Henrywood and Agarwal, Equation (12)

Percentage Accurate: 66.5% → 81.3%

Time: 28.2s

Alternatives: 17

Speedup: 1.4×

Specification

Math

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

FPCore

(FPCore (d h l M D)
 :precision binary64
 (*
  (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
  (- 1.0 (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l)))))

C

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

Fortran

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

Java

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

Python

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

Julia

function code(d, h, l, M, D)
	return Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
end

MATLAB

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

Wolfram

code[d_, h_, l_, M_, D_] := N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 66.5% 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: 81.3% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if l < -1.999999999999994e-310

   1. Initial program 64.3%

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

   3. Simplified 62.0%

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

      1. associate-*r/ 62.2%

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

      2. frac-times 64.4%

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

      3. associate-/l* 62.2%

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

      4. *-commutative 62.2%

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

   6. Applied egg-rr 62.2%

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

      1. *-commutative 62.2%

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

      2. associate-/l* 63.0%

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

      3. associate-*r/ 65.3%

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

      4. *-commutative 65.3%

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

      5. times-frac 63.0%

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

   8. Simplified 63.0%

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

      1. add-sqr-sqrt 22.1%

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

      2. sqrt-div 0.0%

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

      3. sqrt-pow1 0.0%

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

      4. metadata-eval 0.0%

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

      5. pow1 0.0%

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

      6. *-commutative 0.0%

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

      7. frac-times 0.0%

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

      8. *-commutative 0.0%

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

      9. associate-*r/ 0.0%

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

      10. sqrt-div 0.0%

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

      11. sqrt-pow1 0.0%

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

      12. metadata-eval 0.0%

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

      13. pow1 0.0%

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

      14. *-commutative 0.0%

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

      15. frac-times 0.0%

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

      16. *-commutative 0.0%

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

      17. associate-*r/ 0.0%

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

   10. Applied egg-rr 65.4%

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

       1. frac-2neg 65.4%

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

       2. sqrt-div 76.9%

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

   12. Applied egg-rr 76.9%

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

   if -1.999999999999994e-310 < l

   1. Initial program 63.1%

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

   3. Simplified 63.1%

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

      1. sqrt-div 65.7%

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

   6. Applied egg-rr 65.7%

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

      1. sqrt-div 77.6%

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

   8. Applied egg-rr 77.6%

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

4. Final simplification 77.3%

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

Alternative 2: 80.5% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if l < -1.999999999999994e-310

   1. Initial program 64.3%

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

   3. Simplified 62.0%

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

      1. associate-*r/ 62.2%

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

      2. frac-times 64.4%

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

      3. associate-/l* 62.2%

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

      4. *-commutative 62.2%

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

   6. Applied egg-rr 62.2%

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

      1. *-commutative 62.2%

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

      2. associate-/l* 63.0%

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

      3. associate-*r/ 65.3%

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

      4. *-commutative 65.3%

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

      5. times-frac 63.0%

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

   8. Simplified 63.0%

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

      1. add-sqr-sqrt 22.1%

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

      2. sqrt-div 0.0%

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

      3. sqrt-pow1 0.0%

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

      4. metadata-eval 0.0%

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

      5. pow1 0.0%

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

      6. *-commutative 0.0%

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

      7. frac-times 0.0%

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

      8. *-commutative 0.0%

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

      9. associate-*r/ 0.0%

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

      10. sqrt-div 0.0%

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

      11. sqrt-pow1 0.0%

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

      12. metadata-eval 0.0%

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

      13. pow1 0.0%

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

      14. *-commutative 0.0%

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

      15. frac-times 0.0%

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

      16. *-commutative 0.0%

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

      17. associate-*r/ 0.0%

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

   10. Applied egg-rr 65.4%

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

       1. frac-2neg 65.4%

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

       2. sqrt-div 76.9%

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

   12. Applied egg-rr 76.9%

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

   if -1.999999999999994e-310 < l

   1. Initial program 63.1%

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

   3. Simplified 63.8%

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

   6. Applied egg-rr 75.2%

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

      1. unpow1 75.2%

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

      2. associate-*l/ 78.4%

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

      3. associate-/l* 77.6%

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

      4. +-commutative 77.6%

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

      5. associate-*r* 77.6%

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

      6. fma-define 77.6%

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

      7. *-commutative 77.6%

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

      8. associate-*r/ 77.7%

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

      9. *-commutative 77.7%

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

      10. times-frac 77.6%

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

   8. Simplified 77.6%

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

4. Final simplification 77.3%

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

Alternative 3: 79.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \frac{D\_m}{d} \cdot \left(0.5 \cdot M\_m\right)\\ t_1 := 1 - 0.5 \cdot \left(h \cdot \left(t\_0 \cdot \frac{t\_0}{\ell}\right)\right)\\ \mathbf{if}\;h \leq -5 \cdot 10^{-311}:\\ \;\;\;\;t\_1 \cdot \left(\sqrt{\frac{d}{h}} \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\right)\\ \mathbf{elif}\;h \leq 2.6 \cdot 10^{+74}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 + \left(\frac{h}{\ell} \cdot -0.5\right) \cdot {\left(D\_m \cdot \frac{\frac{M\_m}{d}}{2}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \frac{\sqrt{d}}{\sqrt{h}}\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (* (/ D_m d) (* 0.5 M_m)))
        (t_1 (- 1.0 (* 0.5 (* h (* t_0 (/ t_0 l)))))))
   (if (<= h -5e-311)
     (* t_1 (* (sqrt (/ d h)) (/ (sqrt (- d)) (sqrt (- l)))))
     (if (<= h 2.6e+74)
       (*
        (/ d (* (sqrt l) (sqrt h)))
        (+ 1.0 (* (* (/ h l) -0.5) (pow (* D_m (/ (/ M_m d) 2.0)) 2.0))))
       (* t_1 (* (sqrt (/ d l)) (/ (sqrt d) (sqrt h))))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = (D_m / d) * (0.5 * M_m);
	double t_1 = 1.0 - (0.5 * (h * (t_0 * (t_0 / l))));
	double tmp;
	if (h <= -5e-311) {
		tmp = t_1 * (sqrt((d / h)) * (sqrt(-d) / sqrt(-l)));
	} else if (h <= 2.6e+74) {
		tmp = (d / (sqrt(l) * sqrt(h))) * (1.0 + (((h / l) * -0.5) * pow((D_m * ((M_m / d) / 2.0)), 2.0)));
	} else {
		tmp = t_1 * (sqrt((d / l)) * (sqrt(d) / sqrt(h)));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (d_m / d) * (0.5d0 * m_m)
    t_1 = 1.0d0 - (0.5d0 * (h * (t_0 * (t_0 / l))))
    if (h <= (-5d-311)) then
        tmp = t_1 * (sqrt((d / h)) * (sqrt(-d) / sqrt(-l)))
    else if (h <= 2.6d+74) then
        tmp = (d / (sqrt(l) * sqrt(h))) * (1.0d0 + (((h / l) * (-0.5d0)) * ((d_m * ((m_m / d) / 2.0d0)) ** 2.0d0)))
    else
        tmp = t_1 * (sqrt((d / l)) * (sqrt(d) / sqrt(h)))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = (D_m / d) * (0.5 * M_m);
	double t_1 = 1.0 - (0.5 * (h * (t_0 * (t_0 / l))));
	double tmp;
	if (h <= -5e-311) {
		tmp = t_1 * (Math.sqrt((d / h)) * (Math.sqrt(-d) / Math.sqrt(-l)));
	} else if (h <= 2.6e+74) {
		tmp = (d / (Math.sqrt(l) * Math.sqrt(h))) * (1.0 + (((h / l) * -0.5) * Math.pow((D_m * ((M_m / d) / 2.0)), 2.0)));
	} else {
		tmp = t_1 * (Math.sqrt((d / l)) * (Math.sqrt(d) / Math.sqrt(h)));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = (D_m / d) * (0.5 * M_m)
	t_1 = 1.0 - (0.5 * (h * (t_0 * (t_0 / l))))
	tmp = 0
	if h <= -5e-311:
		tmp = t_1 * (math.sqrt((d / h)) * (math.sqrt(-d) / math.sqrt(-l)))
	elif h <= 2.6e+74:
		tmp = (d / (math.sqrt(l) * math.sqrt(h))) * (1.0 + (((h / l) * -0.5) * math.pow((D_m * ((M_m / d) / 2.0)), 2.0)))
	else:
		tmp = t_1 * (math.sqrt((d / l)) * (math.sqrt(d) / math.sqrt(h)))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(Float64(D_m / d) * Float64(0.5 * M_m))
	t_1 = Float64(1.0 - Float64(0.5 * Float64(h * Float64(t_0 * Float64(t_0 / l)))))
	tmp = 0.0
	if (h <= -5e-311)
		tmp = Float64(t_1 * Float64(sqrt(Float64(d / h)) * Float64(sqrt(Float64(-d)) / sqrt(Float64(-l)))));
	elseif (h <= 2.6e+74)
		tmp = Float64(Float64(d / Float64(sqrt(l) * sqrt(h))) * Float64(1.0 + Float64(Float64(Float64(h / l) * -0.5) * (Float64(D_m * Float64(Float64(M_m / d) / 2.0)) ^ 2.0))));
	else
		tmp = Float64(t_1 * Float64(sqrt(Float64(d / l)) * Float64(sqrt(d) / sqrt(h))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = (D_m / d) * (0.5 * M_m);
	t_1 = 1.0 - (0.5 * (h * (t_0 * (t_0 / l))));
	tmp = 0.0;
	if (h <= -5e-311)
		tmp = t_1 * (sqrt((d / h)) * (sqrt(-d) / sqrt(-l)));
	elseif (h <= 2.6e+74)
		tmp = (d / (sqrt(l) * sqrt(h))) * (1.0 + (((h / l) * -0.5) * ((D_m * ((M_m / d) / 2.0)) ^ 2.0)));
	else
		tmp = t_1 * (sqrt((d / l)) * (sqrt(d) / sqrt(h)));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(N[(D$95$m / d), $MachinePrecision] * N[(0.5 * M$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - N[(0.5 * N[(h * N[(t$95$0 * N[(t$95$0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[h, -5e-311], N[(t$95$1 * N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, 2.6e+74], N[(N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision] * N[Power[N[(D$95$m * N[(N[(M$95$m / d), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if h < -5.00000000000023e-311

   1. Initial program 64.3%

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

   3. Simplified 62.0%

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

      1. associate-*r/ 62.2%

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

      2. frac-times 64.4%

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

      3. associate-/l* 62.2%

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

      4. *-commutative 62.2%

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

   6. Applied egg-rr 62.2%

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

      1. *-commutative 62.2%

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

      2. associate-/l* 63.0%

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

      3. associate-*r/ 65.3%

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

      4. *-commutative 65.3%

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

      5. times-frac 63.0%

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

   8. Simplified 63.0%

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

      1. add-sqr-sqrt 22.1%

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

      2. sqrt-div 0.0%

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

      3. sqrt-pow1 0.0%

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

      4. metadata-eval 0.0%

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

      5. pow1 0.0%

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

      6. *-commutative 0.0%

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

      7. frac-times 0.0%

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

      8. *-commutative 0.0%

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

      9. associate-*r/ 0.0%

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

      10. sqrt-div 0.0%

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

      11. sqrt-pow1 0.0%

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

      12. metadata-eval 0.0%

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

      13. pow1 0.0%

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

      14. *-commutative 0.0%

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

      15. frac-times 0.0%

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

      16. *-commutative 0.0%

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

      17. associate-*r/ 0.0%

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

   10. Applied egg-rr 65.4%

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

       1. frac-2neg 65.4%

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

       2. sqrt-div 72.8%

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

   12. Applied egg-rr 72.8%

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

   if -5.00000000000023e-311 < h < 2.6000000000000001e74

   1. Initial program 66.7%

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

   3. Simplified 67.9%

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

   6. Applied egg-rr 83.7%

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

      1. unpow1 83.7%

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

      2. associate-*r* 83.7%

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

      3. *-commutative 83.7%

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

      4. associate-*r/ 82.5%

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

      5. *-commutative 82.5%

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

      6. associate-*r/ 81.3%

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

      7. associate-*r* 81.3%

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

      8. associate-*r* 81.3%

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

      9. associate-/r* 81.3%

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

   8. Simplified 81.3%

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

   if 2.6000000000000001e74 < h

   1. Initial program 57.2%

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

   3. Simplified 57.1%

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

      1. associate-*r/ 63.4%

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

      2. frac-times 63.4%

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

      3. associate-/l* 63.4%

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

      4. *-commutative 63.4%

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

   6. Applied egg-rr 63.4%

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

      1. *-commutative 63.4%

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

      2. associate-/l* 65.3%

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

      3. associate-*r/ 65.4%

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

      4. *-commutative 65.4%

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

      5. times-frac 65.3%

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

   8. Simplified 65.3%

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

      1. add-sqr-sqrt 65.4%

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

      2. sqrt-div 65.4%

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

      3. sqrt-pow1 42.9%

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

      4. metadata-eval 42.9%

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

      5. pow1 42.9%

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

      6. *-commutative 42.9%

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

      7. frac-times 42.9%

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

      8. *-commutative 42.9%

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

      9. associate-*r/ 42.9%

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

      10. sqrt-div 42.9%

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

      11. sqrt-pow1 67.3%

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

      12. metadata-eval 67.3%

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

      13. pow1 67.3%

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

      14. *-commutative 67.3%

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

      15. frac-times 67.3%

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

      16. *-commutative 67.3%

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

      17. associate-*r/ 67.3%

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

   10. Applied egg-rr 67.3%

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

       1. sqrt-div 71.9%

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

   12. Applied egg-rr 71.5%

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

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left(h \cdot \left(\left(\frac{D}{d} \cdot \left(0.5 \cdot M\right)\right) \cdot \frac{\frac{D}{d} \cdot \left(0.5 \cdot M\right)}{\ell}\right)\right)\right) \cdot \left(\sqrt{\frac{d}{h}} \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\right)\\ \mathbf{elif}\;h \leq 2.6 \cdot 10^{+74}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 + \left(\frac{h}{\ell} \cdot -0.5\right) \cdot {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left(h \cdot \left(\left(\frac{D}{d} \cdot \left(0.5 \cdot M\right)\right) \cdot \frac{\frac{D}{d} \cdot \left(0.5 \cdot M\right)}{\ell}\right)\right)\right) \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \frac{\sqrt{d}}{\sqrt{h}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := \frac{D\_m}{d} \cdot \left(0.5 \cdot M\_m\right)\\ t_2 := 1 - 0.5 \cdot \left(h \cdot \left(t\_1 \cdot \frac{t\_1}{\ell}\right)\right)\\ \mathbf{if}\;h \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot t\_0\right) \cdot t\_2\\ \mathbf{elif}\;h \leq 4 \cdot 10^{+72}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 + \left(\frac{h}{\ell} \cdot -0.5\right) \cdot {\left(D\_m \cdot \frac{\frac{M\_m}{d}}{2}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2 \cdot \left(t\_0 \cdot \frac{\sqrt{d}}{\sqrt{h}}\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (sqrt (/ d l)))
        (t_1 (* (/ D_m d) (* 0.5 M_m)))
        (t_2 (- 1.0 (* 0.5 (* h (* t_1 (/ t_1 l)))))))
   (if (<= h -5e-311)
     (* (* (/ (sqrt (- d)) (sqrt (- h))) t_0) t_2)
     (if (<= h 4e+72)
       (*
        (/ d (* (sqrt l) (sqrt h)))
        (+ 1.0 (* (* (/ h l) -0.5) (pow (* D_m (/ (/ M_m d) 2.0)) 2.0))))
       (* t_2 (* t_0 (/ (sqrt d) (sqrt h))))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = sqrt((d / l));
	double t_1 = (D_m / d) * (0.5 * M_m);
	double t_2 = 1.0 - (0.5 * (h * (t_1 * (t_1 / l))));
	double tmp;
	if (h <= -5e-311) {
		tmp = ((sqrt(-d) / sqrt(-h)) * t_0) * t_2;
	} else if (h <= 4e+72) {
		tmp = (d / (sqrt(l) * sqrt(h))) * (1.0 + (((h / l) * -0.5) * pow((D_m * ((M_m / d) / 2.0)), 2.0)));
	} else {
		tmp = t_2 * (t_0 * (sqrt(d) / sqrt(h)));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = sqrt((d / l))
    t_1 = (d_m / d) * (0.5d0 * m_m)
    t_2 = 1.0d0 - (0.5d0 * (h * (t_1 * (t_1 / l))))
    if (h <= (-5d-311)) then
        tmp = ((sqrt(-d) / sqrt(-h)) * t_0) * t_2
    else if (h <= 4d+72) then
        tmp = (d / (sqrt(l) * sqrt(h))) * (1.0d0 + (((h / l) * (-0.5d0)) * ((d_m * ((m_m / d) / 2.0d0)) ** 2.0d0)))
    else
        tmp = t_2 * (t_0 * (sqrt(d) / sqrt(h)))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = Math.sqrt((d / l));
	double t_1 = (D_m / d) * (0.5 * M_m);
	double t_2 = 1.0 - (0.5 * (h * (t_1 * (t_1 / l))));
	double tmp;
	if (h <= -5e-311) {
		tmp = ((Math.sqrt(-d) / Math.sqrt(-h)) * t_0) * t_2;
	} else if (h <= 4e+72) {
		tmp = (d / (Math.sqrt(l) * Math.sqrt(h))) * (1.0 + (((h / l) * -0.5) * Math.pow((D_m * ((M_m / d) / 2.0)), 2.0)));
	} else {
		tmp = t_2 * (t_0 * (Math.sqrt(d) / Math.sqrt(h)));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = math.sqrt((d / l))
	t_1 = (D_m / d) * (0.5 * M_m)
	t_2 = 1.0 - (0.5 * (h * (t_1 * (t_1 / l))))
	tmp = 0
	if h <= -5e-311:
		tmp = ((math.sqrt(-d) / math.sqrt(-h)) * t_0) * t_2
	elif h <= 4e+72:
		tmp = (d / (math.sqrt(l) * math.sqrt(h))) * (1.0 + (((h / l) * -0.5) * math.pow((D_m * ((M_m / d) / 2.0)), 2.0)))
	else:
		tmp = t_2 * (t_0 * (math.sqrt(d) / math.sqrt(h)))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = sqrt(Float64(d / l))
	t_1 = Float64(Float64(D_m / d) * Float64(0.5 * M_m))
	t_2 = Float64(1.0 - Float64(0.5 * Float64(h * Float64(t_1 * Float64(t_1 / l)))))
	tmp = 0.0
	if (h <= -5e-311)
		tmp = Float64(Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-h))) * t_0) * t_2);
	elseif (h <= 4e+72)
		tmp = Float64(Float64(d / Float64(sqrt(l) * sqrt(h))) * Float64(1.0 + Float64(Float64(Float64(h / l) * -0.5) * (Float64(D_m * Float64(Float64(M_m / d) / 2.0)) ^ 2.0))));
	else
		tmp = Float64(t_2 * Float64(t_0 * Float64(sqrt(d) / sqrt(h))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = sqrt((d / l));
	t_1 = (D_m / d) * (0.5 * M_m);
	t_2 = 1.0 - (0.5 * (h * (t_1 * (t_1 / l))));
	tmp = 0.0;
	if (h <= -5e-311)
		tmp = ((sqrt(-d) / sqrt(-h)) * t_0) * t_2;
	elseif (h <= 4e+72)
		tmp = (d / (sqrt(l) * sqrt(h))) * (1.0 + (((h / l) * -0.5) * ((D_m * ((M_m / d) / 2.0)) ^ 2.0)));
	else
		tmp = t_2 * (t_0 * (sqrt(d) / sqrt(h)));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(D$95$m / d), $MachinePrecision] * N[(0.5 * M$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 - N[(0.5 * N[(h * N[(t$95$1 * N[(t$95$1 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[h, -5e-311], N[(N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[h, 4e+72], N[(N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision] * N[Power[N[(D$95$m * N[(N[(M$95$m / d), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$2 * N[(t$95$0 * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if h < -5.00000000000023e-311

