 46.7%

       \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \]
   11. Step-by-step derivation

       1. mul-1-neg 46.7%

          \[\leadsto \color{blue}{-d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

       2. *-commutative 46.7%

          \[\leadsto -\color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]

       3. distribute-rgt-neg-in 46.7%

          \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)} \]

       4. *-commutative 46.7%

          \[\leadsto \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \cdot \left(-d\right) \]

       5. associate-/r* 46.7%

          \[\leadsto \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \cdot \left(-d\right) \]

   12. Simplified 46.7%

       \[\leadsto \color{blue}{\sqrt{\frac{\frac{1}{h}}{\ell}} \cdot \left(-d\right)} \]

   if -1.4e115 < d < -3.0000000000000001e-138 or 1.05e-119 < d < 3.19999999999999994e151

   1. Initial program 75.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 75.5%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 75.5%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 75.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 75.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 75.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 75.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 75.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 75.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 75.4%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]
   4. Step-by-step derivation

      1. expm1-log1p-u 36.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\right)\right)} \]

      2. expm1-udef 23.8%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\right)} - 1} \]

   5. Applied egg-rr 18.8%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 30.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)\right)} \]

      2. expm1-log1p 64.7%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)} \]

      3. sub-neg 64.7%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \color{blue}{\left(1 + \left(-{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)} \]

      4. +-commutative 64.7%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \color{blue}{\left(\left(-{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) + 1\right)} \]

      5. associate-*l/ 66.6%

         \[\leadsto \sqrt{\color{blue}{\frac{d \cdot \frac{d}{\ell}}{h}}} \cdot \left(\left(-{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) + 1\right) \]

      6. associate-/l* 65.3%

         \[\leadsto \sqrt{\color{blue}{\frac{d}{\frac{h}{\frac{d}{\ell}}}}} \cdot \left(\left(-{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) + 1\right) \]

      7. +-commutative 65.3%

         \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \color{blue}{\left(1 + \left(-{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)} \]

      8. sub-neg 65.3%

         \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \color{blue}{\left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)} \]

      9. associate-/l* 65.3%

         \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \color{blue}{\frac{0.5}{\frac{\ell}{h}}}\right) \]

      10. associate-*r/ 65.4%

          \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - \color{blue}{\frac{{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot 0.5}{\frac{\ell}{h}}}\right) \]

      11. associate-*l/ 65.4%

          \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - \color{blue}{\frac{{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}}{\frac{\ell}{h}} \cdot 0.5}\right) \]

      12. associate-/l* 72.2%

          \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - \color{blue}{\frac{{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot h}{\ell}} \cdot 0.5\right) \]

      13. associate-*r/ 65.3%

          \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - \color{blue}{\left({\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\ell}\right)} \cdot 0.5\right) \]

   7. Simplified 64.5%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - 0.5 \cdot \left({\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]

   8. Taylor expanded in D around 0 50.2%

      \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - 0.5 \cdot \color{blue}{\left(0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}\right)}\right) \]
   9. Step-by-step derivation

      1. times-frac 52.0%

         \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \color{blue}{\left(\frac{{D}^{2}}{\ell} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right)}\right)\right) \]

      2. *-commutative 52.0%

         \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{{D}^{2}}{\ell} \cdot \frac{\color{blue}{h \cdot {M}^{2}}}{{d}^{2}}\right)\right)\right) \]

      3. unpow2 52.0%

         \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{\color{blue}{D \cdot D}}{\ell} \cdot \frac{h \cdot {M}^{2}}{{d}^{2}}\right)\right)\right) \]

      4. associate-/l* 56.1%

         \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\color{blue}{\frac{D}{\frac{\ell}{D}}} \cdot \frac{h \cdot {M}^{2}}{{d}^{2}}\right)\right)\right) \]

      5. *-commutative 56.1%

         \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{D}{\frac{\ell}{D}} \cdot \frac{\color{blue}{{M}^{2} \cdot h}}{{d}^{2}}\right)\right)\right) \]

      6. unpow2 56.1%

         \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{D}{\frac{\ell}{D}} \cdot \frac{\color{blue}{\left(M \cdot M\right)} \cdot h}{{d}^{2}}\right)\right)\right) \]

      7. associate-*r* 57.8%

         \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{D}{\frac{\ell}{D}} \cdot \frac{\color{blue}{M \cdot \left(M \cdot h\right)}}{{d}^{2}}\right)\right)\right) \]

      8. unpow2 57.8%

         \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{D}{\frac{\ell}{D}} \cdot \frac{M \cdot \left(M \cdot h\right)}{\color{blue}{d \cdot d}}\right)\right)\right) \]

      9. times-frac 64.5%

         \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{D}{\frac{\ell}{D}} \cdot \color{blue}{\left(\frac{M}{d} \cdot \frac{M \cdot h}{d}\right)}\right)\right)\right) \]

   10. Simplified 64.5%

       \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - 0.5 \cdot \color{blue}{\left(0.25 \cdot \left(\frac{D}{\frac{\ell}{D}} \cdot \left(\frac{M}{d} \cdot \frac{M \cdot h}{d}\right)\right)\right)}\right) \]

   if 1.3500000000000001e-278 < d < 1.05e-119 or 3.19999999999999994e151 < d

   1. Initial program 55.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 55.8%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 55.8%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 55.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 55.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 55.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 55.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 55.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 55.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 55.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]

   4. Taylor expanded in d around inf 55.2%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   5. Step-by-step derivation

      1. *-un-lft-identity 55.2%

         \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot d \]

      2. associate-/r* 55.1%

         \[\leadsto \left(1 \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}}\right) \cdot d \]

   6. Applied egg-rr 55.1%

      \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right)} \cdot d \]
   7. Step-by-step derivation

      1. *-lft-identity 55.1%

         \[\leadsto \color{blue}{\sqrt{\frac{\frac{1}{\ell}}{h}}} \cdot d \]

      2. associate-/l/ 55.2%

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      3. unpow-1 55.2%

         \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \cdot d \]

      4. sqr-pow 55.2%

         \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}}} \cdot d \]

      5. rem-sqrt-square 55.2%

         \[\leadsto \color{blue}{\left|{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}\right|} \cdot d \]

      6. metadata-eval 55.2%

         \[\leadsto \left|{\left(h \cdot \ell\right)}^{\color{blue}{-0.5}}\right| \cdot d \]

      7. sqr-pow 54.9%

         \[\leadsto \left|\color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-0.5}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{-0.5}{2}\right)}}\right| \cdot d \]

      8. fabs-sqr 54.9%

         \[\leadsto \color{blue}{\left({\left(h \cdot \ell\right)}^{\left(\frac{-0.5}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{-0.5}{2}\right)}\right)} \cdot d \]

      9. sqr-pow 55.2%

         \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \cdot d \]

   8. Simplified 55.2%

      \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \cdot d \]
3. Recombined 3 regimes into one program.

4. Final simplification 57.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.4 \cdot 10^{+115}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{h}}{\ell}}\right)\\ \mathbf{elif}\;d \leq -3 \cdot 10^{-138}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{D}{\frac{\ell}{D}} \cdot \left(\frac{M}{d} \cdot \frac{h \cdot M}{d}\right)\right)\right)\right) \cdot \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}}\\ \mathbf{elif}\;d \leq 1.35 \cdot 10^{-278}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{h}}{\ell}}\right)\\ \mathbf{elif}\;d \leq 1.05 \cdot 10^{-119} \lor \neg \left(d \leq 3.2 \cdot 10^{+151}\right):\\ \;\;\;\;d \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{D}{\frac{\ell}{D}} \cdot \left(\frac{M}{d} \cdot \frac{h \cdot M}{d}\right)\right)\right)\right) \cdot \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}}\\ \end{array} \]

Alternative 13: 57.4% accurate, 2.5× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} t_0 := 0.5 \cdot \left(0.25 \cdot \left(\frac{D}{\frac{\ell}{D}} \cdot \left(\frac{M}{d} \cdot \frac{h \cdot M}{d}\right)\right)\right)\\ t_1 := d \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \mathbf{if}\;d \leq -2.9 \cdot 10^{-273}:\\ \;\;\;\;t_1 \cdot \left(-1 + t_0\right)\\ \mathbf{elif}\;d \leq 1.05 \cdot 10^{-119} \lor \neg \left(d \leq 4.9 \cdot 10^{+148}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\left(1 - t_0\right) \cdot \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}}\\ \end{array} \end{array} \]

FPCore

NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* 0.5 (* 0.25 (* (/ D (/ l D)) (* (/ M d) (/ (* h M) d))))))
        (t_1 (* d (pow (* l h) -0.5))))
   (if (<= d -2.9e-273)
     (* t_1 (+ -1.0 t_0))
     (if (or (<= d 1.05e-119) (not (<= d 4.9e+148)))
       t_1
       (* (- 1.0 t_0) (sqrt (/ d (/ h (/ d l)))))))))

