Henrywood and Agarwal, Equation (12)

Percentage Accurate: 66.2% → 84.5%

Time: 22.1s

Alternatives: 23

Speedup: 1.4×

• Could not converge on a ground truth

  1. d = -5.870776150534888e+30
  2. h = -1.649376812391256e+204
  3. l = -1.67401613004005e+19
  4. M = 4.3271074370996713e-302
  5. D = 1.7582595499281745e-134

• 23.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 66.2% 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: 84.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := 0.5 \cdot \frac{h \cdot {\left(M\_m \cdot \left(0.5 \cdot \frac{D\_m}{d}\right)\right)}^{2}}{\ell}\\ t_1 := \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \mathbf{if}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \frac{\sqrt{\frac{-1}{\ell}}}{\sqrt{-h}}\right) \cdot \left(-1 + t\_0\right)\\ \mathbf{elif}\;\ell \leq 2.35 \cdot 10^{+42}:\\ \;\;\;\;\left(1 - t\_0\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \left(1 - 0.5 \cdot {\left(D\_m \cdot \left(\left(M\_m \cdot \frac{0.5}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right)\\ \end{array} \end{array} \]

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 (* 0.5 (/ (* h (pow (* M_m (* 0.5 (/ D_m d))) 2.0)) l)))
        (t_1 (/ d (* (sqrt l) (sqrt h)))))
   (if (<= l -5e-310)
     (* (* d (/ (sqrt (/ -1.0 l)) (sqrt (- h)))) (+ -1.0 t_0))
     (if (<= l 2.35e+42)
       (* (- 1.0 t_0) t_1)
       (*
        t_1
        (-
         1.0
         (* 0.5 (pow (* D_m (* (* M_m (/ 0.5 d)) (sqrt (/ h l)))) 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 = 0.5 * ((h * pow((M_m * (0.5 * (D_m / d))), 2.0)) / l);
	double t_1 = d / (sqrt(l) * sqrt(h));
	double tmp;
	if (l <= -5e-310) {
		tmp = (d * (sqrt((-1.0 / l)) / sqrt(-h))) * (-1.0 + t_0);
	} else if (l <= 2.35e+42) {
		tmp = (1.0 - t_0) * t_1;
	} else {
		tmp = t_1 * (1.0 - (0.5 * pow((D_m * ((M_m * (0.5 / d)) * sqrt((h / l)))), 2.0)));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 0.5d0 * ((h * ((m_m * (0.5d0 * (d_m / d))) ** 2.0d0)) / l)
    t_1 = d / (sqrt(l) * sqrt(h))
    if (l <= (-5d-310)) then
        tmp = (d * (sqrt(((-1.0d0) / l)) / sqrt(-h))) * ((-1.0d0) + t_0)
    else if (l <= 2.35d+42) then
        tmp = (1.0d0 - t_0) * t_1
    else
        tmp = t_1 * (1.0d0 - (0.5d0 * ((d_m * ((m_m * (0.5d0 / d)) * sqrt((h / l)))) ** 2.0d0)))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = 0.5 * ((h * Math.pow((M_m * (0.5 * (D_m / d))), 2.0)) / l);
	double t_1 = d / (Math.sqrt(l) * Math.sqrt(h));
	double tmp;
	if (l <= -5e-310) {
		tmp = (d * (Math.sqrt((-1.0 / l)) / Math.sqrt(-h))) * (-1.0 + t_0);
	} else if (l <= 2.35e+42) {
		tmp = (1.0 - t_0) * t_1;
	} else {
		tmp = t_1 * (1.0 - (0.5 * Math.pow((D_m * ((M_m * (0.5 / d)) * Math.sqrt((h / l)))), 2.0)));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = 0.5 * ((h * math.pow((M_m * (0.5 * (D_m / d))), 2.0)) / l)
	t_1 = d / (math.sqrt(l) * math.sqrt(h))
	tmp = 0
	if l <= -5e-310:
		tmp = (d * (math.sqrt((-1.0 / l)) / math.sqrt(-h))) * (-1.0 + t_0)
	elif l <= 2.35e+42:
		tmp = (1.0 - t_0) * t_1
	else:
		tmp = t_1 * (1.0 - (0.5 * math.pow((D_m * ((M_m * (0.5 / d)) * math.sqrt((h / l)))), 2.0)))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(0.5 * Float64(Float64(h * (Float64(M_m * Float64(0.5 * Float64(D_m / d))) ^ 2.0)) / l))
	t_1 = Float64(d / Float64(sqrt(l) * sqrt(h)))
	tmp = 0.0
	if (l <= -5e-310)
		tmp = Float64(Float64(d * Float64(sqrt(Float64(-1.0 / l)) / sqrt(Float64(-h)))) * Float64(-1.0 + t_0));
	elseif (l <= 2.35e+42)
		tmp = Float64(Float64(1.0 - t_0) * t_1);
	else
		tmp = Float64(t_1 * Float64(1.0 - Float64(0.5 * (Float64(D_m * Float64(Float64(M_m * Float64(0.5 / d)) * sqrt(Float64(h / l)))) ^ 2.0))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = 0.5 * ((h * ((M_m * (0.5 * (D_m / d))) ^ 2.0)) / l);
	t_1 = d / (sqrt(l) * sqrt(h));
	tmp = 0.0;
	if (l <= -5e-310)
		tmp = (d * (sqrt((-1.0 / l)) / sqrt(-h))) * (-1.0 + t_0);
	elseif (l <= 2.35e+42)
		tmp = (1.0 - t_0) * t_1;
	else
		tmp = t_1 * (1.0 - (0.5 * ((D_m * ((M_m * (0.5 / d)) * sqrt((h / l)))) ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(0.5 * N[(N[(h * N[Power[N[(M$95$m * N[(0.5 * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -5e-310], N[(N[(d * N[(N[Sqrt[N[(-1.0 / l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0 + t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 2.35e+42], N[(N[(1.0 - t$95$0), $MachinePrecision] * t$95$1), $MachinePrecision], N[(t$95$1 * N[(1.0 - N[(0.5 * N[Power[N[(D$95$m * N[(N[(M$95$m * N[(0.5 / d), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -4.999999999999985e-310