   1. Initial program 64.3%

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

   3. Simplified 62.0%

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

      1. associate-*r/ 62.2%

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

      2. frac-times 64.4%

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

      3. associate-/l* 62.2%

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

      4. *-commutative 62.2%

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

   6. Applied egg-rr 62.2%

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

      1. *-commutative 62.2%

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

      2. associate-/l* 63.0%

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

      3. associate-*r/ 65.3%

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

      4. *-commutative 65.3%

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

      5. times-frac 63.0%

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

   8. Simplified 63.0%

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

      1. add-sqr-sqrt 22.1%

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

      2. sqrt-div 0.0%

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

      3. sqrt-pow1 0.0%

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

      4. metadata-eval 0.0%

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

      5. pow1 0.0%

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

      6. *-commutative 0.0%

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

      7. frac-times 0.0%

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

      8. *-commutative 0.0%

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

      9. associate-*r/ 0.0%

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

      10. sqrt-div 0.0%

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

      11. sqrt-pow1 0.0%

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

      12. metadata-eval 0.0%

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

      13. pow1 0.0%

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

      14. *-commutative 0.0%

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

      15. frac-times 0.0%

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

      16. *-commutative 0.0%

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

      17. associate-*r/ 0.0%

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

   10. Applied egg-rr 65.4%

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

       1. frac-2neg 65.4%

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

       2. sqrt-div 76.9%

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

   12. Applied egg-rr 76.9%

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

   if -5.00000000000023e-311 < h < 3.99999999999999978e72

   1. Initial program 66.7%

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

   3. Simplified 67.9%

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

   6. Applied egg-rr 83.7%

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

      1. unpow1 83.7%

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

      2. associate-*r* 83.7%

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

      3. *-commutative 83.7%

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

      4. associate-*r/ 82.5%

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

      5. *-commutative 82.5%

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

      6. associate-*r/ 81.3%

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

      7. associate-*r* 81.3%

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

      8. associate-*r* 81.3%

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

      9. associate-/r* 81.3%

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

   8. Simplified 81.3%

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

   if 3.99999999999999978e72 < h

   1. Initial program 57.2%

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

   3. Simplified 57.1%

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

      1. associate-*r/ 63.4%

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

      2. frac-times 63.4%

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

      3. associate-/l* 63.4%

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

      4. *-commutative 63.4%

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

   6. Applied egg-rr 63.4%

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

      1. *-commutative 63.4%

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

      2. associate-/l* 65.3%

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

      3. associate-*r/ 65.4%

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

      4. *-commutative 65.4%

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

      5. times-frac 65.3%

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

   8. Simplified 65.3%

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

      1. add-sqr-sqrt 65.4%

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

      2. sqrt-div 65.4%

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

      3. sqrt-pow1 42.9%

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

      4. metadata-eval 42.9%

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

      5. pow1 42.9%

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

      6. *-commutative 42.9%

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

      7. frac-times 42.9%

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

      8. *-commutative 42.9%

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

      9. associate-*r/ 42.9%

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

      10. sqrt-div 42.9%

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

      11. sqrt-pow1 67.3%

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

      12. metadata-eval 67.3%

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

      13. pow1 67.3%

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

      14. *-commutative 67.3%

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

      15. frac-times 67.3%

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

      16. *-commutative 67.3%

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

      17. associate-*r/ 67.3%

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

   10. Applied egg-rr 67.3%

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

       1. sqrt-div 71.9%

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

   12. Applied egg-rr 71.5%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \left(\left(\frac{D}{d} \cdot \left(0.5 \cdot M\right)\right) \cdot \frac{\frac{D}{d} \cdot \left(0.5 \cdot M\right)}{\ell}\right)\right)\right)\\ \mathbf{elif}\;h \leq 4 \cdot 10^{+72}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 + \left(\frac{h}{\ell} \cdot -0.5\right) \cdot {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left(h \cdot \left(\left(\frac{D}{d} \cdot \left(0.5 \cdot M\right)\right) \cdot \frac{\frac{D}{d} \cdot \left(0.5 \cdot M\right)}{\ell}\right)\right)\right) \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \frac{\sqrt{d}}{\sqrt{h}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \frac{D\_m}{d} \cdot \left(0.5 \cdot M\_m\right)\\ \mathbf{if}\;\ell \leq 2.5 \cdot 10^{-90}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left(h \cdot \left(t\_0 \cdot \frac{t\_0}{\ell}\right)\right)\right) \cdot \left(\sqrt{\frac{d}{h}} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 + \left(\frac{h}{\ell} \cdot -0.5\right) \cdot {\left(D\_m \cdot \frac{\frac{M\_m}{d}}{2}\right)}^{2}\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (* (/ D_m d) (* 0.5 M_m))))
   (if (<= l 2.5e-90)
     (*
      (- 1.0 (* 0.5 (* h (* t_0 (/ t_0 l)))))
      (* (sqrt (/ d h)) (/ 1.0 (sqrt (/ l d)))))
     (*
      (/ d (* (sqrt l) (sqrt h)))
      (+ 1.0 (* (* (/ h l) -0.5) (pow (* D_m (/ (/ M_m d) 2.0)) 2.0)))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = (D_m / d) * (0.5 * M_m);
	double tmp;
	if (l <= 2.5e-90) {
		tmp = (1.0 - (0.5 * (h * (t_0 * (t_0 / l))))) * (sqrt((d / h)) * (1.0 / sqrt((l / d))));
	} else {
		tmp = (d / (sqrt(l) * sqrt(h))) * (1.0 + (((h / l) * -0.5) * pow((D_m * ((M_m / d) / 2.0)), 2.0)));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (d_m / d) * (0.5d0 * m_m)
    if (l <= 2.5d-90) then
        tmp = (1.0d0 - (0.5d0 * (h * (t_0 * (t_0 / l))))) * (sqrt((d / h)) * (1.0d0 / sqrt((l / d))))
    else
        tmp = (d / (sqrt(l) * sqrt(h))) * (1.0d0 + (((h / l) * (-0.5d0)) * ((d_m * ((m_m / d) / 2.0d0)) ** 2.0d0)))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = (D_m / d) * (0.5 * M_m);
	double tmp;
	if (l <= 2.5e-90) {
		tmp = (1.0 - (0.5 * (h * (t_0 * (t_0 / l))))) * (Math.sqrt((d / h)) * (1.0 / Math.sqrt((l / d))));
	} else {
		tmp = (d / (Math.sqrt(l) * Math.sqrt(h))) * (1.0 + (((h / l) * -0.5) * Math.pow((D_m * ((M_m / d) / 2.0)), 2.0)));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = (D_m / d) * (0.5 * M_m)
	tmp = 0
	if l <= 2.5e-90:
		tmp = (1.0 - (0.5 * (h * (t_0 * (t_0 / l))))) * (math.sqrt((d / h)) * (1.0 / math.sqrt((l / d))))
	else:
		tmp = (d / (math.sqrt(l) * math.sqrt(h))) * (1.0 + (((h / l) * -0.5) * math.pow((D_m * ((M_m / d) / 2.0)), 2.0)))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(Float64(D_m / d) * Float64(0.5 * M_m))
	tmp = 0.0
	if (l <= 2.5e-90)
		tmp = Float64(Float64(1.0 - Float64(0.5 * Float64(h * Float64(t_0 * Float64(t_0 / l))))) * Float64(sqrt(Float64(d / h)) * Float64(1.0 / sqrt(Float64(l / d)))));
	else
		tmp = Float64(Float64(d / Float64(sqrt(l) * sqrt(h))) * Float64(1.0 + Float64(Float64(Float64(h / l) * -0.5) * (Float64(D_m * Float64(Float64(M_m / d) / 2.0)) ^ 2.0))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = (D_m / d) * (0.5 * M_m);
	tmp = 0.0;
	if (l <= 2.5e-90)
		tmp = (1.0 - (0.5 * (h * (t_0 * (t_0 / l))))) * (sqrt((d / h)) * (1.0 / sqrt((l / d))));
	else
		tmp = (d / (sqrt(l) * sqrt(h))) * (1.0 + (((h / l) * -0.5) * ((D_m * ((M_m / d) / 2.0)) ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(N[(D$95$m / d), $MachinePrecision] * N[(0.5 * M$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, 2.5e-90], N[(N[(1.0 - N[(0.5 * N[(h * N[(t$95$0 * N[(t$95$0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(1.0 / N[Sqrt[N[(l / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision] * N[Power[N[(D$95$m * N[(N[(M$95$m / d), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < 2.5000000000000001e-90

   1. Initial program 68.0%

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

   3. Simplified 66.4%

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

      1. associate-*r/ 68.8%

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

      2. frac-times 70.4%

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

      3. associate-/l* 68.8%

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

      4. *-commutative 68.8%

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

   6. Applied egg-rr 68.8%

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

      1. *-commutative 68.8%

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

      2. associate-/l* 69.4%

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

      3. associate-*r/ 71.0%

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

      4. *-commutative 71.0%

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

      5. times-frac 69.4%

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

   8. Simplified 69.4%

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

      1. add-sqr-sqrt 39.9%

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

      2. sqrt-div 23.9%

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

      3. sqrt-pow1 16.0%

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

      4. metadata-eval 16.0%

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

      5. pow1 16.0%

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

      6. *-commutative 16.0%

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

      7. frac-times 16.0%

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

      8. *-commutative 16.0%

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

      9. associate-*r/ 16.0%

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

      10. sqrt-div 16.0%

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

      11. sqrt-pow1 24.5%

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

      12. metadata-eval 24.5%

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

      13. pow1 24.5%

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

      14. *-commutative 24.5%

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

      15. frac-times 24.5%

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

      16. *-commutative 24.5%

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

      17. associate-*r/ 24.5%

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

   10. Applied egg-rr 71.7%

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

       1. clear-num 71.7%

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

       2. sqrt-div 71.7%

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

       3. metadata-eval 71.7%

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

   12. Applied egg-rr 71.7%

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

   if 2.5000000000000001e-90 < l

   1. Initial program 54.1%

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

   3. Simplified 55.2%

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

   6. Applied egg-rr 69.9%

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

      1. unpow1 69.9%

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

      2. associate-*r* 69.9%

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

      3. *-commutative 69.9%

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

      4. associate-*r/ 68.8%

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

      5. *-commutative 68.8%

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

      6. associate-*r/ 68.8%

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

      7. associate-*r* 68.8%

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

      8. associate-*r* 68.8%

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

      9. associate-/r* 68.8%

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

   8. Simplified 68.8%

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

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2.5 \cdot 10^{-90}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left(h \cdot \left(\left(\frac{D}{d} \cdot \left(0.5 \cdot M\right)\right) \cdot \frac{\frac{D}{d} \cdot \left(0.5 \cdot M\right)}{\ell}\right)\right)\right) \cdot \left(\sqrt{\frac{d}{h}} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 + \left(\frac{h}{\ell} \cdot -0.5\right) \cdot {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 66.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\ell \cdot h}\\ \mathbf{if}\;\ell \leq -5.2 \cdot 10^{+138}:\\ \;\;\;\;\frac{d}{-t\_0}\\ \mathbf{elif}\;\ell \leq 1.65 \cdot 10^{-307}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \left(\left(\frac{D\_m}{d} \cdot \left(0.5 \cdot M\_m\right)\right) \cdot \left(D\_m \cdot \left(0.5 \cdot \frac{M\_m}{\ell \cdot d}\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \left(\frac{h}{\ell} \cdot -0.5\right) \cdot {\left(D\_m \cdot \frac{\frac{M\_m}{d}}{2}\right)}^{2}\right) \cdot \frac{d}{t\_0}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (sqrt (* l h))))
   (if (<= l -5.2e+138)
     (/ d (- t_0))
     (if (<= l 1.65e-307)
       (*
        (* (sqrt (/ d l)) (sqrt (/ d h)))
        (-
         1.0
         (*
          0.5
          (*
           h
           (* (* (/ D_m d) (* 0.5 M_m)) (* D_m (* 0.5 (/ M_m (* l d)))))))))
       (*
        (+ 1.0 (* (* (/ h l) -0.5) (pow (* D_m (/ (/ M_m d) 2.0)) 2.0)))
        (/ d t_0))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = sqrt((l * h));
	double tmp;
	if (l <= -5.2e+138) {
		tmp = d / -t_0;
	} else if (l <= 1.65e-307) {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - (0.5 * (h * (((D_m / d) * (0.5 * M_m)) * (D_m * (0.5 * (M_m / (l * d))))))));
	} else {
		tmp = (1.0 + (((h / l) * -0.5) * pow((D_m * ((M_m / d) / 2.0)), 2.0))) * (d / t_0);
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((l * h))
    if (l <= (-5.2d+138)) then
        tmp = d / -t_0
    else if (l <= 1.65d-307) then
        tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0d0 - (0.5d0 * (h * (((d_m / d) * (0.5d0 * m_m)) * (d_m * (0.5d0 * (m_m / (l * d))))))))
    else
        tmp = (1.0d0 + (((h / l) * (-0.5d0)) * ((d_m * ((m_m / d) / 2.0d0)) ** 2.0d0))) * (d / t_0)
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = Math.sqrt((l * h));
	double tmp;
	if (l <= -5.2e+138) {
		tmp = d / -t_0;
	} else if (l <= 1.65e-307) {
		tmp = (Math.sqrt((d / l)) * Math.sqrt((d / h))) * (1.0 - (0.5 * (h * (((D_m / d) * (0.5 * M_m)) * (D_m * (0.5 * (M_m / (l * d))))))));
	} else {
		tmp = (1.0 + (((h / l) * -0.5) * Math.pow((D_m * ((M_m / d) / 2.0)), 2.0))) * (d / t_0);
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = math.sqrt((l * h))
	tmp = 0
	if l <= -5.2e+138:
		tmp = d / -t_0
	elif l <= 1.65e-307:
		tmp = (math.sqrt((d / l)) * math.sqrt((d / h))) * (1.0 - (0.5 * (h * (((D_m / d) * (0.5 * M_m)) * (D_m * (0.5 * (M_m / (l * d))))))))
	else:
		tmp = (1.0 + (((h / l) * -0.5) * math.pow((D_m * ((M_m / d) / 2.0)), 2.0))) * (d / t_0)
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = sqrt(Float64(l * h))
	tmp = 0.0
	if (l <= -5.2e+138)
		tmp = Float64(d / Float64(-t_0));
	elseif (l <= 1.65e-307)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * Float64(1.0 - Float64(0.5 * Float64(h * Float64(Float64(Float64(D_m / d) * Float64(0.5 * M_m)) * Float64(D_m * Float64(0.5 * Float64(M_m / Float64(l * d)))))))));
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(Float64(h / l) * -0.5) * (Float64(D_m * Float64(Float64(M_m / d) / 2.0)) ^ 2.0))) * Float64(d / t_0));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = sqrt((l * h));
	tmp = 0.0;
	if (l <= -5.2e+138)
		tmp = d / -t_0;
	elseif (l <= 1.65e-307)
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - (0.5 * (h * (((D_m / d) * (0.5 * M_m)) * (D_m * (0.5 * (M_m / (l * d))))))));
	else
		tmp = (1.0 + (((h / l) * -0.5) * ((D_m * ((M_m / d) / 2.0)) ^ 2.0))) * (d / t_0);
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -5.2e+138], N[(d / (-t$95$0)), $MachinePrecision], If[LessEqual[l, 1.65e-307], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(h * N[(N[(N[(D$95$m / d), $MachinePrecision] * N[(0.5 * M$95$m), $MachinePrecision]), $MachinePrecision] * N[(D$95$m * N[(0.5 * N[(M$95$m / N[(l * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision] * N[Power[N[(D$95$m * N[(N[(M$95$m / d), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(d / t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < -5.2000000000000002e138