C

assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = 0.5 * (0.25 * ((D / (l / D)) * ((M / d) * ((h * M) / d))));
	double t_1 = d * pow((l * h), -0.5);
	double tmp;
	if (d <= -2.9e-273) {
		tmp = t_1 * (-1.0 + t_0);
	} else if ((d <= 1.05e-119) || !(d <= 4.9e+148)) {
		tmp = t_1;
	} else {
		tmp = (1.0 - t_0) * sqrt((d / (h / (d / l))));
	}
	return tmp;
}

Fortran

NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 0.5d0 * (0.25d0 * ((d_1 / (l / d_1)) * ((m / d) * ((h * m) / d))))
    t_1 = d * ((l * h) ** (-0.5d0))
    if (d <= (-2.9d-273)) then
        tmp = t_1 * ((-1.0d0) + t_0)
    else if ((d <= 1.05d-119) .or. (.not. (d <= 4.9d+148))) then
        tmp = t_1
    else
        tmp = (1.0d0 - t_0) * sqrt((d / (h / (d / l))))
    end if
    code = tmp
end function

Java

assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double t_0 = 0.5 * (0.25 * ((D / (l / D)) * ((M / d) * ((h * M) / d))));
	double t_1 = d * Math.pow((l * h), -0.5);
	double tmp;
	if (d <= -2.9e-273) {
		tmp = t_1 * (-1.0 + t_0);
	} else if ((d <= 1.05e-119) || !(d <= 4.9e+148)) {
		tmp = t_1;
	} else {
		tmp = (1.0 - t_0) * Math.sqrt((d / (h / (d / l))));
	}
	return tmp;
}

Python

[M, D] = sort([M, D])
def code(d, h, l, M, D):
	t_0 = 0.5 * (0.25 * ((D / (l / D)) * ((M / d) * ((h * M) / d))))
	t_1 = d * math.pow((l * h), -0.5)
	tmp = 0
	if d <= -2.9e-273:
		tmp = t_1 * (-1.0 + t_0)
	elif (d <= 1.05e-119) or not (d <= 4.9e+148):
		tmp = t_1
	else:
		tmp = (1.0 - t_0) * math.sqrt((d / (h / (d / l))))
	return tmp

Julia

M, D = sort([M, D])
function code(d, h, l, M, D)
	t_0 = Float64(0.5 * Float64(0.25 * Float64(Float64(D / Float64(l / D)) * Float64(Float64(M / d) * Float64(Float64(h * M) / d)))))
	t_1 = Float64(d * (Float64(l * h) ^ -0.5))
	tmp = 0.0
	if (d <= -2.9e-273)
		tmp = Float64(t_1 * Float64(-1.0 + t_0));
	elseif ((d <= 1.05e-119) || !(d <= 4.9e+148))
		tmp = t_1;
	else
		tmp = Float64(Float64(1.0 - t_0) * sqrt(Float64(d / Float64(h / Float64(d / l)))));
	end
	return tmp
end

MATLAB

M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	t_0 = 0.5 * (0.25 * ((D / (l / D)) * ((M / d) * ((h * M) / d))));
	t_1 = d * ((l * h) ^ -0.5);
	tmp = 0.0;
	if (d <= -2.9e-273)
		tmp = t_1 * (-1.0 + t_0);
	elseif ((d <= 1.05e-119) || ~((d <= 4.9e+148)))
		tmp = t_1;
	else
		tmp = (1.0 - t_0) * sqrt((d / (h / (d / l))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(0.5 * N[(0.25 * N[(N[(D / N[(l / D), $MachinePrecision]), $MachinePrecision] * N[(N[(M / d), $MachinePrecision] * N[(N[(h * M), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(d * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -2.9e-273], N[(t$95$1 * N[(-1.0 + t$95$0), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[d, 1.05e-119], N[Not[LessEqual[d, 4.9e+148]], $MachinePrecision]], t$95$1, N[(N[(1.0 - t$95$0), $MachinePrecision] * N[Sqrt[N[(d / N[(h / N[(d / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -2.89999999999999986e-273

   1. Initial program 61.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 61.1%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 61.1%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 61.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 61.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 61.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 61.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 61.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 61.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 61.1%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]
   4. Step-by-step derivation

      1. expm1-log1p-u 37.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\right)\right)} \]

      2. expm1-udef 26.2%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\right)} - 1} \]

   5. Applied egg-rr 20.1%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 30.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)\right)} \]

      2. expm1-log1p 51.7%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)} \]

      3. sub-neg 51.7%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \color{blue}{\left(1 + \left(-{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)} \]

      4. +-commutative 51.7%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \color{blue}{\left(\left(-{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) + 1\right)} \]

      5. associate-*l/ 50.2%

         \[\leadsto \sqrt{\color{blue}{\frac{d \cdot \frac{d}{\ell}}{h}}} \cdot \left(\left(-{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) + 1\right) \]

      6. associate-/l* 51.0%

         \[\leadsto \sqrt{\color{blue}{\frac{d}{\frac{h}{\frac{d}{\ell}}}}} \cdot \left(\left(-{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) + 1\right) \]

      7. +-commutative 51.0%

         \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \color{blue}{\left(1 + \left(-{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)} \]

      8. sub-neg 51.0%

         \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \color{blue}{\left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)} \]

      9. associate-/l* 51.0%

         \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \color{blue}{\frac{0.5}{\frac{\ell}{h}}}\right) \]

      10. associate-*r/ 51.0%

          \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - \color{blue}{\frac{{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot 0.5}{\frac{\ell}{h}}}\right) \]

      11. associate-*l/ 51.0%

          \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - \color{blue}{\frac{{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}}{\frac{\ell}{h}} \cdot 0.5}\right) \]

      12. associate-/l* 55.3%

          \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - \color{blue}{\frac{{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot h}{\ell}} \cdot 0.5\right) \]

      13. associate-*r/ 51.0%

          \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - \color{blue}{\left({\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\ell}\right)} \cdot 0.5\right) \]

   7. Simplified 50.2%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - 0.5 \cdot \left({\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]

   8. Taylor expanded in d around -inf 68.1%

      \[\leadsto \color{blue}{\left(-1 \cdot \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)\right)} \cdot \left(1 - 0.5 \cdot \left({\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
   9. Step-by-step derivation

      1. mul-1-neg 68.1%

         \[\leadsto \color{blue}{\left(-d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot \left(1 - 0.5 \cdot \left({\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      2. associate-/r* 69.0%

         \[\leadsto \left(-d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      3. distribute-rgt-neg-in 69.0%

         \[\leadsto \color{blue}{\left(d \cdot \left(-\sqrt{\frac{\frac{1}{\ell}}{h}}\right)\right)} \cdot \left(1 - 0.5 \cdot \left({\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      4. associate-/l/ 68.1%

         \[\leadsto \left(d \cdot \left(-\sqrt{\color{blue}{\frac{1}{h \cdot \ell}}}\right)\right) \cdot \left(1 - 0.5 \cdot \left({\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      5. unpow-1 68.1%

         \[\leadsto \left(d \cdot \left(-\sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}}\right)\right) \cdot \left(1 - 0.5 \cdot \left({\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      6. sqr-pow 68.1%

         \[\leadsto \left(d \cdot \left(-\sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}}}\right)\right) \cdot \left(1 - 0.5 \cdot \left({\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      7. rem-sqrt-square 68.1%

         \[\leadsto \left(d \cdot \left(-\color{blue}{\left|{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}\right|}\right)\right) \cdot \left(1 - 0.5 \cdot \left({\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 68.1%

         \[\leadsto \left(d \cdot \left(-\left|{\left(h \cdot \ell\right)}^{\color{blue}{-0.5}}\right|\right)\right) \cdot \left(1 - 0.5 \cdot \left({\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      9. sqr-pow 68.0%

         \[\leadsto \left(d \cdot \left(-\left|\color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-0.5}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{-0.5}{2}\right)}}\right|\right)\right) \cdot \left(1 - 0.5 \cdot \left({\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      10. fabs-sqr 68.0%

          \[\leadsto \left(d \cdot \left(-\color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-0.5}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{-0.5}{2}\right)}}\right)\right) \cdot \left(1 - 0.5 \cdot \left({\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      11. sqr-pow 68.1%

          \[\leadsto \left(d \cdot \left(-\color{blue}{{\left(h \cdot \ell\right)}^{-0.5}}\right)\right) \cdot \left(1 - 0.5 \cdot \left({\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   10. Simplified 68.1%

       \[\leadsto \color{blue}{\left(d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\right)} \cdot \left(1 - 0.5 \cdot \left({\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   11. Taylor expanded in D around 0 48.1%

       \[\leadsto \left(d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}\right)}\right) \]
   12. Step-by-step derivation

       1. times-frac 29.3%

          \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \color{blue}{\left(\frac{{D}^{2}}{\ell} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right)}\right)\right) \]

