Henrywood and Agarwal, Equation (12)

Percentage Accurate: 65.7% → 84.4%

Time: 35.2s

Alternatives: 20

Speedup: 1.4×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 65.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 84.4% accurate, 0.6× speedup

Math

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

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 (sqrt (- d))) (t_1 (* D_m (* 0.5 (/ M_m d)))))
   (if (<= h -5e-310)
     (*
      (/ t_0 (sqrt (- h)))
      (*
       (/ t_0 (sqrt (- l)))
       (+ 1.0 (* h (/ (pow (* D_m (* M_m (/ 0.5 d))) 2.0) (/ l -0.5))))))
     (*
      (/ (sqrt d) (sqrt h))
      (* (sqrt (/ d l)) (+ 1.0 (* h (/ t_1 (/ (* l -2.0) t_1)))))))))

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

Derivation

1. Split input into 2 regimes

2. if h < -4.999999999999985e-310

   1. Initial program 67.5%

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

   3. Simplified 66.8%

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

   6. Applied egg-rr 38.5%

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

      1. expm1-def 38.5%

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

      2. expm1-log1p 66.8%

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

      3. *-commutative 66.8%

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

      4. associate-*r/ 69.9%

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

      5. associate-*l/ 70.6%

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

      6. *-commutative 70.6%

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

      7. associate-/l* 70.6%

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

      8. associate-*r* 70.6%

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

      9. *-commutative 70.6%

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

      10. associate-*l/ 71.3%

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

      11. associate-*r/ 71.2%

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

      12. associate-*r/ 71.2%

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

   8. Simplified 71.2%

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

      1. frac-2neg 71.2%

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

      2. sqrt-div 84.4%

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

   10. Applied egg-rr 84.4%

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

       1. frac-2neg 84.4%

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

       2. sqrt-div 87.4%

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

   12. Applied egg-rr 87.4%

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

   if -4.999999999999985e-310 < h

   1. Initial program 63.8%

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

   3. Simplified 63.8%

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

   6. Applied egg-rr 37.5%

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

      1. expm1-def 37.5%

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

      2. expm1-log1p 63.8%

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

      3. *-commutative 63.8%

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

      4. associate-*r/ 66.0%

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

      5. associate-*l/ 67.3%

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

      6. *-commutative 67.3%

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

      7. associate-/l* 67.3%

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

      8. associate-*r* 67.3%

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

      9. *-commutative 67.3%

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

      10. associate-*l/ 67.3%

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

      11. associate-*r/ 67.3%

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

      12. associate-*r/ 67.3%

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

   8. Simplified 67.3%

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

      1. expm1-log1p-u 42.4%

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

      2. expm1-udef 39.3%

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

      3. associate-/r/ 39.3%

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

      4. add-sqr-sqrt 39.3%

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

      5. pow2 39.3%

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

      6. sqrt-pow1 39.3%

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

      7. metadata-eval 39.3%

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

      8. pow1 39.3%

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

      9. associate-*r/ 39.3%

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

   10. Applied egg-rr 39.3%

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

       1. expm1-def 42.4%

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

       2. expm1-log1p 67.3%

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

       3. metadata-eval 67.3%

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

       4. times-frac 67.3%

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

       5. associate-*r/ 67.3%

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

       6. *-commutative 67.3%

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

       7. associate-/l* 66.5%

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

       8. associate-/r/ 67.3%

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

       9. *-commutative 67.3%

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

       10. associate-/r* 67.3%

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

       11. metadata-eval 67.3%

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

   12. Simplified 67.3%

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

       1. *-commutative 67.3%

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

       2. clear-num 67.3%

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

       3. div-inv 67.3%

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

       4. unpow2 67.3%

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

       5. associate-/l* 71.3%

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

       6. *-commutative 71.3%

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

       7. associate-*l/ 71.3%

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

       8. *-un-lft-identity 71.3%

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

       9. times-frac 71.3%

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

       10. metadata-eval 71.3%

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

       11. div-inv 71.3%

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

       12. metadata-eval 71.3%

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

       13. *-commutative 71.3%

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

       14. associate-*l/ 71.3%

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

       15. *-un-lft-identity 71.3%

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

       16. times-frac 71.3%

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

       17. metadata-eval 71.3%

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

   14. Applied egg-rr 71.3%

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

       1. sqrt-div 82.8%

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

       2. div-inv 82.8%

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

   16. Applied egg-rr 82.8%

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

       1. associate-*r/ 82.8%

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

       2. *-rgt-identity 82.8%

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

   18. Simplified 82.8%

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

4. Final simplification 85.2%

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

Alternative 2: 72.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := D_m \cdot \left(0.5 \cdot \frac{M_m}{d}\right)\\ t_2 := \left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \left(0.5 \cdot {\left(\frac{D_m \cdot M_m}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_3 := \sqrt{h \cdot \ell}\\ \mathbf{if}\;t_2 \leq -2 \cdot 10^{-122}:\\ \;\;\;\;\left(t_0 \cdot \left(1 + h \cdot \frac{t_1}{\frac{\ell \cdot -2}{t_1}}\right)\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;t_2 \leq 0:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left(-0.5, \frac{h}{\ell} \cdot {\left(D_m \cdot \left(M_m \cdot \frac{0.5}{d}\right)\right)}^{2}, 1\right)}{t_3}\\ \mathbf{elif}\;t_2 \leq \infty:\\ \;\;\;\;\frac{1}{\sqrt{\frac{h}{d}}} \cdot \left(t_0 \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{M_m}{2} \cdot \frac{D_m}{d}\right)}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{t_3} \cdot \left(1 + -0.5 \cdot \frac{h}{\frac{\ell}{{\left(M_m \cdot \frac{D_m \cdot 0.5}{d}\right)}^{2}}}\right)\\ \end{array} \end{array} \]

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 (sqrt (/ d l)))
        (t_1 (* D_m (* 0.5 (/ M_m d))))
        (t_2
         (*
          (* (pow (/ d h) 0.5) (pow (/ d l) 0.5))
          (- 1.0 (* (* 0.5 (pow (/ (* D_m M_m) (* d 2.0)) 2.0)) (/ h l)))))
        (t_3 (sqrt (* h l))))
   (if (<= t_2 -2e-122)
     (* (* t_0 (+ 1.0 (* h (/ t_1 (/ (* l -2.0) t_1))))) (sqrt (/ d h)))
     (if (<= t_2 0.0)
       (*
        d
        (/ (fma -0.5 (* (/ h l) (pow (* D_m (* M_m (/ 0.5 d))) 2.0)) 1.0) t_3))
       (if (<= t_2 INFINITY)
         (*
          (/ 1.0 (sqrt (/ h d)))
          (*
           t_0
           (+ 1.0 (* (/ h l) (* -0.5 (pow (* (/ M_m 2.0) (/ D_m d)) 2.0))))))
         (*
          (/ d t_3)
          (+
           1.0
           (* -0.5 (/ h (/ l (pow (* M_m (/ (* D_m 0.5) d)) 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 = sqrt((d / l));
	double t_1 = D_m * (0.5 * (M_m / d));
	double t_2 = (pow((d / h), 0.5) * pow((d / l), 0.5)) * (1.0 - ((0.5 * pow(((D_m * M_m) / (d * 2.0)), 2.0)) * (h / l)));
	double t_3 = sqrt((h * l));
	double tmp;
	if (t_2 <= -2e-122) {
		tmp = (t_0 * (1.0 + (h * (t_1 / ((l * -2.0) / t_1))))) * sqrt((d / h));
	} else if (t_2 <= 0.0) {
		tmp = d * (fma(-0.5, ((h / l) * pow((D_m * (M_m * (0.5 / d))), 2.0)), 1.0) / t_3);
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = (1.0 / sqrt((h / d))) * (t_0 * (1.0 + ((h / l) * (-0.5 * pow(((M_m / 2.0) * (D_m / d)), 2.0)))));
	} else {
		tmp = (d / t_3) * (1.0 + (-0.5 * (h / (l / pow((M_m * ((D_m * 0.5) / d)), 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 = sqrt(Float64(d / l))
	t_1 = Float64(D_m * Float64(0.5 * Float64(M_m / d)))
	t_2 = Float64(Float64((Float64(d / h) ^ 0.5) * (Float64(d / l) ^ 0.5)) * Float64(1.0 - Float64(Float64(0.5 * (Float64(Float64(D_m * M_m) / Float64(d * 2.0)) ^ 2.0)) * Float64(h / l))))
	t_3 = sqrt(Float64(h * l))
	tmp = 0.0
	if (t_2 <= -2e-122)
		tmp = Float64(Float64(t_0 * Float64(1.0 + Float64(h * Float64(t_1 / Float64(Float64(l * -2.0) / t_1))))) * sqrt(Float64(d / h)));
	elseif (t_2 <= 0.0)
		tmp = Float64(d * Float64(fma(-0.5, Float64(Float64(h / l) * (Float64(D_m * Float64(M_m * Float64(0.5 / d))) ^ 2.0)), 1.0) / t_3));
	elseif (t_2 <= Inf)
		tmp = Float64(Float64(1.0 / sqrt(Float64(h / d))) * Float64(t_0 * Float64(1.0 + Float64(Float64(h / l) * Float64(-0.5 * (Float64(Float64(M_m / 2.0) * Float64(D_m / d)) ^ 2.0))))));
	else
		tmp = Float64(Float64(d / t_3) * Float64(1.0 + Float64(-0.5 * Float64(h / Float64(l / (Float64(M_m * Float64(Float64(D_m * 0.5) / d)) ^ 2.0))))));
	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[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(D$95$m * N[(0.5 * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Power[N[(d / h), $MachinePrecision], 0.5], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(0.5 * N[Power[N[(N[(D$95$m * M$95$m), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$2, -2e-122], N[(N[(t$95$0 * N[(1.0 + N[(h * N[(t$95$1 / N[(N[(l * -2.0), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.0], N[(d * N[(N[(-0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(D$95$m * N[(M$95$m * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(N[(1.0 / N[Sqrt[N[(h / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(-0.5 * N[Power[N[(N[(M$95$m / 2.0), $MachinePrecision] * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(d / t$95$3), $MachinePrecision] * N[(1.0 + N[(-0.5 * N[(h / N[(l / N[Power[N[(M$95$m * N[(N[(D$95$m * 0.5), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (-.f64 1 (*.f64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)) (/.f64 h l)))) < -2.00000000000000012e-122

   1. Initial program 85.3%

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

   3. Simplified 84.2%

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

   6. Applied egg-rr 0.1%

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

      1. expm1-def 0.1%

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

      2. expm1-log1p 84.2%

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

      3. *-commutative 84.2%

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

      4. associate-*r/ 82.4%

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

      5. associate-*l/ 85.4%

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

      6. *-commutative 85.4%

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

      7. associate-/l* 85.4%

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

      8. associate-*r* 85.4%

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

      9. *-commutative 85.4%

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

      10. associate-*l/ 86.4%

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

      11. associate-*r/ 86.6%

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

      12. associate-*r/ 86.5%

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

   8. Simplified 86.5%

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

      1. expm1-log1p-u 51.9%

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

      2. expm1-udef 44.1%

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

      3. associate-/r/ 44.1%

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

      4. add-sqr-sqrt 44.1%

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

      5. pow2 44.1%

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

      6. sqrt-pow1 44.1%

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

      7. metadata-eval 44.1%

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

      8. pow1 44.1%

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

      9. associate-*r/ 44.1%

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

   10. Applied egg-rr 44.1%

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

       1. expm1-def 51.9%

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

       2. expm1-log1p 86.6%

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

       3. metadata-eval 86.6%

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

       4. times-frac 86.6%

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

       5. associate-*r/ 86.6%

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

       6. *-commutative 86.6%

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

       7. associate-/l* 85.3%

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

       8. associate-/r/ 86.5%

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

       9. *-commutative 86.5%

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

       10. associate-/r* 86.5%

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

       11. metadata-eval 86.5%

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

   12. Simplified 86.5%

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

       1. *-commutative 86.5%

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

       2. clear-num 86.5%

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

       3. div-inv 86.5%

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

       4. unpow2 86.5%

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

       5. associate-/l* 92.3%

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

       6. *-commutative 92.3%

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

       7. associate-*l/ 92.4%

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

       8. *-un-lft-identity 92.4%

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

       9. times-frac 92.4%

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

       10. metadata-eval 92.4%

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

       11. div-inv 92.4%

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

       12. metadata-eval 92.4%

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

       13. *-commutative 92.4%

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

       14. associate-*l/ 92.4%

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

       15. *-un-lft-identity 92.4%

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

       16. times-frac 92.4%

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

       17. metadata-eval 92.4%

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

   14. Applied egg-rr 92.4%

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

   if -2.00000000000000012e-122 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (-.f64 1 (*.f64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)) (/.f64 h l)))) < 0.0

   1. Initial program 50.6%

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

   3. Simplified 50.4%

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

      1. *-commutative 50.4%

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

      2. associate-*l/ 50.4%

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

      3. div-inv 50.4%

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

      4. associate-*l* 50.4%

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

      5. metadata-eval 50.4%

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

   6. Applied egg-rr 50.4%

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

   8. Applied egg-rr 50.4%

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

      1. expm1-def 66.4%

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

      2. expm1-log1p 66.5%

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

      3. associate-*l/ 72.0%

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

      4. *-lft-identity 72.0%

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

      5. times-frac 72.0%

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

      6. rem-square-sqrt 35.3%

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

      7. associate-*r/ 35.3%

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

      8. /-rgt-identity 35.3%

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

      9. rem-square-sqrt 72.0%

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

   10. Simplified 77.3%

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

   if 0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (-.f64 1 (*.f64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)) (/.f64 h l)))) < +inf.0

   1. Initial program 85.5%

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

   3. Simplified 85.5%

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

      1. clear-num 85.3%

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

      2. sqrt-div 86.9%

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

      3. metadata-eval 86.9%

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

   6. Applied egg-rr 86.9%

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

   if +inf.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (-.f64 1 (*.f64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)) (/.f64 h l))))

   1. Initial program 0.0%

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

   3. Simplified 0.1%

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

      1. *-commutative 0.1%

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

      2. associate-*l/ 16.1%

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

      3. div-inv 16.1%

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

      4. associate-*l* 16.1%

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

      5. metadata-eval 16.1%

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

   6. Applied egg-rr 16.1%

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

      1. pow1 16.1%

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

   8. Applied egg-rr 31.2%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \left(0.5 \cdot {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -2 \cdot 10^{-122}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \frac{D \cdot \left(0.5 \cdot \frac{M}{d}\right)}{\frac{\ell \cdot -2}{D \cdot \left(0.5 \cdot \frac{M}{d}\right)}}\right)\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \left(0.5 \cdot {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 0:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left(-0.5, \frac{h}{\ell} \cdot {\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}, 1\right)}{\sqrt{h \cdot \ell}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \left(0.5 \cdot {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq \infty:\\ \;\;\;\;\frac{1}{\sqrt{\frac{h}{d}}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 + -0.5 \cdot \frac{h}{\frac{\ell}{{\left(M \cdot \frac{D \cdot 0.5}{d}\right)}^{2}}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.1% accurate, 1.0× speedup

Math

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

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 (* 0.5 (/ M_m d))))
        (t_1 (+ 1.0 (* h (/ t_0 (/ (* l -2.0) t_0))))))
   (if (<= h -5e-310)
     (* (sqrt (/ d h)) (* (/ (sqrt (- d)) (sqrt (- l))) t_1))
     (* (/ (sqrt d) (sqrt h)) (* (sqrt (/ d l)) t_1)))))

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

Derivation

1. Split input into 2 regimes

2. if h < -4.999999999999985e-310

   1. Initial program 67.5%

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

   3. Simplified 66.8%

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

   6. Applied egg-rr 38.5%

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

      1. expm1-def 38.5%

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

      2. expm1-log1p 66.8%

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

      3. *-commutative 66.8%

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

      4. associate-*r/ 69.9%

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

      5. associate-*l/ 70.6%

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

      6. *-commutative 70.6%

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

      7. associate-/l* 70.6%

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

      8. associate-*r* 70.6%

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

      9. *-commutative 70.6%

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

      10. associate-*l/ 71.3%

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

      11. associate-*r/ 71.2%

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

      12. associate-*r/ 71.2%

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

   8. Simplified 71.2%

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

      1. expm1-log1p-u 71.0%

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

      2. expm1-udef 68.1%

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

      3. associate-/r/ 68.1%

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

      4. add-sqr-sqrt 68.1%

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

      5. pow2 68.1%

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

      6. sqrt-pow1 68.1%

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

      7. metadata-eval 68.1%

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

      8. pow1 68.1%

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

      9. associate-*r/ 68.1%

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

   10. Applied egg-rr 68.1%

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

       1. expm1-def 71.0%

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

       2. expm1-log1p 71.2%

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

       3. metadata-eval 71.2%

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

       4. times-frac 71.2%

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

       5. associate-*r/ 71.2%

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

       6. *-commutative 71.2%

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

       7. associate-/l* 71.2%

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

       8. associate-/r/ 71.2%

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

       9. *-commutative 71.2%

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

       10. associate-/r* 71.2%

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

       11. metadata-eval 71.2%

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

   12. Simplified 71.2%

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

       1. *-commutative 71.2%

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

       2. clear-num 71.2%

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

       3. div-inv 71.2%

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

       4. unpow2 71.2%

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

       5. associate-/l* 73.1%

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

       6. *-commutative 73.1%

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

       7. associate-*l/ 73.1%

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

       8. *-un-lft-identity 73.1%

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

       9. times-frac 73.1%

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

       10. metadata-eval 73.1%

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

       11. div-inv 73.1%

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

       12. metadata-eval 73.1%

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

       13. *-commutative 73.1%

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

       14. associate-*l/ 73.1%

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

       15. *-un-lft-identity 73.1%

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

       16. times-frac 73.1%

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

       17. metadata-eval 73.1%

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

   14. Applied egg-rr 73.1%

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

       1. frac-2neg 84.4%

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

       2. sqrt-div 87.4%

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

   16. Applied egg-rr 76.3%

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

   if -4.999999999999985e-310 < h

   1. Initial program 63.8%

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

   3. Simplified 63.8%

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

   6. Applied egg-rr 37.5%

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

      1. expm1-def 37.5%

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

      2. expm1-log1p 63.8%

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

      3. *-commutative 63.8%

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

      4. associate-*r/ 66.0%

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

      5. associate-*l/ 67.3%

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

      6. *-commutative 67.3%

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

      7. associate-/l* 67.3%

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

      8. associate-*r* 67.3%

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

      9. *-commutative 67.3%

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

      10. associate-*l/ 67.3%

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

      11. associate-*r/ 67.3%

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

      12. associate-*r/ 67.3%

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

   8. Simplified 67.3%

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

      1. expm1-log1p-u 42.4%

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

      2. expm1-udef 39.3%

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

      3. associate-/r/ 39.3%

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

      4. add-sqr-sqrt 39.3%

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

      5. pow2 39.3%

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

      6. sqrt-pow1 39.3%

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

      7. metadata-eval 39.3%

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

      8. pow1 39.3%

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

      9. associate-*r/ 39.3%

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

   10. Applied egg-rr 39.3%

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

       1. expm1-def 42.4%

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

       2. expm1-log1p 67.3%

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

       3. metadata-eval 67.3%

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

       4. times-frac 67.3%

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

       5. associate-*r/ 67.3%

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

       6. *-commutative 67.3%

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

       7. associate-/l* 66.5%

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

       8. associate-/r/ 67.3%

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

       9. *-commutative 67.3%

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

       10. associate-/r* 67.3%

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

       11. metadata-eval 67.3%

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

   12. Simplified 67.3%

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

       1. *-commutative 67.3%

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

       2. clear-num 67.3%

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

       3. div-inv 67.3%

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

       4. unpow2 67.3%

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

       5. associate-/l* 71.3%

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

       6. *-commutative 71.3%

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

       7. associate-*l/ 71.3%

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

       8. *-un-lft-identity 71.3%

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

       9. times-frac 71.3%

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

       10. metadata-eval 71.3%

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

       11. div-inv 71.3%

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

       12. metadata-eval 71.3%

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

       13. *-commutative 71.3%

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

       14. associate-*l/ 71.3%

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

       15. *-un-lft-identity 71.3%

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

       16. times-frac 71.3%

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

       17. metadata-eval 71.3%

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

   14. Applied egg-rr 71.3%

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

       1. sqrt-div 82.8%

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

       2. div-inv 82.8%

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

   16. Applied egg-rr 82.8%

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

       1. associate-*r/ 82.8%

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

       2. *-rgt-identity 82.8%

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

   18. Simplified 82.8%

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

4. Final simplification 79.4%

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

Alternative 4: 80.8% accurate, 1.0× speedup

Math

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

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 (* 0.5 (/ M_m d))))
        (t_1 (* (sqrt (/ d l)) (+ 1.0 (* h (/ t_0 (/ (* l -2.0) t_0)))))))
   (if (<= l -2e-310)
     (* (/ (sqrt (- d)) (sqrt (- h))) t_1)
     (* (/ (sqrt d) (sqrt h)) t_1))))