   1. Initial program 44.9%

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

   3. Simplified 44.5%

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

      1. associate-*r/ 37.2%

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

      2. frac-times 37.2%

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

      3. associate-/l* 37.2%

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

      4. *-commutative 37.2%

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

   6. Applied egg-rr 37.2%

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

      1. *-commutative 37.2%

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

      2. associate-/l* 44.6%

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

      3. associate-*r/ 45.0%

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

      4. *-commutative 45.0%

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

      5. times-frac 44.6%

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

   8. Simplified 44.6%

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

      1. add-sqr-sqrt 20.7%

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

      2. sqrt-div 0.0%

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

      3. sqrt-pow1 0.0%

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

      4. metadata-eval 0.0%

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

      5. pow1 0.0%

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

      6. *-commutative 0.0%

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

      7. frac-times 0.0%

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

      8. *-commutative 0.0%

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

      9. associate-*r/ 0.0%

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

      10. sqrt-div 0.0%

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

      11. sqrt-pow1 0.0%

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

      12. metadata-eval 0.0%

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

      13. pow1 0.0%

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

      14. *-commutative 0.0%

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

      15. frac-times 0.0%

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

      16. *-commutative 0.0%

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

      17. associate-*r/ 0.0%

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

   10. Applied egg-rr 48.5%

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

   11. Taylor expanded in l around -inf 0.0%

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

       1. *-commutative 0.0%

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

       2. unpow2 0.0%

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

       3. rem-square-sqrt 59.0%

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

       4. mul-1-neg 59.0%

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

       5. distribute-lft-neg-in 59.0%

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

       6. rem-exp-log 56.3%

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

       7. exp-neg 56.3%

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

       8. unpow1/2 56.3%

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

       9. exp-prod 56.3%

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

       10. distribute-lft-neg-out 56.3%

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

       11. exp-neg 56.3%

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

       12. exp-to-pow 59.0%

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

       13. unpow1/2 59.0%

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

       14. associate-/l* 59.1%

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

       15. associate-*l/ 59.1%

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

       16. *-rgt-identity 59.1%

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

       17. distribute-neg-frac2 59.1%

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

   13. Simplified 59.1%

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

   if -5.2000000000000002e138 < l < 1.65e-307

   1. Initial program 69.2%

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

   3. Simplified 66.5%

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

      1. associate-*r/ 68.6%

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

      2. frac-times 71.4%

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

      3. associate-/l* 68.6%

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

      4. *-commutative 68.6%

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

   6. Applied egg-rr 68.6%

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

      1. *-commutative 68.6%

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

      2. associate-/l* 67.8%

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

      3. associate-*r/ 70.5%

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

      4. *-commutative 70.5%

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

      5. times-frac 67.8%

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

   8. Simplified 67.8%

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

      1. add-sqr-sqrt 22.5%

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

      2. sqrt-div 0.0%

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

      3. sqrt-pow1 0.0%

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

      4. metadata-eval 0.0%

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

      5. pow1 0.0%

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

      6. *-commutative 0.0%

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

      7. frac-times 0.0%

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

      8. *-commutative 0.0%

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

      9. associate-*r/ 0.0%

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

      10. sqrt-div 0.0%

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

      11. sqrt-pow1 0.0%

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

      12. metadata-eval 0.0%

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

      13. pow1 0.0%

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

      14. *-commutative 0.0%

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

      15. frac-times 0.0%

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

      16. *-commutative 0.0%

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

      17. associate-*r/ 0.0%

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

   10. Applied egg-rr 69.7%

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

   11. Taylor expanded in D around 0 67.8%

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

       1. *-commutative 67.8%

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

       2. associate-/l* 65.9%

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

       3. associate-*l* 65.9%

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

   13. Simplified 65.9%

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

   if 1.65e-307 < l

   1. Initial program 63.1%

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

   3. Simplified 63.8%

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

   6. Applied egg-rr 75.2%

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

      1. unpow1 75.2%

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

      2. associate-*r* 75.2%

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

      3. *-commutative 75.2%

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

      4. associate-*r/ 74.5%

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

      5. *-commutative 74.5%

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

      6. associate-*r/ 73.7%

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

      7. associate-*r* 73.7%

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

      8. associate-*r* 73.7%

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

      9. associate-/r* 73.7%

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

   8. Simplified 73.7%

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

   9. Taylor expanded in h around 0 63.2%

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

4. Final simplification 63.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5.2 \cdot 10^{+138}:\\ \;\;\;\;\frac{d}{-\sqrt{\ell \cdot h}}\\ \mathbf{elif}\;\ell \leq 1.65 \cdot 10^{-307}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \left(\left(\frac{D}{d} \cdot \left(0.5 \cdot M\right)\right) \cdot \left(D \cdot \left(0.5 \cdot \frac{M}{\ell \cdot d}\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \left(\frac{h}{\ell} \cdot -0.5\right) \cdot {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2}\right) \cdot \frac{d}{\sqrt{\ell \cdot h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 62.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\ell \cdot h}\\ \mathbf{if}\;\ell \leq -3.7 \cdot 10^{+80}:\\ \;\;\;\;\frac{d}{-t\_0}\\ \mathbf{elif}\;\ell \leq 1.2 \cdot 10^{-69}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{{\left(\frac{D\_m}{d} \cdot \left(0.5 \cdot M\_m\right)\right)}^{2}}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \left(\frac{h}{\ell} \cdot -0.5\right) \cdot {\left(D\_m \cdot \frac{\frac{M\_m}{d}}{2}\right)}^{2}\right) \cdot \frac{d}{t\_0}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (sqrt (* l h))))
   (if (<= l -3.7e+80)
     (/ d (- t_0))
     (if (<= l 1.2e-69)
       (*
        (sqrt (* (/ d l) (/ d h)))
        (- 1.0 (* 0.5 (* h (/ (pow (* (/ D_m d) (* 0.5 M_m)) 2.0) l)))))
       (*
        (+ 1.0 (* (* (/ h l) -0.5) (pow (* D_m (/ (/ M_m d) 2.0)) 2.0)))
        (/ d t_0))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = sqrt((l * h));
	double tmp;
	if (l <= -3.7e+80) {
		tmp = d / -t_0;
	} else if (l <= 1.2e-69) {
		tmp = sqrt(((d / l) * (d / h))) * (1.0 - (0.5 * (h * (pow(((D_m / d) * (0.5 * M_m)), 2.0) / l))));
	} else {
		tmp = (1.0 + (((h / l) * -0.5) * pow((D_m * ((M_m / d) / 2.0)), 2.0))) * (d / t_0);
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((l * h))
    if (l <= (-3.7d+80)) then
        tmp = d / -t_0
    else if (l <= 1.2d-69) then
        tmp = sqrt(((d / l) * (d / h))) * (1.0d0 - (0.5d0 * (h * ((((d_m / d) * (0.5d0 * m_m)) ** 2.0d0) / l))))
    else
        tmp = (1.0d0 + (((h / l) * (-0.5d0)) * ((d_m * ((m_m / d) / 2.0d0)) ** 2.0d0))) * (d / t_0)
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = Math.sqrt((l * h));
	double tmp;
	if (l <= -3.7e+80) {
		tmp = d / -t_0;
	} else if (l <= 1.2e-69) {
		tmp = Math.sqrt(((d / l) * (d / h))) * (1.0 - (0.5 * (h * (Math.pow(((D_m / d) * (0.5 * M_m)), 2.0) / l))));
	} else {
		tmp = (1.0 + (((h / l) * -0.5) * Math.pow((D_m * ((M_m / d) / 2.0)), 2.0))) * (d / t_0);
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = math.sqrt((l * h))
	tmp = 0
	if l <= -3.7e+80:
		tmp = d / -t_0
	elif l <= 1.2e-69:
		tmp = math.sqrt(((d / l) * (d / h))) * (1.0 - (0.5 * (h * (math.pow(((D_m / d) * (0.5 * M_m)), 2.0) / l))))
	else:
		tmp = (1.0 + (((h / l) * -0.5) * math.pow((D_m * ((M_m / d) / 2.0)), 2.0))) * (d / t_0)
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = sqrt(Float64(l * h))
	tmp = 0.0
	if (l <= -3.7e+80)
		tmp = Float64(d / Float64(-t_0));
	elseif (l <= 1.2e-69)
		tmp = Float64(sqrt(Float64(Float64(d / l) * Float64(d / h))) * Float64(1.0 - Float64(0.5 * Float64(h * Float64((Float64(Float64(D_m / d) * Float64(0.5 * M_m)) ^ 2.0) / l)))));
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(Float64(h / l) * -0.5) * (Float64(D_m * Float64(Float64(M_m / d) / 2.0)) ^ 2.0))) * Float64(d / t_0));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = sqrt((l * h));
	tmp = 0.0;
	if (l <= -3.7e+80)
		tmp = d / -t_0;
	elseif (l <= 1.2e-69)
		tmp = sqrt(((d / l) * (d / h))) * (1.0 - (0.5 * (h * ((((D_m / d) * (0.5 * M_m)) ^ 2.0) / l))));
	else
		tmp = (1.0 + (((h / l) * -0.5) * ((D_m * ((M_m / d) / 2.0)) ^ 2.0))) * (d / t_0);
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -3.7e+80], N[(d / (-t$95$0)), $MachinePrecision], If[LessEqual[l, 1.2e-69], N[(N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(0.5 * N[(h * N[(N[Power[N[(N[(D$95$m / d), $MachinePrecision] * N[(0.5 * M$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision] * N[Power[N[(D$95$m * N[(N[(M$95$m / d), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(d / t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < -3.69999999999999996e80

   1. Initial program 48.5%

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

   3. Simplified 48.3%

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

      1. associate-*r/ 44.5%

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

      2. frac-times 44.5%

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

      3. associate-/l* 44.5%

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

      4. *-commutative 44.5%

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

   6. Applied egg-rr 44.5%

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

      1. *-commutative 44.5%

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

      2. associate-/l* 48.5%

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

      3. associate-*r/ 48.8%

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

      4. *-commutative 48.8%

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

      5. times-frac 48.5%

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

   8. Simplified 48.5%

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

      1. add-sqr-sqrt 26.5%

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

      2. sqrt-div 0.0%

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

      3. sqrt-pow1 0.0%

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

      4. metadata-eval 0.0%

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

      5. pow1 0.0%

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

      6. *-commutative 0.0%

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

      7. frac-times 0.0%

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

      8. *-commutative 0.0%

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

      9. associate-*r/ 0.0%

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

      10. sqrt-div 0.0%

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

      11. sqrt-pow1 0.0%

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

      12. metadata-eval 0.0%

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

      13. pow1 0.0%

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

      14. *-commutative 0.0%

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

      15. frac-times 0.0%

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

      16. *-commutative 0.0%

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

      17. associate-*r/ 0.0%

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

   10. Applied egg-rr 54.8%

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

   11. Taylor expanded in l around -inf 0.0%

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

       1. *-commutative 0.0%

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

       2. unpow2 0.0%

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

       3. rem-square-sqrt 58.5%

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

       4. mul-1-neg 58.5%

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

       5. distribute-lft-neg-in 58.5%

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

       6. rem-exp-log 55.3%

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

       7. exp-neg 55.3%

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

       8. unpow1/2 55.3%

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

       9. exp-prod 55.3%

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

       10. distribute-lft-neg-out 55.3%

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

       11. exp-neg 55.3%

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

       12. exp-to-pow 58.4%

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

       13. unpow1/2 58.4%

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

       14. associate-/l* 58.6%

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

       15. associate-*l/ 58.6%

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

       16. *-rgt-identity 58.6%

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

       17. distribute-neg-frac2 58.6%

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

   13. Simplified 58.6%

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

   if -3.69999999999999996e80 < l < 1.2000000000000001e-69

   1. Initial program 75.7%

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

   3. Simplified 73.6%

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

      1. associate-*r/ 78.2%

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

      2. frac-times 80.3%

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

      3. associate-/l* 78.2%

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

      4. *-commutative 78.2%

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

   6. Applied egg-rr 78.2%

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

      1. *-commutative 78.2%

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

      2. associate-/l* 77.6%

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

      3. associate-*r/ 79.6%

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

      4. *-commutative 79.6%

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

      5. times-frac 77.6%

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

   8. Simplified 77.6%

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

      1. add-sqr-sqrt 46.1%

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

      2. sqrt-div 34.2%

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

      3. sqrt-pow1 23.6%

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

      4. metadata-eval 23.6%

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

      5. pow1 23.6%

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

      6. *-commutative 23.6%

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

      7. frac-times 23.6%

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

      8. *-commutative 23.6%

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

      9. associate-*r/ 23.6%

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

      10. sqrt-div 23.6%

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

      11. sqrt-pow1 34.9%

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

      12. metadata-eval 34.9%

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

      13. pow1 34.9%

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

      14. *-commutative 34.9%

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

      15. frac-times 34.9%

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

      16. *-commutative 34.9%

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

      17. associate-*r/ 34.9%

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

   10. Applied egg-rr 78.3%

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

       1. pow1 78.3%

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

       2. sqrt-unprod 70.7%

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

       3. associate-*r* 70.7%

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

       4. frac-times 69.9%

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

       5. pow2 69.9%

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

       6. *-commutative 69.9%

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

       7. *-un-lft-identity 69.9%

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

   12. Applied egg-rr 69.9%

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

       1. unpow1 69.9%

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

       2. associate-*l* 69.9%

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

       3. *-commutative 69.9%

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

   14. Simplified 69.9%

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

   if 1.2000000000000001e-69 < l

   1. Initial program 52.3%

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

   3. Simplified 53.5%

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

   6. Applied egg-rr 68.7%

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

      1. unpow1 68.7%

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

      2. associate-*r* 68.7%

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

      3. *-commutative 68.7%

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

      4. associate-*r/ 67.6%

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

      5. *-commutative 67.6%

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

      6. associate-*r/ 67.6%

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

      7. associate-*r* 67.6%

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

      8. associate-*r* 67.6%

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

      9. associate-/r* 67.6%

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

   8. Simplified 67.6%

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

   9. Taylor expanded in h around 0 56.0%

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

4. Final simplification 63.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -3.7 \cdot 10^{+80}:\\ \;\;\;\;\frac{d}{-\sqrt{\ell \cdot h}}\\ \mathbf{elif}\;\ell \leq 1.2 \cdot 10^{-69}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{{\left(\frac{D}{d} \cdot \left(0.5 \cdot M\right)\right)}^{2}}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \left(\frac{h}{\ell} \cdot -0.5\right) \cdot {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2}\right) \cdot \frac{d}{\sqrt{\ell \cdot h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 55.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\ell \cdot h}\\ \mathbf{if}\;d \leq -2.4 \cdot 10^{-158}:\\ \;\;\;\;\frac{d}{-t\_0}\\ \mathbf{elif}\;d \leq 9.2 \cdot 10^{-272}:\\ \;\;\;\;d \cdot {\left({\left(\ell \cdot h\right)}^{2}\right)}^{-0.25}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \left(\frac{h}{\ell} \cdot -0.5\right) \cdot {\left(D\_m \cdot \frac{\frac{M\_m}{d}}{2}\right)}^{2}\right) \cdot \frac{d}{t\_0}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (sqrt (* l h))))
   (if (<= d -2.4e-158)
     (/ d (- t_0))
     (if (<= d 9.2e-272)
       (* d (pow (pow (* l h) 2.0) -0.25))
       (*
        (+ 1.0 (* (* (/ h l) -0.5) (pow (* D_m (/ (/ M_m d) 2.0)) 2.0)))
        (/ d t_0))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = sqrt((l * h));
	double tmp;
	if (d <= -2.4e-158) {
		tmp = d / -t_0;
	} else if (d <= 9.2e-272) {
		tmp = d * pow(pow((l * h), 2.0), -0.25);
	} else {
		tmp = (1.0 + (((h / l) * -0.5) * pow((D_m * ((M_m / d) / 2.0)), 2.0))) * (d / t_0);
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((l * h))
    if (d <= (-2.4d-158)) then
        tmp = d / -t_0
    else if (d <= 9.2d-272) then
        tmp = d * (((l * h) ** 2.0d0) ** (-0.25d0))
    else
        tmp = (1.0d0 + (((h / l) * (-0.5d0)) * ((d_m * ((m_m / d) / 2.0d0)) ** 2.0d0))) * (d / t_0)
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = Math.sqrt((l * h));
	double tmp;
	if (d <= -2.4e-158) {
		tmp = d / -t_0;
	} else if (d <= 9.2e-272) {
		tmp = d * Math.pow(Math.pow((l * h), 2.0), -0.25);
	} else {
		tmp = (1.0 + (((h / l) * -0.5) * Math.pow((D_m * ((M_m / d) / 2.0)), 2.0))) * (d / t_0);
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = math.sqrt((l * h))
	tmp = 0
	if d <= -2.4e-158:
		tmp = d / -t_0
	elif d <= 9.2e-272:
		tmp = d * math.pow(math.pow((l * h), 2.0), -0.25)
	else:
		tmp = (1.0 + (((h / l) * -0.5) * math.pow((D_m * ((M_m / d) / 2.0)), 2.0))) * (d / t_0)
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = sqrt(Float64(l * h))
	tmp = 0.0
	if (d <= -2.4e-158)
		tmp = Float64(d / Float64(-t_0));
	elseif (d <= 9.2e-272)
		tmp = Float64(d * ((Float64(l * h) ^ 2.0) ^ -0.25));
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(Float64(h / l) * -0.5) * (Float64(D_m * Float64(Float64(M_m / d) / 2.0)) ^ 2.0))) * Float64(d / t_0));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = sqrt((l * h));
	tmp = 0.0;
	if (d <= -2.4e-158)
		tmp = d / -t_0;
	elseif (d <= 9.2e-272)
		tmp = d * (((l * h) ^ 2.0) ^ -0.25);
	else
		tmp = (1.0 + (((h / l) * -0.5) * ((D_m * ((M_m / d) / 2.0)) ^ 2.0))) * (d / t_0);
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[d, -2.4e-158], N[(d / (-t$95$0)), $MachinePrecision], If[LessEqual[d, 9.2e-272], N[(d * N[Power[N[Power[N[(l * h), $MachinePrecision], 2.0], $MachinePrecision], -0.25], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision] * N[Power[N[(D$95$m * N[(N[(M$95$m / d), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(d / t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if d < -2.40000000000000007e-158