       2. *-commutative 29.3%

          \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{{D}^{2}}{\ell} \cdot \frac{\color{blue}{h \cdot {M}^{2}}}{{d}^{2}}\right)\right)\right) \]

       3. unpow2 29.3%

          \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{\color{blue}{D \cdot D}}{\ell} \cdot \frac{h \cdot {M}^{2}}{{d}^{2}}\right)\right)\right) \]

       4. associate-/l* 31.8%

          \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\color{blue}{\frac{D}{\frac{\ell}{D}}} \cdot \frac{h \cdot {M}^{2}}{{d}^{2}}\right)\right)\right) \]

       5. *-commutative 31.8%

          \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{D}{\frac{\ell}{D}} \cdot \frac{\color{blue}{{M}^{2} \cdot h}}{{d}^{2}}\right)\right)\right) \]

       6. unpow2 31.8%

          \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{D}{\frac{\ell}{D}} \cdot \frac{\color{blue}{\left(M \cdot M\right)} \cdot h}{{d}^{2}}\right)\right)\right) \]

       7. associate-*r* 33.5%

          \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{D}{\frac{\ell}{D}} \cdot \frac{\color{blue}{M \cdot \left(M \cdot h\right)}}{{d}^{2}}\right)\right)\right) \]

       8. unpow2 33.5%

          \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{D}{\frac{\ell}{D}} \cdot \frac{M \cdot \left(M \cdot h\right)}{\color{blue}{d \cdot d}}\right)\right)\right) \]

       9. times-frac 45.2%

          \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{D}{\frac{\ell}{D}} \cdot \color{blue}{\left(\frac{M}{d} \cdot \frac{M \cdot h}{d}\right)}\right)\right)\right) \]

   13. Simplified 64.7%

       \[\leadsto \left(d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(0.25 \cdot \left(\frac{D}{\frac{\ell}{D}} \cdot \left(\frac{M}{d} \cdot \frac{M \cdot h}{d}\right)\right)\right)}\right) \]

   if -2.89999999999999986e-273 < d < 1.05e-119 or 4.9e148 < d

   1. Initial program 52.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 52.0%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 52.0%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 52.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 52.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 52.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 52.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 51.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 51.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 51.9%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]

   4. Taylor expanded in d around inf 49.0%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   5. Step-by-step derivation

      1. *-un-lft-identity 49.0%

         \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot d \]

      2. associate-/r* 49.0%

         \[\leadsto \left(1 \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}}\right) \cdot d \]

   6. Applied egg-rr 49.0%

      \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right)} \cdot d \]
   7. Step-by-step derivation

      1. *-lft-identity 49.0%

         \[\leadsto \color{blue}{\sqrt{\frac{\frac{1}{\ell}}{h}}} \cdot d \]

      2. associate-/l/ 49.0%

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      3. unpow-1 49.0%

         \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \cdot d \]

      4. sqr-pow 49.0%

         \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}}} \cdot d \]

      5. rem-sqrt-square 49.0%

         \[\leadsto \color{blue}{\left|{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}\right|} \cdot d \]

      6. metadata-eval 49.0%

         \[\leadsto \left|{\left(h \cdot \ell\right)}^{\color{blue}{-0.5}}\right| \cdot d \]

      7. sqr-pow 48.8%

         \[\leadsto \left|\color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-0.5}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{-0.5}{2}\right)}}\right| \cdot d \]

      8. fabs-sqr 48.8%

         \[\leadsto \color{blue}{\left({\left(h \cdot \ell\right)}^{\left(\frac{-0.5}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{-0.5}{2}\right)}\right)} \cdot d \]

      9. sqr-pow 49.0%

         \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \cdot d \]

   8. Simplified 49.0%

      \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \cdot d \]

   if 1.05e-119 < d < 4.9e148

   1. Initial program 80.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 80.9%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 80.9%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 80.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 80.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 80.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 80.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 80.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 80.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 80.9%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]
   4. Step-by-step derivation

      1. expm1-log1p-u 31.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\right)\right)} \]

      2. expm1-udef 24.4%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\right)} - 1} \]

   5. Applied egg-rr 16.9%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 24.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)\right)} \]

      2. expm1-log1p 65.8%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)} \]

      3. sub-neg 65.8%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \color{blue}{\left(1 + \left(-{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)} \]

      4. +-commutative 65.8%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \color{blue}{\left(\left(-{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) + 1\right)} \]

      5. associate-*l/ 71.1%

         \[\leadsto \sqrt{\color{blue}{\frac{d \cdot \frac{d}{\ell}}{h}}} \cdot \left(\left(-{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) + 1\right) \]

      6. associate-/l* 66.8%

         \[\leadsto \sqrt{\color{blue}{\frac{d}{\frac{h}{\frac{d}{\ell}}}}} \cdot \left(\left(-{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) + 1\right) \]

      7. +-commutative 66.8%

         \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \color{blue}{\left(1 + \left(-{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)} \]

      8. sub-neg 66.8%

         \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \color{blue}{\left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)} \]

      9. associate-/l* 66.8%

         \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \color{blue}{\frac{0.5}{\frac{\ell}{h}}}\right) \]

      10. associate-*r/ 66.8%

          \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - \color{blue}{\frac{{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot 0.5}{\frac{\ell}{h}}}\right) \]

      11. associate-*l/ 66.8%

          \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - \color{blue}{\frac{{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}}{\frac{\ell}{h}} \cdot 0.5}\right) \]

      12. associate-/l* 70.4%

          \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - \color{blue}{\frac{{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot h}{\ell}} \cdot 0.5\right) \]

      13. associate-*r/ 66.8%

          \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - \color{blue}{\left({\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\ell}\right)} \cdot 0.5\right) \]

   7. Simplified 66.7%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - 0.5 \cdot \left({\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]

   8. Taylor expanded in D around 0 49.6%

      \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - 0.5 \cdot \color{blue}{\left(0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}\right)}\right) \]
   9. Step-by-step derivation

      1. times-frac 51.5%

         \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \color{blue}{\left(\frac{{D}^{2}}{\ell} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right)}\right)\right) \]

      2. *-commutative 51.5%

         \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{{D}^{2}}{\ell} \cdot \frac{\color{blue}{h \cdot {M}^{2}}}{{d}^{2}}\right)\right)\right) \]

      3. unpow2 51.5%

         \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{\color{blue}{D \cdot D}}{\ell} \cdot \frac{h \cdot {M}^{2}}{{d}^{2}}\right)\right)\right) \]

      4. associate-/l* 58.2%

         \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\color{blue}{\frac{D}{\frac{\ell}{D}}} \cdot \frac{h \cdot {M}^{2}}{{d}^{2}}\right)\right)\right) \]

      5. *-commutative 58.2%

         \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{D}{\frac{\ell}{D}} \cdot \frac{\color{blue}{{M}^{2} \cdot h}}{{d}^{2}}\right)\right)\right) \]

      6. unpow2 58.2%

         \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{D}{\frac{\ell}{D}} \cdot \frac{\color{blue}{\left(M \cdot M\right)} \cdot h}{{d}^{2}}\right)\right)\right) \]

      7. associate-*r* 61.6%

         \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{D}{\frac{\ell}{D}} \cdot \frac{\color{blue}{M \cdot \left(M \cdot h\right)}}{{d}^{2}}\right)\right)\right) \]

      8. unpow2 61.6%

         \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{D}{\frac{\ell}{D}} \cdot \frac{M \cdot \left(M \cdot h\right)}{\color{blue}{d \cdot d}}\right)\right)\right) \]

      9. times-frac 66.7%

         \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{D}{\frac{\ell}{D}} \cdot \color{blue}{\left(\frac{M}{d} \cdot \frac{M \cdot h}{d}\right)}\right)\right)\right) \]

   10. Simplified 66.7%

       \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - 0.5 \cdot \color{blue}{\left(0.25 \cdot \left(\frac{D}{\frac{\ell}{D}} \cdot \left(\frac{M}{d} \cdot \frac{M \cdot h}{d}\right)\right)\right)}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 60.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.9 \cdot 10^{-273}:\\ \;\;\;\;\left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right) \cdot \left(-1 + 0.5 \cdot \left(0.25 \cdot \left(\frac{D}{\frac{\ell}{D}} \cdot \left(\frac{M}{d} \cdot \frac{h \cdot M}{d}\right)\right)\right)\right)\\ \mathbf{elif}\;d \leq 1.05 \cdot 10^{-119} \lor \neg \left(d \leq 4.9 \cdot 10^{+148}\right):\\ \;\;\;\;d \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{D}{\frac{\ell}{D}} \cdot \left(\frac{M}{d} \cdot \frac{h \cdot M}{d}\right)\right)\right)\right) \cdot \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}}\\ \end{array} \]