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

Derivation

1. Split input into 2 regimes

2. if l < -1.999999999999994e-310

   1. Initial program 67.5%

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

   3. Simplified 66.8%

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

   6. Applied egg-rr 38.5%

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

      1. expm1-def 38.5%

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

      2. expm1-log1p 66.8%

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

      3. *-commutative 66.8%

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

      4. associate-*r/ 69.9%

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

      5. associate-*l/ 70.6%

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

      6. *-commutative 70.6%

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

      7. associate-/l* 70.6%

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

      8. associate-*r* 70.6%

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

      9. *-commutative 70.6%

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

      10. associate-*l/ 71.3%

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

      11. associate-*r/ 71.2%

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

      12. associate-*r/ 71.2%

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

   8. Simplified 71.2%

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

      1. expm1-log1p-u 71.0%

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

      2. expm1-udef 68.1%

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

      3. associate-/r/ 68.1%

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

      4. add-sqr-sqrt 68.1%

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

      5. pow2 68.1%

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

      6. sqrt-pow1 68.1%

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

      7. metadata-eval 68.1%

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

      8. pow1 68.1%

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

      9. associate-*r/ 68.1%

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

   10. Applied egg-rr 68.1%

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

       1. expm1-def 71.0%

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

       2. expm1-log1p 71.2%

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

       3. metadata-eval 71.2%

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

       4. times-frac 71.2%

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

       5. associate-*r/ 71.2%

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

       6. *-commutative 71.2%

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

       7. associate-/l* 71.2%

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

       8. associate-/r/ 71.2%

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

       9. *-commutative 71.2%

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

       10. associate-/r* 71.2%

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

       11. metadata-eval 71.2%

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

   12. Simplified 71.2%

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

       1. *-commutative 71.2%

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

       2. clear-num 71.2%

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

       3. div-inv 71.2%

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

       4. unpow2 71.2%

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

       5. associate-/l* 73.1%

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

       6. *-commutative 73.1%

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

       7. associate-*l/ 73.1%

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

       8. *-un-lft-identity 73.1%

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

       9. times-frac 73.1%

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

       10. metadata-eval 73.1%

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

       11. div-inv 73.1%

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

       12. metadata-eval 73.1%

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

       13. *-commutative 73.1%

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

       14. associate-*l/ 73.1%

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

       15. *-un-lft-identity 73.1%

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

       16. times-frac 73.1%

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

       17. metadata-eval 73.1%

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

   14. Applied egg-rr 73.1%

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

       1. frac-2neg 71.2%

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

       2. sqrt-div 84.4%

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

   16. Applied egg-rr 86.3%

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

   if -1.999999999999994e-310 < l

   1. Initial program 63.8%

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

   3. Simplified 63.8%

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

   6. Applied egg-rr 37.5%

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

      1. expm1-def 37.5%

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

      2. expm1-log1p 63.8%

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

      3. *-commutative 63.8%

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

      4. associate-*r/ 66.0%

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

      5. associate-*l/ 67.3%

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

      6. *-commutative 67.3%

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

      7. associate-/l* 67.3%

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

      8. associate-*r* 67.3%

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

      9. *-commutative 67.3%

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

      10. associate-*l/ 67.3%

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

      11. associate-*r/ 67.3%

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

      12. associate-*r/ 67.3%

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

   8. Simplified 67.3%

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

      1. expm1-log1p-u 42.4%

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

      2. expm1-udef 39.3%

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

      3. associate-/r/ 39.3%

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

      4. add-sqr-sqrt 39.3%

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

      5. pow2 39.3%

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

      6. sqrt-pow1 39.3%

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

      7. metadata-eval 39.3%

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

      8. pow1 39.3%

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

      9. associate-*r/ 39.3%

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

   10. Applied egg-rr 39.3%

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

       1. expm1-def 42.4%

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

       2. expm1-log1p 67.3%

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

       3. metadata-eval 67.3%

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

       4. times-frac 67.3%

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

       5. associate-*r/ 67.3%

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

       6. *-commutative 67.3%

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

       7. associate-/l* 66.5%

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

       8. associate-/r/ 67.3%

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

       9. *-commutative 67.3%

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

       10. associate-/r* 67.3%

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

       11. metadata-eval 67.3%

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

   12. Simplified 67.3%

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

       1. *-commutative 67.3%

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

       2. clear-num 67.3%

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

       3. div-inv 67.3%

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

       4. unpow2 67.3%

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

       5. associate-/l* 71.3%

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

       6. *-commutative 71.3%

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

       7. associate-*l/ 71.3%

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

       8. *-un-lft-identity 71.3%

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

       9. times-frac 71.3%

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

       10. metadata-eval 71.3%

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

       11. div-inv 71.3%

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

       12. metadata-eval 71.3%

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

       13. *-commutative 71.3%

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

       14. associate-*l/ 71.3%

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

       15. *-un-lft-identity 71.3%

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

       16. times-frac 71.3%

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

       17. metadata-eval 71.3%

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

   14. Applied egg-rr 71.3%

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

       1. sqrt-div 82.8%

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

       2. div-inv 82.8%

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

   16. Applied egg-rr 82.8%

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

       1. associate-*r/ 82.8%

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

       2. *-rgt-identity 82.8%

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

   18. Simplified 82.8%

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

4. Final simplification 84.6%

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

Alternative 5: 73.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := D_m \cdot \left(0.5 \cdot \frac{M_m}{d}\right)\\ t_1 := \sqrt{\frac{d}{h}}\\ t_2 := 1 + h \cdot \frac{t_0}{\frac{\ell \cdot -2}{t_0}}\\ \mathbf{if}\;\ell \leq 1.1 \cdot 10^{-308}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot t_2\right) \cdot t_1\\ \mathbf{elif}\;\ell \leq 3.75 \cdot 10^{+15}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 + -0.5 \cdot \frac{h}{\frac{\ell}{{\left(M_m \cdot \frac{D_m \cdot 0.5}{d}\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(t_2 \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right)\\ \end{array} \end{array} \]

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
 (let* ((t_0 (* D_m (* 0.5 (/ M_m d))))
        (t_1 (sqrt (/ d h)))
        (t_2 (+ 1.0 (* h (/ t_0 (/ (* l -2.0) t_0))))))
   (if (<= l 1.1e-308)
     (* (* (sqrt (/ d l)) t_2) t_1)
     (if (<= l 3.75e+15)
       (*
        (/ d (sqrt (* h l)))
        (+ 1.0 (* -0.5 (/ h (/ l (pow (* M_m (/ (* D_m 0.5) d)) 2.0))))))
       (* t_1 (* t_2 (/ (sqrt d) (sqrt 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 t_0 = D_m * (0.5 * (M_m / d));
	double t_1 = sqrt((d / h));
	double t_2 = 1.0 + (h * (t_0 / ((l * -2.0) / t_0)));
	double tmp;
	if (l <= 1.1e-308) {
		tmp = (sqrt((d / l)) * t_2) * t_1;
	} else if (l <= 3.75e+15) {
		tmp = (d / sqrt((h * l))) * (1.0 + (-0.5 * (h / (l / pow((M_m * ((D_m * 0.5) / d)), 2.0)))));
	} else {
		tmp = t_1 * (t_2 * (sqrt(d) / sqrt(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) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = d_m * (0.5d0 * (m_m / d))
    t_1 = sqrt((d / h))
    t_2 = 1.0d0 + (h * (t_0 / ((l * (-2.0d0)) / t_0)))
    if (l <= 1.1d-308) then
        tmp = (sqrt((d / l)) * t_2) * t_1
    else if (l <= 3.75d+15) then
        tmp = (d / sqrt((h * l))) * (1.0d0 + ((-0.5d0) * (h / (l / ((m_m * ((d_m * 0.5d0) / d)) ** 2.0d0)))))
    else
        tmp = t_1 * (t_2 * (sqrt(d) / sqrt(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 t_0 = D_m * (0.5 * (M_m / d));
	double t_1 = Math.sqrt((d / h));
	double t_2 = 1.0 + (h * (t_0 / ((l * -2.0) / t_0)));
	double tmp;
	if (l <= 1.1e-308) {
		tmp = (Math.sqrt((d / l)) * t_2) * t_1;
	} else if (l <= 3.75e+15) {
		tmp = (d / Math.sqrt((h * l))) * (1.0 + (-0.5 * (h / (l / Math.pow((M_m * ((D_m * 0.5) / d)), 2.0)))));
	} else {
		tmp = t_1 * (t_2 * (Math.sqrt(d) / Math.sqrt(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):
	t_0 = D_m * (0.5 * (M_m / d))
	t_1 = math.sqrt((d / h))
	t_2 = 1.0 + (h * (t_0 / ((l * -2.0) / t_0)))
	tmp = 0
	if l <= 1.1e-308:
		tmp = (math.sqrt((d / l)) * t_2) * t_1
	elif l <= 3.75e+15:
		tmp = (d / math.sqrt((h * l))) * (1.0 + (-0.5 * (h / (l / math.pow((M_m * ((D_m * 0.5) / d)), 2.0)))))
	else:
		tmp = t_1 * (t_2 * (math.sqrt(d) / math.sqrt(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)
	t_0 = Float64(D_m * Float64(0.5 * Float64(M_m / d)))
	t_1 = sqrt(Float64(d / h))
	t_2 = Float64(1.0 + Float64(h * Float64(t_0 / Float64(Float64(l * -2.0) / t_0))))
	tmp = 0.0
	if (l <= 1.1e-308)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * t_2) * t_1);
	elseif (l <= 3.75e+15)
		tmp = Float64(Float64(d / sqrt(Float64(h * l))) * Float64(1.0 + Float64(-0.5 * Float64(h / Float64(l / (Float64(M_m * Float64(Float64(D_m * 0.5) / d)) ^ 2.0))))));
	else
		tmp = Float64(t_1 * Float64(t_2 * Float64(sqrt(d) / sqrt(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)
	t_0 = D_m * (0.5 * (M_m / d));
	t_1 = sqrt((d / h));
	t_2 = 1.0 + (h * (t_0 / ((l * -2.0) / t_0)));
	tmp = 0.0;
	if (l <= 1.1e-308)
		tmp = (sqrt((d / l)) * t_2) * t_1;
	elseif (l <= 3.75e+15)
		tmp = (d / sqrt((h * l))) * (1.0 + (-0.5 * (h / (l / ((M_m * ((D_m * 0.5) / d)) ^ 2.0)))));
	else
		tmp = t_1 * (t_2 * (sqrt(d) / sqrt(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_] := Block[{t$95$0 = N[(D$95$m * N[(0.5 * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(1.0 + N[(h * N[(t$95$0 / N[(N[(l * -2.0), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, 1.1e-308], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * t$95$2), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[l, 3.75e+15], N[(N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(-0.5 * N[(h / N[(l / N[Power[N[(M$95$m * N[(N[(D$95$m * 0.5), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(t$95$2 * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < 1.1000000000000001e-308

   1. Initial program 67.5%

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

   3. Simplified 66.8%

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

   6. Applied egg-rr 38.5%

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

      1. expm1-def 38.5%

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

      2. expm1-log1p 66.8%

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

      3. *-commutative 66.8%

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

      4. associate-*r/ 69.9%

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

      5. associate-*l/ 70.6%

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

      6. *-commutative 70.6%

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

      7. associate-/l* 70.6%

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

      8. associate-*r* 70.6%

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

      9. *-commutative 70.6%

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

      10. associate-*l/ 71.3%

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

      11. associate-*r/ 71.2%

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

      12. associate-*r/ 71.2%

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

   8. Simplified 71.2%

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

      1. expm1-log1p-u 71.0%

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

      2. expm1-udef 68.1%

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

      3. associate-/r/ 68.1%

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

      4. add-sqr-sqrt 68.1%

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

      5. pow2 68.1%

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

      6. sqrt-pow1 68.1%

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

      7. metadata-eval 68.1%

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

      8. pow1 68.1%

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

      9. associate-*r/ 68.1%

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

   10. Applied egg-rr 68.1%

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

       1. expm1-def 71.0%

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

       2. expm1-log1p 71.2%

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

       3. metadata-eval 71.2%

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

       4. times-frac 71.2%

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

       5. associate-*r/ 71.2%

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

       6. *-commutative 71.2%

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

       7. associate-/l* 71.2%

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

       8. associate-/r/ 71.2%

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

       9. *-commutative 71.2%

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

       10. associate-/r* 71.2%

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

       11. metadata-eval 71.2%

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

   12. Simplified 71.2%

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

       1. *-commutative 71.2%

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

       2. clear-num 71.2%

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

       3. div-inv 71.2%

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

       4. unpow2 71.2%

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

       5. associate-/l* 73.1%

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

       6. *-commutative 73.1%

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

       7. associate-*l/ 73.1%

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

       8. *-un-lft-identity 73.1%

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

       9. times-frac 73.1%

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

       10. metadata-eval 73.1%

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

       11. div-inv 73.1%

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

       12. metadata-eval 73.1%

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

       13. *-commutative 73.1%

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

       14. associate-*l/ 73.1%

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

       15. *-un-lft-identity 73.1%

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

       16. times-frac 73.1%

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

       17. metadata-eval 73.1%

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

   14. Applied egg-rr 73.1%

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

   if 1.1000000000000001e-308 < l < 3.75e15

   1. Initial program 72.8%

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

   3. Simplified 72.8%

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

      1. *-commutative 72.8%

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

      2. associate-*l/ 78.7%

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

      3. div-inv 78.7%

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

      4. associate-*l* 78.7%

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

      5. metadata-eval 78.7%

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

   6. Applied egg-rr 78.7%

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

      1. pow1 78.7%

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

   8. Applied egg-rr 90.1%

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

   if 3.75e15 < l

   1. Initial program 52.0%

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

   3. Simplified 52.0%

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

   6. Applied egg-rr 35.2%

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

      1. expm1-def 35.2%

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

      2. expm1-log1p 52.0%

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

      3. *-commutative 52.0%

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

      4. associate-*r/ 51.0%

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

      5. associate-*l/ 53.9%

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

      6. *-commutative 53.9%

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

      7. associate-/l* 53.9%

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

      8. associate-*r* 53.9%

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

      9. *-commutative 53.9%

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

      10. associate-*l/ 53.9%

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

      11. associate-*r/ 54.0%

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

      12. associate-*r/ 54.0%

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

   8. Simplified 54.0%

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

      1. expm1-log1p-u 42.2%

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

      2. expm1-udef 36.7%

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

      3. associate-/r/ 36.7%

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

      4. add-sqr-sqrt 36.7%

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

      5. pow2 36.7%

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

      6. sqrt-pow1 36.7%

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

      7. metadata-eval 36.7%

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

      8. pow1 36.7%

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

      9. associate-*r/ 36.7%

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

   10. Applied egg-rr 36.7%

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

       1. expm1-def 42.2%

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

       2. expm1-log1p 54.0%

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

       3. metadata-eval 54.0%

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

       4. times-frac 54.0%

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

       5. associate-*r/ 54.0%

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

       6. *-commutative 54.0%

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

       7. associate-/l* 54.0%

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

       8. associate-/r/ 54.0%

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

       9. *-commutative 54.0%

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

       10. associate-/r* 54.0%

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

       11. metadata-eval 54.0%

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

   12. Simplified 54.0%

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

       1. *-commutative 54.0%

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

       2. clear-num 54.0%

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

       3. div-inv 54.0%

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

       4. unpow2 54.0%

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

       5. associate-/l* 61.3%

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

       6. *-commutative 61.3%

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

       7. associate-*l/ 61.4%

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

       8. *-un-lft-identity 61.4%

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

       9. times-frac 61.4%

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

       10. metadata-eval 61.4%

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

       11. div-inv 61.4%

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

       12. metadata-eval 61.4%

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

       13. *-commutative 61.4%

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

       14. associate-*l/ 61.3%

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

       15. *-un-lft-identity 61.3%

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

       16. times-frac 61.3%

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

       17. metadata-eval 61.3%

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

   14. Applied egg-rr 61.3%

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

       1. sqrt-div 74.5%

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

       2. div-inv 74.5%

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

   16. Applied egg-rr 74.5%

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

       1. associate-*r/ 74.5%

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

       2. *-rgt-identity 74.5%

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

   18. Simplified 74.5%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.1 \cdot 10^{-308}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \frac{D \cdot \left(0.5 \cdot \frac{M}{d}\right)}{\frac{\ell \cdot -2}{D \cdot \left(0.5 \cdot \frac{M}{d}\right)}}\right)\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;\ell \leq 3.75 \cdot 10^{+15}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 + -0.5 \cdot \frac{h}{\frac{\ell}{{\left(M \cdot \frac{D \cdot 0.5}{d}\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\left(1 + h \cdot \frac{D \cdot \left(0.5 \cdot \frac{M}{d}\right)}{\frac{\ell \cdot -2}{D \cdot \left(0.5 \cdot \frac{M}{d}\right)}}\right) \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := D_m \cdot \left(0.5 \cdot \frac{M_m}{d}\right)\\ t_1 := \sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \frac{t_0}{\frac{\ell \cdot -2}{t_0}}\right)\\ \mathbf{if}\;\ell \leq 3.6 \cdot 10^{-255}:\\ \;\;\;\;t_1 \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot t_1\\ \end{array} \end{array} \]

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 (* 0.5 (/ M_m d))))
        (t_1 (* (sqrt (/ d l)) (+ 1.0 (* h (/ t_0 (/ (* l -2.0) t_0)))))))
   (if (<= l 3.6e-255) (* t_1 (sqrt (/ d h))) (* (/ (sqrt d) (sqrt h)) t_1))))

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 * (0.5 * (M_m / d));
	double t_1 = sqrt((d / l)) * (1.0 + (h * (t_0 / ((l * -2.0) / t_0))));
	double tmp;
	if (l <= 3.6e-255) {
		tmp = t_1 * sqrt((d / h));
	} else {
		tmp = (sqrt(d) / sqrt(h)) * t_1;
	}
	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 * (0.5d0 * (m_m / d))
    t_1 = sqrt((d / l)) * (1.0d0 + (h * (t_0 / ((l * (-2.0d0)) / t_0))))
    if (l <= 3.6d-255) then
        tmp = t_1 * sqrt((d / h))
    else
        tmp = (sqrt(d) / sqrt(h)) * t_1
    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 * (0.5 * (M_m / d));
	double t_1 = Math.sqrt((d / l)) * (1.0 + (h * (t_0 / ((l * -2.0) / t_0))));
	double tmp;
	if (l <= 3.6e-255) {
		tmp = t_1 * Math.sqrt((d / h));
	} else {
		tmp = (Math.sqrt(d) / Math.sqrt(h)) * t_1;
	}
	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 * (0.5 * (M_m / d))
	t_1 = math.sqrt((d / l)) * (1.0 + (h * (t_0 / ((l * -2.0) / t_0))))
	tmp = 0
	if l <= 3.6e-255:
		tmp = t_1 * math.sqrt((d / h))
	else:
		tmp = (math.sqrt(d) / math.sqrt(h)) * t_1
	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(D_m * Float64(0.5 * Float64(M_m / d)))
	t_1 = Float64(sqrt(Float64(d / l)) * Float64(1.0 + Float64(h * Float64(t_0 / Float64(Float64(l * -2.0) / t_0)))))
	tmp = 0.0
	if (l <= 3.6e-255)
		tmp = Float64(t_1 * sqrt(Float64(d / h)));
	else
		tmp = Float64(Float64(sqrt(d) / sqrt(h)) * t_1);
	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 * (0.5 * (M_m / d));
	t_1 = sqrt((d / l)) * (1.0 + (h * (t_0 / ((l * -2.0) / t_0))));
	tmp = 0.0;
	if (l <= 3.6e-255)
		tmp = t_1 * sqrt((d / h));
	else
		tmp = (sqrt(d) / sqrt(h)) * t_1;
	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[(D$95$m * N[(0.5 * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(h * N[(t$95$0 / N[(N[(l * -2.0), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, 3.6e-255], N[(t$95$1 * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if l < 3.6000000000000002e-255