   1. Initial program 74.6%

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

   3. Simplified 71.4%

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

      1. associate-*r/ 71.6%

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

      2. frac-times 74.7%

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

      3. associate-/l* 71.6%

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

      4. *-commutative 71.6%

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

   6. Applied egg-rr 71.6%

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

      1. *-commutative 71.6%

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

      2. associate-/l* 73.8%

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

      3. associate-*r/ 77.0%

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

      4. *-commutative 77.0%

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

      5. times-frac 73.8%

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

   8. Simplified 73.8%

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

      1. add-sqr-sqrt 30.4%

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

      2. sqrt-div 0.0%

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

      3. sqrt-pow1 0.0%

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

      4. metadata-eval 0.0%

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

      5. pow1 0.0%

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

      6. *-commutative 0.0%

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

      7. frac-times 0.0%

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

      8. *-commutative 0.0%

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

      9. associate-*r/ 0.0%

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

      10. sqrt-div 0.0%

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

      11. sqrt-pow1 0.0%

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

      12. metadata-eval 0.0%

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

      13. pow1 0.0%

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

      14. *-commutative 0.0%

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

      15. frac-times 0.0%

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

      16. *-commutative 0.0%

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

      17. associate-*r/ 0.0%

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

   10. Applied egg-rr 76.0%

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

   11. Taylor expanded in l around -inf 0.0%

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

       1. *-commutative 0.0%

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

       2. unpow2 0.0%

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

       3. rem-square-sqrt 50.5%

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

       4. mul-1-neg 50.5%

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

       5. distribute-lft-neg-in 50.5%

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

       6. rem-exp-log 47.9%

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

       7. exp-neg 47.9%

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

       8. unpow1/2 47.9%

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

       9. exp-prod 47.9%

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

       10. distribute-lft-neg-out 47.9%

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

       11. exp-neg 47.9%

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

       12. exp-to-pow 50.4%

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

       13. unpow1/2 50.4%

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

       14. associate-/l* 50.6%

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

       15. associate-*l/ 50.6%

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

       16. *-rgt-identity 50.6%

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

       17. distribute-neg-frac2 50.6%

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

   13. Simplified 50.6%

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

   if -2.40000000000000007e-158 < d < 9.19999999999999955e-272

   1. Initial program 36.1%

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

   3. Simplified 36.1%

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

   5. Taylor expanded in d around inf 15.9%

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

      1. add-cbrt-cube 22.3%

         \[\leadsto d \cdot \color{blue}{\sqrt[3]{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}}} \]

      2. pow1/3 22.3%

         \[\leadsto d \cdot \color{blue}{{\left(\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)}^{0.3333333333333333}} \]

      3. add-sqr-sqrt 22.3%

         \[\leadsto d \cdot {\left(\color{blue}{\frac{1}{h \cdot \ell}} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)}^{0.3333333333333333} \]

      4. pow1 22.3%

         \[\leadsto d \cdot {\left(\color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{1}} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)}^{0.3333333333333333} \]

      5. pow1/2 22.3%

         \[\leadsto d \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1} \cdot \color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{0.5}}\right)}^{0.3333333333333333} \]

      6. pow-prod-up 22.3%

         \[\leadsto d \cdot {\color{blue}{\left({\left(\frac{1}{h \cdot \ell}\right)}^{\left(1 + 0.5\right)}\right)}}^{0.3333333333333333} \]

      7. associate-/r* 22.3%

         \[\leadsto d \cdot {\left({\color{blue}{\left(\frac{\frac{1}{h}}{\ell}\right)}}^{\left(1 + 0.5\right)}\right)}^{0.3333333333333333} \]

      8. metadata-eval 22.3%

         \[\leadsto d \cdot {\left({\left(\frac{\frac{1}{h}}{\ell}\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333} \]

   7. Applied egg-rr 22.3%

      \[\leadsto d \cdot \color{blue}{{\left({\left(\frac{\frac{1}{h}}{\ell}\right)}^{1.5}\right)}^{0.3333333333333333}} \]
   8. Step-by-step derivation

      1. pow-pow 15.9%

         \[\leadsto d \cdot \color{blue}{{\left(\frac{\frac{1}{h}}{\ell}\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} \]

      2. div-inv 15.9%

         \[\leadsto d \cdot {\color{blue}{\left(\frac{1}{h} \cdot \frac{1}{\ell}\right)}}^{\left(1.5 \cdot 0.3333333333333333\right)} \]

      3. metadata-eval 15.9%

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

      4. unpow-prod-down 2.5%

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

      5. pow1/2 2.5%

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

   9. Applied egg-rr 2.5%

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

       1. unpow1/2 2.5%

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

   11. Simplified 2.5%

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

       1. pow1/2 2.5%

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

       2. pow1/2 2.5%

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

       3. inv-pow 2.5%

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

       4. pow-pow 2.5%

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

       5. metadata-eval 2.5%

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

       6. inv-pow 2.5%

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

       7. pow-pow 2.5%

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

       8. metadata-eval 2.5%

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

       9. unpow-prod-down 15.9%

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

       10. sqr-pow 15.9%

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

       11. pow-prod-down 24.5%

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

       12. pow2 24.5%

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

       13. *-commutative 24.5%

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

       14. metadata-eval 24.5%

           \[\leadsto d \cdot {\left({\left(\ell \cdot h\right)}^{2}\right)}^{\color{blue}{-0.25}} \]

   13. Applied egg-rr 24.5%

       \[\leadsto d \cdot \color{blue}{{\left({\left(\ell \cdot h\right)}^{2}\right)}^{-0.25}} \]

   if 9.19999999999999955e-272 < d

   1. Initial program 65.9%

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

   3. Simplified 66.6%

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

   6. Applied egg-rr 78.6%

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

      1. unpow1 78.6%

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

      2. associate-*r* 78.6%

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

      3. *-commutative 78.6%

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

      4. associate-*r/ 77.9%

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

      5. *-commutative 77.9%

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

      6. associate-*r/ 77.1%

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

      7. associate-*r* 77.1%

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

      8. associate-*r* 77.1%

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

      9. associate-/r* 77.1%

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

   8. Simplified 77.1%

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

   9. Taylor expanded in h around 0 66.7%

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

4. Final simplification 53.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.4 \cdot 10^{-158}:\\ \;\;\;\;\frac{d}{-\sqrt{\ell \cdot h}}\\ \mathbf{elif}\;d \leq 9.2 \cdot 10^{-272}:\\ \;\;\;\;d \cdot {\left({\left(\ell \cdot h\right)}^{2}\right)}^{-0.25}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \left(\frac{h}{\ell} \cdot -0.5\right) \cdot {\left(D \cdot \frac{\frac{M}{d}}{2}\right)}^{2}\right) \cdot \frac{d}{\sqrt{\ell \cdot h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 69.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \frac{D\_m}{d} \cdot \left(0.5 \cdot M\_m\right)\\ \left(1 - 0.5 \cdot \left(h \cdot \left(t\_0 \cdot \frac{t\_0}{\ell}\right)\right)\right) \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (* (/ D_m d) (* 0.5 M_m))))
   (*
    (- 1.0 (* 0.5 (* h (* t_0 (/ t_0 l)))))
    (* (sqrt (/ d l)) (sqrt (/ d h))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = (D_m / d) * (0.5 * M_m);
	return (1.0 - (0.5 * (h * (t_0 * (t_0 / l))))) * (sqrt((d / l)) * sqrt((d / h)));
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    t_0 = (d_m / d) * (0.5d0 * m_m)
    code = (1.0d0 - (0.5d0 * (h * (t_0 * (t_0 / l))))) * (sqrt((d / l)) * sqrt((d / h)))
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = (D_m / d) * (0.5 * M_m);
	return (1.0 - (0.5 * (h * (t_0 * (t_0 / l))))) * (Math.sqrt((d / l)) * Math.sqrt((d / h)));
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = (D_m / d) * (0.5 * M_m)
	return (1.0 - (0.5 * (h * (t_0 * (t_0 / l))))) * (math.sqrt((d / l)) * math.sqrt((d / h)))

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(Float64(D_m / d) * Float64(0.5 * M_m))
	return Float64(Float64(1.0 - Float64(0.5 * Float64(h * Float64(t_0 * Float64(t_0 / l))))) * Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))))
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp = code(d, h, l, M_m, D_m)
	t_0 = (D_m / d) * (0.5 * M_m);
	tmp = (1.0 - (0.5 * (h * (t_0 * (t_0 / l))))) * (sqrt((d / l)) * sqrt((d / h)));
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(N[(D$95$m / d), $MachinePrecision] * N[(0.5 * M$95$m), $MachinePrecision]), $MachinePrecision]}, N[(N[(1.0 - N[(0.5 * N[(h * N[(t$95$0 * N[(t$95$0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 63.7%

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

3. Simplified 62.9%

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

   1. associate-*r/ 64.3%

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

   2. frac-times 65.0%

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

   3. associate-/l* 64.3%

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

   4. *-commutative 64.3%

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

6. Applied egg-rr 64.3%

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

   1. *-commutative 64.3%

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

   2. associate-/l* 65.1%

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

   3. associate-*r/ 65.8%

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

   4. *-commutative 65.8%

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

   5. times-frac 65.1%

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

8. Simplified 65.1%

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

   1. add-sqr-sqrt 44.8%

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

   2. sqrt-div 33.8%

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

   3. sqrt-pow1 25.1%

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

   4. metadata-eval 25.1%

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

   5. pow1 25.1%

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

   6. *-commutative 25.1%

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

   7. frac-times 25.1%

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

   8. *-commutative 25.1%

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

   9. associate-*r/ 25.1%

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

   10. sqrt-div 25.1%

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

   11. sqrt-pow1 34.2%

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

   12. metadata-eval 34.2%

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

   13. pow1 34.2%

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

   14. *-commutative 34.2%

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

   15. frac-times 33.8%

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

   16. *-commutative 33.8%

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

   17. associate-*r/ 34.2%

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

10. Applied egg-rr 66.6%

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

11. Final simplification 66.6%

    \[\leadsto \left(1 - 0.5 \cdot \left(h \cdot \left(\left(\frac{D}{d} \cdot \left(0.5 \cdot M\right)\right) \cdot \frac{\frac{D}{d} \cdot \left(0.5 \cdot M\right)}{\ell}\right)\right)\right) \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \]
12. Add Preprocessing

Alternative 10: 46.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -2 \cdot 10^{-158}:\\ \;\;\;\;\frac{d}{-\sqrt{\ell \cdot h}}\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;d \cdot {\left({\left(\ell \cdot h\right)}^{2}\right)}^{-0.25}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left(\sqrt{\frac{1}{h}} \cdot \sqrt{\frac{1}{\ell}}\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= d -2e-158)
   (/ d (- (sqrt (* l h))))
   (if (<= d -5e-310)
     (* d (pow (pow (* l h) 2.0) -0.25))
     (* d (* (sqrt (/ 1.0 h)) (sqrt (/ 1.0 l)))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= -2e-158) {
		tmp = d / -sqrt((l * h));
	} else if (d <= -5e-310) {
		tmp = d * pow(pow((l * h), 2.0), -0.25);
	} else {
		tmp = d * (sqrt((1.0 / h)) * sqrt((1.0 / l)));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (d <= (-2d-158)) then
        tmp = d / -sqrt((l * h))
    else if (d <= (-5d-310)) then
        tmp = d * (((l * h) ** 2.0d0) ** (-0.25d0))
    else
        tmp = d * (sqrt((1.0d0 / h)) * sqrt((1.0d0 / l)))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= -2e-158) {
		tmp = d / -Math.sqrt((l * h));
	} else if (d <= -5e-310) {
		tmp = d * Math.pow(Math.pow((l * h), 2.0), -0.25);
	} else {
		tmp = d * (Math.sqrt((1.0 / h)) * Math.sqrt((1.0 / l)));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if d <= -2e-158:
		tmp = d / -math.sqrt((l * h))
	elif d <= -5e-310:
		tmp = d * math.pow(math.pow((l * h), 2.0), -0.25)
	else:
		tmp = d * (math.sqrt((1.0 / h)) * math.sqrt((1.0 / l)))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (d <= -2e-158)
		tmp = Float64(d / Float64(-sqrt(Float64(l * h))));
	elseif (d <= -5e-310)
		tmp = Float64(d * ((Float64(l * h) ^ 2.0) ^ -0.25));
	else
		tmp = Float64(d * Float64(sqrt(Float64(1.0 / h)) * sqrt(Float64(1.0 / l))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (d <= -2e-158)
		tmp = d / -sqrt((l * h));
	elseif (d <= -5e-310)
		tmp = d * (((l * h) ^ 2.0) ^ -0.25);
	else
		tmp = d * (sqrt((1.0 / h)) * sqrt((1.0 / l)));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[d, -2e-158], N[(d / (-N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], If[LessEqual[d, -5e-310], N[(d * N[Power[N[Power[N[(l * h), $MachinePrecision], 2.0], $MachinePrecision], -0.25], $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Sqrt[N[(1.0 / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(1.0 / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -2.00000000000000013e-158

   1. Initial program 74.6%

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

   3. Simplified 71.4%

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

      1. associate-*r/ 71.6%

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

      2. frac-times 74.7%

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

      3. associate-/l* 71.6%

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

      4. *-commutative 71.6%

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

   6. Applied egg-rr 71.6%

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

      1. *-commutative 71.6%

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

      2. associate-/l* 73.8%

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

      3. associate-*r/ 77.0%

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

      4. *-commutative 77.0%

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

      5. times-frac 73.8%

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

   8. Simplified 73.8%

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

      1. add-sqr-sqrt 30.4%

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

      2. sqrt-div 0.0%

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

      3. sqrt-pow1 0.0%

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

      4. metadata-eval 0.0%

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

      5. pow1 0.0%

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

      6. *-commutative 0.0%

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

      7. frac-times 0.0%

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

      8. *-commutative 0.0%

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

      9. associate-*r/ 0.0%

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

      10. sqrt-div 0.0%

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

      11. sqrt-pow1 0.0%

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

      12. metadata-eval 0.0%

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

      13. pow1 0.0%

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

      14. *-commutative 0.0%

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

      15. frac-times 0.0%

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

      16. *-commutative 0.0%

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

      17. associate-*r/ 0.0%

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

   10. Applied egg-rr 76.0%

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

   11. Taylor expanded in l around -inf 0.0%

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

       1. *-commutative 0.0%

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

       2. unpow2 0.0%

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

       3. rem-square-sqrt 50.5%

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

       4. mul-1-neg 50.5%

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

       5. distribute-lft-neg-in 50.5%

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

       6. rem-exp-log 47.9%

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

       7. exp-neg 47.9%

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

       8. unpow1/2 47.9%

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

       9. exp-prod 47.9%

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

       10. distribute-lft-neg-out 47.9%

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

       11. exp-neg 47.9%

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

       12. exp-to-pow 50.4%

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

       13. unpow1/2 50.4%

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

       14. associate-/l* 50.6%

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

       15. associate-*l/ 50.6%

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

       16. *-rgt-identity 50.6%

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

       17. distribute-neg-frac2 50.6%

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

   13. Simplified 50.6%

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

   if -2.00000000000000013e-158 < d < -4.999999999999985e-310

   1. Initial program 40.0%

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

   3. Simplified 40.0%

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

   5. Taylor expanded in d around inf 18.3%

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

      1. add-cbrt-cube 26.0%

         \[\leadsto d \cdot \color{blue}{\sqrt[3]{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}}} \]

      2. pow1/3 26.0%

         \[\leadsto d \cdot \color{blue}{{\left(\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)}^{0.3333333333333333}} \]

      3. add-sqr-sqrt 26.0%

         \[\leadsto d \cdot {\left(\color{blue}{\frac{1}{h \cdot \ell}} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)}^{0.3333333333333333} \]