Alternative 14: 57.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} t_0 := 0.5 \cdot \left(0.25 \cdot \left(\frac{D}{\frac{\ell}{D}} \cdot \left(\frac{M}{d} \cdot \frac{h \cdot M}{d}\right)\right)\right)\\ \mathbf{if}\;d \leq -2.9 \cdot 10^{-273}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(-1 + t_0\right)\\ \mathbf{elif}\;d \leq 1.05 \cdot 10^{-119} \lor \neg \left(d \leq 9.2 \cdot 10^{+147}\right):\\ \;\;\;\;d \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\left(1 - t_0\right) \cdot \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}}\\ \end{array} \end{array} \]

FPCore

NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* 0.5 (* 0.25 (* (/ D (/ l D)) (* (/ M d) (/ (* h M) d)))))))
   (if (<= d -2.9e-273)
     (* (* d (sqrt (/ 1.0 (* l h)))) (+ -1.0 t_0))
     (if (or (<= d 1.05e-119) (not (<= d 9.2e+147)))
       (* d (pow (* l h) -0.5))
       (* (- 1.0 t_0) (sqrt (/ d (/ h (/ d l)))))))))

C

assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = 0.5 * (0.25 * ((D / (l / D)) * ((M / d) * ((h * M) / d))));
	double tmp;
	if (d <= -2.9e-273) {
		tmp = (d * sqrt((1.0 / (l * h)))) * (-1.0 + t_0);
	} else if ((d <= 1.05e-119) || !(d <= 9.2e+147)) {
		tmp = d * pow((l * h), -0.5);
	} else {
		tmp = (1.0 - t_0) * sqrt((d / (h / (d / l))));
	}
	return tmp;
}

Fortran

NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * (0.25d0 * ((d_1 / (l / d_1)) * ((m / d) * ((h * m) / d))))
    if (d <= (-2.9d-273)) then
        tmp = (d * sqrt((1.0d0 / (l * h)))) * ((-1.0d0) + t_0)
    else if ((d <= 1.05d-119) .or. (.not. (d <= 9.2d+147))) then
        tmp = d * ((l * h) ** (-0.5d0))
    else
        tmp = (1.0d0 - t_0) * sqrt((d / (h / (d / l))))
    end if
    code = tmp
end function

Java

assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double t_0 = 0.5 * (0.25 * ((D / (l / D)) * ((M / d) * ((h * M) / d))));
	double tmp;
	if (d <= -2.9e-273) {
		tmp = (d * Math.sqrt((1.0 / (l * h)))) * (-1.0 + t_0);
	} else if ((d <= 1.05e-119) || !(d <= 9.2e+147)) {
		tmp = d * Math.pow((l * h), -0.5);
	} else {
		tmp = (1.0 - t_0) * Math.sqrt((d / (h / (d / l))));
	}
	return tmp;
}

Python

[M, D] = sort([M, D])
def code(d, h, l, M, D):
	t_0 = 0.5 * (0.25 * ((D / (l / D)) * ((M / d) * ((h * M) / d))))
	tmp = 0
	if d <= -2.9e-273:
		tmp = (d * math.sqrt((1.0 / (l * h)))) * (-1.0 + t_0)
	elif (d <= 1.05e-119) or not (d <= 9.2e+147):
		tmp = d * math.pow((l * h), -0.5)
	else:
		tmp = (1.0 - t_0) * math.sqrt((d / (h / (d / l))))
	return tmp

Julia

M, D = sort([M, D])
function code(d, h, l, M, D)
	t_0 = Float64(0.5 * Float64(0.25 * Float64(Float64(D / Float64(l / D)) * Float64(Float64(M / d) * Float64(Float64(h * M) / d)))))
	tmp = 0.0
	if (d <= -2.9e-273)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(l * h)))) * Float64(-1.0 + t_0));
	elseif ((d <= 1.05e-119) || !(d <= 9.2e+147))
		tmp = Float64(d * (Float64(l * h) ^ -0.5));
	else
		tmp = Float64(Float64(1.0 - t_0) * sqrt(Float64(d / Float64(h / Float64(d / l)))));
	end
	return tmp
end

MATLAB

M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	t_0 = 0.5 * (0.25 * ((D / (l / D)) * ((M / d) * ((h * M) / d))));
	tmp = 0.0;
	if (d <= -2.9e-273)
		tmp = (d * sqrt((1.0 / (l * h)))) * (-1.0 + t_0);
	elseif ((d <= 1.05e-119) || ~((d <= 9.2e+147)))
		tmp = d * ((l * h) ^ -0.5);
	else
		tmp = (1.0 - t_0) * sqrt((d / (h / (d / l))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(0.5 * N[(0.25 * N[(N[(D / N[(l / D), $MachinePrecision]), $MachinePrecision] * N[(N[(M / d), $MachinePrecision] * N[(N[(h * M), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -2.9e-273], N[(N[(d * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-1.0 + t$95$0), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[d, 1.05e-119], N[Not[LessEqual[d, 9.2e+147]], $MachinePrecision]], N[(d * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - t$95$0), $MachinePrecision] * N[Sqrt[N[(d / N[(h / N[(d / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if d < -2.89999999999999986e-273

   1. Initial program 61.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 61.1%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 61.1%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 61.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 61.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 61.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 61.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 61.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 61.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 61.1%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]
   4. Step-by-step derivation

      1. expm1-log1p-u 37.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\right)\right)} \]

      2. expm1-udef 26.2%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\right)} - 1} \]

   5. Applied egg-rr 20.1%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 30.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)\right)} \]

      2. expm1-log1p 51.7%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)} \]

      3. sub-neg 51.7%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \color{blue}{\left(1 + \left(-{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)} \]

      4. +-commutative 51.7%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \color{blue}{\left(\left(-{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) + 1\right)} \]

      5. associate-*l/ 50.2%

         \[\leadsto \sqrt{\color{blue}{\frac{d \cdot \frac{d}{\ell}}{h}}} \cdot \left(\left(-{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) + 1\right) \]

      6. associate-/l* 51.0%

         \[\leadsto \sqrt{\color{blue}{\frac{d}{\frac{h}{\frac{d}{\ell}}}}} \cdot \left(\left(-{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) + 1\right) \]

      7. +-commutative 51.0%

         \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \color{blue}{\left(1 + \left(-{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)} \]

      8. sub-neg 51.0%

         \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \color{blue}{\left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)} \]

      9. associate-/l* 51.0%

         \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \color{blue}{\frac{0.5}{\frac{\ell}{h}}}\right) \]

      10. associate-*r/ 51.0%

          \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - \color{blue}{\frac{{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot 0.5}{\frac{\ell}{h}}}\right) \]

      11. associate-*l/ 51.0%

          \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - \color{blue}{\frac{{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}}{\frac{\ell}{h}} \cdot 0.5}\right) \]

      12. associate-/l* 55.3%

          \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - \color{blue}{\frac{{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot h}{\ell}} \cdot 0.5\right) \]

      13. associate-*r/ 51.0%

          \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - \color{blue}{\left({\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\ell}\right)} \cdot 0.5\right) \]

   7. Simplified 50.2%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - 0.5 \cdot \left({\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]

   8. Taylor expanded in d around -inf 68.1%

      \[\leadsto \color{blue}{\left(-1 \cdot \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)\right)} \cdot \left(1 - 0.5 \cdot \left({\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
   9. Step-by-step derivation

      1. associate-*r* 68.1%

         \[\leadsto \color{blue}{\left(\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot \left(1 - 0.5 \cdot \left({\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      2. mul-1-neg 68.1%

         \[\leadsto \left(\color{blue}{\left(-d\right)} \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      3. *-commutative 68.1%

         \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   10. Simplified 68.1%

       \[\leadsto \color{blue}{\left(\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot \left(1 - 0.5 \cdot \left({\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   11. Taylor expanded in D around 0 48.1%

       \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}\right)}\right) \]
   12. Step-by-step derivation

       1. times-frac 29.3%

          \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \color{blue}{\left(\frac{{D}^{2}}{\ell} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right)}\right)\right) \]

       2. *-commutative 29.3%

          \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{{D}^{2}}{\ell} \cdot \frac{\color{blue}{h \cdot {M}^{2}}}{{d}^{2}}\right)\right)\right) \]

       3. unpow2 29.3%

          \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{\color{blue}{D \cdot D}}{\ell} \cdot \frac{h \cdot {M}^{2}}{{d}^{2}}\right)\right)\right) \]

       4. associate-/l* 31.8%

          \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\color{blue}{\frac{D}{\frac{\ell}{D}}} \cdot \frac{h \cdot {M}^{2}}{{d}^{2}}\right)\right)\right) \]

       5. *-commutative 31.8%

          \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{D}{\frac{\ell}{D}} \cdot \frac{\color{blue}{{M}^{2} \cdot h}}{{d}^{2}}\right)\right)\right) \]

       6. unpow2 31.8%

          \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{D}{\frac{\ell}{D}} \cdot \frac{\color{blue}{\left(M \cdot M\right)} \cdot h}{{d}^{2}}\right)\right)\right) \]