   1. Initial program 66.1%

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

   3. Simplified 65.5%

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

   6. Applied egg-rr 36.8%

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

      1. expm1-def 36.8%

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

      2. expm1-log1p 65.5%

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

      3. *-commutative 65.5%

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

      4. associate-*r/ 69.1%

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

      5. associate-*l/ 69.8%

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

      6. *-commutative 69.8%

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

      7. associate-/l* 69.8%

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

      8. associate-*r* 69.8%

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

      9. *-commutative 69.8%

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

      10. associate-*l/ 70.4%

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

      11. associate-*r/ 70.3%

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

      12. associate-*r/ 70.3%

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

   8. Simplified 70.3%

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

      1. expm1-log1p-u 67.4%

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

      2. expm1-udef 64.7%

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

      3. associate-/r/ 64.7%

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

      4. add-sqr-sqrt 64.7%

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

      5. pow2 64.7%

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

      6. sqrt-pow1 64.7%

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

      7. metadata-eval 64.7%

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

      8. pow1 64.7%

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

      9. associate-*r/ 64.7%

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

   10. Applied egg-rr 64.7%

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

       1. expm1-def 67.4%

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

       2. expm1-log1p 70.3%

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

       3. metadata-eval 70.3%

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

       4. times-frac 70.3%

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

       5. associate-*r/ 70.3%

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

       6. *-commutative 70.3%

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

       7. associate-/l* 69.6%

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

       8. associate-/r/ 70.3%

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

       9. *-commutative 70.3%

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

       10. associate-/r* 70.3%

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

       11. metadata-eval 70.3%

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

   12. Simplified 70.3%

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

       1. *-commutative 70.3%

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

       2. clear-num 70.3%

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

       3. div-inv 70.3%

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

       4. unpow2 70.3%

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

       5. associate-/l* 72.7%

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

       6. *-commutative 72.7%

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

       7. associate-*l/ 72.7%

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

       8. *-un-lft-identity 72.7%

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

       9. times-frac 72.7%

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

       10. metadata-eval 72.7%

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

       11. div-inv 72.7%

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

       12. metadata-eval 72.7%

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

       13. *-commutative 72.7%

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

       14. associate-*l/ 72.7%

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

       15. *-un-lft-identity 72.7%

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

       16. times-frac 72.7%

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

       17. metadata-eval 72.7%

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

   14. Applied egg-rr 72.7%

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

   if 3.6000000000000002e-255 < l

   1. Initial program 65.3%

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

   3. Simplified 65.3%

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

   6. Applied egg-rr 39.7%

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

      1. expm1-def 39.7%

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

      2. expm1-log1p 65.3%

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

      3. *-commutative 65.3%

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

      4. associate-*r/ 66.6%

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

      5. associate-*l/ 68.0%

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

      6. *-commutative 68.0%

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

      7. associate-/l* 68.0%

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

      8. associate-*r* 68.0%

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

      9. *-commutative 68.0%

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

      10. associate-*l/ 68.0%

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

      11. associate-*r/ 68.1%

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

      12. associate-*r/ 68.0%

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

   8. Simplified 68.0%

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

      1. expm1-log1p-u 44.1%

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

      2. expm1-udef 40.6%

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

      3. associate-/r/ 40.6%

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

      4. add-sqr-sqrt 40.6%

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

      5. pow2 40.6%

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

      6. sqrt-pow1 40.6%

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

      7. metadata-eval 40.6%

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

      8. pow1 40.6%

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

      9. associate-*r/ 40.6%

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

   10. Applied egg-rr 40.6%

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

       1. expm1-def 44.1%

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

       2. expm1-log1p 68.1%

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

       3. metadata-eval 68.1%

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

       4. times-frac 68.1%

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

       5. associate-*r/ 68.1%

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

       6. *-commutative 68.1%

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

       7. associate-/l* 68.1%

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

       8. associate-/r/ 68.1%

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

       9. *-commutative 68.1%

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

       10. associate-/r* 68.1%

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

       11. metadata-eval 68.1%

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

   12. Simplified 68.1%

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

       1. *-commutative 68.1%

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

       2. clear-num 68.1%

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

       3. div-inv 68.0%

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

       4. unpow2 68.0%

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

       5. associate-/l* 71.5%

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

       6. *-commutative 71.5%

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

       7. associate-*l/ 71.6%

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

       8. *-un-lft-identity 71.6%

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

       9. times-frac 71.6%

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

       10. metadata-eval 71.6%

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

       11. div-inv 71.6%

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

       12. metadata-eval 71.6%

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

       13. *-commutative 71.6%

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

       14. associate-*l/ 71.6%

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

       15. *-un-lft-identity 71.6%

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

       16. times-frac 71.6%

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

       17. metadata-eval 71.6%

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

   14. Applied egg-rr 71.6%

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

       1. sqrt-div 84.4%

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

       2. div-inv 84.4%

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

   16. Applied egg-rr 84.4%

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

       1. associate-*r/ 84.4%

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

       2. *-rgt-identity 84.4%

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

   18. Simplified 84.4%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 3.6 \cdot 10^{-255}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \frac{D \cdot \left(0.5 \cdot \frac{M}{d}\right)}{\frac{\ell \cdot -2}{D \cdot \left(0.5 \cdot \frac{M}{d}\right)}}\right)\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \frac{D \cdot \left(0.5 \cdot \frac{M}{d}\right)}{\frac{\ell \cdot -2}{D \cdot \left(0.5 \cdot \frac{M}{d}\right)}}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 72.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := D_m \cdot \left(0.5 \cdot \frac{M_m}{d}\right)\\ t_1 := \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \frac{t_0}{\frac{\ell \cdot -2}{t_0}}\right)\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{if}\;\ell \leq -1 \cdot 10^{-306}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 2.6 \cdot 10^{+213}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 + -0.5 \cdot \frac{h}{\frac{\ell}{{\left(M_m \cdot \frac{D_m \cdot 0.5}{d}\right)}^{2}}}\right)\\ \mathbf{elif}\;\ell \leq 7.5 \cdot 10^{+275}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

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
 (let* ((t_0 (* D_m (* 0.5 (/ M_m d))))
        (t_1
         (*
          (* (sqrt (/ d l)) (+ 1.0 (* h (/ t_0 (/ (* l -2.0) t_0)))))
          (sqrt (/ d h)))))
   (if (<= l -1e-306)
     t_1
     (if (<= l 2.6e+213)
       (*
        (/ d (sqrt (* h l)))
        (+ 1.0 (* -0.5 (/ h (/ l (pow (* M_m (/ (* D_m 0.5) d)) 2.0))))))
       (if (<= l 7.5e+275) t_1 (/ d (* (sqrt h) (sqrt 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 t_0 = D_m * (0.5 * (M_m / d));
	double t_1 = (sqrt((d / l)) * (1.0 + (h * (t_0 / ((l * -2.0) / t_0))))) * sqrt((d / h));
	double tmp;
	if (l <= -1e-306) {
		tmp = t_1;
	} else if (l <= 2.6e+213) {
		tmp = (d / sqrt((h * l))) * (1.0 + (-0.5 * (h / (l / pow((M_m * ((D_m * 0.5) / d)), 2.0)))));
	} else if (l <= 7.5e+275) {
		tmp = t_1;
	} else {
		tmp = d / (sqrt(h) * sqrt(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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = d_m * (0.5d0 * (m_m / d))
    t_1 = (sqrt((d / l)) * (1.0d0 + (h * (t_0 / ((l * (-2.0d0)) / t_0))))) * sqrt((d / h))
    if (l <= (-1d-306)) then
        tmp = t_1
    else if (l <= 2.6d+213) then
        tmp = (d / sqrt((h * l))) * (1.0d0 + ((-0.5d0) * (h / (l / ((m_m * ((d_m * 0.5d0) / d)) ** 2.0d0)))))
    else if (l <= 7.5d+275) then
        tmp = t_1
    else
        tmp = d / (sqrt(h) * sqrt(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 t_0 = D_m * (0.5 * (M_m / d));
	double t_1 = (Math.sqrt((d / l)) * (1.0 + (h * (t_0 / ((l * -2.0) / t_0))))) * Math.sqrt((d / h));
	double tmp;
	if (l <= -1e-306) {
		tmp = t_1;
	} else if (l <= 2.6e+213) {
		tmp = (d / Math.sqrt((h * l))) * (1.0 + (-0.5 * (h / (l / Math.pow((M_m * ((D_m * 0.5) / d)), 2.0)))));
	} else if (l <= 7.5e+275) {
		tmp = t_1;
	} else {
		tmp = d / (Math.sqrt(h) * Math.sqrt(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):
	t_0 = D_m * (0.5 * (M_m / d))
	t_1 = (math.sqrt((d / l)) * (1.0 + (h * (t_0 / ((l * -2.0) / t_0))))) * math.sqrt((d / h))
	tmp = 0
	if l <= -1e-306:
		tmp = t_1
	elif l <= 2.6e+213:
		tmp = (d / math.sqrt((h * l))) * (1.0 + (-0.5 * (h / (l / math.pow((M_m * ((D_m * 0.5) / d)), 2.0)))))
	elif l <= 7.5e+275:
		tmp = t_1
	else:
		tmp = d / (math.sqrt(h) * math.sqrt(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)
	t_0 = Float64(D_m * Float64(0.5 * Float64(M_m / d)))
	t_1 = Float64(Float64(sqrt(Float64(d / l)) * Float64(1.0 + Float64(h * Float64(t_0 / Float64(Float64(l * -2.0) / t_0))))) * sqrt(Float64(d / h)))
	tmp = 0.0
	if (l <= -1e-306)
		tmp = t_1;
	elseif (l <= 2.6e+213)
		tmp = Float64(Float64(d / sqrt(Float64(h * l))) * Float64(1.0 + Float64(-0.5 * Float64(h / Float64(l / (Float64(M_m * Float64(Float64(D_m * 0.5) / d)) ^ 2.0))))));
	elseif (l <= 7.5e+275)
		tmp = t_1;
	else
		tmp = Float64(d / Float64(sqrt(h) * sqrt(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)
	t_0 = D_m * (0.5 * (M_m / d));
	t_1 = (sqrt((d / l)) * (1.0 + (h * (t_0 / ((l * -2.0) / t_0))))) * sqrt((d / h));
	tmp = 0.0;
	if (l <= -1e-306)
		tmp = t_1;
	elseif (l <= 2.6e+213)
		tmp = (d / sqrt((h * l))) * (1.0 + (-0.5 * (h / (l / ((M_m * ((D_m * 0.5) / d)) ^ 2.0)))));
	elseif (l <= 7.5e+275)
		tmp = t_1;
	else
		tmp = d / (sqrt(h) * sqrt(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_] := Block[{t$95$0 = N[(D$95$m * N[(0.5 * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(h * N[(t$95$0 / N[(N[(l * -2.0), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -1e-306], t$95$1, If[LessEqual[l, 2.6e+213], N[(N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(-0.5 * N[(h / N[(l / N[Power[N[(M$95$m * N[(N[(D$95$m * 0.5), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 7.5e+275], t$95$1, N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.00000000000000003e-306 or 2.59999999999999999e213 < l < 7.49999999999999978e275

   1. Initial program 65.5%

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

   3. Simplified 64.8%

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

   6. Applied egg-rr 38.5%

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

      1. expm1-def 38.5%

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

      2. expm1-log1p 64.8%

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

      3. *-commutative 64.8%

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

      4. associate-*r/ 67.2%

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

      5. associate-*l/ 68.3%

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

      6. *-commutative 68.3%

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

      7. associate-/l* 68.3%

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

      8. associate-*r* 68.3%

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

      9. *-commutative 68.3%

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

      10. associate-*l/ 68.9%

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

      11. associate-*r/ 68.8%

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

      12. associate-*r/ 68.8%

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

   8. Simplified 68.8%

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

      1. expm1-log1p-u 68.6%

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

      2. expm1-udef 65.5%

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

      3. associate-/r/ 65.5%

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

      4. add-sqr-sqrt 65.5%

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

      5. pow2 65.5%

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

      6. sqrt-pow1 65.5%

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

      7. metadata-eval 65.5%

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

      8. pow1 65.5%

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

      9. associate-*r/ 65.5%

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

   10. Applied egg-rr 65.5%

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

       1. expm1-def 68.6%

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

       2. expm1-log1p 68.8%

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

       3. metadata-eval 68.8%

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

       4. times-frac 68.8%

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

       5. associate-*r/ 68.8%

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

       6. *-commutative 68.8%

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

       7. associate-/l* 68.8%

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

       8. associate-/r/ 68.8%

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

       9. *-commutative 68.8%

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

       10. associate-/r* 68.8%

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

       11. metadata-eval 68.8%

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

   12. Simplified 68.8%

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

       1. *-commutative 68.8%

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

       2. clear-num 68.8%

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

       3. div-inv 68.8%

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

       4. unpow2 68.8%

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

       5. associate-/l* 72.5%

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

       6. *-commutative 72.5%

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

       7. associate-*l/ 72.5%

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

       8. *-un-lft-identity 72.5%

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

       9. times-frac 72.5%

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

       10. metadata-eval 72.5%

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

       11. div-inv 72.5%

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

       12. metadata-eval 72.5%

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

       13. *-commutative 72.5%

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

       14. associate-*l/ 72.5%

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

       15. *-un-lft-identity 72.5%

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

       16. times-frac 72.5%

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

       17. metadata-eval 72.5%

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

   14. Applied egg-rr 72.5%

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

   if -1.00000000000000003e-306 < l < 2.59999999999999999e213

   1. Initial program 69.1%

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

   3. Simplified 69.1%

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

      1. *-commutative 69.1%

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

      2. associate-*l/ 74.3%

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

      3. div-inv 74.3%

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

      4. associate-*l* 74.3%

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

      5. metadata-eval 74.3%

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

   6. Applied egg-rr 74.3%

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

      1. pow1 74.3%

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

   8. Applied egg-rr 84.2%

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

   if 7.49999999999999978e275 < l

   1. Initial program 34.3%

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

   3. Simplified 34.4%

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

      1. frac-2neg 34.8%

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

      2. sqrt-div 0.0%

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

   6. Applied egg-rr 0.0%

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

   7. Taylor expanded in d around inf 56.8%

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

      1. *-commutative 56.8%

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

   9. Simplified 56.8%

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

       1. expm1-log1p-u 53.9%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)\right)} \]

       2. expm1-udef 32.7%

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

       3. sqrt-div 32.7%

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

       4. metadata-eval 32.7%

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

       5. un-div-inv 32.7%

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

   11. Applied egg-rr 32.7%

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

       1. expm1-def 53.9%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{d}{\sqrt{\ell \cdot h}}\right)\right)} \]

       2. expm1-log1p 56.8%

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

   13. Simplified 56.8%

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

       1. sqrt-prod 67.4%

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

   15. Applied egg-rr 67.4%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1 \cdot 10^{-306}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \frac{D \cdot \left(0.5 \cdot \frac{M}{d}\right)}{\frac{\ell \cdot -2}{D \cdot \left(0.5 \cdot \frac{M}{d}\right)}}\right)\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;\ell \leq 2.6 \cdot 10^{+213}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 + -0.5 \cdot \frac{h}{\frac{\ell}{{\left(M \cdot \frac{D \cdot 0.5}{d}\right)}^{2}}}\right)\\ \mathbf{elif}\;\ell \leq 7.5 \cdot 10^{+275}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \frac{D \cdot \left(0.5 \cdot \frac{M}{d}\right)}{\frac{\ell \cdot -2}{D \cdot \left(0.5 \cdot \frac{M}{d}\right)}}\right)\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 68.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := D_m \cdot \left(0.5 \cdot \frac{M_m}{d}\right)\\ \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \frac{t_0}{\frac{\ell \cdot -2}{t_0}}\right)\right) \cdot \sqrt{\frac{d}{h}} \end{array} \end{array} \]

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 (* 0.5 (/ M_m d)))))
   (*
    (* (sqrt (/ d l)) (+ 1.0 (* h (/ t_0 (/ (* l -2.0) t_0)))))
    (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 * (0.5 * (M_m / d));
	return (sqrt((d / l)) * (1.0 + (h * (t_0 / ((l * -2.0) / t_0))))) * 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 * (0.5d0 * (m_m / d))
    code = (sqrt((d / l)) * (1.0d0 + (h * (t_0 / ((l * (-2.0d0)) / t_0))))) * 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 * (0.5 * (M_m / d));
	return (Math.sqrt((d / l)) * (1.0 + (h * (t_0 / ((l * -2.0) / t_0))))) * 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 * (0.5 * (M_m / d))
	return (math.sqrt((d / l)) * (1.0 + (h * (t_0 / ((l * -2.0) / t_0))))) * 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(D_m * Float64(0.5 * Float64(M_m / d)))
	return Float64(Float64(sqrt(Float64(d / l)) * Float64(1.0 + Float64(h * Float64(t_0 / Float64(Float64(l * -2.0) / t_0))))) * 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 * (0.5 * (M_m / d));
	tmp = (sqrt((d / l)) * (1.0 + (h * (t_0 / ((l * -2.0) / t_0))))) * 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[(D$95$m * N[(0.5 * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(h * N[(t$95$0 / N[(N[(l * -2.0), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 65.8%

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

3. Simplified 65.4%

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

6. Applied egg-rr 38.1%

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

   1. expm1-def 38.1%

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

   2. expm1-log1p 65.4%

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

   3. *-commutative 65.4%

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

   4. associate-*r/ 68.1%

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

   5. associate-*l/ 69.0%

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

   6. *-commutative 69.0%

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

   7. associate-/l* 69.0%

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

   8. associate-*r* 69.0%

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

   9. *-commutative 69.0%

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

   10. associate-*l/ 69.4%

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

   11. associate-*r/ 69.4%

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

   12. associate-*r/ 69.4%

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

8. Simplified 69.4%

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

   1. expm1-log1p-u 57.5%

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

   2. expm1-udef 54.5%

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

   3. associate-/r/ 54.5%

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

   4. add-sqr-sqrt 54.5%

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

   5. pow2 54.5%

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

   6. sqrt-pow1 54.5%

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

   7. metadata-eval 54.5%

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

   8. pow1 54.5%

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

   9. associate-*r/ 54.5%

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

10. Applied egg-rr 54.5%

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

    1. expm1-def 57.5%

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

    2. expm1-log1p 69.4%

       \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \color{blue}{\left(\frac{{\left(D \cdot \frac{M \cdot 0.5}{d}\right)}^{2}}{\ell} \cdot -0.5\right)}\right)\right) \]

    3. metadata-eval 69.4%

       \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \left(\frac{{\left(D \cdot \frac{M \cdot 0.5}{d}\right)}^{2}}{\ell} \cdot \color{blue}{\frac{1}{-2}}\right)\right)\right) \]

    4. times-frac 69.4%

       \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \color{blue}{\frac{{\left(D \cdot \frac{M \cdot 0.5}{d}\right)}^{2} \cdot 1}{\ell \cdot -2}}\right)\right) \]

    5. associate-*r/ 69.4%

       \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \color{blue}{\left({\left(D \cdot \frac{M \cdot 0.5}{d}\right)}^{2} \cdot \frac{1}{\ell \cdot -2}\right)}\right)\right) \]

    6. *-commutative 69.4%

       \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \left({\left(D \cdot \frac{\color{blue}{0.5 \cdot M}}{d}\right)}^{2} \cdot \frac{1}{\ell \cdot -2}\right)\right)\right) \]