   1. Initial program 66.7%

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

   3. Simplified 67.4%

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

      1. associate-*r/ 70.6%

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

      2. frac-times 70.6%

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

      3. associate-/l* 70.6%

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

      4. *-un-lft-identity 70.6%

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

      5. times-frac 70.6%

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

      6. metadata-eval 70.6%

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

   6. Applied egg-rr 70.6%

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

      1. clear-num 70.5%

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

      2. sqrt-div 70.5%

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

      3. metadata-eval 70.5%

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

   8. Applied egg-rr 70.5%

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

   9. Taylor expanded in d around -inf 67.1%

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

       1. mul-1-neg 67.1%

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

       2. unpow-1 67.1%

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

       3. metadata-eval 67.1%

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

       4. pow-sqr 67.1%

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

       5. rem-sqrt-square 67.6%

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

       6. metadata-eval 67.6%

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

       7. pow-sqr 67.4%

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

       8. fabs-sqr 67.4%

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

       9. pow-sqr 67.6%

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

       10. metadata-eval 67.6%

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

       11. distribute-rgt-neg-in 67.6%

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

   11. Simplified 67.6%

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

       1. unpow-prod-down 0.0%

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

       2. metadata-eval 0.0%

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

       3. sqrt-pow1 0.0%

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

       4. inv-pow 0.0%

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

       5. metadata-eval 0.0%

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

       6. sqrt-pow1 0.0%

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

       7. inv-pow 0.0%

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

       8. *-commutative 0.0%

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

       9. sqrt-prod 68.7%

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

       10. div-inv 68.7%

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

       11. frac-2neg 68.7%

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

       12. sqrt-div 84.0%

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

       13. distribute-neg-frac 84.0%

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

       14. metadata-eval 84.0%

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

   13. Applied egg-rr 84.0%

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

   if -4.999999999999985e-310 < l < 2.34999999999999993e42

   1. Initial program 70.5%

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

   3. Simplified 70.4%

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

      1. associate-*r/ 79.6%

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

      2. frac-times 79.7%

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

      3. associate-/l* 79.6%

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

      4. *-un-lft-identity 79.6%

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

      5. times-frac 79.6%

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

      6. metadata-eval 79.6%

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

   6. Applied egg-rr 79.6%

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

      1. *-commutative 79.6%

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

      2. sqrt-div 85.4%

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

      3. sqrt-div 91.9%

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

      4. frac-times 91.8%

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

      5. add-sqr-sqrt 92.0%

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

   8. Applied egg-rr 92.0%

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

   if 2.34999999999999993e42 < l

   1. Initial program 62.9%

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

   3. Simplified 61.1%

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

      1. *-commutative 61.0%

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

      2. sqrt-div 68.7%

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

      3. sqrt-div 81.5%

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

      4. frac-times 81.6%

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

      5. add-sqr-sqrt 81.9%

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

   6. Applied egg-rr 81.8%

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

      1. add-sqr-sqrt 81.8%

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

      2. pow2 81.8%

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

      3. sqrt-prod 81.8%

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

      4. sqrt-pow1 86.0%

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

      5. metadata-eval 86.0%

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

      6. pow1 86.0%

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

      7. frac-times 85.7%

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

      8. associate-/l* 86.0%

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

      9. *-un-lft-identity 86.0%

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

      10. times-frac 86.0%

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

      11. metadata-eval 86.0%

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

   8. Applied egg-rr 86.0%

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

      1. associate-*r* 86.0%

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

      2. *-commutative 86.0%

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

      3. associate-*l/ 85.7%

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

      4. associate-/l* 88.0%

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

      5. *-commutative 88.0%

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

      6. associate-*r/ 88.0%

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

      7. associate-*l* 88.0%

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

      8. associate-*r/ 88.0%

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

      9. *-commutative 88.0%

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

      10. associate-/l* 88.0%

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

   10. Simplified 88.0%

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

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \frac{\sqrt{\frac{-1}{\ell}}}{\sqrt{-h}}\right) \cdot \left(-1 + 0.5 \cdot \frac{h \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell}\right)\\ \mathbf{elif}\;\ell \leq 2.35 \cdot 10^{+42}:\\ \;\;\;\;\left(1 - 0.5 \cdot \frac{h \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell}\right) \cdot \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\left(M \cdot \frac{0.5}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := 0.5 \cdot \frac{h \cdot {\left(M\_m \cdot \left(0.5 \cdot \frac{D\_m}{d}\right)\right)}^{2}}{\ell}\\ t_1 := \sqrt{\ell} \cdot \sqrt{h}\\ \mathbf{if}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \frac{\sqrt{\frac{-1}{\ell}}}{\sqrt{-h}}\right) \cdot \left(-1 + t\_0\right)\\ \mathbf{elif}\;\ell \leq 7.2 \cdot 10^{-84}:\\ \;\;\;\;\left(1 - t\_0\right) \cdot \frac{d}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left({\left(D\_m \cdot \frac{M\_m}{d \cdot 2}\right)}^{2}, \frac{h \cdot -0.5}{\ell}, 1\right)}{t\_1}\\ \end{array} \end{array} \]

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 (* 0.5 (/ (* h (pow (* M_m (* 0.5 (/ D_m d))) 2.0)) l)))
        (t_1 (* (sqrt l) (sqrt h))))
   (if (<= l -5e-310)
     (* (* d (/ (sqrt (/ -1.0 l)) (sqrt (- h)))) (+ -1.0 t_0))
     (if (<= l 7.2e-84)
       (* (- 1.0 t_0) (/ d t_1))
       (*
        d
        (/
         (fma (pow (* D_m (/ M_m (* d 2.0))) 2.0) (/ (* h -0.5) l) 1.0)
         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 = 0.5 * ((h * pow((M_m * (0.5 * (D_m / d))), 2.0)) / l);
	double t_1 = sqrt(l) * sqrt(h);
	double tmp;
	if (l <= -5e-310) {
		tmp = (d * (sqrt((-1.0 / l)) / sqrt(-h))) * (-1.0 + t_0);
	} else if (l <= 7.2e-84) {
		tmp = (1.0 - t_0) * (d / t_1);
	} else {
		tmp = d * (fma(pow((D_m * (M_m / (d * 2.0))), 2.0), ((h * -0.5) / l), 1.0) / 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 = Float64(0.5 * Float64(Float64(h * (Float64(M_m * Float64(0.5 * Float64(D_m / d))) ^ 2.0)) / l))
	t_1 = Float64(sqrt(l) * sqrt(h))
	tmp = 0.0
	if (l <= -5e-310)
		tmp = Float64(Float64(d * Float64(sqrt(Float64(-1.0 / l)) / sqrt(Float64(-h)))) * Float64(-1.0 + t_0));
	elseif (l <= 7.2e-84)
		tmp = Float64(Float64(1.0 - t_0) * Float64(d / t_1));
	else
		tmp = Float64(d * Float64(fma((Float64(D_m * Float64(M_m / Float64(d * 2.0))) ^ 2.0), Float64(Float64(h * -0.5) / l), 1.0) / t_1));
	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[(0.5 * N[(N[(h * N[Power[N[(M$95$m * N[(0.5 * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -5e-310], N[(N[(d * N[(N[Sqrt[N[(-1.0 / l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0 + t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 7.2e-84], N[(N[(1.0 - t$95$0), $MachinePrecision] * N[(d / t$95$1), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[(N[Power[N[(D$95$m * N[(M$95$m / N[(d * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(h * -0.5), $MachinePrecision] / l), $MachinePrecision] + 1.0), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -4.999999999999985e-310