      4. pow1 26.0%

         \[\leadsto d \cdot {\left(\color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{1}} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)}^{0.3333333333333333} \]

      5. pow1/2 26.0%

         \[\leadsto d \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1} \cdot \color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{0.5}}\right)}^{0.3333333333333333} \]

      6. pow-prod-up 26.0%

         \[\leadsto d \cdot {\color{blue}{\left({\left(\frac{1}{h \cdot \ell}\right)}^{\left(1 + 0.5\right)}\right)}}^{0.3333333333333333} \]

      7. associate-/r* 26.0%

         \[\leadsto d \cdot {\left({\color{blue}{\left(\frac{\frac{1}{h}}{\ell}\right)}}^{\left(1 + 0.5\right)}\right)}^{0.3333333333333333} \]

      8. metadata-eval 26.0%

         \[\leadsto d \cdot {\left({\left(\frac{\frac{1}{h}}{\ell}\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333} \]

   7. Applied egg-rr 26.0%

      \[\leadsto d \cdot \color{blue}{{\left({\left(\frac{\frac{1}{h}}{\ell}\right)}^{1.5}\right)}^{0.3333333333333333}} \]
   8. Step-by-step derivation

      1. pow-pow 18.3%

         \[\leadsto d \cdot \color{blue}{{\left(\frac{\frac{1}{h}}{\ell}\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} \]

      2. div-inv 18.3%

         \[\leadsto d \cdot {\color{blue}{\left(\frac{1}{h} \cdot \frac{1}{\ell}\right)}}^{\left(1.5 \cdot 0.3333333333333333\right)} \]

      3. metadata-eval 18.3%

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

      4. unpow-prod-down 0.0%

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

      5. pow1/2 0.0%

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

   9. Applied egg-rr 0.0%

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

       1. unpow1/2 0.0%

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

   11. Simplified 0.0%

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

       1. pow1/2 0.0%

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

       2. pow1/2 0.0%

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

       3. inv-pow 0.0%

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

       4. pow-pow 0.0%

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

       5. metadata-eval 0.0%

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

       6. inv-pow 0.0%

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

       7. pow-pow 0.0%

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

       8. metadata-eval 0.0%

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

       9. unpow-prod-down 18.3%

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

       10. sqr-pow 18.3%

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

       11. pow-prod-down 28.6%

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

       12. pow2 28.6%

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

       13. *-commutative 28.6%

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

       14. metadata-eval 28.6%

           \[\leadsto d \cdot {\left({\left(\ell \cdot h\right)}^{2}\right)}^{\color{blue}{-0.25}} \]

   13. Applied egg-rr 28.6%

       \[\leadsto d \cdot \color{blue}{{\left({\left(\ell \cdot h\right)}^{2}\right)}^{-0.25}} \]

   if -4.999999999999985e-310 < d

   1. Initial program 63.1%

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

   3. Simplified 63.8%

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

   5. Taylor expanded in d around inf 35.9%

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

      1. add-cbrt-cube 31.4%

         \[\leadsto d \cdot \color{blue}{\sqrt[3]{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}}} \]

      2. pow1/3 29.9%

         \[\leadsto d \cdot \color{blue}{{\left(\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)}^{0.3333333333333333}} \]

      3. add-sqr-sqrt 29.9%

         \[\leadsto d \cdot {\left(\color{blue}{\frac{1}{h \cdot \ell}} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)}^{0.3333333333333333} \]

      4. pow1 29.9%

         \[\leadsto d \cdot {\left(\color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{1}} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)}^{0.3333333333333333} \]

      5. pow1/2 29.9%

         \[\leadsto d \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1} \cdot \color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{0.5}}\right)}^{0.3333333333333333} \]

      6. pow-prod-up 29.9%

         \[\leadsto d \cdot {\color{blue}{\left({\left(\frac{1}{h \cdot \ell}\right)}^{\left(1 + 0.5\right)}\right)}}^{0.3333333333333333} \]

      7. associate-/r* 29.9%

         \[\leadsto d \cdot {\left({\color{blue}{\left(\frac{\frac{1}{h}}{\ell}\right)}}^{\left(1 + 0.5\right)}\right)}^{0.3333333333333333} \]

      8. metadata-eval 29.9%

         \[\leadsto d \cdot {\left({\left(\frac{\frac{1}{h}}{\ell}\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333} \]

   7. Applied egg-rr 29.9%

      \[\leadsto d \cdot \color{blue}{{\left({\left(\frac{\frac{1}{h}}{\ell}\right)}^{1.5}\right)}^{0.3333333333333333}} \]
   8. Step-by-step derivation

      1. pow-pow 37.2%

         \[\leadsto d \cdot \color{blue}{{\left(\frac{\frac{1}{h}}{\ell}\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} \]

      2. div-inv 37.2%

         \[\leadsto d \cdot {\color{blue}{\left(\frac{1}{h} \cdot \frac{1}{\ell}\right)}}^{\left(1.5 \cdot 0.3333333333333333\right)} \]

      3. metadata-eval 37.2%

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

      4. unpow-prod-down 44.6%

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

      5. pow1/2 44.6%

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

   9. Applied egg-rr 44.6%

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

       1. unpow1/2 44.6%

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

   11. Simplified 44.6%

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

4. Final simplification 44.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2 \cdot 10^{-158}:\\ \;\;\;\;\frac{d}{-\sqrt{\ell \cdot h}}\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;d \cdot {\left({\left(\ell \cdot h\right)}^{2}\right)}^{-0.25}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left(\sqrt{\frac{1}{h}} \cdot \sqrt{\frac{1}{\ell}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 46.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -7.6 \cdot 10^{-153}:\\ \;\;\;\;\frac{d}{-\sqrt{\ell \cdot h}}\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;d \cdot {\left({\left(\ell \cdot h\right)}^{2}\right)}^{-0.25}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= d -7.6e-153)
   (/ d (- (sqrt (* l h))))
   (if (<= d -5e-310)
     (* d (pow (pow (* l h) 2.0) -0.25))
     (* d (* (pow l -0.5) (pow h -0.5))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= -7.6e-153) {
		tmp = d / -sqrt((l * h));
	} else if (d <= -5e-310) {
		tmp = d * pow(pow((l * h), 2.0), -0.25);
	} else {
		tmp = d * (pow(l, -0.5) * pow(h, -0.5));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (d <= (-7.6d-153)) then
        tmp = d / -sqrt((l * h))
    else if (d <= (-5d-310)) then
        tmp = d * (((l * h) ** 2.0d0) ** (-0.25d0))
    else
        tmp = d * ((l ** (-0.5d0)) * (h ** (-0.5d0)))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= -7.6e-153) {
		tmp = d / -Math.sqrt((l * h));
	} else if (d <= -5e-310) {
		tmp = d * Math.pow(Math.pow((l * h), 2.0), -0.25);
	} else {
		tmp = d * (Math.pow(l, -0.5) * Math.pow(h, -0.5));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if d <= -7.6e-153:
		tmp = d / -math.sqrt((l * h))
	elif d <= -5e-310:
		tmp = d * math.pow(math.pow((l * h), 2.0), -0.25)
	else:
		tmp = d * (math.pow(l, -0.5) * math.pow(h, -0.5))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (d <= -7.6e-153)
		tmp = Float64(d / Float64(-sqrt(Float64(l * h))));
	elseif (d <= -5e-310)
		tmp = Float64(d * ((Float64(l * h) ^ 2.0) ^ -0.25));
	else
		tmp = Float64(d * Float64((l ^ -0.5) * (h ^ -0.5)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (d <= -7.6e-153)
		tmp = d / -sqrt((l * h));
	elseif (d <= -5e-310)
		tmp = d * (((l * h) ^ 2.0) ^ -0.25);
	else
		tmp = d * ((l ^ -0.5) * (h ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[d, -7.6e-153], N[(d / (-N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], If[LessEqual[d, -5e-310], N[(d * N[Power[N[Power[N[(l * h), $MachinePrecision], 2.0], $MachinePrecision], -0.25], $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[l, -0.5], $MachinePrecision] * N[Power[h, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -7.60000000000000046e-153

   1. Initial program 74.6%

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

   3. Simplified 71.4%

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

      1. associate-*r/ 71.6%

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

      2. frac-times 74.7%

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

      3. associate-/l* 71.6%

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

      4. *-commutative 71.6%

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

   6. Applied egg-rr 71.6%

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

      1. *-commutative 71.6%

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

      2. associate-/l* 73.8%

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

      3. associate-*r/ 77.0%

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

      4. *-commutative 77.0%

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

      5. times-frac 73.8%

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

   8. Simplified 73.8%

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

      1. add-sqr-sqrt 30.4%

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

      2. sqrt-div 0.0%

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

      3. sqrt-pow1 0.0%

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

      4. metadata-eval 0.0%

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

      5. pow1 0.0%

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

      6. *-commutative 0.0%

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

      7. frac-times 0.0%

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

      8. *-commutative 0.0%

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

      9. associate-*r/ 0.0%

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

      10. sqrt-div 0.0%

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

      11. sqrt-pow1 0.0%

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

      12. metadata-eval 0.0%

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

      13. pow1 0.0%

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

      14. *-commutative 0.0%

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

      15. frac-times 0.0%

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

      16. *-commutative 0.0%

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

      17. associate-*r/ 0.0%

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

   10. Applied egg-rr 76.0%

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

   11. Taylor expanded in l around -inf 0.0%

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

       1. *-commutative 0.0%

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

       2. unpow2 0.0%

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

       3. rem-square-sqrt 50.5%

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

       4. mul-1-neg 50.5%

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

       5. distribute-lft-neg-in 50.5%

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

       6. rem-exp-log 47.9%

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

       7. exp-neg 47.9%

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

       8. unpow1/2 47.9%

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

       9. exp-prod 47.9%

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

       10. distribute-lft-neg-out 47.9%

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

       11. exp-neg 47.9%

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

       12. exp-to-pow 50.4%

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

       13. unpow1/2 50.4%

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

       14. associate-/l* 50.6%

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

       15. associate-*l/ 50.6%

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

       16. *-rgt-identity 50.6%

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

       17. distribute-neg-frac2 50.6%

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

   13. Simplified 50.6%

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

   if -7.60000000000000046e-153 < d < -4.999999999999985e-310

   1. Initial program 40.0%

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

   3. Simplified 40.0%

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

   5. Taylor expanded in d around inf 18.3%

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

      1. add-cbrt-cube 26.0%

         \[\leadsto d \cdot \color{blue}{\sqrt[3]{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}}} \]

      2. pow1/3 26.0%

         \[\leadsto d \cdot \color{blue}{{\left(\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)}^{0.3333333333333333}} \]

      3. add-sqr-sqrt 26.0%

         \[\leadsto d \cdot {\left(\color{blue}{\frac{1}{h \cdot \ell}} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)}^{0.3333333333333333} \]

      4. pow1 26.0%

         \[\leadsto d \cdot {\left(\color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{1}} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)}^{0.3333333333333333} \]

      5. pow1/2 26.0%

         \[\leadsto d \cdot {\left({\left(\frac{1}{h \cdot \ell}\right)}^{1} \cdot \color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{0.5}}\right)}^{0.3333333333333333} \]

      6. pow-prod-up 26.0%

         \[\leadsto d \cdot {\color{blue}{\left({\left(\frac{1}{h \cdot \ell}\right)}^{\left(1 + 0.5\right)}\right)}}^{0.3333333333333333} \]

      7. associate-/r* 26.0%

         \[\leadsto d \cdot {\left({\color{blue}{\left(\frac{\frac{1}{h}}{\ell}\right)}}^{\left(1 + 0.5\right)}\right)}^{0.3333333333333333} \]

      8. metadata-eval 26.0%

         \[\leadsto d \cdot {\left({\left(\frac{\frac{1}{h}}{\ell}\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333} \]

   7. Applied egg-rr 26.0%

      \[\leadsto d \cdot \color{blue}{{\left({\left(\frac{\frac{1}{h}}{\ell}\right)}^{1.5}\right)}^{0.3333333333333333}} \]
   8. Step-by-step derivation

      1. pow-pow 18.3%

         \[\leadsto d \cdot \color{blue}{{\left(\frac{\frac{1}{h}}{\ell}\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} \]

      2. div-inv 18.3%

         \[\leadsto d \cdot {\color{blue}{\left(\frac{1}{h} \cdot \frac{1}{\ell}\right)}}^{\left(1.5 \cdot 0.3333333333333333\right)} \]

      3. metadata-eval 18.3%

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

      4. unpow-prod-down 0.0%

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

      5. pow1/2 0.0%

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

   9. Applied egg-rr 0.0%

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

       1. unpow1/2 0.0%

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

   11. Simplified 0.0%

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

       1. pow1/2 0.0%

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

       2. pow1/2 0.0%

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

       3. inv-pow 0.0%

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

       4. pow-pow 0.0%

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

       5. metadata-eval 0.0%

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

       6. inv-pow 0.0%

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

       7. pow-pow 0.0%

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

       8. metadata-eval 0.0%

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

       9. unpow-prod-down 18.3%

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

       10. sqr-pow 18.3%

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

       11. pow-prod-down 28.6%

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

       12. pow2 28.6%

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

       13. *-commutative 28.6%

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

       14. metadata-eval 28.6%

           \[\leadsto d \cdot {\left({\left(\ell \cdot h\right)}^{2}\right)}^{\color{blue}{-0.25}} \]

   13. Applied egg-rr 28.6%

       \[\leadsto d \cdot \color{blue}{{\left({\left(\ell \cdot h\right)}^{2}\right)}^{-0.25}} \]

   if -4.999999999999985e-310 < d

   1. Initial program 63.1%

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

   3. Simplified 63.8%

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

   6. Applied egg-rr 40.8%

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

   7. Taylor expanded in d around inf 35.9%

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

      1. unpow-1 35.9%

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

      2. metadata-eval 35.9%

         \[\leadsto d \cdot \sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \]

      3. pow-sqr 35.9%

         \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}} \]

      4. rem-sqrt-square 36.3%

         \[\leadsto d \cdot \color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|} \]

      5. rem-square-sqrt 36.2%

         \[\leadsto d \cdot \left|\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right| \]

      6. fabs-sqr 36.2%

         \[\leadsto d \cdot \color{blue}{\left(\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}\right)} \]

      7. rem-square-sqrt 36.3%

         \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

   9. Simplified 36.3%

      \[\leadsto \color{blue}{d \cdot {\left(h \cdot \ell\right)}^{-0.5}} \]
   10. Step-by-step derivation

       1. *-commutative 36.3%

          \[\leadsto d \cdot {\color{blue}{\left(\ell \cdot h\right)}}^{-0.5} \]

       2. unpow-prod-down 44.6%

          \[\leadsto d \cdot \color{blue}{\left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)} \]

   11. Applied egg-rr 44.6%

       \[\leadsto d \cdot \color{blue}{\left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 44.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -7.6 \cdot 10^{-153}:\\ \;\;\;\;\frac{d}{-\sqrt{\ell \cdot h}}\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;d \cdot {\left({\left(\ell \cdot h\right)}^{2}\right)}^{-0.25}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 46.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;h \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\frac{d}{-\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= h -5e-311)
   (/ d (- (sqrt (* l h))))
   (* d (* (pow l -0.5) (pow h -0.5)))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (h <= -5e-311) {
		tmp = d / -sqrt((l * h));
	} else {
		tmp = d * (pow(l, -0.5) * pow(h, -0.5));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (h <= (-5d-311)) then
        tmp = d / -sqrt((l * h))
    else
        tmp = d * ((l ** (-0.5d0)) * (h ** (-0.5d0)))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (h <= -5e-311) {
		tmp = d / -Math.sqrt((l * h));
	} else {
		tmp = d * (Math.pow(l, -0.5) * Math.pow(h, -0.5));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if h <= -5e-311:
		tmp = d / -math.sqrt((l * h))
	else:
		tmp = d * (math.pow(l, -0.5) * math.pow(h, -0.5))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (h <= -5e-311)
		tmp = Float64(d / Float64(-sqrt(Float64(l * h))));
	else
		tmp = Float64(d * Float64((l ^ -0.5) * (h ^ -0.5)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (h <= -5e-311)
		tmp = d / -sqrt((l * h));
	else
		tmp = d * ((l ^ -0.5) * (h ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[h, -5e-311], N[(d / (-N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(d * N[(N[Power[l, -0.5], $MachinePrecision] * N[Power[h, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if h < -5.00000000000023e-311

   1. Initial program 64.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 62.0%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r/ 62.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot h}{\ell}}\right) \]

      2. frac-times 64.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot h}{\ell}\right) \]

      3. associate-/l* 62.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)}}^{2} \cdot h}{\ell}\right) \]

      4. *-commutative 62.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(M \cdot \frac{D}{\color{blue}{d \cdot 2}}\right)}^{2} \cdot h}{\ell}\right) \]

   6. Applied egg-rr 62.2%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot h}{\ell}}\right) \]
   7. Step-by-step derivation

      1. *-commutative 62.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{\color{blue}{h \cdot {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}}}{\ell}\right) \]

      2. associate-/l* 63.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(h \cdot \frac{{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}}{\ell}\right)}\right) \]

      3. associate-*r/ 65.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{{\color{blue}{\left(\frac{M \cdot D}{d \cdot 2}\right)}}^{2}}{\ell}\right)\right) \]

      4. *-commutative 65.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{{\left(\frac{\color{blue}{D \cdot M}}{d \cdot 2}\right)}^{2}}{\ell}\right)\right) \]