    7. associate-/l* 69.0%

       \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \left({\left(D \cdot \color{blue}{\frac{0.5}{\frac{d}{M}}}\right)}^{2} \cdot \frac{1}{\ell \cdot -2}\right)\right)\right) \]

    8. associate-/r/ 69.4%

       \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \left({\left(D \cdot \color{blue}{\left(\frac{0.5}{d} \cdot M\right)}\right)}^{2} \cdot \frac{1}{\ell \cdot -2}\right)\right)\right) \]

    9. *-commutative 69.4%

       \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \left({\left(D \cdot \left(\frac{0.5}{d} \cdot M\right)\right)}^{2} \cdot \frac{1}{\color{blue}{-2 \cdot \ell}}\right)\right)\right) \]

    10. associate-/r* 69.4%

        \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \left({\left(D \cdot \left(\frac{0.5}{d} \cdot M\right)\right)}^{2} \cdot \color{blue}{\frac{\frac{1}{-2}}{\ell}}\right)\right)\right) \]

    11. metadata-eval 69.4%

        \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \left({\left(D \cdot \left(\frac{0.5}{d} \cdot M\right)\right)}^{2} \cdot \frac{\color{blue}{-0.5}}{\ell}\right)\right)\right) \]

12. Simplified 69.4%

    \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \color{blue}{\left({\left(D \cdot \left(\frac{0.5}{d} \cdot M\right)\right)}^{2} \cdot \frac{-0.5}{\ell}\right)}\right)\right) \]
13. Step-by-step derivation

    1. *-commutative 69.4%

       \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \left({\left(D \cdot \color{blue}{\left(M \cdot \frac{0.5}{d}\right)}\right)}^{2} \cdot \frac{-0.5}{\ell}\right)\right)\right) \]

    2. clear-num 69.4%

       \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \left({\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot \color{blue}{\frac{1}{\frac{\ell}{-0.5}}}\right)\right)\right) \]

    3. div-inv 69.4%

       \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \color{blue}{\frac{{\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}}{\frac{\ell}{-0.5}}}\right)\right) \]

    4. unpow2 69.4%

       \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \frac{\color{blue}{\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right) \cdot \left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}}{\frac{\ell}{-0.5}}\right)\right) \]

    5. associate-/l* 72.2%

       \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \color{blue}{\frac{D \cdot \left(M \cdot \frac{0.5}{d}\right)}{\frac{\frac{\ell}{-0.5}}{D \cdot \left(M \cdot \frac{0.5}{d}\right)}}}\right)\right) \]

    6. *-commutative 72.2%

       \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \frac{D \cdot \color{blue}{\left(\frac{0.5}{d} \cdot M\right)}}{\frac{\frac{\ell}{-0.5}}{D \cdot \left(M \cdot \frac{0.5}{d}\right)}}\right)\right) \]

    7. associate-*l/ 72.2%

       \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \frac{D \cdot \color{blue}{\frac{0.5 \cdot M}{d}}}{\frac{\frac{\ell}{-0.5}}{D \cdot \left(M \cdot \frac{0.5}{d}\right)}}\right)\right) \]

    8. *-un-lft-identity 72.2%

       \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \frac{D \cdot \frac{0.5 \cdot M}{\color{blue}{1 \cdot d}}}{\frac{\frac{\ell}{-0.5}}{D \cdot \left(M \cdot \frac{0.5}{d}\right)}}\right)\right) \]

    9. times-frac 72.2%

       \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \frac{D \cdot \color{blue}{\left(\frac{0.5}{1} \cdot \frac{M}{d}\right)}}{\frac{\frac{\ell}{-0.5}}{D \cdot \left(M \cdot \frac{0.5}{d}\right)}}\right)\right) \]

    10. metadata-eval 72.2%

        \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \frac{D \cdot \left(\color{blue}{0.5} \cdot \frac{M}{d}\right)}{\frac{\frac{\ell}{-0.5}}{D \cdot \left(M \cdot \frac{0.5}{d}\right)}}\right)\right) \]

    11. div-inv 72.2%

        \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \frac{D \cdot \left(0.5 \cdot \frac{M}{d}\right)}{\frac{\color{blue}{\ell \cdot \frac{1}{-0.5}}}{D \cdot \left(M \cdot \frac{0.5}{d}\right)}}\right)\right) \]

    12. metadata-eval 72.2%

        \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \frac{D \cdot \left(0.5 \cdot \frac{M}{d}\right)}{\frac{\ell \cdot \color{blue}{-2}}{D \cdot \left(M \cdot \frac{0.5}{d}\right)}}\right)\right) \]

    13. *-commutative 72.2%

        \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \frac{D \cdot \left(0.5 \cdot \frac{M}{d}\right)}{\frac{\ell \cdot -2}{D \cdot \color{blue}{\left(\frac{0.5}{d} \cdot M\right)}}}\right)\right) \]

    14. associate-*l/ 72.2%

        \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \frac{D \cdot \left(0.5 \cdot \frac{M}{d}\right)}{\frac{\ell \cdot -2}{D \cdot \color{blue}{\frac{0.5 \cdot M}{d}}}}\right)\right) \]

    15. *-un-lft-identity 72.2%

        \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \frac{D \cdot \left(0.5 \cdot \frac{M}{d}\right)}{\frac{\ell \cdot -2}{D \cdot \frac{0.5 \cdot M}{\color{blue}{1 \cdot d}}}}\right)\right) \]

    16. times-frac 72.2%

        \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \frac{D \cdot \left(0.5 \cdot \frac{M}{d}\right)}{\frac{\ell \cdot -2}{D \cdot \color{blue}{\left(\frac{0.5}{1} \cdot \frac{M}{d}\right)}}}\right)\right) \]

    17. metadata-eval 72.2%

        \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \frac{D \cdot \left(0.5 \cdot \frac{M}{d}\right)}{\frac{\ell \cdot -2}{D \cdot \left(\color{blue}{0.5} \cdot \frac{M}{d}\right)}}\right)\right) \]

14. Applied egg-rr 72.2%

    \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \color{blue}{\frac{D \cdot \left(0.5 \cdot \frac{M}{d}\right)}{\frac{\ell \cdot -2}{D \cdot \left(0.5 \cdot \frac{M}{d}\right)}}}\right)\right) \]

15. Final simplification 72.2%

    \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \frac{D \cdot \left(0.5 \cdot \frac{M}{d}\right)}{\frac{\ell \cdot -2}{D \cdot \left(0.5 \cdot \frac{M}{d}\right)}}\right)\right) \cdot \sqrt{\frac{d}{h}} \]
16. Add Preprocessing

Alternative 9: 63.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \frac{D_m \cdot \left(0.5 \cdot \frac{M_m}{d}\right)}{\frac{-4}{M_m} \cdot \left(d \cdot \frac{\ell}{D_m}\right)}\right)\right) \end{array} \]

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
 (*
  (sqrt (/ d h))
  (*
   (sqrt (/ d l))
   (+
    1.0
    (* h (/ (* D_m (* 0.5 (/ M_m d))) (* (/ -4.0 M_m) (* d (/ l D_m)))))))))

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 sqrt((d / h)) * (sqrt((d / l)) * (1.0 + (h * ((D_m * (0.5 * (M_m / d))) / ((-4.0 / M_m) * (d * (l / D_m)))))));
}

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 = sqrt((d / h)) * (sqrt((d / l)) * (1.0d0 + (h * ((d_m * (0.5d0 * (m_m / d))) / (((-4.0d0) / m_m) * (d * (l / d_m)))))))
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 Math.sqrt((d / h)) * (Math.sqrt((d / l)) * (1.0 + (h * ((D_m * (0.5 * (M_m / d))) / ((-4.0 / M_m) * (d * (l / D_m)))))));
}

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 math.sqrt((d / h)) * (math.sqrt((d / l)) * (1.0 + (h * ((D_m * (0.5 * (M_m / d))) / ((-4.0 / M_m) * (d * (l / D_m)))))))

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(sqrt(Float64(d / h)) * Float64(sqrt(Float64(d / l)) * Float64(1.0 + Float64(h * Float64(Float64(D_m * Float64(0.5 * Float64(M_m / d))) / Float64(Float64(-4.0 / M_m) * Float64(d * Float64(l / D_m))))))))
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 = sqrt((d / h)) * (sqrt((d / l)) * (1.0 + (h * ((D_m * (0.5 * (M_m / d))) / ((-4.0 / M_m) * (d * (l / D_m)))))));
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[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(h * N[(N[(D$95$m * N[(0.5 * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(-4.0 / M$95$m), $MachinePrecision] * N[(d * N[(l / D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 65.8%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

3. Simplified 65.4%

   \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot -0.5\right)\right)\right)} \]
4. Add Preprocessing

6. Applied egg-rr 38.1%

   \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{h}{\ell} \cdot \left({\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot -0.5\right)\right)} - 1\right)}\right)\right) \]
7. Step-by-step derivation

   1. expm1-def 38.1%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{h}{\ell} \cdot \left({\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot -0.5\right)\right)\right)}\right)\right) \]

   2. expm1-log1p 65.4%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \color{blue}{\frac{h}{\ell} \cdot \left({\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot -0.5\right)}\right)\right) \]

   3. *-commutative 65.4%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \color{blue}{\left({\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot -0.5\right) \cdot \frac{h}{\ell}}\right)\right) \]

   4. associate-*r/ 68.1%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \color{blue}{\frac{\left({\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot -0.5\right) \cdot h}{\ell}}\right)\right) \]

   5. associate-*l/ 69.0%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \color{blue}{\frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot -0.5}{\ell} \cdot h}\right)\right) \]

   6. *-commutative 69.0%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \color{blue}{h \cdot \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot -0.5}{\ell}}\right)\right) \]

   7. associate-/l* 69.0%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \color{blue}{\frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\frac{\ell}{-0.5}}}\right)\right) \]

   8. associate-*r* 69.0%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \frac{{\color{blue}{\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}}^{2}}{\frac{\ell}{-0.5}}\right)\right) \]

   9. *-commutative 69.0%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \frac{{\color{blue}{\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}}^{2}}{\frac{\ell}{-0.5}}\right)\right) \]

   10. associate-*l/ 69.4%

       \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \frac{{\color{blue}{\left(\frac{D \cdot \left(M \cdot 0.5\right)}{d}\right)}}^{2}}{\frac{\ell}{-0.5}}\right)\right) \]

   11. associate-*r/ 69.4%

       \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \frac{{\color{blue}{\left(D \cdot \frac{M \cdot 0.5}{d}\right)}}^{2}}{\frac{\ell}{-0.5}}\right)\right) \]

   12. associate-*r/ 69.4%

       \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \frac{{\left(D \cdot \color{blue}{\left(M \cdot \frac{0.5}{d}\right)}\right)}^{2}}{\frac{\ell}{-0.5}}\right)\right) \]

8. Simplified 69.4%

   \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \color{blue}{h \cdot \frac{{\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}}{\frac{\ell}{-0.5}}}\right)\right) \]
9. Step-by-step derivation

   1. expm1-log1p-u 57.5%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}}{\frac{\ell}{-0.5}}\right)\right)}\right)\right) \]

   2. expm1-udef 54.5%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{{\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}}{\frac{\ell}{-0.5}}\right)} - 1\right)}\right)\right) \]

   3. associate-/r/ 54.5%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{{\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}}{\ell} \cdot -0.5}\right)} - 1\right)\right)\right) \]

   4. add-sqr-sqrt 54.5%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{{\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}} \cdot \sqrt{{\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}}}}{\ell} \cdot -0.5\right)} - 1\right)\right)\right) \]

   5. pow2 54.5%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{{\left(\sqrt{{\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}}\right)}^{2}}}{\ell} \cdot -0.5\right)} - 1\right)\right)\right) \]

   6. sqrt-pow1 54.5%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \left(e^{\mathsf{log1p}\left(\frac{{\color{blue}{\left({\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{\left(\frac{2}{2}\right)}\right)}}^{2}}{\ell} \cdot -0.5\right)} - 1\right)\right)\right) \]

   7. metadata-eval 54.5%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \left(e^{\mathsf{log1p}\left(\frac{{\left({\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{\color{blue}{1}}\right)}^{2}}{\ell} \cdot -0.5\right)} - 1\right)\right)\right) \]

   8. pow1 54.5%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \left(e^{\mathsf{log1p}\left(\frac{{\color{blue}{\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}}^{2}}{\ell} \cdot -0.5\right)} - 1\right)\right)\right) \]

   9. associate-*r/ 54.5%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \left(e^{\mathsf{log1p}\left(\frac{{\left(D \cdot \color{blue}{\frac{M \cdot 0.5}{d}}\right)}^{2}}{\ell} \cdot -0.5\right)} - 1\right)\right)\right) \]

10. Applied egg-rr 54.5%

    \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{{\left(D \cdot \frac{M \cdot 0.5}{d}\right)}^{2}}{\ell} \cdot -0.5\right)} - 1\right)}\right)\right) \]
11. Step-by-step derivation

    1. expm1-def 57.5%

       \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\left(D \cdot \frac{M \cdot 0.5}{d}\right)}^{2}}{\ell} \cdot -0.5\right)\right)}\right)\right) \]

    2. expm1-log1p 69.4%

       \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \color{blue}{\left(\frac{{\left(D \cdot \frac{M \cdot 0.5}{d}\right)}^{2}}{\ell} \cdot -0.5\right)}\right)\right) \]

    3. metadata-eval 69.4%

       \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \left(\frac{{\left(D \cdot \frac{M \cdot 0.5}{d}\right)}^{2}}{\ell} \cdot \color{blue}{\frac{1}{-2}}\right)\right)\right) \]

    4. times-frac 69.4%

       \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \color{blue}{\frac{{\left(D \cdot \frac{M \cdot 0.5}{d}\right)}^{2} \cdot 1}{\ell \cdot -2}}\right)\right) \]

    5. associate-*r/ 69.4%

       \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \color{blue}{\left({\left(D \cdot \frac{M \cdot 0.5}{d}\right)}^{2} \cdot \frac{1}{\ell \cdot -2}\right)}\right)\right) \]

    6. *-commutative 69.4%

       \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \left({\left(D \cdot \frac{\color{blue}{0.5 \cdot M}}{d}\right)}^{2} \cdot \frac{1}{\ell \cdot -2}\right)\right)\right) \]

    7. associate-/l* 69.0%

       \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \left({\left(D \cdot \color{blue}{\frac{0.5}{\frac{d}{M}}}\right)}^{2} \cdot \frac{1}{\ell \cdot -2}\right)\right)\right) \]

    8. associate-/r/ 69.4%

       \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \left({\left(D \cdot \color{blue}{\left(\frac{0.5}{d} \cdot M\right)}\right)}^{2} \cdot \frac{1}{\ell \cdot -2}\right)\right)\right) \]

    9. *-commutative 69.4%

       \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \left({\left(D \cdot \left(\frac{0.5}{d} \cdot M\right)\right)}^{2} \cdot \frac{1}{\color{blue}{-2 \cdot \ell}}\right)\right)\right) \]

    10. associate-/r* 69.4%

        \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \left({\left(D \cdot \left(\frac{0.5}{d} \cdot M\right)\right)}^{2} \cdot \color{blue}{\frac{\frac{1}{-2}}{\ell}}\right)\right)\right) \]

    11. metadata-eval 69.4%

        \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \left({\left(D \cdot \left(\frac{0.5}{d} \cdot M\right)\right)}^{2} \cdot \frac{\color{blue}{-0.5}}{\ell}\right)\right)\right) \]

12. Simplified 69.4%

    \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \color{blue}{\left({\left(D \cdot \left(\frac{0.5}{d} \cdot M\right)\right)}^{2} \cdot \frac{-0.5}{\ell}\right)}\right)\right) \]
13. Step-by-step derivation

    1. *-commutative 69.4%

       \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \left({\left(D \cdot \color{blue}{\left(M \cdot \frac{0.5}{d}\right)}\right)}^{2} \cdot \frac{-0.5}{\ell}\right)\right)\right) \]

    2. clear-num 69.4%

       \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \left({\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot \color{blue}{\frac{1}{\frac{\ell}{-0.5}}}\right)\right)\right) \]

    3. div-inv 69.4%

       \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \color{blue}{\frac{{\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}}{\frac{\ell}{-0.5}}}\right)\right) \]

    4. unpow2 69.4%

       \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \frac{\color{blue}{\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right) \cdot \left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}}{\frac{\ell}{-0.5}}\right)\right) \]

    5. associate-/l* 72.2%

       \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \color{blue}{\frac{D \cdot \left(M \cdot \frac{0.5}{d}\right)}{\frac{\frac{\ell}{-0.5}}{D \cdot \left(M \cdot \frac{0.5}{d}\right)}}}\right)\right) \]

    6. *-commutative 72.2%

       \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \frac{D \cdot \color{blue}{\left(\frac{0.5}{d} \cdot M\right)}}{\frac{\frac{\ell}{-0.5}}{D \cdot \left(M \cdot \frac{0.5}{d}\right)}}\right)\right) \]

    7. associate-*l/ 72.2%

       \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \frac{D \cdot \color{blue}{\frac{0.5 \cdot M}{d}}}{\frac{\frac{\ell}{-0.5}}{D \cdot \left(M \cdot \frac{0.5}{d}\right)}}\right)\right) \]

    8. *-un-lft-identity 72.2%

       \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \frac{D \cdot \frac{0.5 \cdot M}{\color{blue}{1 \cdot d}}}{\frac{\frac{\ell}{-0.5}}{D \cdot \left(M \cdot \frac{0.5}{d}\right)}}\right)\right) \]

    9. times-frac 72.2%

       \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \frac{D \cdot \color{blue}{\left(\frac{0.5}{1} \cdot \frac{M}{d}\right)}}{\frac{\frac{\ell}{-0.5}}{D \cdot \left(M \cdot \frac{0.5}{d}\right)}}\right)\right) \]

    10. metadata-eval 72.2%

        \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \frac{D \cdot \left(\color{blue}{0.5} \cdot \frac{M}{d}\right)}{\frac{\frac{\ell}{-0.5}}{D \cdot \left(M \cdot \frac{0.5}{d}\right)}}\right)\right) \]

    11. div-inv 72.2%

        \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \frac{D \cdot \left(0.5 \cdot \frac{M}{d}\right)}{\frac{\color{blue}{\ell \cdot \frac{1}{-0.5}}}{D \cdot \left(M \cdot \frac{0.5}{d}\right)}}\right)\right) \]

    12. metadata-eval 72.2%

        \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \frac{D \cdot \left(0.5 \cdot \frac{M}{d}\right)}{\frac{\ell \cdot \color{blue}{-2}}{D \cdot \left(M \cdot \frac{0.5}{d}\right)}}\right)\right) \]

    13. *-commutative 72.2%

        \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \frac{D \cdot \left(0.5 \cdot \frac{M}{d}\right)}{\frac{\ell \cdot -2}{D \cdot \color{blue}{\left(\frac{0.5}{d} \cdot M\right)}}}\right)\right) \]

    14. associate-*l/ 72.2%

        \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \frac{D \cdot \left(0.5 \cdot \frac{M}{d}\right)}{\frac{\ell \cdot -2}{D \cdot \color{blue}{\frac{0.5 \cdot M}{d}}}}\right)\right) \]

    15. *-un-lft-identity 72.2%

        \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \frac{D \cdot \left(0.5 \cdot \frac{M}{d}\right)}{\frac{\ell \cdot -2}{D \cdot \frac{0.5 \cdot M}{\color{blue}{1 \cdot d}}}}\right)\right) \]

    16. times-frac 72.2%

        \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \frac{D \cdot \left(0.5 \cdot \frac{M}{d}\right)}{\frac{\ell \cdot -2}{D \cdot \color{blue}{\left(\frac{0.5}{1} \cdot \frac{M}{d}\right)}}}\right)\right) \]

    17. metadata-eval 72.2%

        \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \frac{D \cdot \left(0.5 \cdot \frac{M}{d}\right)}{\frac{\ell \cdot -2}{D \cdot \left(\color{blue}{0.5} \cdot \frac{M}{d}\right)}}\right)\right) \]