   1. Initial program 66.7%

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

   3. Simplified 67.4%

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

      1. associate-*r/ 70.6%

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

      2. frac-times 70.6%

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

      3. associate-/l* 70.6%

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

      4. *-un-lft-identity 70.6%

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

      5. times-frac 70.6%

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

      6. metadata-eval 70.6%

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

   6. Applied egg-rr 70.6%

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

      1. clear-num 70.5%

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

      2. sqrt-div 70.5%

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

      3. metadata-eval 70.5%

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

   8. Applied egg-rr 70.5%

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

   9. Taylor expanded in d around -inf 67.1%

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

       1. mul-1-neg 67.1%

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

       2. unpow-1 67.1%

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

       3. metadata-eval 67.1%

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

       4. pow-sqr 67.1%

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

       5. rem-sqrt-square 67.6%

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

       6. metadata-eval 67.6%

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

       7. pow-sqr 67.4%

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

       8. fabs-sqr 67.4%

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

       9. pow-sqr 67.6%

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

       10. metadata-eval 67.6%

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

       11. distribute-rgt-neg-in 67.6%

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

   11. Simplified 67.6%

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

       1. unpow-prod-down 0.0%

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

       2. metadata-eval 0.0%

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

       3. sqrt-pow1 0.0%

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

       4. inv-pow 0.0%

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

       5. metadata-eval 0.0%

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

       6. sqrt-pow1 0.0%

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

       7. inv-pow 0.0%

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

       8. *-commutative 0.0%

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

       9. sqrt-prod 68.7%

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

       10. div-inv 68.7%

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

       11. frac-2neg 68.7%

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

       12. sqrt-div 84.0%

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

       13. distribute-neg-frac 84.0%

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

       14. metadata-eval 84.0%

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

   13. Applied egg-rr 84.0%

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

   if -4.999999999999985e-310 < l < 7.20000000000000007e-84

   1. Initial program 71.1%

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

   3. Simplified 71.0%

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

      1. associate-*r/ 83.9%

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

      2. frac-times 84.0%

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

      3. associate-/l* 83.9%

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

      4. *-un-lft-identity 83.9%

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

      5. times-frac 83.9%

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

      6. metadata-eval 83.9%

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

   6. Applied egg-rr 83.9%

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

      1. *-commutative 83.9%

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

      2. sqrt-div 90.5%

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

      3. sqrt-div 95.9%

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

      4. frac-times 95.8%

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

      5. add-sqr-sqrt 96.1%

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

   8. Applied egg-rr 96.1%

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

   if 7.20000000000000007e-84 < l

   1. Initial program 64.8%

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

   3. Simplified 63.5%

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

      1. pow1 63.5%

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

   6. Applied egg-rr 81.9%

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

      1. unpow1 81.9%

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

      2. associate-*l/ 86.0%

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

      3. associate-/l* 86.1%

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

      4. *-commutative 86.1%

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

      5. metadata-eval 86.1%

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

      6. times-frac 86.1%

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

      7. *-rgt-identity 86.1%

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

      8. associate-/l* 86.1%

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

      9. *-commutative 86.1%

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

      10. associate-/l* 86.1%

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

      11. *-commutative 86.1%

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

      12. *-commutative 86.1%

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

   8. Simplified 86.1%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \frac{\sqrt{\frac{-1}{\ell}}}{\sqrt{-h}}\right) \cdot \left(-1 + 0.5 \cdot \frac{h \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell}\right)\\ \mathbf{elif}\;\ell \leq 7.2 \cdot 10^{-84}:\\ \;\;\;\;\left(1 - 0.5 \cdot \frac{h \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell}\right) \cdot \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left({\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}, \frac{h \cdot -0.5}{\ell}, 1\right)}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 77.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := {\left(M\_m \cdot \left(0.5 \cdot \frac{D\_m}{d}\right)\right)}^{2}\\ t_1 := 0.5 \cdot \frac{h \cdot t\_0}{\ell}\\ \mathbf{if}\;d \leq -3.8 \cdot 10^{+201}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + t\_0 \cdot \frac{h \cdot -0.5}{\ell}\right)\right)\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right) \cdot \left(-1 + t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - t\_1\right) \cdot \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (pow (* M_m (* 0.5 (/ D_m d))) 2.0))
        (t_1 (* 0.5 (/ (* h t_0) l))))
   (if (<= d -3.8e+201)
     (* (sqrt (/ d h)) (* (sqrt (/ d l)) (+ 1.0 (* t_0 (/ (* h -0.5) l)))))
     (if (<= d -5e-310)
       (* (* d (pow (* l h) -0.5)) (+ -1.0 t_1))
       (* (- 1.0 t_1) (/ d (* (sqrt l) (sqrt h))))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = pow((M_m * (0.5 * (D_m / d))), 2.0);
	double t_1 = 0.5 * ((h * t_0) / l);
	double tmp;
	if (d <= -3.8e+201) {
		tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0 + (t_0 * ((h * -0.5) / l))));
	} else if (d <= -5e-310) {
		tmp = (d * pow((l * h), -0.5)) * (-1.0 + t_1);
	} else {
		tmp = (1.0 - t_1) * (d / (sqrt(l) * sqrt(h)));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (m_m * (0.5d0 * (d_m / d))) ** 2.0d0
    t_1 = 0.5d0 * ((h * t_0) / l)
    if (d <= (-3.8d+201)) then
        tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0d0 + (t_0 * ((h * (-0.5d0)) / l))))
    else if (d <= (-5d-310)) then
        tmp = (d * ((l * h) ** (-0.5d0))) * ((-1.0d0) + t_1)
    else
        tmp = (1.0d0 - t_1) * (d / (sqrt(l) * sqrt(h)))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = Math.pow((M_m * (0.5 * (D_m / d))), 2.0);
	double t_1 = 0.5 * ((h * t_0) / l);
	double tmp;
	if (d <= -3.8e+201) {
		tmp = Math.sqrt((d / h)) * (Math.sqrt((d / l)) * (1.0 + (t_0 * ((h * -0.5) / l))));
	} else if (d <= -5e-310) {
		tmp = (d * Math.pow((l * h), -0.5)) * (-1.0 + t_1);
	} else {
		tmp = (1.0 - t_1) * (d / (Math.sqrt(l) * Math.sqrt(h)));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = math.pow((M_m * (0.5 * (D_m / d))), 2.0)
	t_1 = 0.5 * ((h * t_0) / l)
	tmp = 0
	if d <= -3.8e+201:
		tmp = math.sqrt((d / h)) * (math.sqrt((d / l)) * (1.0 + (t_0 * ((h * -0.5) / l))))
	elif d <= -5e-310:
		tmp = (d * math.pow((l * h), -0.5)) * (-1.0 + t_1)
	else:
		tmp = (1.0 - t_1) * (d / (math.sqrt(l) * math.sqrt(h)))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(M_m * Float64(0.5 * Float64(D_m / d))) ^ 2.0
	t_1 = Float64(0.5 * Float64(Float64(h * t_0) / l))
	tmp = 0.0
	if (d <= -3.8e+201)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(sqrt(Float64(d / l)) * Float64(1.0 + Float64(t_0 * Float64(Float64(h * -0.5) / l)))));
	elseif (d <= -5e-310)
		tmp = Float64(Float64(d * (Float64(l * h) ^ -0.5)) * Float64(-1.0 + t_1));
	else
		tmp = Float64(Float64(1.0 - t_1) * Float64(d / Float64(sqrt(l) * sqrt(h))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = (M_m * (0.5 * (D_m / d))) ^ 2.0;
	t_1 = 0.5 * ((h * t_0) / l);
	tmp = 0.0;
	if (d <= -3.8e+201)
		tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0 + (t_0 * ((h * -0.5) / l))));
	elseif (d <= -5e-310)
		tmp = (d * ((l * h) ^ -0.5)) * (-1.0 + t_1);
	else
		tmp = (1.0 - t_1) * (d / (sqrt(l) * sqrt(h)));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[Power[N[(M$95$m * N[(0.5 * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[(N[(h * t$95$0), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -3.8e+201], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(t$95$0 * N[(N[(h * -0.5), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -5e-310], N[(N[(d * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] * N[(-1.0 + t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - t$95$1), $MachinePrecision] * N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -3.79999999999999995e201

   1. Initial program 89.6%

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

   3. Simplified 93.3%

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

      1. fma-undefine 93.3%

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

      2. metadata-eval 93.3%

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

      3. associate-*r/ 89.6%

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

      4. times-frac 89.6%

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

      5. *-un-lft-identity 89.6%

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

      6. associate-/l* 93.3%

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

      7. *-un-lft-identity 93.3%

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

      8. times-frac 93.3%

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

      9. metadata-eval 93.3%

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

      10. associate-*r/ 93.3%

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

   6. Applied egg-rr 93.3%

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

   if -3.79999999999999995e201 < d < -4.999999999999985e-310

   1. Initial program 60.7%

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

   3. Simplified 60.6%

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

      1. associate-*r/ 64.6%

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

      2. frac-times 65.6%

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

      3. associate-/l* 64.6%

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

      4. *-un-lft-identity 64.6%

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

      5. times-frac 64.6%

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

      6. metadata-eval 64.6%

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

   6. Applied egg-rr 64.6%

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

      1. clear-num 64.6%

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

      2. sqrt-div 64.5%

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

      3. metadata-eval 64.5%

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

   8. Applied egg-rr 64.5%

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

   9. Taylor expanded in d around -inf 64.2%

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

       1. mul-1-neg 64.2%

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

       2. unpow-1 64.2%

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

       3. metadata-eval 64.2%

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

       4. pow-sqr 64.2%

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

       5. rem-sqrt-square 64.8%

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

       6. metadata-eval 64.8%

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

       7. pow-sqr 64.6%

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

       8. fabs-sqr 64.6%

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

       9. pow-sqr 64.8%

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

       10. metadata-eval 64.8%

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

       11. distribute-rgt-neg-in 64.8%

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

   11. Simplified 64.8%

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

   if -4.999999999999985e-310 < d

   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 66.7%

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

      1. associate-*r/ 72.2%

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

      2. frac-times 73.0%

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

      3. associate-/l* 72.2%

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

      4. *-un-lft-identity 72.2%

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

      5. times-frac 72.2%

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

      6. metadata-eval 72.2%

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

   6. Applied egg-rr 72.2%

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

      1. *-commutative 72.2%

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

      2. sqrt-div 78.8%

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

      3. sqrt-div 87.8%

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

      4. frac-times 87.7%

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

      5. add-sqr-sqrt 88.0%

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

   8. Applied egg-rr 88.0%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -3.8 \cdot 10^{+201}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \frac{h \cdot -0.5}{\ell}\right)\right)\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right) \cdot \left(-1 + 0.5 \cdot \frac{h \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - 0.5 \cdot \frac{h \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell}\right) \cdot \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := 0.5 \cdot \frac{h \cdot {\left(M\_m \cdot \left(0.5 \cdot \frac{D\_m}{d}\right)\right)}^{2}}{\ell}\\ \mathbf{if}\;\ell \leq -9.8 \cdot 10^{-238}:\\ \;\;\;\;\left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right) \cdot \left(-1 + t\_0\right)\\ \mathbf{elif}\;\ell \leq 1.25 \cdot 10^{-159}:\\ \;\;\;\;\left(1 - t\_0\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 - 0.125 \cdot \left(\frac{h}{\ell} \cdot {\left(D\_m \cdot \frac{M\_m}{d}\right)}^{2}\right)\right)\\ \end{array} \end{array} \]