      5. times-frac 63.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{{\color{blue}{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}}^{2}}{\ell}\right)\right) \]

   8. Simplified 63.0%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(h \cdot \frac{{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}}{\ell}\right)}\right) \]
   9. Step-by-step derivation

      1. add-sqr-sqrt 22.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \color{blue}{\left(\sqrt{\frac{{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}}{\ell}} \cdot \sqrt{\frac{{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}}{\ell}}\right)}\right)\right) \]

      2. sqrt-div 0.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \left(\color{blue}{\frac{\sqrt{{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}}}{\sqrt{\ell}}} \cdot \sqrt{\frac{{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}}{\ell}}\right)\right)\right) \]

      3. sqrt-pow1 0.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \left(\frac{\color{blue}{{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{\left(\frac{2}{2}\right)}}}{\sqrt{\ell}} \cdot \sqrt{\frac{{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}}{\ell}}\right)\right)\right) \]

      4. metadata-eval 0.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \left(\frac{{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{\color{blue}{1}}}{\sqrt{\ell}} \cdot \sqrt{\frac{{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}}{\ell}}\right)\right)\right) \]

      5. pow1 0.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \left(\frac{\color{blue}{\frac{D}{d} \cdot \frac{M}{2}}}{\sqrt{\ell}} \cdot \sqrt{\frac{{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}}{\ell}}\right)\right)\right) \]

      6. *-commutative 0.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \left(\frac{\color{blue}{\frac{M}{2} \cdot \frac{D}{d}}}{\sqrt{\ell}} \cdot \sqrt{\frac{{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}}{\ell}}\right)\right)\right) \]

      7. frac-times 0.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \left(\frac{\color{blue}{\frac{M \cdot D}{2 \cdot d}}}{\sqrt{\ell}} \cdot \sqrt{\frac{{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}}{\ell}}\right)\right)\right) \]

      8. *-commutative 0.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \left(\frac{\frac{M \cdot D}{\color{blue}{d \cdot 2}}}{\sqrt{\ell}} \cdot \sqrt{\frac{{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}}{\ell}}\right)\right)\right) \]

      9. associate-*r/ 0.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \left(\frac{\color{blue}{M \cdot \frac{D}{d \cdot 2}}}{\sqrt{\ell}} \cdot \sqrt{\frac{{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}}{\ell}}\right)\right)\right) \]

      10. sqrt-div 0.0%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \left(\frac{M \cdot \frac{D}{d \cdot 2}}{\sqrt{\ell}} \cdot \color{blue}{\frac{\sqrt{{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}}}{\sqrt{\ell}}}\right)\right)\right) \]

      11. sqrt-pow1 0.0%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \left(\frac{M \cdot \frac{D}{d \cdot 2}}{\sqrt{\ell}} \cdot \frac{\color{blue}{{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{\left(\frac{2}{2}\right)}}}{\sqrt{\ell}}\right)\right)\right) \]

      12. metadata-eval 0.0%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \left(\frac{M \cdot \frac{D}{d \cdot 2}}{\sqrt{\ell}} \cdot \frac{{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{\color{blue}{1}}}{\sqrt{\ell}}\right)\right)\right) \]

      13. pow1 0.0%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \left(\frac{M \cdot \frac{D}{d \cdot 2}}{\sqrt{\ell}} \cdot \frac{\color{blue}{\frac{D}{d} \cdot \frac{M}{2}}}{\sqrt{\ell}}\right)\right)\right) \]

      14. *-commutative 0.0%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \left(\frac{M \cdot \frac{D}{d \cdot 2}}{\sqrt{\ell}} \cdot \frac{\color{blue}{\frac{M}{2} \cdot \frac{D}{d}}}{\sqrt{\ell}}\right)\right)\right) \]

      15. frac-times 0.0%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \left(\frac{M \cdot \frac{D}{d \cdot 2}}{\sqrt{\ell}} \cdot \frac{\color{blue}{\frac{M \cdot D}{2 \cdot d}}}{\sqrt{\ell}}\right)\right)\right) \]

      16. *-commutative 0.0%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \left(\frac{M \cdot \frac{D}{d \cdot 2}}{\sqrt{\ell}} \cdot \frac{\frac{M \cdot D}{\color{blue}{d \cdot 2}}}{\sqrt{\ell}}\right)\right)\right) \]

      17. associate-*r/ 0.0%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \left(\frac{M \cdot \frac{D}{d \cdot 2}}{\sqrt{\ell}} \cdot \frac{\color{blue}{M \cdot \frac{D}{d \cdot 2}}}{\sqrt{\ell}}\right)\right)\right) \]

   10. Applied egg-rr 65.4%

       \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \color{blue}{\left(\frac{\frac{D}{d} \cdot \left(M \cdot 0.5\right)}{1} \cdot \frac{\frac{D}{d} \cdot \left(M \cdot 0.5\right)}{\ell}\right)}\right)\right) \]

   11. Taylor expanded in l around -inf 0.0%

       \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   12. Step-by-step derivation

       1. *-commutative 0.0%

          \[\leadsto \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot d\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

       2. unpow2 0.0%

          \[\leadsto \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

       3. rem-square-sqrt 40.5%

          \[\leadsto \left(\color{blue}{-1} \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

       4. mul-1-neg 40.5%

          \[\leadsto \color{blue}{\left(-d\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

       5. distribute-lft-neg-in 40.5%

          \[\leadsto \color{blue}{-d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

       6. rem-exp-log 38.5%

          \[\leadsto -d \cdot \sqrt{\frac{1}{\color{blue}{e^{\log \left(h \cdot \ell\right)}}}} \]

       7. exp-neg 38.5%

          \[\leadsto -d \cdot \sqrt{\color{blue}{e^{-\log \left(h \cdot \ell\right)}}} \]

       8. unpow1/2 38.5%

          \[\leadsto -d \cdot \color{blue}{{\left(e^{-\log \left(h \cdot \ell\right)}\right)}^{0.5}} \]

       9. exp-prod 38.5%

          \[\leadsto -d \cdot \color{blue}{e^{\left(-\log \left(h \cdot \ell\right)\right) \cdot 0.5}} \]

       10. distribute-lft-neg-out 38.5%

           \[\leadsto -d \cdot e^{\color{blue}{-\log \left(h \cdot \ell\right) \cdot 0.5}} \]

       11. exp-neg 38.5%

           \[\leadsto -d \cdot \color{blue}{\frac{1}{e^{\log \left(h \cdot \ell\right) \cdot 0.5}}} \]

       12. exp-to-pow 40.4%

           \[\leadsto -d \cdot \frac{1}{\color{blue}{{\left(h \cdot \ell\right)}^{0.5}}} \]

       13. unpow1/2 40.4%

           \[\leadsto -d \cdot \frac{1}{\color{blue}{\sqrt{h \cdot \ell}}} \]

       14. associate-/l* 40.5%

           \[\leadsto -\color{blue}{\frac{d \cdot 1}{\sqrt{h \cdot \ell}}} \]

       15. associate-*l/ 40.5%

           \[\leadsto -\color{blue}{\frac{d}{\sqrt{h \cdot \ell}} \cdot 1} \]

       16. *-rgt-identity 40.5%

           \[\leadsto -\color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]

       17. distribute-neg-frac2 40.5%

           \[\leadsto \color{blue}{\frac{d}{-\sqrt{h \cdot \ell}}} \]

   13. Simplified 40.5%

       \[\leadsto \color{blue}{\frac{d}{-\sqrt{h \cdot \ell}}} \]

   if -5.00000000000023e-311 < h

   1. Initial program 63.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 63.8%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing

   6. Applied egg-rr 40.8%

      \[\leadsto \color{blue}{{\left(\sqrt{\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}\right)\right)}\right)}^{2}} \]

   7. Taylor expanded in d around inf 35.9%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   8. Step-by-step derivation

      1. unpow-1 35.9%

         \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \]

      2. metadata-eval 35.9%

         \[\leadsto d \cdot \sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \]

      3. pow-sqr 35.9%

         \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}} \]

      4. rem-sqrt-square 36.3%

         \[\leadsto d \cdot \color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|} \]

      5. rem-square-sqrt 36.2%

         \[\leadsto d \cdot \left|\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right| \]

      6. fabs-sqr 36.2%

         \[\leadsto d \cdot \color{blue}{\left(\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}\right)} \]

      7. rem-square-sqrt 36.3%

         \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

   9. Simplified 36.3%

      \[\leadsto \color{blue}{d \cdot {\left(h \cdot \ell\right)}^{-0.5}} \]
   10. Step-by-step derivation

       1. *-commutative 36.3%

          \[\leadsto d \cdot {\color{blue}{\left(\ell \cdot h\right)}}^{-0.5} \]

       2. unpow-prod-down 44.6%

          \[\leadsto d \cdot \color{blue}{\left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)} \]

   11. Applied egg-rr 44.6%

       \[\leadsto d \cdot \color{blue}{\left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 42.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\frac{d}{-\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 46.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;h \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\frac{d}{-\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= h -5e-311) (/ d (- (sqrt (* l h)))) (/ d (* (sqrt l) (sqrt h)))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (h <= -5e-311) {
		tmp = d / -sqrt((l * h));
	} else {
		tmp = d / (sqrt(l) * sqrt(h));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (h <= (-5d-311)) then
        tmp = d / -sqrt((l * h))
    else
        tmp = d / (sqrt(l) * sqrt(h))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (h <= -5e-311) {
		tmp = d / -Math.sqrt((l * h));
	} else {
		tmp = d / (Math.sqrt(l) * Math.sqrt(h));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if h <= -5e-311:
		tmp = d / -math.sqrt((l * h))
	else:
		tmp = d / (math.sqrt(l) * math.sqrt(h))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (h <= -5e-311)
		tmp = Float64(d / Float64(-sqrt(Float64(l * h))));
	else
		tmp = Float64(d / Float64(sqrt(l) * sqrt(h)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (h <= -5e-311)
		tmp = d / -sqrt((l * h));
	else
		tmp = d / (sqrt(l) * sqrt(h));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[h, -5e-311], N[(d / (-N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if h < -5.00000000000023e-311

   1. Initial program 64.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 62.0%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r/ 62.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot h}{\ell}}\right) \]

      2. frac-times 64.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot h}{\ell}\right) \]

      3. associate-/l* 62.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)}}^{2} \cdot h}{\ell}\right) \]

      4. *-commutative 62.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(M \cdot \frac{D}{\color{blue}{d \cdot 2}}\right)}^{2} \cdot h}{\ell}\right) \]

   6. Applied egg-rr 62.2%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot h}{\ell}}\right) \]
   7. Step-by-step derivation

      1. *-commutative 62.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{\color{blue}{h \cdot {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}}}{\ell}\right) \]

      2. associate-/l* 63.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(h \cdot \frac{{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}}{\ell}\right)}\right) \]

      3. associate-*r/ 65.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{{\color{blue}{\left(\frac{M \cdot D}{d \cdot 2}\right)}}^{2}}{\ell}\right)\right) \]

      4. *-commutative 65.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{{\left(\frac{\color{blue}{D \cdot M}}{d \cdot 2}\right)}^{2}}{\ell}\right)\right) \]

      5. times-frac 63.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{{\color{blue}{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}}^{2}}{\ell}\right)\right) \]

   8. Simplified 63.0%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(h \cdot \frac{{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}}{\ell}\right)}\right) \]
   9. Step-by-step derivation

      1. add-sqr-sqrt 22.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \color{blue}{\left(\sqrt{\frac{{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}}{\ell}} \cdot \sqrt{\frac{{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}}{\ell}}\right)}\right)\right) \]

      2. sqrt-div 0.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \left(\color{blue}{\frac{\sqrt{{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}}}{\sqrt{\ell}}} \cdot \sqrt{\frac{{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}}{\ell}}\right)\right)\right) \]

      3. sqrt-pow1 0.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \left(\frac{\color{blue}{{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{\left(\frac{2}{2}\right)}}}{\sqrt{\ell}} \cdot \sqrt{\frac{{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}}{\ell}}\right)\right)\right) \]

      4. metadata-eval 0.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \left(\frac{{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{\color{blue}{1}}}{\sqrt{\ell}} \cdot \sqrt{\frac{{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}}{\ell}}\right)\right)\right) \]

      5. pow1 0.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \left(\frac{\color{blue}{\frac{D}{d} \cdot \frac{M}{2}}}{\sqrt{\ell}} \cdot \sqrt{\frac{{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}}{\ell}}\right)\right)\right) \]

      6. *-commutative 0.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \left(\frac{\color{blue}{\frac{M}{2} \cdot \frac{D}{d}}}{\sqrt{\ell}} \cdot \sqrt{\frac{{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}}{\ell}}\right)\right)\right) \]

      7. frac-times 0.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \left(\frac{\color{blue}{\frac{M \cdot D}{2 \cdot d}}}{\sqrt{\ell}} \cdot \sqrt{\frac{{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}}{\ell}}\right)\right)\right) \]

      8. *-commutative 0.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \left(\frac{\frac{M \cdot D}{\color{blue}{d \cdot 2}}}{\sqrt{\ell}} \cdot \sqrt{\frac{{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}}{\ell}}\right)\right)\right) \]

      9. associate-*r/ 0.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \left(\frac{\color{blue}{M \cdot \frac{D}{d \cdot 2}}}{\sqrt{\ell}} \cdot \sqrt{\frac{{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}}{\ell}}\right)\right)\right) \]

      10. sqrt-div 0.0%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \left(\frac{M \cdot \frac{D}{d \cdot 2}}{\sqrt{\ell}} \cdot \color{blue}{\frac{\sqrt{{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}}}{\sqrt{\ell}}}\right)\right)\right) \]

      11. sqrt-pow1 0.0%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \left(\frac{M \cdot \frac{D}{d \cdot 2}}{\sqrt{\ell}} \cdot \frac{\color{blue}{{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{\left(\frac{2}{2}\right)}}}{\sqrt{\ell}}\right)\right)\right) \]

      12. metadata-eval 0.0%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \left(\frac{M \cdot \frac{D}{d \cdot 2}}{\sqrt{\ell}} \cdot \frac{{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{\color{blue}{1}}}{\sqrt{\ell}}\right)\right)\right) \]

      13. pow1 0.0%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \left(\frac{M \cdot \frac{D}{d \cdot 2}}{\sqrt{\ell}} \cdot \frac{\color{blue}{\frac{D}{d} \cdot \frac{M}{2}}}{\sqrt{\ell}}\right)\right)\right) \]

      14. *-commutative 0.0%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \left(\frac{M \cdot \frac{D}{d \cdot 2}}{\sqrt{\ell}} \cdot \frac{\color{blue}{\frac{M}{2} \cdot \frac{D}{d}}}{\sqrt{\ell}}\right)\right)\right) \]

      15. frac-times 0.0%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \left(\frac{M \cdot \frac{D}{d \cdot 2}}{\sqrt{\ell}} \cdot \frac{\color{blue}{\frac{M \cdot D}{2 \cdot d}}}{\sqrt{\ell}}\right)\right)\right) \]

      16. *-commutative 0.0%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \left(\frac{M \cdot \frac{D}{d \cdot 2}}{\sqrt{\ell}} \cdot \frac{\frac{M \cdot D}{\color{blue}{d \cdot 2}}}{\sqrt{\ell}}\right)\right)\right) \]

      17. associate-*r/ 0.0%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \left(\frac{M \cdot \frac{D}{d \cdot 2}}{\sqrt{\ell}} \cdot \frac{\color{blue}{M \cdot \frac{D}{d \cdot 2}}}{\sqrt{\ell}}\right)\right)\right) \]

   10. Applied egg-rr 65.4%

       \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \color{blue}{\left(\frac{\frac{D}{d} \cdot \left(M \cdot 0.5\right)}{1} \cdot \frac{\frac{D}{d} \cdot \left(M \cdot 0.5\right)}{\ell}\right)}\right)\right) \]

   11. Taylor expanded in l around -inf 0.0%

       \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   12. Step-by-step derivation

       1. *-commutative 0.0%

          \[\leadsto \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot d\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

       2. unpow2 0.0%

          \[\leadsto \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

       3. rem-square-sqrt 40.5%

          \[\leadsto \left(\color{blue}{-1} \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

       4. mul-1-neg 40.5%

          \[\leadsto \color{blue}{\left(-d\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

       5. distribute-lft-neg-in 40.5%

          \[\leadsto \color{blue}{-d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

       6. rem-exp-log 38.5%

          \[\leadsto -d \cdot \sqrt{\frac{1}{\color{blue}{e^{\log \left(h \cdot \ell\right)}}}} \]

       7. exp-neg 38.5%

          \[\leadsto -d \cdot \sqrt{\color{blue}{e^{-\log \left(h \cdot \ell\right)}}} \]

       8. unpow1/2 38.5%

          \[\leadsto -d \cdot \color{blue}{{\left(e^{-\log \left(h \cdot \ell\right)}\right)}^{0.5}} \]

       9. exp-prod 38.5%

          \[\leadsto -d \cdot \color{blue}{e^{\left(-\log \left(h \cdot \ell\right)\right) \cdot 0.5}} \]

       10. distribute-lft-neg-out 38.5%

           \[\leadsto -d \cdot e^{\color{blue}{-\log \left(h \cdot \ell\right) \cdot 0.5}} \]

       11. exp-neg 38.5%

           \[\leadsto -d \cdot \color{blue}{\frac{1}{e^{\log \left(h \cdot \ell\right) \cdot 0.5}}} \]