14. Applied egg-rr 72.2%

    \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \color{blue}{\frac{D \cdot \left(0.5 \cdot \frac{M}{d}\right)}{\frac{\ell \cdot -2}{D \cdot \left(0.5 \cdot \frac{M}{d}\right)}}}\right)\right) \]

15. Taylor expanded in l around 0 68.1%

    \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \frac{D \cdot \left(0.5 \cdot \frac{M}{d}\right)}{\color{blue}{-4 \cdot \frac{d \cdot \ell}{D \cdot M}}}\right)\right) \]
16. Step-by-step derivation

    1. associate-*r/ 68.1%

       \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \frac{D \cdot \left(0.5 \cdot \frac{M}{d}\right)}{\color{blue}{\frac{-4 \cdot \left(d \cdot \ell\right)}{D \cdot M}}}\right)\right) \]

    2. times-frac 67.2%

       \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \frac{D \cdot \left(0.5 \cdot \frac{M}{d}\right)}{\color{blue}{\frac{-4}{D} \cdot \frac{d \cdot \ell}{M}}}\right)\right) \]

    3. metadata-eval 67.2%

       \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \frac{D \cdot \left(0.5 \cdot \frac{M}{d}\right)}{\frac{\color{blue}{\frac{-2}{0.5}}}{D} \cdot \frac{d \cdot \ell}{M}}\right)\right) \]

    4. associate-/r* 67.2%

       \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \frac{D \cdot \left(0.5 \cdot \frac{M}{d}\right)}{\color{blue}{\frac{-2}{0.5 \cdot D}} \cdot \frac{d \cdot \ell}{M}}\right)\right) \]

    5. *-commutative 67.2%

       \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \frac{D \cdot \left(0.5 \cdot \frac{M}{d}\right)}{\frac{-2}{\color{blue}{D \cdot 0.5}} \cdot \frac{d \cdot \ell}{M}}\right)\right) \]

    6. *-commutative 67.2%

       \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \frac{D \cdot \left(0.5 \cdot \frac{M}{d}\right)}{\frac{-2}{D \cdot 0.5} \cdot \frac{\color{blue}{\ell \cdot d}}{M}}\right)\right) \]

    7. associate-*l/ 69.6%

       \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \frac{D \cdot \left(0.5 \cdot \frac{M}{d}\right)}{\frac{-2}{D \cdot 0.5} \cdot \color{blue}{\left(\frac{\ell}{M} \cdot d\right)}}\right)\right) \]

    8. associate-/r/ 70.6%

       \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \frac{D \cdot \left(0.5 \cdot \frac{M}{d}\right)}{\frac{-2}{D \cdot 0.5} \cdot \color{blue}{\frac{\ell}{\frac{M}{d}}}}\right)\right) \]

    9. associate-*r/ 71.0%

       \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \frac{D \cdot \left(0.5 \cdot \frac{M}{d}\right)}{\color{blue}{\frac{\frac{-2}{D \cdot 0.5} \cdot \ell}{\frac{M}{d}}}}\right)\right) \]

    10. associate-*l/ 71.0%

        \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \frac{D \cdot \left(0.5 \cdot \frac{M}{d}\right)}{\frac{\color{blue}{\frac{-2 \cdot \ell}{D \cdot 0.5}}}{\frac{M}{d}}}\right)\right) \]

    11. *-commutative 71.0%

        \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \frac{D \cdot \left(0.5 \cdot \frac{M}{d}\right)}{\frac{\frac{\color{blue}{\ell \cdot -2}}{D \cdot 0.5}}{\frac{M}{d}}}\right)\right) \]

    12. rem-square-sqrt 38.2%

        \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \frac{D \cdot \left(0.5 \cdot \frac{M}{d}\right)}{\frac{\frac{\color{blue}{\sqrt{\ell \cdot -2} \cdot \sqrt{\ell \cdot -2}}}{D \cdot 0.5}}{\frac{M}{d}}}\right)\right) \]

    13. associate-*l/ 38.2%

        \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \frac{D \cdot \left(0.5 \cdot \frac{M}{d}\right)}{\frac{\color{blue}{\frac{\sqrt{\ell \cdot -2}}{D \cdot 0.5} \cdot \sqrt{\ell \cdot -2}}}{\frac{M}{d}}}\right)\right) \]

    14. associate-/r/ 38.2%

        \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \frac{D \cdot \left(0.5 \cdot \frac{M}{d}\right)}{\color{blue}{\frac{\frac{\sqrt{\ell \cdot -2}}{D \cdot 0.5} \cdot \sqrt{\ell \cdot -2}}{M} \cdot d}}\right)\right) \]

17. Simplified 70.0%

    \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \frac{D \cdot \left(0.5 \cdot \frac{M}{d}\right)}{\color{blue}{\frac{-4}{M} \cdot \left(d \cdot \frac{\ell}{D}\right)}}\right)\right) \]

18. Final simplification 70.0%

    \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \frac{D \cdot \left(0.5 \cdot \frac{M}{d}\right)}{\frac{-4}{M} \cdot \left(d \cdot \frac{\ell}{D}\right)}\right)\right) \]
19. Add Preprocessing

Alternative 10: 55.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -3.2 \cdot 10^{-186}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{elif}\;\ell \leq 5.8 \cdot 10^{-290}:\\ \;\;\;\;\frac{d}{\sqrt[3]{{\left(h \cdot \ell\right)}^{1.5}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(M_m \cdot \frac{0.5}{\frac{d}{D_m}}\right)}^{2}\right)\right)\\ \end{array} \end{array} \]

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
 (if (<= l -3.2e-186)
   (* d (- (sqrt (/ 1.0 (* h l)))))
   (if (<= l 5.8e-290)
     (/ d (cbrt (pow (* h l) 1.5)))
     (*
      (/ d (sqrt (* h l)))
      (+ 1.0 (* -0.5 (* (/ h l) (pow (* M_m (/ 0.5 (/ d D_m))) 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 tmp;
	if (l <= -3.2e-186) {
		tmp = d * -sqrt((1.0 / (h * l)));
	} else if (l <= 5.8e-290) {
		tmp = d / cbrt(pow((h * l), 1.5));
	} else {
		tmp = (d / sqrt((h * l))) * (1.0 + (-0.5 * ((h / l) * pow((M_m * (0.5 / (d / D_m))), 2.0))));
	}
	return tmp;
}

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 <= -3.2e-186) {
		tmp = d * -Math.sqrt((1.0 / (h * l)));
	} else if (l <= 5.8e-290) {
		tmp = d / Math.cbrt(Math.pow((h * l), 1.5));
	} else {
		tmp = (d / Math.sqrt((h * l))) * (1.0 + (-0.5 * ((h / l) * Math.pow((M_m * (0.5 / (d / D_m))), 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)
	tmp = 0.0
	if (l <= -3.2e-186)
		tmp = Float64(d * Float64(-sqrt(Float64(1.0 / Float64(h * l)))));
	elseif (l <= 5.8e-290)
		tmp = Float64(d / cbrt((Float64(h * l) ^ 1.5)));
	else
		tmp = Float64(Float64(d / sqrt(Float64(h * l))) * Float64(1.0 + Float64(-0.5 * Float64(Float64(h / l) * (Float64(M_m * Float64(0.5 / Float64(d / D_m))) ^ 2.0)))));
	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_] := If[LessEqual[l, -3.2e-186], N[(d * (-N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], If[LessEqual[l, 5.8e-290], N[(d / N[Power[N[Power[N[(h * l), $MachinePrecision], 1.5], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(-0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(M$95$m * N[(0.5 / N[(d / D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -3.2e-186

   1. Initial program 66.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 65.5%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. frac-2neg 69.6%

         \[\leadsto \sqrt{\color{blue}{\frac{-d}{-h}}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \frac{{\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}}{\frac{\ell}{-0.5}}\right)\right) \]

      2. sqrt-div 83.0%

         \[\leadsto \color{blue}{\frac{\sqrt{-d}}{\sqrt{-h}}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \frac{{\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}}{\frac{\ell}{-0.5}}\right)\right) \]

   6. Applied egg-rr 78.0%

      \[\leadsto \left(\color{blue}{\frac{\sqrt{-d}}{\sqrt{-h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   7. Taylor expanded in d around -inf 48.0%

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   8. Step-by-step derivation

      1. mul-1-neg 48.0%

         \[\leadsto \color{blue}{-d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      2. *-commutative 48.0%

         \[\leadsto -\color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      3. distribute-rgt-neg-in 48.0%

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-d\right)} \]

      4. *-commutative 48.0%

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot \left(-d\right) \]

   9. Simplified 48.0%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)} \]

   if -3.2e-186 < l < 5.79999999999999989e-290

   1. Initial program 66.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 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
   5. Step-by-step derivation

      1. frac-2neg 76.4%

         \[\leadsto \sqrt{\color{blue}{\frac{-d}{-h}}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \frac{{\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}}{\frac{\ell}{-0.5}}\right)\right) \]

      2. sqrt-div 76.1%

         \[\leadsto \color{blue}{\frac{\sqrt{-d}}{\sqrt{-h}}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \frac{{\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}}{\frac{\ell}{-0.5}}\right)\right) \]

   6. Applied egg-rr 71.4%

      \[\leadsto \left(\color{blue}{\frac{\sqrt{-d}}{\sqrt{-h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   7. Taylor expanded in d around inf 43.9%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   8. Step-by-step derivation

      1. *-commutative 43.9%

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

   9. Simplified 43.9%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]
   10. Step-by-step derivation

       1. expm1-log1p-u 9.6%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)\right)} \]

       2. expm1-udef 9.4%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} - 1} \]

       3. sqrt-div 9.4%

          \[\leadsto e^{\mathsf{log1p}\left(d \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\ell \cdot h}}}\right)} - 1 \]

       4. metadata-eval 9.4%

          \[\leadsto e^{\mathsf{log1p}\left(d \cdot \frac{\color{blue}{1}}{\sqrt{\ell \cdot h}}\right)} - 1 \]

       5. un-div-inv 9.4%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{d}{\sqrt{\ell \cdot h}}}\right)} - 1 \]

   11. Applied egg-rr 9.4%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{d}{\sqrt{\ell \cdot h}}\right)} - 1} \]
   12. Step-by-step derivation

       1. expm1-def 9.6%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{d}{\sqrt{\ell \cdot h}}\right)\right)} \]

       2. expm1-log1p 44.0%

          \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]

   13. Simplified 44.0%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
   14. Step-by-step derivation

       1. add-cbrt-cube 57.6%

          \[\leadsto \frac{d}{\color{blue}{\sqrt[3]{\left(\sqrt{\ell \cdot h} \cdot \sqrt{\ell \cdot h}\right) \cdot \sqrt{\ell \cdot h}}}} \]

       2. pow1/3 57.1%

          \[\leadsto \frac{d}{\color{blue}{{\left(\left(\sqrt{\ell \cdot h} \cdot \sqrt{\ell \cdot h}\right) \cdot \sqrt{\ell \cdot h}\right)}^{0.3333333333333333}}} \]

       3. add-sqr-sqrt 57.1%

          \[\leadsto \frac{d}{{\left(\color{blue}{\left(\ell \cdot h\right)} \cdot \sqrt{\ell \cdot h}\right)}^{0.3333333333333333}} \]

       4. pow1 57.1%

          \[\leadsto \frac{d}{{\left(\color{blue}{{\left(\ell \cdot h\right)}^{1}} \cdot \sqrt{\ell \cdot h}\right)}^{0.3333333333333333}} \]

       5. pow1/2 57.1%

          \[\leadsto \frac{d}{{\left({\left(\ell \cdot h\right)}^{1} \cdot \color{blue}{{\left(\ell \cdot h\right)}^{0.5}}\right)}^{0.3333333333333333}} \]

       6. pow-prod-up 57.1%

          \[\leadsto \frac{d}{{\color{blue}{\left({\left(\ell \cdot h\right)}^{\left(1 + 0.5\right)}\right)}}^{0.3333333333333333}} \]

       7. metadata-eval 57.1%

          \[\leadsto \frac{d}{{\left({\left(\ell \cdot h\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333}} \]

   15. Applied egg-rr 57.1%

       \[\leadsto \frac{d}{\color{blue}{{\left({\left(\ell \cdot h\right)}^{1.5}\right)}^{0.3333333333333333}}} \]
   16. Step-by-step derivation

       1. unpow1/3 57.6%

          \[\leadsto \frac{d}{\color{blue}{\sqrt[3]{{\left(\ell \cdot h\right)}^{1.5}}}} \]

   17. Simplified 57.6%

       \[\leadsto \frac{d}{\color{blue}{\sqrt[3]{{\left(\ell \cdot h\right)}^{1.5}}}} \]

   if 5.79999999999999989e-290 < l

   1. Initial program 65.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 65.2%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 65.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}\right)}\right) \]

      2. associate-*l/ 67.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{h \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}}{\ell}}\right) \]

      3. div-inv 67.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{h \cdot {\left(\color{blue}{\left(M \cdot \frac{1}{2}\right)} \cdot \frac{D}{d}\right)}^{2}}{\ell}\right) \]

      4. associate-*l* 67.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{h \cdot {\color{blue}{\left(M \cdot \left(\frac{1}{2} \cdot \frac{D}{d}\right)\right)}}^{2}}{\ell}\right) \]

      5. metadata-eval 67.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{h \cdot {\left(M \cdot \left(\color{blue}{0.5} \cdot \frac{D}{d}\right)\right)}^{2}}{\ell}\right) \]

   6. Applied egg-rr 67.3%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{h \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell}}\right) \]

   8. Applied egg-rr 32.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{d}{\sqrt{\ell \cdot h}} \cdot \left(1 + -0.5 \cdot \frac{h}{\frac{\ell}{{\left(M \cdot \frac{0.5 \cdot D}{d}\right)}^{2}}}\right)\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 42.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{d}{\sqrt{\ell \cdot h}} \cdot \left(1 + -0.5 \cdot \frac{h}{\frac{\ell}{{\left(M \cdot \frac{0.5 \cdot D}{d}\right)}^{2}}}\right)\right)\right)} \]

      2. expm1-log1p 75.8%

         \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}} \cdot \left(1 + -0.5 \cdot \frac{h}{\frac{\ell}{{\left(M \cdot \frac{0.5 \cdot D}{d}\right)}^{2}}}\right)} \]

      3. associate-/r/ 69.9%

         \[\leadsto \frac{d}{\sqrt{\ell \cdot h}} \cdot \left(1 + -0.5 \cdot \color{blue}{\left(\frac{h}{\ell} \cdot {\left(M \cdot \frac{0.5 \cdot D}{d}\right)}^{2}\right)}\right) \]

      4. associate-/l* 70.0%

         \[\leadsto \frac{d}{\sqrt{\ell \cdot h}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(M \cdot \color{blue}{\frac{0.5}{\frac{d}{D}}}\right)}^{2}\right)\right) \]

   10. Simplified 70.0%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(M \cdot \frac{0.5}{\frac{d}{D}}\right)}^{2}\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 58.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -3.2 \cdot 10^{-186}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{elif}\;\ell \leq 5.8 \cdot 10^{-290}:\\ \;\;\;\;\frac{d}{\sqrt[3]{{\left(h \cdot \ell\right)}^{1.5}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(M \cdot \frac{0.5}{\frac{d}{D}}\right)}^{2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 61.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;h \leq -2.25 \cdot 10^{-122}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D_m}{d} \cdot \left(M_m \cdot 0.5\right)\right)}^{2}\right)\right)\\ \mathbf{elif}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(M_m \cdot \frac{0.5}{\frac{d}{D_m}}\right)}^{2}\right)\right)\\ \end{array} \end{array} \]

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
 (if (<= h -2.25e-122)
   (*
    (sqrt (* (/ d l) (/ d h)))
    (+ 1.0 (* -0.5 (* (/ h l) (pow (* (/ D_m d) (* M_m 0.5)) 2.0)))))
   (if (<= h -5e-310)
     (* d (- (sqrt (/ 1.0 (* h l)))))
     (*
      (/ d (sqrt (* h l)))
      (+ 1.0 (* -0.5 (* (/ h l) (pow (* M_m (/ 0.5 (/ d D_m))) 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 tmp;
	if (h <= -2.25e-122) {
		tmp = sqrt(((d / l) * (d / h))) * (1.0 + (-0.5 * ((h / l) * pow(((D_m / d) * (M_m * 0.5)), 2.0))));
	} else if (h <= -5e-310) {
		tmp = d * -sqrt((1.0 / (h * l)));
	} else {
		tmp = (d / sqrt((h * l))) * (1.0 + (-0.5 * ((h / l) * pow((M_m * (0.5 / (d / D_m))), 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) :: tmp
    if (h <= (-2.25d-122)) then
        tmp = sqrt(((d / l) * (d / h))) * (1.0d0 + ((-0.5d0) * ((h / l) * (((d_m / d) * (m_m * 0.5d0)) ** 2.0d0))))
    else if (h <= (-5d-310)) then
        tmp = d * -sqrt((1.0d0 / (h * l)))
    else
        tmp = (d / sqrt((h * l))) * (1.0d0 + ((-0.5d0) * ((h / l) * ((m_m * (0.5d0 / (d / d_m))) ** 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 tmp;
	if (h <= -2.25e-122) {
		tmp = Math.sqrt(((d / l) * (d / h))) * (1.0 + (-0.5 * ((h / l) * Math.pow(((D_m / d) * (M_m * 0.5)), 2.0))));
	} else if (h <= -5e-310) {
		tmp = d * -Math.sqrt((1.0 / (h * l)));
	} else {
		tmp = (d / Math.sqrt((h * l))) * (1.0 + (-0.5 * ((h / l) * Math.pow((M_m * (0.5 / (d / D_m))), 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):
	tmp = 0
	if h <= -2.25e-122:
		tmp = math.sqrt(((d / l) * (d / h))) * (1.0 + (-0.5 * ((h / l) * math.pow(((D_m / d) * (M_m * 0.5)), 2.0))))
	elif h <= -5e-310:
		tmp = d * -math.sqrt((1.0 / (h * l)))
	else:
		tmp = (d / math.sqrt((h * l))) * (1.0 + (-0.5 * ((h / l) * math.pow((M_m * (0.5 / (d / D_m))), 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)
	tmp = 0.0
	if (h <= -2.25e-122)
		tmp = Float64(sqrt(Float64(Float64(d / l) * Float64(d / h))) * Float64(1.0 + Float64(-0.5 * Float64(Float64(h / l) * (Float64(Float64(D_m / d) * Float64(M_m * 0.5)) ^ 2.0)))));
	elseif (h <= -5e-310)
		tmp = Float64(d * Float64(-sqrt(Float64(1.0 / Float64(h * l)))));
	else
		tmp = Float64(Float64(d / sqrt(Float64(h * l))) * Float64(1.0 + Float64(-0.5 * Float64(Float64(h / l) * (Float64(M_m * Float64(0.5 / Float64(d / D_m))) ^ 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)
	tmp = 0.0;
	if (h <= -2.25e-122)
		tmp = sqrt(((d / l) * (d / h))) * (1.0 + (-0.5 * ((h / l) * (((D_m / d) * (M_m * 0.5)) ^ 2.0))));
	elseif (h <= -5e-310)
		tmp = d * -sqrt((1.0 / (h * l)));
	else
		tmp = (d / sqrt((h * l))) * (1.0 + (-0.5 * ((h / l) * ((M_m * (0.5 / (d / D_m))) ^ 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_] := If[LessEqual[h, -2.25e-122], N[(N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(-0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D$95$m / d), $MachinePrecision] * N[(M$95$m * 0.5), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, -5e-310], N[(d * (-N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(-0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(M$95$m * N[(0.5 / N[(d / D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if h < -2.2499999999999999e-122

   1. Initial program 68.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 68.5%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. frac-2neg 71.8%

         \[\leadsto \sqrt{\color{blue}{\frac{-d}{-h}}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \frac{{\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}}{\frac{\ell}{-0.5}}\right)\right) \]

      2. sqrt-div 83.9%

         \[\leadsto \color{blue}{\frac{\sqrt{-d}}{\sqrt{-h}}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \frac{{\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}}{\frac{\ell}{-0.5}}\right)\right) \]