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
 (let* ((t_0 (* 0.5 (/ (* h (pow (* M_m (* 0.5 (/ D_m d))) 2.0)) l))))
   (if (<= l -9.8e-238)
     (* (* d (pow (* l h) -0.5)) (+ -1.0 t_0))
     (if (<= l 1.25e-159)
       (* (- 1.0 t_0) (sqrt (* (/ d h) (/ d l))))
       (*
        (/ d (* (sqrt l) (sqrt h)))
        (- 1.0 (* 0.125 (* (/ h l) (pow (* D_m (/ M_m 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 = 0.5 * ((h * pow((M_m * (0.5 * (D_m / d))), 2.0)) / l);
	double tmp;
	if (l <= -9.8e-238) {
		tmp = (d * pow((l * h), -0.5)) * (-1.0 + t_0);
	} else if (l <= 1.25e-159) {
		tmp = (1.0 - t_0) * sqrt(((d / h) * (d / l)));
	} else {
		tmp = (d / (sqrt(l) * sqrt(h))) * (1.0 - (0.125 * ((h / l) * pow((D_m * (M_m / d)), 2.0))));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * ((h * ((m_m * (0.5d0 * (d_m / d))) ** 2.0d0)) / l)
    if (l <= (-9.8d-238)) then
        tmp = (d * ((l * h) ** (-0.5d0))) * ((-1.0d0) + t_0)
    else if (l <= 1.25d-159) then
        tmp = (1.0d0 - t_0) * sqrt(((d / h) * (d / l)))
    else
        tmp = (d / (sqrt(l) * sqrt(h))) * (1.0d0 - (0.125d0 * ((h / l) * ((d_m * (m_m / d)) ** 2.0d0))))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = 0.5 * ((h * Math.pow((M_m * (0.5 * (D_m / d))), 2.0)) / l);
	double tmp;
	if (l <= -9.8e-238) {
		tmp = (d * Math.pow((l * h), -0.5)) * (-1.0 + t_0);
	} else if (l <= 1.25e-159) {
		tmp = (1.0 - t_0) * Math.sqrt(((d / h) * (d / l)));
	} else {
		tmp = (d / (Math.sqrt(l) * Math.sqrt(h))) * (1.0 - (0.125 * ((h / l) * Math.pow((D_m * (M_m / d)), 2.0))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = 0.5 * ((h * math.pow((M_m * (0.5 * (D_m / d))), 2.0)) / l)
	tmp = 0
	if l <= -9.8e-238:
		tmp = (d * math.pow((l * h), -0.5)) * (-1.0 + t_0)
	elif l <= 1.25e-159:
		tmp = (1.0 - t_0) * math.sqrt(((d / h) * (d / l)))
	else:
		tmp = (d / (math.sqrt(l) * math.sqrt(h))) * (1.0 - (0.125 * ((h / l) * math.pow((D_m * (M_m / 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 = Float64(0.5 * Float64(Float64(h * (Float64(M_m * Float64(0.5 * Float64(D_m / d))) ^ 2.0)) / l))
	tmp = 0.0
	if (l <= -9.8e-238)
		tmp = Float64(Float64(d * (Float64(l * h) ^ -0.5)) * Float64(-1.0 + t_0));
	elseif (l <= 1.25e-159)
		tmp = Float64(Float64(1.0 - t_0) * sqrt(Float64(Float64(d / h) * Float64(d / l))));
	else
		tmp = Float64(Float64(d / Float64(sqrt(l) * sqrt(h))) * Float64(1.0 - Float64(0.125 * Float64(Float64(h / l) * (Float64(D_m * Float64(M_m / d)) ^ 2.0)))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = 0.5 * ((h * ((M_m * (0.5 * (D_m / d))) ^ 2.0)) / l);
	tmp = 0.0;
	if (l <= -9.8e-238)
		tmp = (d * ((l * h) ^ -0.5)) * (-1.0 + t_0);
	elseif (l <= 1.25e-159)
		tmp = (1.0 - t_0) * sqrt(((d / h) * (d / l)));
	else
		tmp = (d / (sqrt(l) * sqrt(h))) * (1.0 - (0.125 * ((h / l) * ((D_m * (M_m / d)) ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(0.5 * N[(N[(h * N[Power[N[(M$95$m * N[(0.5 * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -9.8e-238], N[(N[(d * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] * N[(-1.0 + t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.25e-159], N[(N[(1.0 - t$95$0), $MachinePrecision] * N[Sqrt[N[(N[(d / h), $MachinePrecision] * N[(d / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.125 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(D$95$m * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < -9.7999999999999996e-238