       12. exp-to-pow 40.4%

           \[\leadsto -d \cdot \frac{1}{\color{blue}{{\left(h \cdot \ell\right)}^{0.5}}} \]

       13. unpow1/2 40.4%

           \[\leadsto -d \cdot \frac{1}{\color{blue}{\sqrt{h \cdot \ell}}} \]

       14. associate-/l* 40.5%

           \[\leadsto -\color{blue}{\frac{d \cdot 1}{\sqrt{h \cdot \ell}}} \]

       15. associate-*l/ 40.5%

           \[\leadsto -\color{blue}{\frac{d}{\sqrt{h \cdot \ell}} \cdot 1} \]

       16. *-rgt-identity 40.5%

           \[\leadsto -\color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]

       17. distribute-neg-frac2 40.5%

           \[\leadsto \color{blue}{\frac{d}{-\sqrt{h \cdot \ell}}} \]

   13. Simplified 40.5%

       \[\leadsto \color{blue}{\frac{d}{-\sqrt{h \cdot \ell}}} \]

   if -5.00000000000023e-311 < h

   1. Initial program 63.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 63.8%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 35.9%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. sqrt-div 36.3%

         \[\leadsto d \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{h \cdot \ell}}} \]

      2. metadata-eval 36.3%

         \[\leadsto d \cdot \frac{\color{blue}{1}}{\sqrt{h \cdot \ell}} \]

      3. sqrt-unprod 44.5%

         \[\leadsto d \cdot \frac{1}{\color{blue}{\sqrt{h} \cdot \sqrt{\ell}}} \]

      4. div-inv 44.6%

         \[\leadsto \color{blue}{\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}} \]

      5. sqrt-unprod 36.4%

         \[\leadsto \frac{d}{\color{blue}{\sqrt{h \cdot \ell}}} \]

   7. Applied egg-rr 36.4%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
   8. Step-by-step derivation

      1. sqrt-prod 44.6%

         \[\leadsto \frac{d}{\color{blue}{\sqrt{h} \cdot \sqrt{\ell}}} \]

      2. *-commutative 44.6%

         \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell} \cdot \sqrt{h}}} \]

   9. Applied egg-rr 44.6%

      \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell} \cdot \sqrt{h}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 42.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\frac{d}{-\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 42.3% accurate, 3.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 2.25 \cdot 10^{-204}:\\ \;\;\;\;\frac{d}{-\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= l 2.25e-204) (/ d (- (sqrt (* l h)))) (* d (sqrt (/ (/ 1.0 h) l)))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= 2.25e-204) {
		tmp = d / -sqrt((l * h));
	} else {
		tmp = d * sqrt(((1.0 / h) / l));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (l <= 2.25d-204) then
        tmp = d / -sqrt((l * h))
    else
        tmp = d * sqrt(((1.0d0 / h) / l))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= 2.25e-204) {
		tmp = d / -Math.sqrt((l * h));
	} else {
		tmp = d * Math.sqrt(((1.0 / h) / l));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if l <= 2.25e-204:
		tmp = d / -math.sqrt((l * h))
	else:
		tmp = d * math.sqrt(((1.0 / h) / l))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (l <= 2.25e-204)
		tmp = Float64(d / Float64(-sqrt(Float64(l * h))));
	else
		tmp = Float64(d * sqrt(Float64(Float64(1.0 / h) / l)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (l <= 2.25e-204)
		tmp = d / -sqrt((l * h));
	else
		tmp = d * sqrt(((1.0 / h) / l));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[l, 2.25e-204], N[(d / (-N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(d * N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 2.24999999999999987e-204

   1. Initial program 67.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 65.2%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r/ 66.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot h}{\ell}}\right) \]

      2. frac-times 67.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot h}{\ell}\right) \]

      3. associate-/l* 66.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)}}^{2} \cdot h}{\ell}\right) \]

      4. *-commutative 66.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(M \cdot \frac{D}{\color{blue}{d \cdot 2}}\right)}^{2} \cdot h}{\ell}\right) \]

   6. Applied egg-rr 66.1%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot h}{\ell}}\right) \]
   7. Step-by-step derivation

      1. *-commutative 66.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{\color{blue}{h \cdot {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}}}{\ell}\right) \]

      2. associate-/l* 66.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(h \cdot \frac{{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}}{\ell}\right)}\right) \]

      3. associate-*r/ 68.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{{\color{blue}{\left(\frac{M \cdot D}{d \cdot 2}\right)}}^{2}}{\ell}\right)\right) \]

      4. *-commutative 68.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{{\left(\frac{\color{blue}{D \cdot M}}{d \cdot 2}\right)}^{2}}{\ell}\right)\right) \]

      5. times-frac 66.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{{\color{blue}{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}}^{2}}{\ell}\right)\right) \]

   8. Simplified 66.8%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(h \cdot \frac{{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}}{\ell}\right)}\right) \]
   9. Step-by-step derivation

      1. add-sqr-sqrt 32.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \color{blue}{\left(\sqrt{\frac{{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}}{\ell}} \cdot \sqrt{\frac{{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}}{\ell}}\right)}\right)\right) \]

      2. sqrt-div 13.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \left(\color{blue}{\frac{\sqrt{{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}}}{\sqrt{\ell}}} \cdot \sqrt{\frac{{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}}{\ell}}\right)\right)\right) \]

      3. sqrt-pow1 8.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \left(\frac{\color{blue}{{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{\left(\frac{2}{2}\right)}}}{\sqrt{\ell}} \cdot \sqrt{\frac{{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}}{\ell}}\right)\right)\right) \]

      4. metadata-eval 8.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \left(\frac{{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{\color{blue}{1}}}{\sqrt{\ell}} \cdot \sqrt{\frac{{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}}{\ell}}\right)\right)\right) \]

      5. pow1 8.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \left(\frac{\color{blue}{\frac{D}{d} \cdot \frac{M}{2}}}{\sqrt{\ell}} \cdot \sqrt{\frac{{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}}{\ell}}\right)\right)\right) \]

      6. *-commutative 8.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \left(\frac{\color{blue}{\frac{M}{2} \cdot \frac{D}{d}}}{\sqrt{\ell}} \cdot \sqrt{\frac{{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}}{\ell}}\right)\right)\right) \]

      7. frac-times 8.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \left(\frac{\color{blue}{\frac{M \cdot D}{2 \cdot d}}}{\sqrt{\ell}} \cdot \sqrt{\frac{{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}}{\ell}}\right)\right)\right) \]

      8. *-commutative 8.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \left(\frac{\frac{M \cdot D}{\color{blue}{d \cdot 2}}}{\sqrt{\ell}} \cdot \sqrt{\frac{{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}}{\ell}}\right)\right)\right) \]

      9. associate-*r/ 8.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \left(\frac{\color{blue}{M \cdot \frac{D}{d \cdot 2}}}{\sqrt{\ell}} \cdot \sqrt{\frac{{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}}{\ell}}\right)\right)\right) \]

      10. sqrt-div 8.0%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \left(\frac{M \cdot \frac{D}{d \cdot 2}}{\sqrt{\ell}} \cdot \color{blue}{\frac{\sqrt{{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}}}{\sqrt{\ell}}}\right)\right)\right) \]

      11. sqrt-pow1 13.4%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \left(\frac{M \cdot \frac{D}{d \cdot 2}}{\sqrt{\ell}} \cdot \frac{\color{blue}{{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{\left(\frac{2}{2}\right)}}}{\sqrt{\ell}}\right)\right)\right) \]

      12. metadata-eval 13.4%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \left(\frac{M \cdot \frac{D}{d \cdot 2}}{\sqrt{\ell}} \cdot \frac{{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{\color{blue}{1}}}{\sqrt{\ell}}\right)\right)\right) \]

      13. pow1 13.4%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \left(\frac{M \cdot \frac{D}{d \cdot 2}}{\sqrt{\ell}} \cdot \frac{\color{blue}{\frac{D}{d} \cdot \frac{M}{2}}}{\sqrt{\ell}}\right)\right)\right) \]

      14. *-commutative 13.4%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \left(\frac{M \cdot \frac{D}{d \cdot 2}}{\sqrt{\ell}} \cdot \frac{\color{blue}{\frac{M}{2} \cdot \frac{D}{d}}}{\sqrt{\ell}}\right)\right)\right) \]

      15. frac-times 13.4%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \left(\frac{M \cdot \frac{D}{d \cdot 2}}{\sqrt{\ell}} \cdot \frac{\color{blue}{\frac{M \cdot D}{2 \cdot d}}}{\sqrt{\ell}}\right)\right)\right) \]

      16. *-commutative 13.4%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \left(\frac{M \cdot \frac{D}{d \cdot 2}}{\sqrt{\ell}} \cdot \frac{\frac{M \cdot D}{\color{blue}{d \cdot 2}}}{\sqrt{\ell}}\right)\right)\right) \]

      17. associate-*r/ 13.4%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \left(\frac{M \cdot \frac{D}{d \cdot 2}}{\sqrt{\ell}} \cdot \frac{\color{blue}{M \cdot \frac{D}{d \cdot 2}}}{\sqrt{\ell}}\right)\right)\right) \]

   10. Applied egg-rr 68.7%

       \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \color{blue}{\left(\frac{\frac{D}{d} \cdot \left(M \cdot 0.5\right)}{1} \cdot \frac{\frac{D}{d} \cdot \left(M \cdot 0.5\right)}{\ell}\right)}\right)\right) \]

   11. Taylor expanded in l around -inf 0.0%

       \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   12. Step-by-step derivation

       1. *-commutative 0.0%

          \[\leadsto \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot d\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

       2. unpow2 0.0%

          \[\leadsto \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

       3. rem-square-sqrt 41.1%

          \[\leadsto \left(\color{blue}{-1} \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

       4. mul-1-neg 41.1%

          \[\leadsto \color{blue}{\left(-d\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

       5. distribute-lft-neg-in 41.1%

          \[\leadsto \color{blue}{-d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

       6. rem-exp-log 39.4%

          \[\leadsto -d \cdot \sqrt{\frac{1}{\color{blue}{e^{\log \left(h \cdot \ell\right)}}}} \]

       7. exp-neg 39.4%

          \[\leadsto -d \cdot \sqrt{\color{blue}{e^{-\log \left(h \cdot \ell\right)}}} \]

       8. unpow1/2 39.4%

          \[\leadsto -d \cdot \color{blue}{{\left(e^{-\log \left(h \cdot \ell\right)}\right)}^{0.5}} \]

       9. exp-prod 39.4%

          \[\leadsto -d \cdot \color{blue}{e^{\left(-\log \left(h \cdot \ell\right)\right) \cdot 0.5}} \]

       10. distribute-lft-neg-out 39.4%

           \[\leadsto -d \cdot e^{\color{blue}{-\log \left(h \cdot \ell\right) \cdot 0.5}} \]

       11. exp-neg 39.4%

           \[\leadsto -d \cdot \color{blue}{\frac{1}{e^{\log \left(h \cdot \ell\right) \cdot 0.5}}} \]

       12. exp-to-pow 41.0%

           \[\leadsto -d \cdot \frac{1}{\color{blue}{{\left(h \cdot \ell\right)}^{0.5}}} \]

       13. unpow1/2 41.0%

           \[\leadsto -d \cdot \frac{1}{\color{blue}{\sqrt{h \cdot \ell}}} \]

       14. associate-/l* 41.1%

           \[\leadsto -\color{blue}{\frac{d \cdot 1}{\sqrt{h \cdot \ell}}} \]

       15. associate-*l/ 41.1%

           \[\leadsto -\color{blue}{\frac{d}{\sqrt{h \cdot \ell}} \cdot 1} \]

       16. *-rgt-identity 41.1%

           \[\leadsto -\color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]

       17. distribute-neg-frac2 41.1%

           \[\leadsto \color{blue}{\frac{d}{-\sqrt{h \cdot \ell}}} \]

   13. Simplified 41.1%

       \[\leadsto \color{blue}{\frac{d}{-\sqrt{h \cdot \ell}}} \]

   if 2.24999999999999987e-204 < l

   1. Initial program 58.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 59.6%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 38.8%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. associate-/r* 40.4%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \]

   7. Simplified 40.4%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 40.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2.25 \cdot 10^{-204}:\\ \;\;\;\;\frac{d}{-\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 42.2% accurate, 3.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 3.1 \cdot 10^{-203}:\\ \;\;\;\;\frac{d}{-\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= l 3.1e-203) (/ d (- (sqrt (* l h)))) (* d (sqrt (/ (/ 1.0 l) h)))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= 3.1e-203) {
		tmp = d / -sqrt((l * h));
	} else {
		tmp = d * sqrt(((1.0 / l) / h));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (l <= 3.1d-203) then
        tmp = d / -sqrt((l * h))
    else
        tmp = d * sqrt(((1.0d0 / l) / h))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= 3.1e-203) {
		tmp = d / -Math.sqrt((l * h));
	} else {
		tmp = d * Math.sqrt(((1.0 / l) / h));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if l <= 3.1e-203:
		tmp = d / -math.sqrt((l * h))
	else:
		tmp = d * math.sqrt(((1.0 / l) / h))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (l <= 3.1e-203)
		tmp = Float64(d / Float64(-sqrt(Float64(l * h))));
	else
		tmp = Float64(d * sqrt(Float64(Float64(1.0 / l) / h)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (l <= 3.1e-203)
		tmp = d / -sqrt((l * h));
	else
		tmp = d * sqrt(((1.0 / l) / h));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[l, 3.1e-203], N[(d / (-N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(d * N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 3.09999999999999977e-203

   1. Initial program 67.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 65.2%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r/ 66.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot h}{\ell}}\right) \]

      2. frac-times 67.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot h}{\ell}\right) \]

      3. associate-/l* 66.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)}}^{2} \cdot h}{\ell}\right) \]

      4. *-commutative 66.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(M \cdot \frac{D}{\color{blue}{d \cdot 2}}\right)}^{2} \cdot h}{\ell}\right) \]

   6. Applied egg-rr 66.1%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot h}{\ell}}\right) \]
   7. Step-by-step derivation

      1. *-commutative 66.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{\color{blue}{h \cdot {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}}}{\ell}\right) \]

      2. associate-/l* 66.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(h \cdot \frac{{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}}{\ell}\right)}\right) \]

      3. associate-*r/ 68.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{{\color{blue}{\left(\frac{M \cdot D}{d \cdot 2}\right)}}^{2}}{\ell}\right)\right) \]

      4. *-commutative 68.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{{\left(\frac{\color{blue}{D \cdot M}}{d \cdot 2}\right)}^{2}}{\ell}\right)\right) \]

      5. times-frac 66.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{{\color{blue}{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}}^{2}}{\ell}\right)\right) \]

   8. Simplified 66.8%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(h \cdot \frac{{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}}{\ell}\right)}\right) \]
   9. Step-by-step derivation

      1. add-sqr-sqrt 32.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \color{blue}{\left(\sqrt{\frac{{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}}{\ell}} \cdot \sqrt{\frac{{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}}{\ell}}\right)}\right)\right) \]

      2. sqrt-div 13.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \left(\color{blue}{\frac{\sqrt{{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}}}{\sqrt{\ell}}} \cdot \sqrt{\frac{{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}}{\ell}}\right)\right)\right) \]

      3. sqrt-pow1 8.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \left(\frac{\color{blue}{{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{\left(\frac{2}{2}\right)}}}{\sqrt{\ell}} \cdot \sqrt{\frac{{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}}{\ell}}\right)\right)\right) \]

      4. metadata-eval 8.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \left(\frac{{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{\color{blue}{1}}}{\sqrt{\ell}} \cdot \sqrt{\frac{{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}}{\ell}}\right)\right)\right) \]

      5. pow1 8.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \left(\frac{\color{blue}{\frac{D}{d} \cdot \frac{M}{2}}}{\sqrt{\ell}} \cdot \sqrt{\frac{{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}}{\ell}}\right)\right)\right) \]

      6. *-commutative 8.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \left(\frac{\color{blue}{\frac{M}{2} \cdot \frac{D}{d}}}{\sqrt{\ell}} \cdot \sqrt{\frac{{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}}{\ell}}\right)\right)\right) \]

      7. frac-times 8.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \left(\frac{\color{blue}{\frac{M \cdot D}{2 \cdot d}}}{\sqrt{\ell}} \cdot \sqrt{\frac{{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}}{\ell}}\right)\right)\right) \]

      8. *-commutative 8.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \left(\frac{\frac{M \cdot D}{\color{blue}{d \cdot 2}}}{\sqrt{\ell}} \cdot \sqrt{\frac{{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}}{\ell}}\right)\right)\right) \]

      9. associate-*r/ 8.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \left(\frac{\color{blue}{M \cdot \frac{D}{d \cdot 2}}}{\sqrt{\ell}} \cdot \sqrt{\frac{{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}}{\ell}}\right)\right)\right) \]

      10. sqrt-div 8.0%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \left(\frac{M \cdot \frac{D}{d \cdot 2}}{\sqrt{\ell}} \cdot \color{blue}{\frac{\sqrt{{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}}}{\sqrt{\ell}}}\right)\right)\right) \]

      11. sqrt-pow1 13.4%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \left(\frac{M \cdot \frac{D}{d \cdot 2}}{\sqrt{\ell}} \cdot \frac{\color{blue}{{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{\left(\frac{2}{2}\right)}}}{\sqrt{\ell}}\right)\right)\right) \]

      12. metadata-eval 13.4%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \left(\frac{M \cdot \frac{D}{d \cdot 2}}{\sqrt{\ell}} \cdot \frac{{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{\color{blue}{1}}}{\sqrt{\ell}}\right)\right)\right) \]