   6. Applied egg-rr 79.4%

      \[\leadsto \left(\color{blue}{\frac{\sqrt{-d}}{\sqrt{-h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
   7. Step-by-step derivation

      1. expm1-log1p-u 37.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)\right)} \]

      2. expm1-udef 27.0%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)} - 1} \]

   8. Applied egg-rr 24.1%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \left({\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 31.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \left({\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)\right)} \]

      2. expm1-log1p 59.3%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \left({\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]

      3. *-commutative 59.3%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \left({\color{blue}{\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   10. Simplified 59.3%

       \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]

   if -2.2499999999999999e-122 < h < -4.999999999999985e-310

   1. Initial program 65.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 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. Step-by-step derivation

      1. frac-2neg 70.0%

         \[\leadsto \sqrt{\color{blue}{\frac{-d}{-h}}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \frac{{\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}}{\frac{\ell}{-0.5}}\right)\right) \]

      2. sqrt-div 85.2%

         \[\leadsto \color{blue}{\frac{\sqrt{-d}}{\sqrt{-h}}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \frac{{\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}}{\frac{\ell}{-0.5}}\right)\right) \]

   6. Applied egg-rr 79.0%

      \[\leadsto \left(\color{blue}{\frac{\sqrt{-d}}{\sqrt{-h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   7. Taylor expanded in d around -inf 60.7%

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   8. Step-by-step derivation

      1. mul-1-neg 60.7%

         \[\leadsto \color{blue}{-d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      2. *-commutative 60.7%

         \[\leadsto -\color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      3. distribute-rgt-neg-in 60.7%

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-d\right)} \]

      4. *-commutative 60.7%

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot \left(-d\right) \]

   9. Simplified 60.7%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)} \]

   if -4.999999999999985e-310 < h

   1. Initial program 63.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 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. Step-by-step derivation

      1. *-commutative 63.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}\right)}\right) \]

      2. associate-*l/ 66.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{h \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}}{\ell}}\right) \]

      3. div-inv 66.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{h \cdot {\left(\color{blue}{\left(M \cdot \frac{1}{2}\right)} \cdot \frac{D}{d}\right)}^{2}}{\ell}\right) \]

      4. associate-*l* 66.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{h \cdot {\color{blue}{\left(M \cdot \left(\frac{1}{2} \cdot \frac{D}{d}\right)\right)}}^{2}}{\ell}\right) \]

      5. metadata-eval 66.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{h \cdot {\left(M \cdot \left(\color{blue}{0.5} \cdot \frac{D}{d}\right)\right)}^{2}}{\ell}\right) \]

   6. Applied egg-rr 66.8%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{h \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell}}\right) \]

   8. Applied egg-rr 32.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{d}{\sqrt{\ell \cdot h}} \cdot \left(1 + -0.5 \cdot \frac{h}{\frac{\ell}{{\left(M \cdot \frac{0.5 \cdot D}{d}\right)}^{2}}}\right)\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 42.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{d}{\sqrt{\ell \cdot h}} \cdot \left(1 + -0.5 \cdot \frac{h}{\frac{\ell}{{\left(M \cdot \frac{0.5 \cdot D}{d}\right)}^{2}}}\right)\right)\right)} \]

      2. expm1-log1p 75.8%

         \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}} \cdot \left(1 + -0.5 \cdot \frac{h}{\frac{\ell}{{\left(M \cdot \frac{0.5 \cdot D}{d}\right)}^{2}}}\right)} \]

      3. associate-/r/ 68.5%

         \[\leadsto \frac{d}{\sqrt{\ell \cdot h}} \cdot \left(1 + -0.5 \cdot \color{blue}{\left(\frac{h}{\ell} \cdot {\left(M \cdot \frac{0.5 \cdot D}{d}\right)}^{2}\right)}\right) \]

      4. associate-/l* 68.5%

         \[\leadsto \frac{d}{\sqrt{\ell \cdot h}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(M \cdot \color{blue}{\frac{0.5}{\frac{d}{D}}}\right)}^{2}\right)\right) \]

   10. Simplified 68.5%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(M \cdot \frac{0.5}{\frac{d}{D}}\right)}^{2}\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 63.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -2.25 \cdot 10^{-122}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2}\right)\right)\\ \mathbf{elif}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(M \cdot \frac{0.5}{\frac{d}{D}}\right)}^{2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 46.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -3 \cdot 10^{-190}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{d}{\sqrt[3]{{\left(h \cdot \ell\right)}^{1.5}}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \end{array} \]

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 (<= l -3e-190)
   (* d (- (sqrt (/ 1.0 (* h l)))))
   (if (<= l -2e-310)
     (/ d (cbrt (pow (* h l) 1.5)))
     (* 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 (l <= -3e-190) {
		tmp = d * -sqrt((1.0 / (h * l)));
	} else if (l <= -2e-310) {
		tmp = d / cbrt(pow((h * l), 1.5));
	} else {
		tmp = d * (pow(l, -0.5) * pow(h, -0.5));
	}
	return tmp;
}

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 <= -3e-190) {
		tmp = d * -Math.sqrt((1.0 / (h * l)));
	} else if (l <= -2e-310) {
		tmp = d / Math.cbrt(Math.pow((h * l), 1.5));
	} 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 (l <= -3e-190)
		tmp = Float64(d * Float64(-sqrt(Float64(1.0 / Float64(h * l)))));
	elseif (l <= -2e-310)
		tmp = Float64(d / cbrt((Float64(h * l) ^ 1.5)));
	else
		tmp = Float64(d * Float64((l ^ -0.5) * (h ^ -0.5)));
	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_] := If[LessEqual[l, -3e-190], N[(d * (-N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], If[LessEqual[l, -2e-310], N[(d / N[Power[N[Power[N[(h * l), $MachinePrecision], 1.5], $MachinePrecision], 1/3], $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 l < -2.9999999999999998e-190

   1. Initial program 66.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 65.5%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. frac-2neg 69.6%

         \[\leadsto \sqrt{\color{blue}{\frac{-d}{-h}}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \frac{{\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}}{\frac{\ell}{-0.5}}\right)\right) \]

      2. sqrt-div 83.0%

         \[\leadsto \color{blue}{\frac{\sqrt{-d}}{\sqrt{-h}}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \frac{{\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}}{\frac{\ell}{-0.5}}\right)\right) \]

   6. Applied egg-rr 78.0%

      \[\leadsto \left(\color{blue}{\frac{\sqrt{-d}}{\sqrt{-h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   7. Taylor expanded in d around -inf 48.0%

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   8. Step-by-step derivation

      1. mul-1-neg 48.0%

         \[\leadsto \color{blue}{-d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      2. *-commutative 48.0%

         \[\leadsto -\color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      3. distribute-rgt-neg-in 48.0%

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-d\right)} \]

      4. *-commutative 48.0%

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot \left(-d\right) \]

   9. Simplified 48.0%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)} \]

   if -2.9999999999999998e-190 < l < -1.999999999999994e-310

   1. Initial program 76.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 76.4%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. frac-2neg 82.3%

         \[\leadsto \sqrt{\color{blue}{\frac{-d}{-h}}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \frac{{\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}}{\frac{\ell}{-0.5}}\right)\right) \]

      2. sqrt-div 94.0%

         \[\leadsto \color{blue}{\frac{\sqrt{-d}}{\sqrt{-h}}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \frac{{\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}}{\frac{\ell}{-0.5}}\right)\right) \]

   6. Applied egg-rr 88.1%

      \[\leadsto \left(\color{blue}{\frac{\sqrt{-d}}{\sqrt{-h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   7. Taylor expanded in d around inf 42.5%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   8. Step-by-step derivation

      1. *-commutative 42.5%

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

   9. Simplified 42.5%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]
   10. Step-by-step derivation

       1. expm1-log1p-u 0.7%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)\right)} \]

       2. expm1-udef 0.5%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} - 1} \]

       3. sqrt-div 0.5%

          \[\leadsto e^{\mathsf{log1p}\left(d \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\ell \cdot h}}}\right)} - 1 \]

       4. metadata-eval 0.5%

          \[\leadsto e^{\mathsf{log1p}\left(d \cdot \frac{\color{blue}{1}}{\sqrt{\ell \cdot h}}\right)} - 1 \]

       5. un-div-inv 0.5%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{d}{\sqrt{\ell \cdot h}}}\right)} - 1 \]

   11. Applied egg-rr 0.5%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{d}{\sqrt{\ell \cdot h}}\right)} - 1} \]
   12. Step-by-step derivation

       1. expm1-def 0.7%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{d}{\sqrt{\ell \cdot h}}\right)\right)} \]

       2. expm1-log1p 42.5%

          \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]

   13. Simplified 42.5%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
   14. Step-by-step derivation

       1. add-cbrt-cube 59.4%

          \[\leadsto \frac{d}{\color{blue}{\sqrt[3]{\left(\sqrt{\ell \cdot h} \cdot \sqrt{\ell \cdot h}\right) \cdot \sqrt{\ell \cdot h}}}} \]

       2. pow1/3 59.4%

          \[\leadsto \frac{d}{\color{blue}{{\left(\left(\sqrt{\ell \cdot h} \cdot \sqrt{\ell \cdot h}\right) \cdot \sqrt{\ell \cdot h}\right)}^{0.3333333333333333}}} \]

       3. add-sqr-sqrt 59.4%

          \[\leadsto \frac{d}{{\left(\color{blue}{\left(\ell \cdot h\right)} \cdot \sqrt{\ell \cdot h}\right)}^{0.3333333333333333}} \]

       4. pow1 59.4%

          \[\leadsto \frac{d}{{\left(\color{blue}{{\left(\ell \cdot h\right)}^{1}} \cdot \sqrt{\ell \cdot h}\right)}^{0.3333333333333333}} \]

       5. pow1/2 59.4%

          \[\leadsto \frac{d}{{\left({\left(\ell \cdot h\right)}^{1} \cdot \color{blue}{{\left(\ell \cdot h\right)}^{0.5}}\right)}^{0.3333333333333333}} \]

       6. pow-prod-up 59.4%

          \[\leadsto \frac{d}{{\color{blue}{\left({\left(\ell \cdot h\right)}^{\left(1 + 0.5\right)}\right)}}^{0.3333333333333333}} \]

       7. metadata-eval 59.4%

          \[\leadsto \frac{d}{{\left({\left(\ell \cdot h\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333}} \]

   15. Applied egg-rr 59.4%

       \[\leadsto \frac{d}{\color{blue}{{\left({\left(\ell \cdot h\right)}^{1.5}\right)}^{0.3333333333333333}}} \]
   16. Step-by-step derivation

       1. unpow1/3 59.4%

          \[\leadsto \frac{d}{\color{blue}{\sqrt[3]{{\left(\ell \cdot h\right)}^{1.5}}}} \]

   17. Simplified 59.4%

       \[\leadsto \frac{d}{\color{blue}{\sqrt[3]{{\left(\ell \cdot h\right)}^{1.5}}}} \]

   if -1.999999999999994e-310 < l

   1. Initial program 63.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 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. Step-by-step derivation

      1. frac-2neg 67.3%

         \[\leadsto \sqrt{\color{blue}{\frac{-d}{-h}}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \frac{{\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}}{\frac{\ell}{-0.5}}\right)\right) \]

      2. sqrt-div 0.0%

         \[\leadsto \color{blue}{\frac{\sqrt{-d}}{\sqrt{-h}}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \frac{{\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}}{\frac{\ell}{-0.5}}\right)\right) \]

   6. Applied egg-rr 0.0%

      \[\leadsto \left(\color{blue}{\frac{\sqrt{-d}}{\sqrt{-h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   7. Taylor expanded in d around inf 45.4%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   8. Step-by-step derivation

      1. *-commutative 45.4%

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

   9. Simplified 45.4%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]
   10. Step-by-step derivation

       1. add-log-exp 13.3%

          \[\leadsto d \cdot \color{blue}{\log \left(e^{\sqrt{\frac{1}{\ell \cdot h}}}\right)} \]

       2. inv-pow 13.3%

          \[\leadsto d \cdot \log \left(e^{\sqrt{\color{blue}{{\left(\ell \cdot h\right)}^{-1}}}}\right) \]

       3. sqrt-pow1 13.3%

          \[\leadsto d \cdot \log \left(e^{\color{blue}{{\left(\ell \cdot h\right)}^{\left(\frac{-1}{2}\right)}}}\right) \]

       4. metadata-eval 13.3%

          \[\leadsto d \cdot \log \left(e^{{\left(\ell \cdot h\right)}^{\color{blue}{-0.5}}}\right) \]

   11. Applied egg-rr 13.3%

       \[\leadsto d \cdot \color{blue}{\log \left(e^{{\left(\ell \cdot h\right)}^{-0.5}}\right)} \]
   12. Step-by-step derivation

       1. rem-log-exp 45.4%

          \[\leadsto d \cdot \color{blue}{{\left(\ell \cdot h\right)}^{-0.5}} \]

       2. unpow-prod-down 52.1%

          \[\leadsto d \cdot \color{blue}{\left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)} \]

   13. Applied egg-rr 52.1%

       \[\leadsto d \cdot \color{blue}{\left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 50.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -3 \cdot 10^{-190}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{d}{\sqrt[3]{{\left(h \cdot \ell\right)}^{1.5}}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 46.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -2.4 \cdot 10^{-190}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{d}{\sqrt[3]{{\left(h \cdot \ell\right)}^{1.5}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

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 -2.4e-190)
   (* d (- (sqrt (/ 1.0 (* h l)))))
   (if (<= l -2e-310)
     (/ d (cbrt (pow (* h l) 1.5)))
     (/ d (* (sqrt h) (sqrt 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 <= -2.4e-190) {
		tmp = d * -sqrt((1.0 / (h * l)));
	} else if (l <= -2e-310) {
		tmp = d / cbrt(pow((h * l), 1.5));
	} else {
		tmp = d / (sqrt(h) * sqrt(l));
	}
	return tmp;
}

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 <= -2.4e-190) {
		tmp = d * -Math.sqrt((1.0 / (h * l)));
	} else if (l <= -2e-310) {
		tmp = d / Math.cbrt(Math.pow((h * l), 1.5));
	} else {
		tmp = d / (Math.sqrt(h) * Math.sqrt(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 <= -2.4e-190)
		tmp = Float64(d * Float64(-sqrt(Float64(1.0 / Float64(h * l)))));
	elseif (l <= -2e-310)
		tmp = Float64(d / cbrt((Float64(h * l) ^ 1.5)));
	else
		tmp = Float64(d / Float64(sqrt(h) * sqrt(l)));
	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_] := If[LessEqual[l, -2.4e-190], N[(d * (-N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], If[LessEqual[l, -2e-310], N[(d / N[Power[N[Power[N[(h * l), $MachinePrecision], 1.5], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -2.4e-190

   1. Initial program 66.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 65.5%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. frac-2neg 69.6%

         \[\leadsto \sqrt{\color{blue}{\frac{-d}{-h}}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \frac{{\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}}{\frac{\ell}{-0.5}}\right)\right) \]

      2. sqrt-div 83.0%

         \[\leadsto \color{blue}{\frac{\sqrt{-d}}{\sqrt{-h}}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \frac{{\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}}{\frac{\ell}{-0.5}}\right)\right) \]

   6. Applied egg-rr 78.0%

      \[\leadsto \left(\color{blue}{\frac{\sqrt{-d}}{\sqrt{-h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   7. Taylor expanded in d around -inf 48.0%

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   8. Step-by-step derivation

      1. mul-1-neg 48.0%

         \[\leadsto \color{blue}{-d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      2. *-commutative 48.0%

         \[\leadsto -\color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      3. distribute-rgt-neg-in 48.0%

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-d\right)} \]

      4. *-commutative 48.0%

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot \left(-d\right) \]

   9. Simplified 48.0%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)} \]

   if -2.4e-190 < l < -1.999999999999994e-310

   1. Initial program 76.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 76.4%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. frac-2neg 82.3%

         \[\leadsto \sqrt{\color{blue}{\frac{-d}{-h}}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \frac{{\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}}{\frac{\ell}{-0.5}}\right)\right) \]

      2. sqrt-div 94.0%

         \[\leadsto \color{blue}{\frac{\sqrt{-d}}{\sqrt{-h}}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \frac{{\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}}{\frac{\ell}{-0.5}}\right)\right) \]

   6. Applied egg-rr 88.1%

      \[\leadsto \left(\color{blue}{\frac{\sqrt{-d}}{\sqrt{-h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   7. Taylor expanded in d around inf 42.5%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   8. Step-by-step derivation

      1. *-commutative 42.5%

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

   9. Simplified 42.5%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]
   10. Step-by-step derivation

       1. expm1-log1p-u 0.7%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)\right)} \]

       2. expm1-udef 0.5%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} - 1} \]

       3. sqrt-div 0.5%

          \[\leadsto e^{\mathsf{log1p}\left(d \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\ell \cdot h}}}\right)} - 1 \]

       4. metadata-eval 0.5%

          \[\leadsto e^{\mathsf{log1p}\left(d \cdot \frac{\color{blue}{1}}{\sqrt{\ell \cdot h}}\right)} - 1 \]

       5. un-div-inv 0.5%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{d}{\sqrt{\ell \cdot h}}}\right)} - 1 \]

   11. Applied egg-rr 0.5%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{d}{\sqrt{\ell \cdot h}}\right)} - 1} \]
   12. Step-by-step derivation

       1. expm1-def 0.7%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{d}{\sqrt{\ell \cdot h}}\right)\right)} \]

       2. expm1-log1p 42.5%

          \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]

   13. Simplified 42.5%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
   14. Step-by-step derivation

       1. add-cbrt-cube 59.4%

          \[\leadsto \frac{d}{\color{blue}{\sqrt[3]{\left(\sqrt{\ell \cdot h} \cdot \sqrt{\ell \cdot h}\right) \cdot \sqrt{\ell \cdot h}}}} \]

       2. pow1/3 59.4%

          \[\leadsto \frac{d}{\color{blue}{{\left(\left(\sqrt{\ell \cdot h} \cdot \sqrt{\ell \cdot h}\right) \cdot \sqrt{\ell \cdot h}\right)}^{0.3333333333333333}}} \]

       3. add-sqr-sqrt 59.4%

          \[\leadsto \frac{d}{{\left(\color{blue}{\left(\ell \cdot h\right)} \cdot \sqrt{\ell \cdot h}\right)}^{0.3333333333333333}} \]

       4. pow1 59.4%

          \[\leadsto \frac{d}{{\left(\color{blue}{{\left(\ell \cdot h\right)}^{1}} \cdot \sqrt{\ell \cdot h}\right)}^{0.3333333333333333}} \]

       5. pow1/2 59.4%

          \[\leadsto \frac{d}{{\left({\left(\ell \cdot h\right)}^{1} \cdot \color{blue}{{\left(\ell \cdot h\right)}^{0.5}}\right)}^{0.3333333333333333}} \]

       6. pow-prod-up 59.4%

          \[\leadsto \frac{d}{{\color{blue}{\left({\left(\ell \cdot h\right)}^{\left(1 + 0.5\right)}\right)}}^{0.3333333333333333}} \]

       7. metadata-eval 59.4%

          \[\leadsto \frac{d}{{\left({\left(\ell \cdot h\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333}} \]

   15. Applied egg-rr 59.4%

       \[\leadsto \frac{d}{\color{blue}{{\left({\left(\ell \cdot h\right)}^{1.5}\right)}^{0.3333333333333333}}} \]
   16. Step-by-step derivation

       1. unpow1/3 59.4%

          \[\leadsto \frac{d}{\color{blue}{\sqrt[3]{{\left(\ell \cdot h\right)}^{1.5}}}} \]

   17. Simplified 59.4%

       \[\leadsto \frac{d}{\color{blue}{\sqrt[3]{{\left(\ell \cdot h\right)}^{1.5}}}} \]

   if -1.999999999999994e-310 < l

   1. Initial program 63.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 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. Step-by-step derivation

      1. frac-2neg 67.3%

         \[\leadsto \sqrt{\color{blue}{\frac{-d}{-h}}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \frac{{\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}}{\frac{\ell}{-0.5}}\right)\right) \]