   1. Initial program 62.7%

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

   3. Simplified 63.5%

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

      1. associate-*r/ 67.3%

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

      2. frac-times 67.3%

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

      3. associate-/l* 67.3%

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

      4. *-un-lft-identity 67.3%

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

      5. times-frac 67.3%

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

      6. metadata-eval 67.3%

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

   6. Applied egg-rr 67.3%

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

      1. clear-num 67.2%

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

      2. sqrt-div 67.2%

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

      3. metadata-eval 67.2%

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

   8. Applied egg-rr 67.2%

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

   9. Taylor expanded in d around -inf 65.7%

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

       1. mul-1-neg 65.7%

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

       2. unpow-1 65.7%

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

       3. metadata-eval 65.7%

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

       4. pow-sqr 65.7%

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

       5. rem-sqrt-square 66.2%

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

       6. metadata-eval 66.2%

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

       7. pow-sqr 66.1%

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

       8. fabs-sqr 66.1%

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

       9. pow-sqr 66.2%

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

       10. metadata-eval 66.2%

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

       11. distribute-rgt-neg-in 66.2%

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

   11. Simplified 66.2%

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

   if -9.7999999999999996e-238 < l < 1.25000000000000008e-159

   1. Initial program 81.5%

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

   3. Simplified 81.4%

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

      1. associate-*r/ 90.8%

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

      2. frac-times 90.9%

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

      3. associate-/l* 90.8%

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

      4. *-un-lft-identity 90.8%

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

      5. times-frac 90.8%

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

      6. metadata-eval 90.8%

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

   6. Applied egg-rr 90.8%

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

      1. *-commutative 90.8%

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

      2. sqrt-unprod 89.3%

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

   8. Applied egg-rr 89.3%

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

   if 1.25000000000000008e-159 < l

   1. Initial program 64.1%

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

   3. Simplified 63.0%

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

      1. *-commutative 65.1%

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

      2. sqrt-div 72.1%

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

      3. sqrt-div 84.5%

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

      4. frac-times 84.4%

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

      5. add-sqr-sqrt 84.7%

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

   6. Applied egg-rr 82.6%

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

   7. Taylor expanded in M around 0 46.7%

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

      1. associate-*r* 48.9%

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

      2. times-frac 47.9%

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

      3. associate-/l* 48.9%

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

      4. unpow2 48.9%

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

      5. unpow2 48.9%

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

      6. unpow2 48.9%

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

      7. times-frac 69.6%

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

      8. swap-sqr 82.6%

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

      9. associate-/l* 81.5%

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

      10. associate-/l* 82.6%

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

      11. unpow2 82.6%

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

      12. associate-/l* 82.6%

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

   9. Simplified 82.6%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -9.8 \cdot 10^{-238}:\\ \;\;\;\;\left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right) \cdot \left(-1 + 0.5 \cdot \frac{h \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell}\right)\\ \mathbf{elif}\;\ell \leq 1.25 \cdot 10^{-159}:\\ \;\;\;\;\left(1 - 0.5 \cdot \frac{h \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell}\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 - 0.125 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right) \cdot \left(-1 + 0.5 \cdot \frac{h \cdot {\left(M\_m \cdot \left(0.5 \cdot \frac{D\_m}{d}\right)\right)}^{2}}{\ell}\right)\\ \mathbf{elif}\;\ell \leq 3.6 \cdot 10^{+192}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left(-0.5, h \cdot \frac{0.25 \cdot {\left(D\_m \cdot \frac{M\_m}{d}\right)}^{2}}{\ell}, 1\right)}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= l -5e-310)
   (*
    (* d (pow (* l h) -0.5))
    (+ -1.0 (* 0.5 (/ (* h (pow (* M_m (* 0.5 (/ D_m d))) 2.0)) l))))
   (if (<= l 3.6e+192)
     (*
      d
      (/
       (fma -0.5 (* h (/ (* 0.25 (pow (* D_m (/ M_m d)) 2.0)) l)) 1.0)
       (sqrt (* l h))))
     (/ (/ d (sqrt l)) (sqrt h)))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= -5e-310) {
		tmp = (d * pow((l * h), -0.5)) * (-1.0 + (0.5 * ((h * pow((M_m * (0.5 * (D_m / d))), 2.0)) / l)));
	} else if (l <= 3.6e+192) {
		tmp = d * (fma(-0.5, (h * ((0.25 * pow((D_m * (M_m / d)), 2.0)) / l)), 1.0) / sqrt((l * h)));
	} else {
		tmp = (d / sqrt(l)) / sqrt(h);
	}
	return tmp;
}

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (l <= -5e-310)
		tmp = Float64(Float64(d * (Float64(l * h) ^ -0.5)) * Float64(-1.0 + Float64(0.5 * Float64(Float64(h * (Float64(M_m * Float64(0.5 * Float64(D_m / d))) ^ 2.0)) / l))));
	elseif (l <= 3.6e+192)
		tmp = Float64(d * Float64(fma(-0.5, Float64(h * Float64(Float64(0.25 * (Float64(D_m * Float64(M_m / d)) ^ 2.0)) / l)), 1.0) / sqrt(Float64(l * h))));
	else
		tmp = Float64(Float64(d / sqrt(l)) / sqrt(h));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[l, -5e-310], N[(N[(d * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] * N[(-1.0 + N[(0.5 * N[(N[(h * N[Power[N[(M$95$m * N[(0.5 * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 3.6e+192], N[(d * N[(N[(-0.5 * N[(h * N[(N[(0.25 * N[Power[N[(D$95$m * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(d / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -4.999999999999985e-310

   1. Initial program 66.7%

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

   3. Simplified 67.4%

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

      1. associate-*r/ 70.6%

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

      2. frac-times 70.6%

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

      3. associate-/l* 70.6%

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

      4. *-un-lft-identity 70.6%

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

      5. times-frac 70.6%

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

      6. metadata-eval 70.6%

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

   6. Applied egg-rr 70.6%

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

      1. clear-num 70.5%

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

      2. sqrt-div 70.5%

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

      3. metadata-eval 70.5%

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

   8. Applied egg-rr 70.5%

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

   9. Taylor expanded in d around -inf 67.1%

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

       1. mul-1-neg 67.1%

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

       2. unpow-1 67.1%

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

       3. metadata-eval 67.1%

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

       4. pow-sqr 67.1%

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

       5. rem-sqrt-square 67.6%

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

       6. metadata-eval 67.6%

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

       7. pow-sqr 67.4%

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

       8. fabs-sqr 67.4%

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

       9. pow-sqr 67.6%

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

       10. metadata-eval 67.6%

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

       11. distribute-rgt-neg-in 67.6%

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

   11. Simplified 67.6%

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

   if -4.999999999999985e-310 < l < 3.6000000000000002e192

   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 68.4%

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

      1. *-commutative 75.0%

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

      2. sqrt-div 82.0%

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

      3. sqrt-div 87.5%

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

      4. frac-times 87.4%

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

      5. add-sqr-sqrt 87.6%

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

   6. Applied egg-rr 79.2%

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

      1. associate-*l/ 82.0%

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

      2. cancel-sign-sub-inv 82.0%

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

      3. metadata-eval 82.0%

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

      4. *-commutative 82.0%

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

      5. div-inv 82.0%

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

      6. metadata-eval 82.0%

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

      7. sqrt-unprod 72.7%

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

      8. *-commutative 72.7%

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

   8. Applied egg-rr 72.7%

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

      1. associate-/l* 72.7%

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

   10. Simplified 81.1%

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

   if 3.6000000000000002e192 < l

   1. Initial program 62.9%

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

   3. Simplified 58.4%

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

   5. Taylor expanded in d around inf 58.4%

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

      1. sqrt-div 58.3%

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

      2. metadata-eval 58.3%

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

      3. *-commutative 58.3%

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

      4. sqrt-unprod 89.9%

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

      5. div-inv 89.9%

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

      6. associate-/r* 90.2%

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

   7. Applied egg-rr 90.2%

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

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right) \cdot \left(-1 + 0.5 \cdot \frac{h \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell}\right)\\ \mathbf{elif}\;\ell \leq 3.6 \cdot 10^{+192}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left(-0.5, h \cdot \frac{0.25 \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}}{\ell}, 1\right)}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 83.5% accurate, 1.0× speedup

Math

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

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (* 0.5 (/ (* h (pow (* M_m (* 0.5 (/ D_m d))) 2.0)) l))))
   (if (<= l -5e-310)
     (* (* d (/ (sqrt (/ -1.0 l)) (sqrt (- h)))) (+ -1.0 t_0))
     (* (- 1.0 t_0) (/ d (* (sqrt l) (sqrt h)))))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if l < -4.999999999999985e-310