       7. associate-*r* 33.5%

          \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{D}{\frac{\ell}{D}} \cdot \frac{\color{blue}{M \cdot \left(M \cdot h\right)}}{{d}^{2}}\right)\right)\right) \]

       8. unpow2 33.5%

          \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{D}{\frac{\ell}{D}} \cdot \frac{M \cdot \left(M \cdot h\right)}{\color{blue}{d \cdot d}}\right)\right)\right) \]

       9. times-frac 45.2%

          \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{D}{\frac{\ell}{D}} \cdot \color{blue}{\left(\frac{M}{d} \cdot \frac{M \cdot h}{d}\right)}\right)\right)\right) \]

   13. Simplified 64.7%

       \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(0.25 \cdot \left(\frac{D}{\frac{\ell}{D}} \cdot \left(\frac{M}{d} \cdot \frac{M \cdot h}{d}\right)\right)\right)}\right) \]

   if -2.89999999999999986e-273 < d < 1.05e-119 or 9.1999999999999997e147 < d

   1. Initial program 52.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 52.0%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 52.0%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 52.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 52.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 52.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 52.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 51.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 51.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 51.9%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]

   4. Taylor expanded in d around inf 49.0%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   5. Step-by-step derivation

      1. *-un-lft-identity 49.0%

         \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot d \]

      2. associate-/r* 49.0%

         \[\leadsto \left(1 \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}}\right) \cdot d \]

   6. Applied egg-rr 49.0%

      \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right)} \cdot d \]
   7. Step-by-step derivation

      1. *-lft-identity 49.0%

         \[\leadsto \color{blue}{\sqrt{\frac{\frac{1}{\ell}}{h}}} \cdot d \]

      2. associate-/l/ 49.0%

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      3. unpow-1 49.0%

         \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \cdot d \]

      4. sqr-pow 49.0%

         \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}}} \cdot d \]

      5. rem-sqrt-square 49.0%

         \[\leadsto \color{blue}{\left|{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}\right|} \cdot d \]

      6. metadata-eval 49.0%

         \[\leadsto \left|{\left(h \cdot \ell\right)}^{\color{blue}{-0.5}}\right| \cdot d \]

      7. sqr-pow 48.8%

         \[\leadsto \left|\color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-0.5}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{-0.5}{2}\right)}}\right| \cdot d \]

      8. fabs-sqr 48.8%

         \[\leadsto \color{blue}{\left({\left(h \cdot \ell\right)}^{\left(\frac{-0.5}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{-0.5}{2}\right)}\right)} \cdot d \]

      9. sqr-pow 49.0%

         \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \cdot d \]

   8. Simplified 49.0%

      \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \cdot d \]

   if 1.05e-119 < d < 9.1999999999999997e147

   1. Initial program 80.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 80.9%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 80.9%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 80.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 80.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 80.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 80.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 80.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 80.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 80.9%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]
   4. Step-by-step derivation

      1. expm1-log1p-u 31.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\right)\right)} \]

      2. expm1-udef 24.4%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\right)} - 1} \]

   5. Applied egg-rr 16.9%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 24.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)\right)} \]

      2. expm1-log1p 65.8%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)} \]

      3. sub-neg 65.8%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \color{blue}{\left(1 + \left(-{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)} \]

      4. +-commutative 65.8%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \color{blue}{\left(\left(-{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) + 1\right)} \]

      5. associate-*l/ 71.1%

         \[\leadsto \sqrt{\color{blue}{\frac{d \cdot \frac{d}{\ell}}{h}}} \cdot \left(\left(-{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) + 1\right) \]

      6. associate-/l* 66.8%

         \[\leadsto \sqrt{\color{blue}{\frac{d}{\frac{h}{\frac{d}{\ell}}}}} \cdot \left(\left(-{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) + 1\right) \]

      7. +-commutative 66.8%

         \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \color{blue}{\left(1 + \left(-{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)} \]

      8. sub-neg 66.8%

         \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \color{blue}{\left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)} \]

      9. associate-/l* 66.8%

         \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \color{blue}{\frac{0.5}{\frac{\ell}{h}}}\right) \]

      10. associate-*r/ 66.8%

          \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - \color{blue}{\frac{{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot 0.5}{\frac{\ell}{h}}}\right) \]

      11. associate-*l/ 66.8%

          \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - \color{blue}{\frac{{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}}{\frac{\ell}{h}} \cdot 0.5}\right) \]

      12. associate-/l* 70.4%

          \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - \color{blue}{\frac{{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot h}{\ell}} \cdot 0.5\right) \]

      13. associate-*r/ 66.8%

          \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - \color{blue}{\left({\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\ell}\right)} \cdot 0.5\right) \]

   7. Simplified 66.7%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - 0.5 \cdot \left({\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]

   8. Taylor expanded in D around 0 49.6%

      \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - 0.5 \cdot \color{blue}{\left(0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}\right)}\right) \]
   9. Step-by-step derivation

      1. times-frac 51.5%

         \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \color{blue}{\left(\frac{{D}^{2}}{\ell} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right)}\right)\right) \]

      2. *-commutative 51.5%

         \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{{D}^{2}}{\ell} \cdot \frac{\color{blue}{h \cdot {M}^{2}}}{{d}^{2}}\right)\right)\right) \]

      3. unpow2 51.5%

         \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{\color{blue}{D \cdot D}}{\ell} \cdot \frac{h \cdot {M}^{2}}{{d}^{2}}\right)\right)\right) \]

      4. associate-/l* 58.2%

         \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\color{blue}{\frac{D}{\frac{\ell}{D}}} \cdot \frac{h \cdot {M}^{2}}{{d}^{2}}\right)\right)\right) \]

      5. *-commutative 58.2%

         \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{D}{\frac{\ell}{D}} \cdot \frac{\color{blue}{{M}^{2} \cdot h}}{{d}^{2}}\right)\right)\right) \]

      6. unpow2 58.2%

         \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{D}{\frac{\ell}{D}} \cdot \frac{\color{blue}{\left(M \cdot M\right)} \cdot h}{{d}^{2}}\right)\right)\right) \]

      7. associate-*r* 61.6%

         \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{D}{\frac{\ell}{D}} \cdot \frac{\color{blue}{M \cdot \left(M \cdot h\right)}}{{d}^{2}}\right)\right)\right) \]

      8. unpow2 61.6%

         \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{D}{\frac{\ell}{D}} \cdot \frac{M \cdot \left(M \cdot h\right)}{\color{blue}{d \cdot d}}\right)\right)\right) \]

      9. times-frac 66.7%

         \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{D}{\frac{\ell}{D}} \cdot \color{blue}{\left(\frac{M}{d} \cdot \frac{M \cdot h}{d}\right)}\right)\right)\right) \]

   10. Simplified 66.7%

       \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - 0.5 \cdot \color{blue}{\left(0.25 \cdot \left(\frac{D}{\frac{\ell}{D}} \cdot \left(\frac{M}{d} \cdot \frac{M \cdot h}{d}\right)\right)\right)}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 60.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.9 \cdot 10^{-273}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(-1 + 0.5 \cdot \left(0.25 \cdot \left(\frac{D}{\frac{\ell}{D}} \cdot \left(\frac{M}{d} \cdot \frac{h \cdot M}{d}\right)\right)\right)\right)\\ \mathbf{elif}\;d \leq 1.05 \cdot 10^{-119} \lor \neg \left(d \leq 9.2 \cdot 10^{+147}\right):\\ \;\;\;\;d \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{D}{\frac{\ell}{D}} \cdot \left(\frac{M}{d} \cdot \frac{h \cdot M}{d}\right)\right)\right)\right) \cdot \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}}\\ \end{array} \]

Alternative 15: 42.9% accurate, 3.0× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.1 \cdot 10^{-194}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \end{array} \end{array} \]

FPCore

NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (if (<= l 1.1e-194)
   (* (- d) (sqrt (/ 1.0 (* l h))))
   (* d (sqrt (/ (/ 1.0 l) h)))))

C

assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= 1.1e-194) {
		tmp = -d * sqrt((1.0 / (l * h)));
	} else {
		tmp = d * sqrt(((1.0 / l) / h));
	}
	return tmp;
}

Fortran

NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (l <= 1.1d-194) then
        tmp = -d * sqrt((1.0d0 / (l * h)))
    else
        tmp = d * sqrt(((1.0d0 / l) / h))
    end if
    code = tmp
end function

Java

assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= 1.1e-194) {
		tmp = -d * Math.sqrt((1.0 / (l * h)));
	} else {
		tmp = d * Math.sqrt(((1.0 / l) / h));
	}
	return tmp;
}

Python

[M, D] = sort([M, D])
def code(d, h, l, M, D):
	tmp = 0
	if l <= 1.1e-194:
		tmp = -d * math.sqrt((1.0 / (l * h)))
	else:
		tmp = d * math.sqrt(((1.0 / l) / h))
	return tmp