      2. sqrt-div 0.0%

         \[\leadsto \color{blue}{\frac{\sqrt{-d}}{\sqrt{-h}}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \frac{{\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}}{\frac{\ell}{-0.5}}\right)\right) \]

   6. Applied egg-rr 0.0%

      \[\leadsto \left(\color{blue}{\frac{\sqrt{-d}}{\sqrt{-h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   7. Taylor expanded in d around inf 45.4%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   8. Step-by-step derivation

      1. *-commutative 45.4%

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

   9. Simplified 45.4%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]
   10. Step-by-step derivation

       1. expm1-log1p-u 43.7%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)\right)} \]

       2. expm1-udef 33.3%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} - 1} \]

       3. sqrt-div 33.3%

          \[\leadsto e^{\mathsf{log1p}\left(d \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\ell \cdot h}}}\right)} - 1 \]

       4. metadata-eval 33.3%

          \[\leadsto e^{\mathsf{log1p}\left(d \cdot \frac{\color{blue}{1}}{\sqrt{\ell \cdot h}}\right)} - 1 \]

       5. un-div-inv 33.3%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{d}{\sqrt{\ell \cdot h}}}\right)} - 1 \]

   11. Applied egg-rr 33.3%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{d}{\sqrt{\ell \cdot h}}\right)} - 1} \]
   12. Step-by-step derivation

       1. expm1-def 43.7%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{d}{\sqrt{\ell \cdot h}}\right)\right)} \]

       2. expm1-log1p 45.4%

          \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]

   13. Simplified 45.4%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
   14. Step-by-step derivation

       1. sqrt-prod 52.1%

          \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell} \cdot \sqrt{h}}} \]

   15. Applied egg-rr 52.1%

       \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell} \cdot \sqrt{h}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 50.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.4 \cdot 10^{-190}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{d}{\sqrt[3]{{\left(h \cdot \ell\right)}^{1.5}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 46.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.15 \cdot 10^{-238}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

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 -1.15e-238)
   (* d (- (sqrt (/ 1.0 (* h l)))))
   (if (<= l -2e-310) (/ d (sqrt (* h l))) (/ d (* (sqrt h) (sqrt 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 <= -1.15e-238) {
		tmp = d * -sqrt((1.0 / (h * l)));
	} else if (l <= -2e-310) {
		tmp = d / sqrt((h * l));
	} else {
		tmp = d / (sqrt(h) * sqrt(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 <= (-1.15d-238)) then
        tmp = d * -sqrt((1.0d0 / (h * l)))
    else if (l <= (-2d-310)) then
        tmp = d / sqrt((h * l))
    else
        tmp = d / (sqrt(h) * sqrt(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 <= -1.15e-238) {
		tmp = d * -Math.sqrt((1.0 / (h * l)));
	} else if (l <= -2e-310) {
		tmp = d / Math.sqrt((h * l));
	} else {
		tmp = d / (Math.sqrt(h) * Math.sqrt(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 <= -1.15e-238:
		tmp = d * -math.sqrt((1.0 / (h * l)))
	elif l <= -2e-310:
		tmp = d / math.sqrt((h * l))
	else:
		tmp = d / (math.sqrt(h) * math.sqrt(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 <= -1.15e-238)
		tmp = Float64(d * Float64(-sqrt(Float64(1.0 / Float64(h * l)))));
	elseif (l <= -2e-310)
		tmp = Float64(d / sqrt(Float64(h * l)));
	else
		tmp = Float64(d / Float64(sqrt(h) * sqrt(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 <= -1.15e-238)
		tmp = d * -sqrt((1.0 / (h * l)));
	elseif (l <= -2e-310)
		tmp = d / sqrt((h * l));
	else
		tmp = d / (sqrt(h) * sqrt(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, -1.15e-238], N[(d * (-N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], If[LessEqual[l, -2e-310], N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.15000000000000002e-238

   1. Initial program 67.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 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
   5. Step-by-step derivation

      1. frac-2neg 70.5%

         \[\leadsto \sqrt{\color{blue}{\frac{-d}{-h}}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \frac{{\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}}{\frac{\ell}{-0.5}}\right)\right) \]

      2. sqrt-div 83.9%

         \[\leadsto \color{blue}{\frac{\sqrt{-d}}{\sqrt{-h}}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \frac{{\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}}{\frac{\ell}{-0.5}}\right)\right) \]

   6. Applied egg-rr 79.2%

      \[\leadsto \left(\color{blue}{\frac{\sqrt{-d}}{\sqrt{-h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   7. Taylor expanded in d around -inf 46.9%

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   8. Step-by-step derivation

      1. mul-1-neg 46.9%

         \[\leadsto \color{blue}{-d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      2. *-commutative 46.9%

         \[\leadsto -\color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      3. distribute-rgt-neg-in 46.9%

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-d\right)} \]

      4. *-commutative 46.9%

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot \left(-d\right) \]

   9. Simplified 46.9%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)} \]

   if -1.15000000000000002e-238 < l < -1.999999999999994e-310

   1. Initial program 70.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 70.0%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. frac-2neg 80.0%

         \[\leadsto \sqrt{\color{blue}{\frac{-d}{-h}}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \frac{{\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}}{\frac{\ell}{-0.5}}\right)\right) \]

      2. sqrt-div 90.0%

         \[\leadsto \color{blue}{\frac{\sqrt{-d}}{\sqrt{-h}}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \frac{{\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}}{\frac{\ell}{-0.5}}\right)\right) \]

   6. Applied egg-rr 80.0%

      \[\leadsto \left(\color{blue}{\frac{\sqrt{-d}}{\sqrt{-h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   7. Taylor expanded in d around inf 70.4%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   8. Step-by-step derivation

      1. *-commutative 70.4%

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

   9. Simplified 70.4%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]
   10. Step-by-step derivation

       1. expm1-log1p-u 0.2%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)\right)} \]

       2. expm1-udef 0.2%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} - 1} \]

       3. sqrt-div 0.2%

          \[\leadsto e^{\mathsf{log1p}\left(d \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\ell \cdot h}}}\right)} - 1 \]

       4. metadata-eval 0.2%

          \[\leadsto e^{\mathsf{log1p}\left(d \cdot \frac{\color{blue}{1}}{\sqrt{\ell \cdot h}}\right)} - 1 \]

       5. un-div-inv 0.2%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{d}{\sqrt{\ell \cdot h}}}\right)} - 1 \]

   11. Applied egg-rr 0.2%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{d}{\sqrt{\ell \cdot h}}\right)} - 1} \]
   12. Step-by-step derivation

       1. expm1-def 0.2%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{d}{\sqrt{\ell \cdot h}}\right)\right)} \]

       2. expm1-log1p 70.4%

          \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]

   13. Simplified 70.4%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]

   if -1.999999999999994e-310 < l

   1. Initial program 63.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 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. Step-by-step derivation

      1. frac-2neg 67.3%

         \[\leadsto \sqrt{\color{blue}{\frac{-d}{-h}}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \frac{{\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}}{\frac{\ell}{-0.5}}\right)\right) \]

      2. sqrt-div 0.0%

         \[\leadsto \color{blue}{\frac{\sqrt{-d}}{\sqrt{-h}}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \frac{{\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}}{\frac{\ell}{-0.5}}\right)\right) \]

   6. Applied egg-rr 0.0%

      \[\leadsto \left(\color{blue}{\frac{\sqrt{-d}}{\sqrt{-h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   7. Taylor expanded in d around inf 45.4%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   8. Step-by-step derivation

      1. *-commutative 45.4%

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

   9. Simplified 45.4%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]
   10. Step-by-step derivation

       1. expm1-log1p-u 43.7%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)\right)} \]

       2. expm1-udef 33.3%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} - 1} \]

       3. sqrt-div 33.3%

          \[\leadsto e^{\mathsf{log1p}\left(d \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\ell \cdot h}}}\right)} - 1 \]

       4. metadata-eval 33.3%

          \[\leadsto e^{\mathsf{log1p}\left(d \cdot \frac{\color{blue}{1}}{\sqrt{\ell \cdot h}}\right)} - 1 \]

       5. un-div-inv 33.3%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{d}{\sqrt{\ell \cdot h}}}\right)} - 1 \]

   11. Applied egg-rr 33.3%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{d}{\sqrt{\ell \cdot h}}\right)} - 1} \]
   12. Step-by-step derivation

       1. expm1-def 43.7%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{d}{\sqrt{\ell \cdot h}}\right)\right)} \]

       2. expm1-log1p 45.4%

          \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]

   13. Simplified 45.4%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
   14. Step-by-step derivation

       1. sqrt-prod 52.1%

          \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell} \cdot \sqrt{h}}} \]

   15. Applied egg-rr 52.1%

       \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell} \cdot \sqrt{h}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 50.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.15 \cdot 10^{-238}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 45.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -4.4 \cdot 10^{-258}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{elif}\;\ell \leq 4.3 \cdot 10^{-297}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

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 -4.4e-258)
   (* d (- (sqrt (/ 1.0 (* h l)))))
   (if (<= l 4.3e-297) (/ d (sqrt (* h l))) (/ (/ d (sqrt h)) (sqrt 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 <= -4.4e-258) {
		tmp = d * -sqrt((1.0 / (h * l)));
	} else if (l <= 4.3e-297) {
		tmp = d / sqrt((h * l));
	} else {
		tmp = (d / sqrt(h)) / sqrt(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 <= (-4.4d-258)) then
        tmp = d * -sqrt((1.0d0 / (h * l)))
    else if (l <= 4.3d-297) then
        tmp = d / sqrt((h * l))
    else
        tmp = (d / sqrt(h)) / sqrt(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 <= -4.4e-258) {
		tmp = d * -Math.sqrt((1.0 / (h * l)));
	} else if (l <= 4.3e-297) {
		tmp = d / Math.sqrt((h * l));
	} else {
		tmp = (d / Math.sqrt(h)) / Math.sqrt(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 <= -4.4e-258:
		tmp = d * -math.sqrt((1.0 / (h * l)))
	elif l <= 4.3e-297:
		tmp = d / math.sqrt((h * l))
	else:
		tmp = (d / math.sqrt(h)) / math.sqrt(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 <= -4.4e-258)
		tmp = Float64(d * Float64(-sqrt(Float64(1.0 / Float64(h * l)))));
	elseif (l <= 4.3e-297)
		tmp = Float64(d / sqrt(Float64(h * l)));
	else
		tmp = Float64(Float64(d / sqrt(h)) / sqrt(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 <= -4.4e-258)
		tmp = d * -sqrt((1.0 / (h * l)));
	elseif (l <= 4.3e-297)
		tmp = d / sqrt((h * l));
	else
		tmp = (d / sqrt(h)) / sqrt(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, -4.4e-258], N[(d * (-N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], If[LessEqual[l, 4.3e-297], N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(d / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -4.40000000000000031e-258

   1. Initial program 67.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 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
   5. Step-by-step derivation

      1. frac-2neg 70.5%

         \[\leadsto \sqrt{\color{blue}{\frac{-d}{-h}}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \frac{{\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}}{\frac{\ell}{-0.5}}\right)\right) \]

      2. sqrt-div 83.9%

         \[\leadsto \color{blue}{\frac{\sqrt{-d}}{\sqrt{-h}}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \frac{{\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}}{\frac{\ell}{-0.5}}\right)\right) \]

   6. Applied egg-rr 79.2%

      \[\leadsto \left(\color{blue}{\frac{\sqrt{-d}}{\sqrt{-h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   7. Taylor expanded in d around -inf 46.9%

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   8. Step-by-step derivation

      1. mul-1-neg 46.9%

         \[\leadsto \color{blue}{-d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      2. *-commutative 46.9%

         \[\leadsto -\color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      3. distribute-rgt-neg-in 46.9%

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-d\right)} \]

      4. *-commutative 46.9%

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot \left(-d\right) \]

   9. Simplified 46.9%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)} \]

   if -4.40000000000000031e-258 < l < 4.3000000000000003e-297

   1. Initial program 63.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 63.6%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. frac-2neg 73.2%

         \[\leadsto \sqrt{\color{blue}{\frac{-d}{-h}}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \frac{{\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}}{\frac{\ell}{-0.5}}\right)\right) \]

      2. sqrt-div 81.8%

         \[\leadsto \color{blue}{\frac{\sqrt{-d}}{\sqrt{-h}}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \frac{{\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}}{\frac{\ell}{-0.5}}\right)\right) \]

   6. Applied egg-rr 72.7%

      \[\leadsto \left(\color{blue}{\frac{\sqrt{-d}}{\sqrt{-h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   7. Taylor expanded in d around inf 72.9%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   8. Step-by-step derivation

      1. *-commutative 72.9%

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

   9. Simplified 72.9%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]
   10. Step-by-step derivation

       1. expm1-log1p-u 8.3%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)\right)} \]

       2. expm1-udef 8.2%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} - 1} \]

       3. sqrt-div 8.2%

          \[\leadsto e^{\mathsf{log1p}\left(d \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\ell \cdot h}}}\right)} - 1 \]

       4. metadata-eval 8.2%

          \[\leadsto e^{\mathsf{log1p}\left(d \cdot \frac{\color{blue}{1}}{\sqrt{\ell \cdot h}}\right)} - 1 \]

       5. un-div-inv 8.2%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{d}{\sqrt{\ell \cdot h}}}\right)} - 1 \]

   11. Applied egg-rr 8.2%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{d}{\sqrt{\ell \cdot h}}\right)} - 1} \]
   12. Step-by-step derivation

       1. expm1-def 8.3%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{d}{\sqrt{\ell \cdot h}}\right)\right)} \]

       2. expm1-log1p 73.1%

          \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]

   13. Simplified 73.1%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]

   if 4.3000000000000003e-297 < l

   1. Initial program 64.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 64.4%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. frac-2neg 67.8%

         \[\leadsto \sqrt{\color{blue}{\frac{-d}{-h}}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \frac{{\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}}{\frac{\ell}{-0.5}}\right)\right) \]

      2. sqrt-div 0.0%

         \[\leadsto \color{blue}{\frac{\sqrt{-d}}{\sqrt{-h}}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \frac{{\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}}{\frac{\ell}{-0.5}}\right)\right) \]

   6. Applied egg-rr 0.0%

      \[\leadsto \left(\color{blue}{\frac{\sqrt{-d}}{\sqrt{-h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   7. Taylor expanded in d around inf 45.0%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   8. Step-by-step derivation

      1. *-commutative 45.0%

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

   9. Simplified 45.0%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]
   10. Step-by-step derivation

       1. expm1-log1p-u 43.3%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)\right)} \]

       2. expm1-udef 32.8%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} - 1} \]

       3. sqrt-div 32.8%

          \[\leadsto e^{\mathsf{log1p}\left(d \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\ell \cdot h}}}\right)} - 1 \]

       4. metadata-eval 32.8%

          \[\leadsto e^{\mathsf{log1p}\left(d \cdot \frac{\color{blue}{1}}{\sqrt{\ell \cdot h}}\right)} - 1 \]

       5. un-div-inv 32.8%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{d}{\sqrt{\ell \cdot h}}}\right)} - 1 \]

   11. Applied egg-rr 32.8%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{d}{\sqrt{\ell \cdot h}}\right)} - 1} \]
   12. Step-by-step derivation

       1. expm1-def 43.3%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{d}{\sqrt{\ell \cdot h}}\right)\right)} \]

       2. expm1-log1p 45.0%

          \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]

   13. Simplified 45.0%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
   14. Step-by-step derivation

       1. *-un-lft-identity 45.0%

          \[\leadsto \frac{\color{blue}{1 \cdot d}}{\sqrt{\ell \cdot h}} \]

       2. sqrt-prod 51.7%

          \[\leadsto \frac{1 \cdot d}{\color{blue}{\sqrt{\ell} \cdot \sqrt{h}}} \]

       3. times-frac 51.7%

          \[\leadsto \color{blue}{\frac{1}{\sqrt{\ell}} \cdot \frac{d}{\sqrt{h}}} \]

   15. Applied egg-rr 51.7%

       \[\leadsto \color{blue}{\frac{1}{\sqrt{\ell}} \cdot \frac{d}{\sqrt{h}}} \]
   16. Step-by-step derivation

       1. associate-*l/ 51.7%

          \[\leadsto \color{blue}{\frac{1 \cdot \frac{d}{\sqrt{h}}}{\sqrt{\ell}}} \]

       2. *-lft-identity 51.7%

          \[\leadsto \frac{\color{blue}{\frac{d}{\sqrt{h}}}}{\sqrt{\ell}} \]

   17. Simplified 51.7%

       \[\leadsto \color{blue}{\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 50.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4.4 \cdot 10^{-258}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{elif}\;\ell \leq 4.3 \cdot 10^{-297}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 42.8% accurate, 3.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 2.3 \cdot 10^{-250}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{\ell} \cdot \frac{1}{h}}\\ \end{array} \end{array} \]

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 (<= d 2.3e-250)
   (* d (- (sqrt (/ 1.0 (* h l)))))
   (* d (sqrt (* (/ 1.0 l) (/ 1.0 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 (d <= 2.3e-250) {
		tmp = d * -sqrt((1.0 / (h * l)));
	} else {
		tmp = d * sqrt(((1.0 / l) * (1.0 / 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 (d <= 2.3d-250) then
        tmp = d * -sqrt((1.0d0 / (h * l)))
    else
        tmp = d * sqrt(((1.0d0 / l) * (1.0d0 / 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 (d <= 2.3e-250) {
		tmp = d * -Math.sqrt((1.0 / (h * l)));
	} else {
		tmp = d * Math.sqrt(((1.0 / l) * (1.0 / 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 d <= 2.3e-250:
		tmp = d * -math.sqrt((1.0 / (h * l)))
	else:
		tmp = d * math.sqrt(((1.0 / l) * (1.0 / 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 (d <= 2.3e-250)
		tmp = Float64(d * Float64(-sqrt(Float64(1.0 / Float64(h * l)))));
	else
		tmp = Float64(d * sqrt(Float64(Float64(1.0 / l) * Float64(1.0 / 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 (d <= 2.3e-250)
		tmp = d * -sqrt((1.0 / (h * l)));
	else
		tmp = d * sqrt(((1.0 / l) * (1.0 / 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[d, 2.3e-250], N[(d * (-N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(d * N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] * N[(1.0 / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 2.2999999999999999e-250

   1. Initial program 64.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 63.6%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. frac-2neg 67.5%

         \[\leadsto \sqrt{\color{blue}{\frac{-d}{-h}}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \frac{{\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}}{\frac{\ell}{-0.5}}\right)\right) \]

      2. sqrt-div 75.9%

         \[\leadsto \color{blue}{\frac{\sqrt{-d}}{\sqrt{-h}}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \frac{{\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}}{\frac{\ell}{-0.5}}\right)\right) \]

   6. Applied egg-rr 71.3%

      \[\leadsto \left(\color{blue}{\frac{\sqrt{-d}}{\sqrt{-h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   7. Taylor expanded in d around -inf 43.3%

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   8. Step-by-step derivation

      1. mul-1-neg 43.3%

         \[\leadsto \color{blue}{-d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      2. *-commutative 43.3%

         \[\leadsto -\color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      3. distribute-rgt-neg-in 43.3%

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-d\right)} \]

      4. *-commutative 43.3%

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot \left(-d\right) \]

   9. Simplified 43.3%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)} \]

   if 2.2999999999999999e-250 < d

   1. Initial program 68.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 68.0%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. frac-2neg 72.0%

         \[\leadsto \sqrt{\color{blue}{\frac{-d}{-h}}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \frac{{\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}}{\frac{\ell}{-0.5}}\right)\right) \]

      2. sqrt-div 0.0%

         \[\leadsto \color{blue}{\frac{\sqrt{-d}}{\sqrt{-h}}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \frac{{\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}}{\frac{\ell}{-0.5}}\right)\right) \]

   6. Applied egg-rr 0.0%

      \[\leadsto \left(\color{blue}{\frac{\sqrt{-d}}{\sqrt{-h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   7. Taylor expanded in d around inf 48.8%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   8. Step-by-step derivation