   1. Initial program 66.7%

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

   3. Simplified 67.4%

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

      1. associate-*r/ 70.6%

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

      2. frac-times 70.6%

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

      3. associate-/l* 70.6%

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

      4. *-un-lft-identity 70.6%

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

      5. times-frac 70.6%

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

      6. metadata-eval 70.6%

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

   6. Applied egg-rr 70.6%

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

      1. clear-num 70.5%

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

      2. sqrt-div 70.5%

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

      3. metadata-eval 70.5%

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

   8. Applied egg-rr 70.5%

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

   9. Taylor expanded in d around -inf 67.1%

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

       1. mul-1-neg 67.1%

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

       2. unpow-1 67.1%

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

       3. metadata-eval 67.1%

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

       4. pow-sqr 67.1%

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

       5. rem-sqrt-square 67.6%

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

       6. metadata-eval 67.6%

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

       7. pow-sqr 67.4%

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

       8. fabs-sqr 67.4%

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

       9. pow-sqr 67.6%

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

       10. metadata-eval 67.6%

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

       11. distribute-rgt-neg-in 67.6%

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

   11. Simplified 67.6%

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

       1. unpow-prod-down 0.0%

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

       2. metadata-eval 0.0%

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

       3. sqrt-pow1 0.0%

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

       4. inv-pow 0.0%

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

       5. metadata-eval 0.0%

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

       6. sqrt-pow1 0.0%

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

       7. inv-pow 0.0%

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

       8. *-commutative 0.0%

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

       9. sqrt-prod 68.7%

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

       10. div-inv 68.7%

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

       11. frac-2neg 68.7%

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

       12. sqrt-div 84.0%

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

       13. distribute-neg-frac 84.0%

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

       14. metadata-eval 84.0%

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

   13. Applied egg-rr 84.0%

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

   if -4.999999999999985e-310 < l

   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 66.7%

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

      1. associate-*r/ 72.2%

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

      2. frac-times 73.0%

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

      3. associate-/l* 72.2%

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

      4. *-un-lft-identity 72.2%

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

      5. times-frac 72.2%

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

      6. metadata-eval 72.2%

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

   6. Applied egg-rr 72.2%

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

      1. *-commutative 72.2%

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

      2. sqrt-div 78.8%

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

      3. sqrt-div 87.8%

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

      4. frac-times 87.7%

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

      5. add-sqr-sqrt 88.0%

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

   8. Applied egg-rr 88.0%

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

4. Final simplification 86.0%

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

Alternative 7: 75.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := 1 - 0.5 \cdot \frac{h \cdot {\left(M\_m \cdot \left(0.5 \cdot \frac{D\_m}{d}\right)\right)}^{2}}{\ell}\\ \mathbf{if}\;\ell \leq 7.8 \cdot 10^{-292}:\\ \;\;\;\;t\_0 \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (- 1.0 (* 0.5 (/ (* h (pow (* M_m (* 0.5 (/ D_m d))) 2.0)) l)))))
   (if (<= l 7.8e-292)
     (* t_0 (* (sqrt (/ d h)) (sqrt (/ d l))))
     (* t_0 (/ d (* (sqrt l) (sqrt h)))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = 1.0 - (0.5 * ((h * pow((M_m * (0.5 * (D_m / d))), 2.0)) / l));
	double tmp;
	if (l <= 7.8e-292) {
		tmp = t_0 * (sqrt((d / h)) * sqrt((d / l)));
	} else {
		tmp = t_0 * (d / (sqrt(l) * sqrt(h)));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 - (0.5d0 * ((h * ((m_m * (0.5d0 * (d_m / d))) ** 2.0d0)) / l))
    if (l <= 7.8d-292) then
        tmp = t_0 * (sqrt((d / h)) * sqrt((d / l)))
    else
        tmp = t_0 * (d / (sqrt(l) * sqrt(h)))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = 1.0 - (0.5 * ((h * Math.pow((M_m * (0.5 * (D_m / d))), 2.0)) / l));
	double tmp;
	if (l <= 7.8e-292) {
		tmp = t_0 * (Math.sqrt((d / h)) * Math.sqrt((d / l)));
	} else {
		tmp = t_0 * (d / (Math.sqrt(l) * Math.sqrt(h)));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = 1.0 - (0.5 * ((h * math.pow((M_m * (0.5 * (D_m / d))), 2.0)) / l))
	tmp = 0
	if l <= 7.8e-292:
		tmp = t_0 * (math.sqrt((d / h)) * math.sqrt((d / l)))
	else:
		tmp = t_0 * (d / (math.sqrt(l) * math.sqrt(h)))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(1.0 - Float64(0.5 * Float64(Float64(h * (Float64(M_m * Float64(0.5 * Float64(D_m / d))) ^ 2.0)) / l)))
	tmp = 0.0
	if (l <= 7.8e-292)
		tmp = Float64(t_0 * Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l))));
	else
		tmp = Float64(t_0 * Float64(d / Float64(sqrt(l) * sqrt(h))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = 1.0 - (0.5 * ((h * ((M_m * (0.5 * (D_m / d))) ^ 2.0)) / l));
	tmp = 0.0;
	if (l <= 7.8e-292)
		tmp = t_0 * (sqrt((d / h)) * sqrt((d / l)));
	else
		tmp = t_0 * (d / (sqrt(l) * sqrt(h)));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(1.0 - N[(0.5 * N[(N[(h * N[Power[N[(M$95$m * N[(0.5 * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, 7.8e-292], N[(t$95$0 * N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < 7.8e-292

   1. Initial program 66.7%

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

   3. Simplified 67.4%

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

      1. associate-*r/ 71.9%

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

      2. frac-times 71.8%

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

      3. associate-/l* 71.9%

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

      4. *-un-lft-identity 71.9%

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

      5. times-frac 71.9%

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

      6. metadata-eval 71.9%

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

   6. Applied egg-rr 71.9%

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

   if 7.8e-292 < l

   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 66.7%

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

      1. associate-*r/ 70.8%

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

      2. frac-times 71.7%

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

      3. associate-/l* 70.8%

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

      4. *-un-lft-identity 70.8%

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

      5. times-frac 70.8%

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

      6. metadata-eval 70.8%

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

   6. Applied egg-rr 70.8%

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

      1. *-commutative 70.8%

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

      2. sqrt-div 77.7%

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

      3. sqrt-div 87.2%

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

      4. frac-times 87.1%

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

      5. add-sqr-sqrt 87.4%

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

   8. Applied egg-rr 87.4%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 7.8 \cdot 10^{-292}:\\ \;\;\;\;\left(1 - 0.5 \cdot \frac{h \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell}\right) \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - 0.5 \cdot \frac{h \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell}\right) \cdot \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 75.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;h \leq 3.7 \cdot 10^{-300}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{0.25 \cdot {\left(D\_m \cdot \frac{M\_m}{d}\right)}^{2}}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - 0.5 \cdot \frac{h \cdot {\left(M\_m \cdot \left(0.5 \cdot \frac{D\_m}{d}\right)\right)}^{2}}{\ell}\right) \cdot \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= h 3.7e-300)
   (*
    (* (sqrt (/ d h)) (sqrt (/ d l)))
    (- 1.0 (* 0.5 (* h (/ (* 0.25 (pow (* D_m (/ M_m d)) 2.0)) l)))))
   (*
    (- 1.0 (* 0.5 (/ (* h (pow (* M_m (* 0.5 (/ D_m d))) 2.0)) l)))
    (/ d (* (sqrt l) (sqrt h))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (h <= 3.7e-300) {
		tmp = (sqrt((d / h)) * sqrt((d / l))) * (1.0 - (0.5 * (h * ((0.25 * pow((D_m * (M_m / d)), 2.0)) / l))));
	} else {
		tmp = (1.0 - (0.5 * ((h * pow((M_m * (0.5 * (D_m / d))), 2.0)) / l))) * (d / (sqrt(l) * sqrt(h)));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (h <= 3.7d-300) then
        tmp = (sqrt((d / h)) * sqrt((d / l))) * (1.0d0 - (0.5d0 * (h * ((0.25d0 * ((d_m * (m_m / d)) ** 2.0d0)) / l))))
    else
        tmp = (1.0d0 - (0.5d0 * ((h * ((m_m * (0.5d0 * (d_m / d))) ** 2.0d0)) / l))) * (d / (sqrt(l) * sqrt(h)))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (h <= 3.7e-300) {
		tmp = (Math.sqrt((d / h)) * Math.sqrt((d / l))) * (1.0 - (0.5 * (h * ((0.25 * Math.pow((D_m * (M_m / d)), 2.0)) / l))));
	} else {
		tmp = (1.0 - (0.5 * ((h * Math.pow((M_m * (0.5 * (D_m / d))), 2.0)) / l))) * (d / (Math.sqrt(l) * Math.sqrt(h)));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if h <= 3.7e-300:
		tmp = (math.sqrt((d / h)) * math.sqrt((d / l))) * (1.0 - (0.5 * (h * ((0.25 * math.pow((D_m * (M_m / d)), 2.0)) / l))))
	else:
		tmp = (1.0 - (0.5 * ((h * math.pow((M_m * (0.5 * (D_m / d))), 2.0)) / l))) * (d / (math.sqrt(l) * math.sqrt(h)))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (h <= 3.7e-300)
		tmp = Float64(Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l))) * Float64(1.0 - Float64(0.5 * Float64(h * Float64(Float64(0.25 * (Float64(D_m * Float64(M_m / d)) ^ 2.0)) / l)))));
	else
		tmp = Float64(Float64(1.0 - Float64(0.5 * Float64(Float64(h * (Float64(M_m * Float64(0.5 * Float64(D_m / d))) ^ 2.0)) / l))) * Float64(d / Float64(sqrt(l) * sqrt(h))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (h <= 3.7e-300)
		tmp = (sqrt((d / h)) * sqrt((d / l))) * (1.0 - (0.5 * (h * ((0.25 * ((D_m * (M_m / d)) ^ 2.0)) / l))));
	else
		tmp = (1.0 - (0.5 * ((h * ((M_m * (0.5 * (D_m / d))) ^ 2.0)) / l))) * (d / (sqrt(l) * sqrt(h)));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[h, 3.7e-300], N[(N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(h * N[(N[(0.25 * N[Power[N[(D$95$m * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[(0.5 * N[(N[(h * N[Power[N[(M$95$m * N[(0.5 * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if h < 3.7000000000000001e-300