Julia

M, D = sort([M, D])
function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= 1.1e-194)
		tmp = Float64(Float64(-d) * sqrt(Float64(1.0 / Float64(l * h))));
	else
		tmp = Float64(d * sqrt(Float64(Float64(1.0 / l) / h)));
	end
	return tmp
end

MATLAB

M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (l <= 1.1e-194)
		tmp = -d * sqrt((1.0 / (l * h)));
	else
		tmp = d * sqrt(((1.0 / l) / h));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := If[LessEqual[l, 1.1e-194], N[((-d) * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d * N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.1000000000000001e-194

   1. Initial program 63.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 63.1%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 63.1%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 63.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 63.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 63.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 63.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 63.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 63.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 63.1%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]
   4. Step-by-step derivation

      1. expm1-log1p-u 34.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\right)\right)} \]

      2. expm1-udef 25.2%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\right)} - 1} \]

   5. Applied egg-rr 19.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 27.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)\right)} \]

      2. expm1-log1p 54.9%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)} \]

      3. sub-neg 54.9%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \color{blue}{\left(1 + \left(-{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)} \]

      4. +-commutative 54.9%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \color{blue}{\left(\left(-{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) + 1\right)} \]

      5. associate-*l/ 53.0%

         \[\leadsto \sqrt{\color{blue}{\frac{d \cdot \frac{d}{\ell}}{h}}} \cdot \left(\left(-{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) + 1\right) \]

      6. associate-/l* 53.7%

         \[\leadsto \sqrt{\color{blue}{\frac{d}{\frac{h}{\frac{d}{\ell}}}}} \cdot \left(\left(-{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) + 1\right) \]

      7. +-commutative 53.7%

         \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \color{blue}{\left(1 + \left(-{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)} \]

      8. sub-neg 53.7%

         \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \color{blue}{\left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)} \]

      9. associate-/l* 53.6%

         \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \color{blue}{\frac{0.5}{\frac{\ell}{h}}}\right) \]

      10. associate-*r/ 53.7%

          \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - \color{blue}{\frac{{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot 0.5}{\frac{\ell}{h}}}\right) \]

      11. associate-*l/ 53.7%

          \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - \color{blue}{\frac{{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}}{\frac{\ell}{h}} \cdot 0.5}\right) \]

      12. associate-/l* 58.5%

          \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - \color{blue}{\frac{{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot h}{\ell}} \cdot 0.5\right) \]

      13. associate-*r/ 53.7%

          \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - \color{blue}{\left({\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\ell}\right)} \cdot 0.5\right) \]

   7. Simplified 53.0%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - 0.5 \cdot \left({\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]

   8. Taylor expanded in d around -inf 55.3%

      \[\leadsto \color{blue}{\left(-1 \cdot \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)\right)} \cdot \left(1 - 0.5 \cdot \left({\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
   9. Step-by-step derivation

      1. associate-*r* 55.3%

         \[\leadsto \color{blue}{\left(\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot \left(1 - 0.5 \cdot \left({\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      2. mul-1-neg 55.3%

         \[\leadsto \left(\color{blue}{\left(-d\right)} \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      3. *-commutative 55.3%

         \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   10. Simplified 55.3%

       \[\leadsto \color{blue}{\left(\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot \left(1 - 0.5 \cdot \left({\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   11. Taylor expanded in d around inf 42.2%

       \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \]
   12. Step-by-step derivation

       1. associate-*r* 42.2%

          \[\leadsto \color{blue}{\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

       2. neg-mul-1 42.2%

          \[\leadsto \color{blue}{\left(-d\right)} \cdot \sqrt{\frac{1}{\ell \cdot h}} \]

       3. *-commutative 42.2%

          \[\leadsto \left(-d\right) \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \]

   13. Simplified 42.2%

       \[\leadsto \color{blue}{\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

   if 1.1000000000000001e-194 < l

   1. Initial program 62.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 62.2%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 62.2%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 62.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 62.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 62.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 62.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 62.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 62.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 62.2%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]
   4. Step-by-step derivation

      1. expm1-log1p-u 35.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\right)\right)} \]

      2. expm1-udef 29.7%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\right)} - 1} \]

   5. Applied egg-rr 22.6%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 26.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)\right)} \]

      2. expm1-log1p 46.0%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)} \]

      3. sub-neg 46.0%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \color{blue}{\left(1 + \left(-{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)} \]

      4. +-commutative 46.0%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \color{blue}{\left(\left(-{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) + 1\right)} \]

      5. associate-*l/ 48.8%

         \[\leadsto \sqrt{\color{blue}{\frac{d \cdot \frac{d}{\ell}}{h}}} \cdot \left(\left(-{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) + 1\right) \]

      6. associate-/l* 48.3%

         \[\leadsto \sqrt{\color{blue}{\frac{d}{\frac{h}{\frac{d}{\ell}}}}} \cdot \left(\left(-{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) + 1\right) \]

      7. +-commutative 48.3%

         \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \color{blue}{\left(1 + \left(-{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)} \]

      8. sub-neg 48.3%

         \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \color{blue}{\left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)} \]

      9. associate-/l* 48.3%

         \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \color{blue}{\frac{0.5}{\frac{\ell}{h}}}\right) \]

      10. associate-*r/ 48.3%

          \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - \color{blue}{\frac{{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot 0.5}{\frac{\ell}{h}}}\right) \]

      11. associate-*l/ 48.3%

          \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - \color{blue}{\frac{{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}}{\frac{\ell}{h}} \cdot 0.5}\right) \]

      12. associate-/l* 50.3%

          \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - \color{blue}{\frac{{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot h}{\ell}} \cdot 0.5\right) \]

      13. associate-*r/ 48.3%

          \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - \color{blue}{\left({\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\ell}\right)} \cdot 0.5\right) \]

   7. Simplified 48.3%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - 0.5 \cdot \left({\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   8. Step-by-step derivation

      1. associate-*r/ 5.2%

         \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{{\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2} \cdot h}{\ell}}\right) \]

      2. div-inv 5.2%

         \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(D \cdot \color{blue}{\left(\frac{M}{d} \cdot \frac{1}{2}\right)}\right)}^{2} \cdot h}{\ell}\right) \]

      3. metadata-eval 5.2%

         \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(D \cdot \left(\frac{M}{d} \cdot \color{blue}{0.5}\right)\right)}^{2} \cdot h}{\ell}\right) \]

   9. Applied egg-rr 50.4%

      \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{{\left(D \cdot \left(\frac{M}{d} \cdot 0.5\right)\right)}^{2} \cdot h}{\ell}}\right) \]

   10. Taylor expanded in d around inf 45.4%

       \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   11. Step-by-step derivation

       1. *-commutative 45.4%

          \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

       2. associate-/r* 45.7%

          \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \]

   12. Simplified 45.7%

       \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 43.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.1 \cdot 10^{-194}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \end{array} \]

Alternative 16: 43.0% accurate, 3.0× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{if}\;\ell \leq 1.06 \cdot 10^{-194}:\\ \;\;\;\;\left(-d\right) \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;d \cdot t_0\\ \end{array} \end{array} \]

FPCore

NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ (/ 1.0 l) h))))
   (if (<= l 1.06e-194) (* (- d) t_0) (* d t_0))))

C

assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt(((1.0 / l) / h));
	double tmp;
	if (l <= 1.06e-194) {
		tmp = -d * t_0;
	} else {
		tmp = d * t_0;
	}
	return tmp;
}

Fortran

NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(((1.0d0 / l) / h))
    if (l <= 1.06d-194) then
        tmp = -d * t_0
    else
        tmp = d * t_0
    end if
    code = tmp
end function

Java

assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt(((1.0 / l) / h));
	double tmp;
	if (l <= 1.06e-194) {
		tmp = -d * t_0;
	} else {
		tmp = d * t_0;
	}
	return tmp;
}

Python

[M, D] = sort([M, D])
def code(d, h, l, M, D):
	t_0 = math.sqrt(((1.0 / l) / h))
	tmp = 0
	if l <= 1.06e-194:
		tmp = -d * t_0
	else:
		tmp = d * t_0
	return tmp

Julia

M, D = sort([M, D])
function code(d, h, l, M, D)
	t_0 = sqrt(Float64(Float64(1.0 / l) / h))
	tmp = 0.0
	if (l <= 1.06e-194)
		tmp = Float64(Float64(-d) * t_0);
	else
		tmp = Float64(d * t_0);
	end
	return tmp
end

MATLAB

M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt(((1.0 / l) / h));
	tmp = 0.0;
	if (l <= 1.06e-194)
		tmp = -d * t_0;
	else
		tmp = d * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, 1.06e-194], N[((-d) * t$95$0), $MachinePrecision], N[(d * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < 1.06000000000000002e-194

   1. Initial program 63.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 63.1%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 63.1%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 63.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 63.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 63.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 63.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 63.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 63.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 63.1%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]
   4. Step-by-step derivation