      1. *-commutative 48.8%

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

   9. Simplified 48.8%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]
   10. Step-by-step derivation

       1. associate-/r* 49.2%

          \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \]

       2. div-inv 49.2%

          \[\leadsto d \cdot \sqrt{\color{blue}{\frac{1}{\ell} \cdot \frac{1}{h}}} \]

   11. Applied egg-rr 49.2%

       \[\leadsto d \cdot \sqrt{\color{blue}{\frac{1}{\ell} \cdot \frac{1}{h}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 45.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 2.3 \cdot 10^{-250}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{\ell} \cdot \frac{1}{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 42.7% accurate, 3.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 1.6 \cdot 10^{-246}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{h \cdot \ell}}{d}}\\ \end{array} \end{array} \]

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 1.6e-246)
   (* d (- (sqrt (/ 1.0 (* h l)))))
   (/ 1.0 (/ (sqrt (* h l)) d))))

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 <= 1.6e-246) {
		tmp = d * -sqrt((1.0 / (h * l)));
	} else {
		tmp = 1.0 / (sqrt((h * l)) / d);
	}
	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 <= 1.6d-246) then
        tmp = d * -sqrt((1.0d0 / (h * l)))
    else
        tmp = 1.0d0 / (sqrt((h * l)) / d)
    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 <= 1.6e-246) {
		tmp = d * -Math.sqrt((1.0 / (h * l)));
	} else {
		tmp = 1.0 / (Math.sqrt((h * l)) / d);
	}
	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 <= 1.6e-246:
		tmp = d * -math.sqrt((1.0 / (h * l)))
	else:
		tmp = 1.0 / (math.sqrt((h * l)) / d)
	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 <= 1.6e-246)
		tmp = Float64(d * Float64(-sqrt(Float64(1.0 / Float64(h * l)))));
	else
		tmp = Float64(1.0 / Float64(sqrt(Float64(h * l)) / d));
	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 <= 1.6e-246)
		tmp = d * -sqrt((1.0 / (h * l)));
	else
		tmp = 1.0 / (sqrt((h * l)) / d);
	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, 1.6e-246], N[(d * (-N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(1.0 / N[(N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 1.6e-246

   1. Initial program 64.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 63.6%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. frac-2neg 67.5%

         \[\leadsto \sqrt{\color{blue}{\frac{-d}{-h}}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \frac{{\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}}{\frac{\ell}{-0.5}}\right)\right) \]

      2. sqrt-div 75.9%

         \[\leadsto \color{blue}{\frac{\sqrt{-d}}{\sqrt{-h}}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \frac{{\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}}{\frac{\ell}{-0.5}}\right)\right) \]

   6. Applied egg-rr 71.3%

      \[\leadsto \left(\color{blue}{\frac{\sqrt{-d}}{\sqrt{-h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   7. Taylor expanded in d around -inf 43.3%

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   8. Step-by-step derivation

      1. mul-1-neg 43.3%

         \[\leadsto \color{blue}{-d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      2. *-commutative 43.3%

         \[\leadsto -\color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      3. distribute-rgt-neg-in 43.3%

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-d\right)} \]

      4. *-commutative 43.3%

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot \left(-d\right) \]

   9. Simplified 43.3%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)} \]

   if 1.6e-246 < d

   1. Initial program 68.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 68.0%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. frac-2neg 72.0%

         \[\leadsto \sqrt{\color{blue}{\frac{-d}{-h}}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \frac{{\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}}{\frac{\ell}{-0.5}}\right)\right) \]

      2. sqrt-div 0.0%

         \[\leadsto \color{blue}{\frac{\sqrt{-d}}{\sqrt{-h}}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \frac{{\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}}{\frac{\ell}{-0.5}}\right)\right) \]

   6. Applied egg-rr 0.0%

      \[\leadsto \left(\color{blue}{\frac{\sqrt{-d}}{\sqrt{-h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   7. Taylor expanded in d around inf 48.8%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   8. Step-by-step derivation

      1. *-commutative 48.8%

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

   9. Simplified 48.8%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]
   10. Step-by-step derivation

       1. add-log-exp 13.0%

          \[\leadsto d \cdot \color{blue}{\log \left(e^{\sqrt{\frac{1}{\ell \cdot h}}}\right)} \]

       2. inv-pow 13.0%

          \[\leadsto d \cdot \log \left(e^{\sqrt{\color{blue}{{\left(\ell \cdot h\right)}^{-1}}}}\right) \]

       3. sqrt-pow1 13.0%

          \[\leadsto d \cdot \log \left(e^{\color{blue}{{\left(\ell \cdot h\right)}^{\left(\frac{-1}{2}\right)}}}\right) \]

       4. metadata-eval 13.0%

          \[\leadsto d \cdot \log \left(e^{{\left(\ell \cdot h\right)}^{\color{blue}{-0.5}}}\right) \]

   11. Applied egg-rr 13.0%

       \[\leadsto d \cdot \color{blue}{\log \left(e^{{\left(\ell \cdot h\right)}^{-0.5}}\right)} \]
   12. Step-by-step derivation

       1. rem-log-exp 48.8%

          \[\leadsto d \cdot \color{blue}{{\left(\ell \cdot h\right)}^{-0.5}} \]

       2. metadata-eval 48.8%

          \[\leadsto d \cdot {\left(\ell \cdot h\right)}^{\color{blue}{\left(-0.5\right)}} \]

       3. pow-flip 48.8%

          \[\leadsto d \cdot \color{blue}{\frac{1}{{\left(\ell \cdot h\right)}^{0.5}}} \]

       4. pow1/2 48.8%

          \[\leadsto d \cdot \frac{1}{\color{blue}{\sqrt{\ell \cdot h}}} \]

       5. div-inv 48.8%

          \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]

       6. clear-num 48.9%

          \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{\ell \cdot h}}{d}}} \]

   13. Applied egg-rr 48.9%

       \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{\ell \cdot h}}{d}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 45.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 1.6 \cdot 10^{-246}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{h \cdot \ell}}{d}}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 38.1% accurate, 3.1× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -5 \cdot 10^{-238}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}}\\ \end{array} \end{array} \]

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 -5e-238) (sqrt (* (/ d l) (/ d h))) (/ d (sqrt (* 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 <= -5e-238) {
		tmp = sqrt(((d / l) * (d / h)));
	} else {
		tmp = d / sqrt((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 <= (-5d-238)) then
        tmp = sqrt(((d / l) * (d / h)))
    else
        tmp = d / sqrt((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 <= -5e-238) {
		tmp = Math.sqrt(((d / l) * (d / h)));
	} else {
		tmp = d / Math.sqrt((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 <= -5e-238:
		tmp = math.sqrt(((d / l) * (d / h)))
	else:
		tmp = d / math.sqrt((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 <= -5e-238)
		tmp = sqrt(Float64(Float64(d / l) * Float64(d / h)));
	else
		tmp = Float64(d / sqrt(Float64(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 <= -5e-238)
		tmp = sqrt(((d / l) * (d / h)));
	else
		tmp = d / sqrt((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, -5e-238], N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -5e-238

   1. Initial program 67.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 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
   5. Step-by-step derivation

      1. frac-2neg 70.5%

         \[\leadsto \sqrt{\color{blue}{\frac{-d}{-h}}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \frac{{\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}}{\frac{\ell}{-0.5}}\right)\right) \]

      2. sqrt-div 83.9%

         \[\leadsto \color{blue}{\frac{\sqrt{-d}}{\sqrt{-h}}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \frac{{\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}}{\frac{\ell}{-0.5}}\right)\right) \]

   6. Applied egg-rr 79.2%

      \[\leadsto \left(\color{blue}{\frac{\sqrt{-d}}{\sqrt{-h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   7. Taylor expanded in d around inf 10.5%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   8. Step-by-step derivation

      1. *-commutative 10.5%

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

   9. Simplified 10.5%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]
   10. Step-by-step derivation

       1. expm1-log1p-u 6.5%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)\right)} \]

       2. expm1-udef 6.5%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} - 1} \]

       3. sqrt-div 6.5%

          \[\leadsto e^{\mathsf{log1p}\left(d \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\ell \cdot h}}}\right)} - 1 \]

       4. metadata-eval 6.5%

          \[\leadsto e^{\mathsf{log1p}\left(d \cdot \frac{\color{blue}{1}}{\sqrt{\ell \cdot h}}\right)} - 1 \]

       5. un-div-inv 6.5%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{d}{\sqrt{\ell \cdot h}}}\right)} - 1 \]

   11. Applied egg-rr 6.5%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{d}{\sqrt{\ell \cdot h}}\right)} - 1} \]
   12. Step-by-step derivation

       1. expm1-def 6.5%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{d}{\sqrt{\ell \cdot h}}\right)\right)} \]

       2. expm1-log1p 9.8%

          \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]

   13. Simplified 9.8%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
   14. Step-by-step derivation

       1. add-sqr-sqrt 0.0%

          \[\leadsto \frac{\color{blue}{\sqrt{d} \cdot \sqrt{d}}}{\sqrt{\ell \cdot h}} \]

       2. sqrt-prod 0.0%

          \[\leadsto \frac{\sqrt{d} \cdot \sqrt{d}}{\color{blue}{\sqrt{\ell} \cdot \sqrt{h}}} \]

       3. frac-times 0.0%

          \[\leadsto \color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \frac{\sqrt{d}}{\sqrt{h}}} \]

       4. sqrt-div 0.0%

          \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \frac{\sqrt{d}}{\sqrt{h}} \]

       5. sqrt-div 42.2%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}} \]

       6. sqrt-unprod 34.3%

          \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} \]

   15. Applied egg-rr 34.3%

       \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} \]

   if -5e-238 < l

   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 64.3%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. frac-2neg 68.3%

         \[\leadsto \sqrt{\color{blue}{\frac{-d}{-h}}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \frac{{\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}}{\frac{\ell}{-0.5}}\right)\right) \]

      2. sqrt-div 6.9%

         \[\leadsto \color{blue}{\frac{\sqrt{-d}}{\sqrt{-h}}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \frac{{\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}}{\frac{\ell}{-0.5}}\right)\right) \]

   6. Applied egg-rr 6.1%

      \[\leadsto \left(\color{blue}{\frac{\sqrt{-d}}{\sqrt{-h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   7. Taylor expanded in d around inf 47.3%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   8. Step-by-step derivation

      1. *-commutative 47.3%

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

   9. Simplified 47.3%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]
   10. Step-by-step derivation

       1. expm1-log1p-u 40.4%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)\right)} \]

       2. expm1-udef 30.8%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} - 1} \]

       3. sqrt-div 30.8%

          \[\leadsto e^{\mathsf{log1p}\left(d \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\ell \cdot h}}}\right)} - 1 \]

       4. metadata-eval 30.8%

          \[\leadsto e^{\mathsf{log1p}\left(d \cdot \frac{\color{blue}{1}}{\sqrt{\ell \cdot h}}\right)} - 1 \]

       5. un-div-inv 30.8%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{d}{\sqrt{\ell \cdot h}}}\right)} - 1 \]

   11. Applied egg-rr 30.8%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{d}{\sqrt{\ell \cdot h}}\right)} - 1} \]
   12. Step-by-step derivation

       1. expm1-def 40.4%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{d}{\sqrt{\ell \cdot h}}\right)\right)} \]

       2. expm1-log1p 47.3%

          \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]

   13. Simplified 47.3%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 41.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5 \cdot 10^{-238}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 38.0% accurate, 3.1× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -3 \cdot 10^{-237}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{h \cdot \ell}}{d}}\\ \end{array} \end{array} \]

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 -3e-237) (sqrt (* (/ d l) (/ d h))) (/ 1.0 (/ (sqrt (* h l)) d))))

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 <= -3e-237) {
		tmp = sqrt(((d / l) * (d / h)));
	} else {
		tmp = 1.0 / (sqrt((h * l)) / d);
	}
	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 <= (-3d-237)) then
        tmp = sqrt(((d / l) * (d / h)))
    else
        tmp = 1.0d0 / (sqrt((h * l)) / d)
    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 <= -3e-237) {
		tmp = Math.sqrt(((d / l) * (d / h)));
	} else {
		tmp = 1.0 / (Math.sqrt((h * l)) / d);
	}
	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 <= -3e-237:
		tmp = math.sqrt(((d / l) * (d / h)))
	else:
		tmp = 1.0 / (math.sqrt((h * l)) / d)
	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 <= -3e-237)
		tmp = sqrt(Float64(Float64(d / l) * Float64(d / h)));
	else
		tmp = Float64(1.0 / Float64(sqrt(Float64(h * l)) / d));
	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 <= -3e-237)
		tmp = sqrt(((d / l) * (d / h)));
	else
		tmp = 1.0 / (sqrt((h * l)) / d);
	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, -3e-237], N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(1.0 / N[(N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -3.00000000000000024e-237

   1. Initial program 67.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 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
   5. Step-by-step derivation

      1. frac-2neg 70.5%

         \[\leadsto \sqrt{\color{blue}{\frac{-d}{-h}}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \frac{{\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}}{\frac{\ell}{-0.5}}\right)\right) \]

      2. sqrt-div 83.9%

         \[\leadsto \color{blue}{\frac{\sqrt{-d}}{\sqrt{-h}}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \frac{{\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}}{\frac{\ell}{-0.5}}\right)\right) \]

   6. Applied egg-rr 79.2%

      \[\leadsto \left(\color{blue}{\frac{\sqrt{-d}}{\sqrt{-h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   7. Taylor expanded in d around inf 10.5%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   8. Step-by-step derivation

      1. *-commutative 10.5%

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

   9. Simplified 10.5%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]
   10. Step-by-step derivation

       1. expm1-log1p-u 6.5%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)\right)} \]

       2. expm1-udef 6.5%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} - 1} \]

       3. sqrt-div 6.5%

          \[\leadsto e^{\mathsf{log1p}\left(d \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\ell \cdot h}}}\right)} - 1 \]

       4. metadata-eval 6.5%

          \[\leadsto e^{\mathsf{log1p}\left(d \cdot \frac{\color{blue}{1}}{\sqrt{\ell \cdot h}}\right)} - 1 \]

       5. un-div-inv 6.5%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{d}{\sqrt{\ell \cdot h}}}\right)} - 1 \]

   11. Applied egg-rr 6.5%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{d}{\sqrt{\ell \cdot h}}\right)} - 1} \]
   12. Step-by-step derivation

       1. expm1-def 6.5%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{d}{\sqrt{\ell \cdot h}}\right)\right)} \]

       2. expm1-log1p 9.8%

          \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]

   13. Simplified 9.8%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
   14. Step-by-step derivation

       1. add-sqr-sqrt 0.0%

          \[\leadsto \frac{\color{blue}{\sqrt{d} \cdot \sqrt{d}}}{\sqrt{\ell \cdot h}} \]

       2. sqrt-prod 0.0%

          \[\leadsto \frac{\sqrt{d} \cdot \sqrt{d}}{\color{blue}{\sqrt{\ell} \cdot \sqrt{h}}} \]

       3. frac-times 0.0%

          \[\leadsto \color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \frac{\sqrt{d}}{\sqrt{h}}} \]

       4. sqrt-div 0.0%

          \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \frac{\sqrt{d}}{\sqrt{h}} \]

       5. sqrt-div 42.2%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}} \]

       6. sqrt-unprod 34.3%

          \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} \]

   15. Applied egg-rr 34.3%

       \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} \]

   if -3.00000000000000024e-237 < l

   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 64.3%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. frac-2neg 68.3%

         \[\leadsto \sqrt{\color{blue}{\frac{-d}{-h}}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \frac{{\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}}{\frac{\ell}{-0.5}}\right)\right) \]

      2. sqrt-div 6.9%

         \[\leadsto \color{blue}{\frac{\sqrt{-d}}{\sqrt{-h}}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \frac{{\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}}{\frac{\ell}{-0.5}}\right)\right) \]

   6. Applied egg-rr 6.1%

      \[\leadsto \left(\color{blue}{\frac{\sqrt{-d}}{\sqrt{-h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   7. Taylor expanded in d around inf 47.3%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   8. Step-by-step derivation

      1. *-commutative 47.3%

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

   9. Simplified 47.3%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]
   10. Step-by-step derivation

       1. add-log-exp 17.6%

          \[\leadsto d \cdot \color{blue}{\log \left(e^{\sqrt{\frac{1}{\ell \cdot h}}}\right)} \]

       2. inv-pow 17.6%

          \[\leadsto d \cdot \log \left(e^{\sqrt{\color{blue}{{\left(\ell \cdot h\right)}^{-1}}}}\right) \]

       3. sqrt-pow1 17.6%

          \[\leadsto d \cdot \log \left(e^{\color{blue}{{\left(\ell \cdot h\right)}^{\left(\frac{-1}{2}\right)}}}\right) \]

       4. metadata-eval 17.6%

          \[\leadsto d \cdot \log \left(e^{{\left(\ell \cdot h\right)}^{\color{blue}{-0.5}}}\right) \]

   11. Applied egg-rr 17.6%

       \[\leadsto d \cdot \color{blue}{\log \left(e^{{\left(\ell \cdot h\right)}^{-0.5}}\right)} \]
   12. Step-by-step derivation

       1. rem-log-exp 47.3%

          \[\leadsto d \cdot \color{blue}{{\left(\ell \cdot h\right)}^{-0.5}} \]

       2. metadata-eval 47.3%

          \[\leadsto d \cdot {\left(\ell \cdot h\right)}^{\color{blue}{\left(-0.5\right)}} \]

       3. pow-flip 47.3%

          \[\leadsto d \cdot \color{blue}{\frac{1}{{\left(\ell \cdot h\right)}^{0.5}}} \]

       4. pow1/2 47.3%

          \[\leadsto d \cdot \frac{1}{\color{blue}{\sqrt{\ell \cdot h}}} \]

       5. div-inv 47.3%

          \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]

       6. clear-num 47.4%

          \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{\ell \cdot h}}{d}}} \]

   13. Applied egg-rr 47.4%

       \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{\ell \cdot h}}{d}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 41.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -3 \cdot 10^{-237}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{h \cdot \ell}}{d}}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 26.6% accurate, 3.3× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \frac{d}{\sqrt{h \cdot \ell}} \end{array} \]

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 (* 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) {
	return d / sqrt((h * l));
}

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((h * l))
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((h * l));
}

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((h * l))

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(h * l)))
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((h * l));
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[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 65.8%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

3. Simplified 65.4%

   \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. frac-2neg 69.4%

      \[\leadsto \sqrt{\color{blue}{\frac{-d}{-h}}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \frac{{\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}}{\frac{\ell}{-0.5}}\right)\right) \]

   2. sqrt-div 44.5%

      \[\leadsto \color{blue}{\frac{\sqrt{-d}}{\sqrt{-h}}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \frac{{\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}}{\frac{\ell}{-0.5}}\right)\right) \]

6. Applied egg-rr 41.8%

   \[\leadsto \left(\color{blue}{\frac{\sqrt{-d}}{\sqrt{-h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

7. Taylor expanded in d around inf 29.4%

   \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
8. Step-by-step derivation

   1. *-commutative 29.4%

      \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

9. Simplified 29.4%

   \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]
10. Step-by-step derivation

    1. expm1-log1p-u 23.8%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)\right)} \]

    2. expm1-udef 18.9%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} - 1} \]

    3. sqrt-div 18.9%

       \[\leadsto e^{\mathsf{log1p}\left(d \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\ell \cdot h}}}\right)} - 1 \]

    4. metadata-eval 18.9%

       \[\leadsto e^{\mathsf{log1p}\left(d \cdot \frac{\color{blue}{1}}{\sqrt{\ell \cdot h}}\right)} - 1 \]

    5. un-div-inv 18.9%

       \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{d}{\sqrt{\ell \cdot h}}}\right)} - 1 \]

11. Applied egg-rr 18.9%

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{d}{\sqrt{\ell \cdot h}}\right)} - 1} \]
12. Step-by-step derivation

    1. expm1-def 23.8%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{d}{\sqrt{\ell \cdot h}}\right)\right)} \]

    2. expm1-log1p 29.0%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]

13. Simplified 29.0%

    \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]

14. Final simplification 29.0%

    \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024011 
(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)))))