   1. Initial program 67.0%

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

   3. Simplified 67.6%

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

      1. associate-*r/ 70.8%

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

      2. frac-times 70.8%

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

      3. associate-/l* 70.8%

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

      4. *-un-lft-identity 70.8%

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

      5. times-frac 70.8%

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

      6. metadata-eval 70.8%

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

   6. Applied egg-rr 70.8%

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

      1. *-un-lft-identity 70.8%

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

      2. associate-/l* 67.6%

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

      3. associate-*r* 67.6%

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

      4. metadata-eval 67.6%

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

      5. div-inv 67.6%

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

      6. *-commutative 67.6%

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

      7. div-inv 67.6%

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

      8. metadata-eval 67.6%

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

   8. Applied egg-rr 67.6%

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

      1. *-lft-identity 67.6%

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

      2. *-commutative 67.6%

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

      3. associate-*l/ 70.8%

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

      4. associate-/l* 69.4%

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

      5. unpow2 69.4%

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

      6. associate-*l/ 68.7%

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

      7. associate-*r* 68.7%

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

      8. associate-*l/ 68.7%

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

      9. *-commutative 68.7%

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

      10. associate-*l/ 69.4%

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

      11. associate-*r* 69.4%

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

      12. associate-*l/ 69.4%

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

      13. *-commutative 69.4%

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

      14. swap-sqr 69.4%

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

      15. metadata-eval 69.4%

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

      16. unpow2 69.4%

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

      17. associate-/l* 69.4%

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

   10. Simplified 69.4%

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

   if 3.7000000000000001e-300 < h

   1. Initial program 67.2%

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

   3. Simplified 66.5%

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

      1. associate-*r/ 72.0%

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

      2. frac-times 72.8%

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

      3. associate-/l* 72.0%

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

      4. *-un-lft-identity 72.0%

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

      5. times-frac 72.0%

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

      6. metadata-eval 72.0%

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

   6. Applied egg-rr 72.0%

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

      1. *-commutative 72.0%

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

      2. sqrt-div 78.6%

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

      3. sqrt-div 87.7%

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

      4. frac-times 87.6%

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

      5. add-sqr-sqrt 87.9%

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

   8. Applied egg-rr 87.9%

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

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq 3.7 \cdot 10^{-300}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{0.25 \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - 0.5 \cdot \frac{h \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell}\right) \cdot \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 77.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := 0.5 \cdot \frac{h \cdot {\left(M\_m \cdot \left(0.5 \cdot \frac{D\_m}{d}\right)\right)}^{2}}{\ell}\\ \mathbf{if}\;h \leq 3.7 \cdot 10^{-300}:\\ \;\;\;\;\left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right) \cdot \left(-1 + t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - t\_0\right) \cdot \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (* 0.5 (/ (* h (pow (* M_m (* 0.5 (/ D_m d))) 2.0)) l))))
   (if (<= h 3.7e-300)
     (* (* d (pow (* l h) -0.5)) (+ -1.0 t_0))
     (* (- 1.0 t_0) (/ d (* (sqrt l) (sqrt h)))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = 0.5 * ((h * pow((M_m * (0.5 * (D_m / d))), 2.0)) / l);
	double tmp;
	if (h <= 3.7e-300) {
		tmp = (d * pow((l * h), -0.5)) * (-1.0 + t_0);
	} else {
		tmp = (1.0 - t_0) * (d / (sqrt(l) * sqrt(h)));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * ((h * ((m_m * (0.5d0 * (d_m / d))) ** 2.0d0)) / l)
    if (h <= 3.7d-300) then
        tmp = (d * ((l * h) ** (-0.5d0))) * ((-1.0d0) + t_0)
    else
        tmp = (1.0d0 - t_0) * (d / (sqrt(l) * sqrt(h)))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = 0.5 * ((h * Math.pow((M_m * (0.5 * (D_m / d))), 2.0)) / l);
	double tmp;
	if (h <= 3.7e-300) {
		tmp = (d * Math.pow((l * h), -0.5)) * (-1.0 + t_0);
	} else {
		tmp = (1.0 - t_0) * (d / (Math.sqrt(l) * Math.sqrt(h)));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = 0.5 * ((h * math.pow((M_m * (0.5 * (D_m / d))), 2.0)) / l)
	tmp = 0
	if h <= 3.7e-300:
		tmp = (d * math.pow((l * h), -0.5)) * (-1.0 + t_0)
	else:
		tmp = (1.0 - t_0) * (d / (math.sqrt(l) * math.sqrt(h)))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(0.5 * Float64(Float64(h * (Float64(M_m * Float64(0.5 * Float64(D_m / d))) ^ 2.0)) / l))
	tmp = 0.0
	if (h <= 3.7e-300)
		tmp = Float64(Float64(d * (Float64(l * h) ^ -0.5)) * Float64(-1.0 + t_0));
	else
		tmp = Float64(Float64(1.0 - t_0) * Float64(d / Float64(sqrt(l) * sqrt(h))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = 0.5 * ((h * ((M_m * (0.5 * (D_m / d))) ^ 2.0)) / l);
	tmp = 0.0;
	if (h <= 3.7e-300)
		tmp = (d * ((l * h) ^ -0.5)) * (-1.0 + t_0);
	else
		tmp = (1.0 - t_0) * (d / (sqrt(l) * sqrt(h)));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(0.5 * N[(N[(h * N[Power[N[(M$95$m * N[(0.5 * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[h, 3.7e-300], N[(N[(d * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] * N[(-1.0 + t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - t$95$0), $MachinePrecision] * N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if h < 3.7000000000000001e-300