      1. expm1-log1p-u 34.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\right)\right)} \]

      2. expm1-udef 25.2%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\right)} - 1} \]

   5. Applied egg-rr 19.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 27.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)\right)} \]

      2. expm1-log1p 54.9%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)} \]

      3. sub-neg 54.9%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \color{blue}{\left(1 + \left(-{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)} \]

      4. +-commutative 54.9%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \color{blue}{\left(\left(-{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) + 1\right)} \]

      5. associate-*l/ 53.0%

         \[\leadsto \sqrt{\color{blue}{\frac{d \cdot \frac{d}{\ell}}{h}}} \cdot \left(\left(-{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) + 1\right) \]

      6. associate-/l* 53.7%

         \[\leadsto \sqrt{\color{blue}{\frac{d}{\frac{h}{\frac{d}{\ell}}}}} \cdot \left(\left(-{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) + 1\right) \]

      7. +-commutative 53.7%

         \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \color{blue}{\left(1 + \left(-{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)} \]

      8. sub-neg 53.7%

         \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \color{blue}{\left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)} \]

      9. associate-/l* 53.6%

         \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \color{blue}{\frac{0.5}{\frac{\ell}{h}}}\right) \]

      10. associate-*r/ 53.7%

          \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - \color{blue}{\frac{{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot 0.5}{\frac{\ell}{h}}}\right) \]

      11. associate-*l/ 53.7%

          \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - \color{blue}{\frac{{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}}{\frac{\ell}{h}} \cdot 0.5}\right) \]

      12. associate-/l* 58.5%

          \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - \color{blue}{\frac{{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot h}{\ell}} \cdot 0.5\right) \]

      13. associate-*r/ 53.7%

          \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - \color{blue}{\left({\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\ell}\right)} \cdot 0.5\right) \]

   7. Simplified 53.0%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - 0.5 \cdot \left({\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   8. Step-by-step derivation

      1. associate-*r/ 62.6%

         \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{{\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2} \cdot h}{\ell}}\right) \]

      2. div-inv 62.6%

         \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(D \cdot \color{blue}{\left(\frac{M}{d} \cdot \frac{1}{2}\right)}\right)}^{2} \cdot h}{\ell}\right) \]

      3. metadata-eval 62.6%

         \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(D \cdot \left(\frac{M}{d} \cdot \color{blue}{0.5}\right)\right)}^{2} \cdot h}{\ell}\right) \]

   9. Applied egg-rr 57.8%

      \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{{\left(D \cdot \left(\frac{M}{d} \cdot 0.5\right)\right)}^{2} \cdot h}{\ell}}\right) \]

   10. Taylor expanded in d around -inf 42.2%

       \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \]
   11. Step-by-step derivation

       1. associate-*r* 42.2%

          \[\leadsto \color{blue}{\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

       2. neg-mul-1 42.2%

          \[\leadsto \color{blue}{\left(-d\right)} \cdot \sqrt{\frac{1}{\ell \cdot h}} \]

       3. associate-/r* 42.2%

          \[\leadsto \left(-d\right) \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \]

   12. Simplified 42.2%

       \[\leadsto \color{blue}{\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}} \]

   if 1.06000000000000002e-194 < l

   1. Initial program 62.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 62.2%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 62.2%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 62.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 62.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 62.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 62.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 62.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 62.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 62.2%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]
   4. Step-by-step derivation

      1. expm1-log1p-u 35.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\right)\right)} \]

      2. expm1-udef 29.7%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\right)} - 1} \]

   5. Applied egg-rr 22.6%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 26.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)\right)} \]

      2. expm1-log1p 46.0%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)} \]

      3. sub-neg 46.0%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \color{blue}{\left(1 + \left(-{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)} \]

      4. +-commutative 46.0%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \color{blue}{\left(\left(-{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) + 1\right)} \]

      5. associate-*l/ 48.8%

         \[\leadsto \sqrt{\color{blue}{\frac{d \cdot \frac{d}{\ell}}{h}}} \cdot \left(\left(-{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) + 1\right) \]

      6. associate-/l* 48.3%

         \[\leadsto \sqrt{\color{blue}{\frac{d}{\frac{h}{\frac{d}{\ell}}}}} \cdot \left(\left(-{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) + 1\right) \]

      7. +-commutative 48.3%

         \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \color{blue}{\left(1 + \left(-{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)} \]

      8. sub-neg 48.3%

         \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \color{blue}{\left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)} \]

      9. associate-/l* 48.3%

         \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \color{blue}{\frac{0.5}{\frac{\ell}{h}}}\right) \]

      10. associate-*r/ 48.3%

          \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - \color{blue}{\frac{{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot 0.5}{\frac{\ell}{h}}}\right) \]

      11. associate-*l/ 48.3%

          \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - \color{blue}{\frac{{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}}{\frac{\ell}{h}} \cdot 0.5}\right) \]

      12. associate-/l* 50.3%

          \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - \color{blue}{\frac{{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot h}{\ell}} \cdot 0.5\right) \]

      13. associate-*r/ 48.3%

          \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - \color{blue}{\left({\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\ell}\right)} \cdot 0.5\right) \]

   7. Simplified 48.3%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - 0.5 \cdot \left({\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   8. Step-by-step derivation

      1. associate-*r/ 5.2%

         \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{{\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2} \cdot h}{\ell}}\right) \]

      2. div-inv 5.2%

         \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(D \cdot \color{blue}{\left(\frac{M}{d} \cdot \frac{1}{2}\right)}\right)}^{2} \cdot h}{\ell}\right) \]

      3. metadata-eval 5.2%

         \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(D \cdot \left(\frac{M}{d} \cdot \color{blue}{0.5}\right)\right)}^{2} \cdot h}{\ell}\right) \]

   9. Applied egg-rr 50.4%

      \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{{\left(D \cdot \left(\frac{M}{d} \cdot 0.5\right)\right)}^{2} \cdot h}{\ell}}\right) \]

   10. Taylor expanded in d around inf 45.4%

       \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   11. Step-by-step derivation

       1. *-commutative 45.4%

          \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

       2. associate-/r* 45.7%

          \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \]

   12. Simplified 45.7%

       \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 43.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.06 \cdot 10^{-194}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \end{array} \]

Alternative 17: 26.8% accurate, 3.1× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}} \end{array} \]

FPCore

NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D) :precision binary64 (* d (sqrt (/ (/ 1.0 h) l))))

C

assert(M < D);
double code(double d, double h, double l, double M, double D) {
	return d * sqrt(((1.0 / h) / l));
}

Fortran

NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    code = d * sqrt(((1.0d0 / h) / l))
end function

Java

assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	return d * Math.sqrt(((1.0 / h) / l));
}

Python

[M, D] = sort([M, D])
def code(d, h, l, M, D):
	return d * math.sqrt(((1.0 / h) / l))

Julia

M, D = sort([M, D])
function code(d, h, l, M, D)
	return Float64(d * sqrt(Float64(Float64(1.0 / h) / l)))
end

MATLAB

M, D = num2cell(sort([M, D])){:}
function tmp = code(d, h, l, M, D)
	tmp = d * sqrt(((1.0 / h) / l));
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := N[(d * N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.7%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
2. Step-by-step derivation

   1. metadata-eval 62.7%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. unpow1/2 62.7%

      \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. metadata-eval 62.7%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. unpow1/2 62.7%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   5. *-commutative 62.7%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

   6. associate-*l* 62.7%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

   7. times-frac 62.7%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

   8. metadata-eval 62.7%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

3. Simplified 62.7%

   \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]
4. Step-by-step derivation

   1. expm1-log1p-u 34.9%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\right)\right)} \]

   2. expm1-udef 27.1%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\right)} - 1} \]

5. Applied egg-rr 20.9%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)} - 1} \]
6. Step-by-step derivation

   1. expm1-def 27.2%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)\right)} \]

   2. expm1-log1p 51.2%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)} \]

   3. sub-neg 51.2%

      \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \color{blue}{\left(1 + \left(-{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)} \]

   4. +-commutative 51.2%

      \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \color{blue}{\left(\left(-{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) + 1\right)} \]

   5. associate-*l/ 51.2%

      \[\leadsto \sqrt{\color{blue}{\frac{d \cdot \frac{d}{\ell}}{h}}} \cdot \left(\left(-{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) + 1\right) \]

   6. associate-/l* 51.4%

      \[\leadsto \sqrt{\color{blue}{\frac{d}{\frac{h}{\frac{d}{\ell}}}}} \cdot \left(\left(-{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) + 1\right) \]

   7. +-commutative 51.4%

      \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \color{blue}{\left(1 + \left(-{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)} \]

   8. sub-neg 51.4%

      \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \color{blue}{\left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)} \]

   9. associate-/l* 51.4%

      \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \color{blue}{\frac{0.5}{\frac{\ell}{h}}}\right) \]