      13. pow1 13.4%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \left(\frac{M \cdot \frac{D}{d \cdot 2}}{\sqrt{\ell}} \cdot \frac{\color{blue}{\frac{D}{d} \cdot \frac{M}{2}}}{\sqrt{\ell}}\right)\right)\right) \]

      14. *-commutative 13.4%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \left(\frac{M \cdot \frac{D}{d \cdot 2}}{\sqrt{\ell}} \cdot \frac{\color{blue}{\frac{M}{2} \cdot \frac{D}{d}}}{\sqrt{\ell}}\right)\right)\right) \]

      15. frac-times 13.4%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \left(\frac{M \cdot \frac{D}{d \cdot 2}}{\sqrt{\ell}} \cdot \frac{\color{blue}{\frac{M \cdot D}{2 \cdot d}}}{\sqrt{\ell}}\right)\right)\right) \]

      16. *-commutative 13.4%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \left(\frac{M \cdot \frac{D}{d \cdot 2}}{\sqrt{\ell}} \cdot \frac{\frac{M \cdot D}{\color{blue}{d \cdot 2}}}{\sqrt{\ell}}\right)\right)\right) \]

      17. associate-*r/ 13.4%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \left(\frac{M \cdot \frac{D}{d \cdot 2}}{\sqrt{\ell}} \cdot \frac{\color{blue}{M \cdot \frac{D}{d \cdot 2}}}{\sqrt{\ell}}\right)\right)\right) \]

   10. Applied egg-rr 68.7%

       \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \color{blue}{\left(\frac{\frac{D}{d} \cdot \left(M \cdot 0.5\right)}{1} \cdot \frac{\frac{D}{d} \cdot \left(M \cdot 0.5\right)}{\ell}\right)}\right)\right) \]

   11. Taylor expanded in l around -inf 0.0%

       \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   12. Step-by-step derivation

       1. *-commutative 0.0%

          \[\leadsto \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot d\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

       2. unpow2 0.0%

          \[\leadsto \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

       3. rem-square-sqrt 41.1%

          \[\leadsto \left(\color{blue}{-1} \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

       4. mul-1-neg 41.1%

          \[\leadsto \color{blue}{\left(-d\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

       5. distribute-lft-neg-in 41.1%

          \[\leadsto \color{blue}{-d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

       6. rem-exp-log 39.4%

          \[\leadsto -d \cdot \sqrt{\frac{1}{\color{blue}{e^{\log \left(h \cdot \ell\right)}}}} \]

       7. exp-neg 39.4%

          \[\leadsto -d \cdot \sqrt{\color{blue}{e^{-\log \left(h \cdot \ell\right)}}} \]

       8. unpow1/2 39.4%

          \[\leadsto -d \cdot \color{blue}{{\left(e^{-\log \left(h \cdot \ell\right)}\right)}^{0.5}} \]

       9. exp-prod 39.4%

          \[\leadsto -d \cdot \color{blue}{e^{\left(-\log \left(h \cdot \ell\right)\right) \cdot 0.5}} \]

       10. distribute-lft-neg-out 39.4%

           \[\leadsto -d \cdot e^{\color{blue}{-\log \left(h \cdot \ell\right) \cdot 0.5}} \]

       11. exp-neg 39.4%

           \[\leadsto -d \cdot \color{blue}{\frac{1}{e^{\log \left(h \cdot \ell\right) \cdot 0.5}}} \]

       12. exp-to-pow 41.0%

           \[\leadsto -d \cdot \frac{1}{\color{blue}{{\left(h \cdot \ell\right)}^{0.5}}} \]

       13. unpow1/2 41.0%

           \[\leadsto -d \cdot \frac{1}{\color{blue}{\sqrt{h \cdot \ell}}} \]

       14. associate-/l* 41.1%

           \[\leadsto -\color{blue}{\frac{d \cdot 1}{\sqrt{h \cdot \ell}}} \]

       15. associate-*l/ 41.1%

           \[\leadsto -\color{blue}{\frac{d}{\sqrt{h \cdot \ell}} \cdot 1} \]

       16. *-rgt-identity 41.1%

           \[\leadsto -\color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]

       17. distribute-neg-frac2 41.1%

           \[\leadsto \color{blue}{\frac{d}{-\sqrt{h \cdot \ell}}} \]

   13. Simplified 41.1%

       \[\leadsto \color{blue}{\frac{d}{-\sqrt{h \cdot \ell}}} \]

   if 3.09999999999999977e-203 < l

   1. Initial program 58.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 59.6%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 38.8%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. *-un-lft-identity 38.8%

         \[\leadsto d \cdot \sqrt{\color{blue}{1 \cdot \frac{1}{h \cdot \ell}}} \]

      2. associate-/r* 40.4%

         \[\leadsto d \cdot \sqrt{1 \cdot \color{blue}{\frac{\frac{1}{h}}{\ell}}} \]

   7. Applied egg-rr 40.4%

      \[\leadsto d \cdot \sqrt{\color{blue}{1 \cdot \frac{\frac{1}{h}}{\ell}}} \]
   8. Step-by-step derivation

      1. *-lft-identity 40.4%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \]

      2. associate-/l/ 38.8%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{1}{\ell \cdot h}}} \]

      3. associate-/r* 40.4%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \]

   9. Simplified 40.4%

      \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 40.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 3.1 \cdot 10^{-203}:\\ \;\;\;\;\frac{d}{-\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 42.2% accurate, 3.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\ell \cdot h}\\ \mathbf{if}\;\ell \leq 1.15 \cdot 10^{-204}:\\ \;\;\;\;\frac{d}{-t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{t\_0}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (sqrt (* l h)))) (if (<= l 1.15e-204) (/ d (- t_0)) (/ d t_0))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = sqrt((l * h));
	double tmp;
	if (l <= 1.15e-204) {
		tmp = d / -t_0;
	} else {
		tmp = d / t_0;
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((l * h))
    if (l <= 1.15d-204) then
        tmp = d / -t_0
    else
        tmp = d / t_0
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = Math.sqrt((l * h));
	double tmp;
	if (l <= 1.15e-204) {
		tmp = d / -t_0;
	} else {
		tmp = d / t_0;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = math.sqrt((l * h))
	tmp = 0
	if l <= 1.15e-204:
		tmp = d / -t_0
	else:
		tmp = d / t_0
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = sqrt(Float64(l * h))
	tmp = 0.0
	if (l <= 1.15e-204)
		tmp = Float64(d / Float64(-t_0));
	else
		tmp = Float64(d / t_0);
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = sqrt((l * h));
	tmp = 0.0;
	if (l <= 1.15e-204)
		tmp = d / -t_0;
	else
		tmp = d / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, 1.15e-204], N[(d / (-t$95$0)), $MachinePrecision], N[(d / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < 1.15e-204

   1. Initial program 67.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 65.2%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r/ 66.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot h}{\ell}}\right) \]

      2. frac-times 67.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot h}{\ell}\right) \]

      3. associate-/l* 66.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)}}^{2} \cdot h}{\ell}\right) \]

      4. *-commutative 66.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(M \cdot \frac{D}{\color{blue}{d \cdot 2}}\right)}^{2} \cdot h}{\ell}\right) \]

   6. Applied egg-rr 66.1%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot h}{\ell}}\right) \]
   7. Step-by-step derivation

      1. *-commutative 66.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{\color{blue}{h \cdot {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}}}{\ell}\right) \]

      2. associate-/l* 66.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(h \cdot \frac{{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}}{\ell}\right)}\right) \]

      3. associate-*r/ 68.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{{\color{blue}{\left(\frac{M \cdot D}{d \cdot 2}\right)}}^{2}}{\ell}\right)\right) \]

      4. *-commutative 68.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{{\left(\frac{\color{blue}{D \cdot M}}{d \cdot 2}\right)}^{2}}{\ell}\right)\right) \]

      5. times-frac 66.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{{\color{blue}{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}}^{2}}{\ell}\right)\right) \]

   8. Simplified 66.8%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(h \cdot \frac{{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}}{\ell}\right)}\right) \]
   9. Step-by-step derivation

      1. add-sqr-sqrt 32.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \color{blue}{\left(\sqrt{\frac{{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}}{\ell}} \cdot \sqrt{\frac{{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}}{\ell}}\right)}\right)\right) \]

      2. sqrt-div 13.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \left(\color{blue}{\frac{\sqrt{{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}}}{\sqrt{\ell}}} \cdot \sqrt{\frac{{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}}{\ell}}\right)\right)\right) \]

      3. sqrt-pow1 8.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \left(\frac{\color{blue}{{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{\left(\frac{2}{2}\right)}}}{\sqrt{\ell}} \cdot \sqrt{\frac{{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}}{\ell}}\right)\right)\right) \]

      4. metadata-eval 8.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \left(\frac{{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{\color{blue}{1}}}{\sqrt{\ell}} \cdot \sqrt{\frac{{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}}{\ell}}\right)\right)\right) \]

      5. pow1 8.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \left(\frac{\color{blue}{\frac{D}{d} \cdot \frac{M}{2}}}{\sqrt{\ell}} \cdot \sqrt{\frac{{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}}{\ell}}\right)\right)\right) \]

      6. *-commutative 8.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \left(\frac{\color{blue}{\frac{M}{2} \cdot \frac{D}{d}}}{\sqrt{\ell}} \cdot \sqrt{\frac{{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}}{\ell}}\right)\right)\right) \]

      7. frac-times 8.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \left(\frac{\color{blue}{\frac{M \cdot D}{2 \cdot d}}}{\sqrt{\ell}} \cdot \sqrt{\frac{{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}}{\ell}}\right)\right)\right) \]

      8. *-commutative 8.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \left(\frac{\frac{M \cdot D}{\color{blue}{d \cdot 2}}}{\sqrt{\ell}} \cdot \sqrt{\frac{{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}}{\ell}}\right)\right)\right) \]

      9. associate-*r/ 8.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \left(\frac{\color{blue}{M \cdot \frac{D}{d \cdot 2}}}{\sqrt{\ell}} \cdot \sqrt{\frac{{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}}{\ell}}\right)\right)\right) \]

      10. sqrt-div 8.0%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \left(\frac{M \cdot \frac{D}{d \cdot 2}}{\sqrt{\ell}} \cdot \color{blue}{\frac{\sqrt{{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}}}{\sqrt{\ell}}}\right)\right)\right) \]

      11. sqrt-pow1 13.4%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \left(\frac{M \cdot \frac{D}{d \cdot 2}}{\sqrt{\ell}} \cdot \frac{\color{blue}{{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{\left(\frac{2}{2}\right)}}}{\sqrt{\ell}}\right)\right)\right) \]

      12. metadata-eval 13.4%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \left(\frac{M \cdot \frac{D}{d \cdot 2}}{\sqrt{\ell}} \cdot \frac{{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{\color{blue}{1}}}{\sqrt{\ell}}\right)\right)\right) \]

      13. pow1 13.4%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \left(\frac{M \cdot \frac{D}{d \cdot 2}}{\sqrt{\ell}} \cdot \frac{\color{blue}{\frac{D}{d} \cdot \frac{M}{2}}}{\sqrt{\ell}}\right)\right)\right) \]

      14. *-commutative 13.4%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \left(\frac{M \cdot \frac{D}{d \cdot 2}}{\sqrt{\ell}} \cdot \frac{\color{blue}{\frac{M}{2} \cdot \frac{D}{d}}}{\sqrt{\ell}}\right)\right)\right) \]

      15. frac-times 13.4%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \left(\frac{M \cdot \frac{D}{d \cdot 2}}{\sqrt{\ell}} \cdot \frac{\color{blue}{\frac{M \cdot D}{2 \cdot d}}}{\sqrt{\ell}}\right)\right)\right) \]

      16. *-commutative 13.4%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \left(\frac{M \cdot \frac{D}{d \cdot 2}}{\sqrt{\ell}} \cdot \frac{\frac{M \cdot D}{\color{blue}{d \cdot 2}}}{\sqrt{\ell}}\right)\right)\right) \]

      17. associate-*r/ 13.4%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \left(\frac{M \cdot \frac{D}{d \cdot 2}}{\sqrt{\ell}} \cdot \frac{\color{blue}{M \cdot \frac{D}{d \cdot 2}}}{\sqrt{\ell}}\right)\right)\right) \]

   10. Applied egg-rr 68.7%

       \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \color{blue}{\left(\frac{\frac{D}{d} \cdot \left(M \cdot 0.5\right)}{1} \cdot \frac{\frac{D}{d} \cdot \left(M \cdot 0.5\right)}{\ell}\right)}\right)\right) \]

   11. Taylor expanded in l around -inf 0.0%

       \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   12. Step-by-step derivation

       1. *-commutative 0.0%

          \[\leadsto \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot d\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

       2. unpow2 0.0%

          \[\leadsto \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

       3. rem-square-sqrt 41.1%

          \[\leadsto \left(\color{blue}{-1} \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

       4. mul-1-neg 41.1%

          \[\leadsto \color{blue}{\left(-d\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

       5. distribute-lft-neg-in 41.1%

          \[\leadsto \color{blue}{-d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

       6. rem-exp-log 39.4%

          \[\leadsto -d \cdot \sqrt{\frac{1}{\color{blue}{e^{\log \left(h \cdot \ell\right)}}}} \]

       7. exp-neg 39.4%

          \[\leadsto -d \cdot \sqrt{\color{blue}{e^{-\log \left(h \cdot \ell\right)}}} \]

       8. unpow1/2 39.4%

          \[\leadsto -d \cdot \color{blue}{{\left(e^{-\log \left(h \cdot \ell\right)}\right)}^{0.5}} \]

       9. exp-prod 39.4%

          \[\leadsto -d \cdot \color{blue}{e^{\left(-\log \left(h \cdot \ell\right)\right) \cdot 0.5}} \]

       10. distribute-lft-neg-out 39.4%

           \[\leadsto -d \cdot e^{\color{blue}{-\log \left(h \cdot \ell\right) \cdot 0.5}} \]

       11. exp-neg 39.4%

           \[\leadsto -d \cdot \color{blue}{\frac{1}{e^{\log \left(h \cdot \ell\right) \cdot 0.5}}} \]

       12. exp-to-pow 41.0%

           \[\leadsto -d \cdot \frac{1}{\color{blue}{{\left(h \cdot \ell\right)}^{0.5}}} \]

       13. unpow1/2 41.0%

           \[\leadsto -d \cdot \frac{1}{\color{blue}{\sqrt{h \cdot \ell}}} \]

       14. associate-/l* 41.1%

           \[\leadsto -\color{blue}{\frac{d \cdot 1}{\sqrt{h \cdot \ell}}} \]

       15. associate-*l/ 41.1%

           \[\leadsto -\color{blue}{\frac{d}{\sqrt{h \cdot \ell}} \cdot 1} \]

       16. *-rgt-identity 41.1%

           \[\leadsto -\color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]

       17. distribute-neg-frac2 41.1%

           \[\leadsto \color{blue}{\frac{d}{-\sqrt{h \cdot \ell}}} \]

   13. Simplified 41.1%

       \[\leadsto \color{blue}{\frac{d}{-\sqrt{h \cdot \ell}}} \]

   if 1.15e-204 < l

   1. Initial program 58.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 59.6%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 38.8%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. sqrt-div 39.3%

         \[\leadsto d \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{h \cdot \ell}}} \]

      2. metadata-eval 39.3%

         \[\leadsto d \cdot \frac{\color{blue}{1}}{\sqrt{h \cdot \ell}} \]

      3. sqrt-unprod 47.6%

         \[\leadsto d \cdot \frac{1}{\color{blue}{\sqrt{h} \cdot \sqrt{\ell}}} \]

      4. div-inv 47.6%

         \[\leadsto \color{blue}{\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}} \]

      5. sqrt-unprod 39.4%

         \[\leadsto \frac{d}{\color{blue}{\sqrt{h \cdot \ell}}} \]

   7. Applied egg-rr 39.4%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 40.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.15 \cdot 10^{-204}:\\ \;\;\;\;\frac{d}{-\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 26.0% accurate, 3.2× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \frac{d}{\sqrt{\ell \cdot h}} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m) :precision binary64 (/ d (sqrt (* l h))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	return d / sqrt((l * h));
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    code = d / sqrt((l * h))
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	return d / Math.sqrt((l * h));
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	return d / math.sqrt((l * h))

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	return Float64(d / sqrt(Float64(l * h)))
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp = code(d, h, l, M_m, D_m)
	tmp = d / sqrt((l * h));
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 63.7%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

3. Simplified 62.9%

   \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in d around inf 23.6%

   \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
6. Step-by-step derivation

   1. sqrt-div 23.8%

      \[\leadsto d \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{h \cdot \ell}}} \]

   2. metadata-eval 23.8%

      \[\leadsto d \cdot \frac{\color{blue}{1}}{\sqrt{h \cdot \ell}} \]

   3. sqrt-unprod 22.4%

      \[\leadsto d \cdot \frac{1}{\color{blue}{\sqrt{h} \cdot \sqrt{\ell}}} \]

   4. div-inv 22.5%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}} \]

   5. sqrt-unprod 23.8%

      \[\leadsto \frac{d}{\color{blue}{\sqrt{h \cdot \ell}}} \]

7. Applied egg-rr 23.8%

   \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]

8. Final simplification 23.8%

   \[\leadsto \frac{d}{\sqrt{\ell \cdot h}} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024076 
(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)))))