   10. associate-*r/ 51.4%

       \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - \color{blue}{\frac{{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot 0.5}{\frac{\ell}{h}}}\right) \]

   11. associate-*l/ 51.4%

       \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - \color{blue}{\frac{{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}}{\frac{\ell}{h}} \cdot 0.5}\right) \]

   12. associate-/l* 55.1%

       \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - \color{blue}{\frac{{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot h}{\ell}} \cdot 0.5\right) \]

   13. associate-*r/ 51.4%

       \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - \color{blue}{\left({\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\ell}\right)} \cdot 0.5\right) \]

7. Simplified 51.0%

   \[\leadsto \color{blue}{\sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - 0.5 \cdot \left({\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
8. Step-by-step derivation

   1. associate-*r/ 38.6%

      \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{{\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2} \cdot h}{\ell}}\right) \]

   2. div-inv 38.6%

      \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(D \cdot \color{blue}{\left(\frac{M}{d} \cdot \frac{1}{2}\right)}\right)}^{2} \cdot h}{\ell}\right) \]

   3. metadata-eval 38.6%

      \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(D \cdot \left(\frac{M}{d} \cdot \color{blue}{0.5}\right)\right)}^{2} \cdot h}{\ell}\right) \]

9. Applied egg-rr 54.7%

   \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{{\left(D \cdot \left(\frac{M}{d} \cdot 0.5\right)\right)}^{2} \cdot h}{\ell}}\right) \]

10. Taylor expanded in d around inf 25.7%

    \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
11. Step-by-step derivation

    1. *-commutative 25.7%

       \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

    2. *-commutative 25.7%

       \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \]

    3. associate-/r* 25.8%

       \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \]

12. Simplified 25.8%

    \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \]

13. Final simplification 25.8%

    \[\leadsto d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}} \]

Alternative 18: 26.8% accurate, 3.1× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}} \end{array} \]

FPCore

NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D) :precision binary64 (* d (sqrt (/ (/ 1.0 l) h))))

C

assert(M < D);
double code(double d, double h, double l, double M, double D) {
	return d * sqrt(((1.0 / l) / h));
}

Fortran

NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    code = d * sqrt(((1.0d0 / l) / h))
end function

Java

assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	return d * Math.sqrt(((1.0 / l) / h));
}

Python

[M, D] = sort([M, D])
def code(d, h, l, M, D):
	return d * math.sqrt(((1.0 / l) / h))

Julia

M, D = sort([M, D])
function code(d, h, l, M, D)
	return Float64(d * sqrt(Float64(Float64(1.0 / l) / h)))
end

MATLAB

M, D = num2cell(sort([M, D])){:}
function tmp = code(d, h, l, M, D)
	tmp = d * sqrt(((1.0 / l) / h));
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := N[(d * N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.7%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
2. Step-by-step derivation

   1. metadata-eval 62.7%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. unpow1/2 62.7%

      \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. metadata-eval 62.7%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. unpow1/2 62.7%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   5. *-commutative 62.7%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

   6. associate-*l* 62.7%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

   7. times-frac 62.7%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

   8. metadata-eval 62.7%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

3. Simplified 62.7%

   \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]
4. Step-by-step derivation

   1. expm1-log1p-u 34.9%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\right)\right)} \]

   2. expm1-udef 27.1%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\right)} - 1} \]

5. Applied egg-rr 20.9%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)} - 1} \]
6. Step-by-step derivation

   1. expm1-def 27.2%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)\right)} \]

   2. expm1-log1p 51.2%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)} \]

   3. sub-neg 51.2%

      \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \color{blue}{\left(1 + \left(-{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)} \]

   4. +-commutative 51.2%

      \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \color{blue}{\left(\left(-{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) + 1\right)} \]

   5. associate-*l/ 51.2%

      \[\leadsto \sqrt{\color{blue}{\frac{d \cdot \frac{d}{\ell}}{h}}} \cdot \left(\left(-{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) + 1\right) \]

   6. associate-/l* 51.4%

      \[\leadsto \sqrt{\color{blue}{\frac{d}{\frac{h}{\frac{d}{\ell}}}}} \cdot \left(\left(-{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right) + 1\right) \]

   7. +-commutative 51.4%

      \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \color{blue}{\left(1 + \left(-{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)\right)} \]

   8. sub-neg 51.4%

      \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \color{blue}{\left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{0.5 \cdot h}{\ell}\right)} \]

   9. associate-/l* 51.4%

      \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \color{blue}{\frac{0.5}{\frac{\ell}{h}}}\right) \]

   10. associate-*r/ 51.4%

       \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - \color{blue}{\frac{{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot 0.5}{\frac{\ell}{h}}}\right) \]

   11. associate-*l/ 51.4%

       \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - \color{blue}{\frac{{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}}{\frac{\ell}{h}} \cdot 0.5}\right) \]

   12. associate-/l* 55.1%

       \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - \color{blue}{\frac{{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot h}{\ell}} \cdot 0.5\right) \]

   13. associate-*r/ 51.4%

       \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - \color{blue}{\left({\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\ell}\right)} \cdot 0.5\right) \]

7. Simplified 51.0%

   \[\leadsto \color{blue}{\sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - 0.5 \cdot \left({\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
8. Step-by-step derivation

   1. associate-*r/ 38.6%

      \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{{\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2} \cdot h}{\ell}}\right) \]

   2. div-inv 38.6%

      \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(D \cdot \color{blue}{\left(\frac{M}{d} \cdot \frac{1}{2}\right)}\right)}^{2} \cdot h}{\ell}\right) \]

   3. metadata-eval 38.6%

      \[\leadsto \left(\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(D \cdot \left(\frac{M}{d} \cdot \color{blue}{0.5}\right)\right)}^{2} \cdot h}{\ell}\right) \]

9. Applied egg-rr 54.7%

   \[\leadsto \sqrt{\frac{d}{\frac{h}{\frac{d}{\ell}}}} \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{{\left(D \cdot \left(\frac{M}{d} \cdot 0.5\right)\right)}^{2} \cdot h}{\ell}}\right) \]

10. Taylor expanded in d around inf 25.7%

    \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
11. Step-by-step derivation

    1. *-commutative 25.7%

       \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

    2. associate-/r* 25.8%

       \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \]

12. Simplified 25.8%

    \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}} \]

13. Final simplification 25.8%

    \[\leadsto d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}} \]

Alternative 19: 26.6% accurate, 3.2× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ \frac{d}{\sqrt{\ell \cdot h}} \end{array} \]

FPCore

NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D) :precision binary64 (/ d (sqrt (* l h))))

C

assert(M < D);
double code(double d, double h, double l, double M, double D) {
	return d / sqrt((l * h));
}

Fortran

NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    code = d / sqrt((l * h))
end function

Java

assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	return d / Math.sqrt((l * h));
}

Python

[M, D] = sort([M, D])
def code(d, h, l, M, D):
	return d / math.sqrt((l * h))

Julia

M, D = sort([M, D])
function code(d, h, l, M, D)
	return Float64(d / sqrt(Float64(l * h)))
end

MATLAB

M, D = num2cell(sort([M, D])){:}
function tmp = code(d, h, l, M, D)
	tmp = d / sqrt((l * h));
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.7%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
2. Step-by-step derivation

   1. metadata-eval 62.7%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. unpow1/2 62.7%

      \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. metadata-eval 62.7%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. unpow1/2 62.7%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   5. *-commutative 62.7%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

   6. associate-*l* 62.7%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

   7. times-frac 62.7%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

   8. metadata-eval 62.7%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

3. Simplified 62.7%

   \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]

4. Taylor expanded in d around inf 25.7%

   \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
5. Step-by-step derivation

   1. sqrt-div 25.7%

      \[\leadsto \color{blue}{\frac{\sqrt{1}}{\sqrt{\ell \cdot h}}} \cdot d \]

   2. metadata-eval 25.7%

      \[\leadsto \frac{\color{blue}{1}}{\sqrt{\ell \cdot h}} \cdot d \]

   3. *-commutative 25.7%

      \[\leadsto \frac{1}{\sqrt{\color{blue}{h \cdot \ell}}} \cdot d \]

6. Applied egg-rr 25.7%

   \[\leadsto \color{blue}{\frac{1}{\sqrt{h \cdot \ell}}} \cdot d \]
7. Step-by-step derivation

   1. associate-*l/ 25.7%

      \[\leadsto \color{blue}{\frac{1 \cdot d}{\sqrt{h \cdot \ell}}} \]

   2. *-un-lft-identity 25.7%

      \[\leadsto \frac{\color{blue}{d}}{\sqrt{h \cdot \ell}} \]

8. Applied egg-rr 25.7%

   \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]

9. Final simplification 25.7%

   \[\leadsto \frac{d}{\sqrt{\ell \cdot h}} \]

Reproduce

herbie shell --seed 2023178 
(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)))))