   1. Initial program 67.0%

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

   3. Simplified 67.6%

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

      1. associate-*r/ 70.8%

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

      2. frac-times 70.8%

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

      3. associate-/l* 70.8%

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

      4. *-un-lft-identity 70.8%

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

      5. times-frac 70.8%

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

      6. metadata-eval 70.8%

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

   6. Applied egg-rr 70.8%

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

      1. clear-num 70.8%

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

      2. sqrt-div 70.7%

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

      3. metadata-eval 70.7%

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

   8. Applied egg-rr 70.7%

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

   9. Taylor expanded in d around -inf 66.6%

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

       1. mul-1-neg 66.6%

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

       2. unpow-1 66.6%

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

       3. metadata-eval 66.6%

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

       4. pow-sqr 66.6%

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

       5. rem-sqrt-square 67.1%

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

       6. metadata-eval 67.1%

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

       7. pow-sqr 66.9%

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

       8. fabs-sqr 66.9%

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

       9. pow-sqr 67.1%

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

       10. metadata-eval 67.1%

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

       11. distribute-rgt-neg-in 67.1%

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

   11. Simplified 67.1%

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

   if 3.7000000000000001e-300 < h

   1. Initial program 67.2%

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

   3. Simplified 66.5%

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

      1. associate-*r/ 72.0%

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

      2. frac-times 72.8%

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

      3. associate-/l* 72.0%

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

      4. *-un-lft-identity 72.0%

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

      5. times-frac 72.0%

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

      6. metadata-eval 72.0%

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

   6. Applied egg-rr 72.0%

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

      1. *-commutative 72.0%

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

      2. sqrt-div 78.6%

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

      3. sqrt-div 87.7%

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

      4. frac-times 87.6%

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

      5. add-sqr-sqrt 87.9%

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

   8. Applied egg-rr 87.9%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq 3.7 \cdot 10^{-300}:\\ \;\;\;\;\left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right) \cdot \left(-1 + 0.5 \cdot \frac{h \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - 0.5 \cdot \frac{h \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell}\right) \cdot \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 67.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := 1 - 0.5 \cdot \frac{h \cdot {\left(M\_m \cdot \left(0.5 \cdot \frac{D\_m}{d}\right)\right)}^{2}}{\ell}\\ \mathbf{if}\;\ell \leq -2.4 \cdot 10^{+103}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{h}}{\ell}}\right)\\ \mathbf{elif}\;\ell \leq 7.8 \cdot 10^{-292}:\\ \;\;\;\;t\_0 \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{elif}\;\ell \leq 2.4 \cdot 10^{+192}:\\ \;\;\;\;t\_0 \cdot \left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (- 1.0 (* 0.5 (/ (* h (pow (* M_m (* 0.5 (/ D_m d))) 2.0)) l)))))
   (if (<= l -2.4e+103)
     (* d (- (sqrt (/ (/ 1.0 h) l))))
     (if (<= l 7.8e-292)
       (* t_0 (sqrt (* (/ d h) (/ d l))))
       (if (<= l 2.4e+192)
         (* t_0 (* d (pow (* l h) -0.5)))
         (/ (/ d (sqrt l)) (sqrt h)))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = 1.0 - (0.5 * ((h * pow((M_m * (0.5 * (D_m / d))), 2.0)) / l));
	double tmp;
	if (l <= -2.4e+103) {
		tmp = d * -sqrt(((1.0 / h) / l));
	} else if (l <= 7.8e-292) {
		tmp = t_0 * sqrt(((d / h) * (d / l)));
	} else if (l <= 2.4e+192) {
		tmp = t_0 * (d * pow((l * h), -0.5));
	} else {
		tmp = (d / sqrt(l)) / sqrt(h);
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 - (0.5d0 * ((h * ((m_m * (0.5d0 * (d_m / d))) ** 2.0d0)) / l))
    if (l <= (-2.4d+103)) then
        tmp = d * -sqrt(((1.0d0 / h) / l))
    else if (l <= 7.8d-292) then
        tmp = t_0 * sqrt(((d / h) * (d / l)))
    else if (l <= 2.4d+192) then
        tmp = t_0 * (d * ((l * h) ** (-0.5d0)))
    else
        tmp = (d / sqrt(l)) / sqrt(h)
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = 1.0 - (0.5 * ((h * Math.pow((M_m * (0.5 * (D_m / d))), 2.0)) / l));
	double tmp;
	if (l <= -2.4e+103) {
		tmp = d * -Math.sqrt(((1.0 / h) / l));
	} else if (l <= 7.8e-292) {
		tmp = t_0 * Math.sqrt(((d / h) * (d / l)));
	} else if (l <= 2.4e+192) {
		tmp = t_0 * (d * Math.pow((l * h), -0.5));
	} else {
		tmp = (d / Math.sqrt(l)) / Math.sqrt(h);
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = 1.0 - (0.5 * ((h * math.pow((M_m * (0.5 * (D_m / d))), 2.0)) / l))
	tmp = 0
	if l <= -2.4e+103:
		tmp = d * -math.sqrt(((1.0 / h) / l))
	elif l <= 7.8e-292:
		tmp = t_0 * math.sqrt(((d / h) * (d / l)))
	elif l <= 2.4e+192:
		tmp = t_0 * (d * math.pow((l * h), -0.5))
	else:
		tmp = (d / math.sqrt(l)) / math.sqrt(h)
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(1.0 - Float64(0.5 * Float64(Float64(h * (Float64(M_m * Float64(0.5 * Float64(D_m / d))) ^ 2.0)) / l)))
	tmp = 0.0
	if (l <= -2.4e+103)
		tmp = Float64(d * Float64(-sqrt(Float64(Float64(1.0 / h) / l))));
	elseif (l <= 7.8e-292)
		tmp = Float64(t_0 * sqrt(Float64(Float64(d / h) * Float64(d / l))));
	elseif (l <= 2.4e+192)
		tmp = Float64(t_0 * Float64(d * (Float64(l * h) ^ -0.5)));
	else
		tmp = Float64(Float64(d / sqrt(l)) / sqrt(h));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = 1.0 - (0.5 * ((h * ((M_m * (0.5 * (D_m / d))) ^ 2.0)) / l));
	tmp = 0.0;
	if (l <= -2.4e+103)
		tmp = d * -sqrt(((1.0 / h) / l));
	elseif (l <= 7.8e-292)
		tmp = t_0 * sqrt(((d / h) * (d / l)));
	elseif (l <= 2.4e+192)
		tmp = t_0 * (d * ((l * h) ^ -0.5));
	else
		tmp = (d / sqrt(l)) / sqrt(h);
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(1.0 - N[(0.5 * N[(N[(h * N[Power[N[(M$95$m * N[(0.5 * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -2.4e+103], N[(d * (-N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], If[LessEqual[l, 7.8e-292], N[(t$95$0 * N[Sqrt[N[(N[(d / h), $MachinePrecision] * N[(d / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 2.4e+192], N[(t$95$0 * N[(d * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(d / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -2.3999999999999998e103

   1. Initial program 42.1%

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

   3. Simplified 44.2%

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

   5. Taylor expanded in d around inf 2.9%

      \[\leadsto \color{blue}{d \cdot \left(\sqrt{\frac{1}{h \cdot \ell}} + -0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative 2.9%

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

      2. fma-define 2.9%

         \[\leadsto d \cdot \color{blue}{\mathsf{fma}\left(-0.125, \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}} \cdot \sqrt{\frac{h}{{\ell}^{3}}}, \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      3. *-commutative 2.9%

         \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}}}, \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      4. *-commutative 2.9%

         \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot \frac{\color{blue}{{M}^{2} \cdot {D}^{2}}}{{d}^{2}}, \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      5. associate-/l* 2.9%

         \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot \color{blue}{\left({M}^{2} \cdot \frac{{D}^{2}}{{d}^{2}}\right)}, \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      6. unpow2 2.9%

         \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\color{blue}{\left(M \cdot M\right)} \cdot \frac{{D}^{2}}{{d}^{2}}\right), \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      7. unpow2 2.9%

         \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\left(M \cdot M\right) \cdot \frac{\color{blue}{D \cdot D}}{{d}^{2}}\right), \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      8. unpow2 2.9%

         \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\left(M \cdot M\right) \cdot \frac{D \cdot D}{\color{blue}{d \cdot d}}\right), \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      9. times-frac 3.3%

         \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\left(M \cdot M\right) \cdot \color{blue}{\left(\frac{D}{d} \cdot \frac{D}{d}\right)}\right), \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      10. swap-sqr 3.3%

          \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot \color{blue}{\left(\left(M \cdot \frac{D}{d}\right) \cdot \left(M \cdot \frac{D}{d}\right)\right)}, \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      11. unpow2 3.3%

          \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot \color{blue}{{\left(M \cdot \frac{D}{d}\right)}^{2}}, \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      12. associate-*r/ 3.3%

          \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot {\color{blue}{\left(\frac{M \cdot D}{d}\right)}}^{2}, \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      13. *-commutative 3.3%

          \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot {\left(\frac{\color{blue}{D \cdot M}}{d}\right)}^{2}, \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      14. associate-/l* 3.3%

          \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot {\color{blue}{\left(D \cdot \frac{M}{d}\right)}}^{2}, \sqrt{\frac{1}{h \cdot \ell}}\right) \]

   7. Simplified 3.3%

      \[\leadsto \color{blue}{d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}, \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

   8. Taylor expanded in h around 0 4.8%

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

      1. associate-/r* 4.7%

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

      2. unpow1/2 4.7%

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

      3. associate-/r* 4.8%

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

      4. rem-exp-log 4.8%

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

      5. exp-neg 4.8%

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

      6. exp-prod 4.8%

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

      7. distribute-lft-neg-out 4.8%

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

      8. distribute-rgt-neg-in 4.8%

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

      9. metadata-eval 4.8%

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

      10. exp-to-pow 4.8%

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

   10. Simplified 4.8%

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

       1. unpow-prod-down 0.0%

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

       2. metadata-eval 0.0%

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

       3. sqrt-pow1 0.0%

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

       4. inv-pow 0.0%

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

       5. metadata-eval 0.0%

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

       6. sqrt-pow1 0.0%

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

       7. inv-pow 0.0%

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

       8. sqrt-prod 4.7%

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

       9. div-inv 4.7%

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

       10. sqrt-div 0.0%

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

       11. inv-pow 0.0%

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

       12. sqrt-pow1 0.0%

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

       13. metadata-eval 0.0%

           \[\leadsto d \cdot \frac{{h}^{\color{blue}{-0.5}}}{\sqrt{\ell}} \]

   12. Applied egg-rr 0.0%

       \[\leadsto d \cdot \color{blue}{\frac{{h}^{-0.5}}{\sqrt{\ell}}} \]

   13. Taylor expanded in h around -inf 0.0%

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

       1. *-commutative 0.0%

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

       2. unpow2 0.0%

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

       3. rem-square-sqrt 51.6%

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

       4. mul-1-neg 51.6%

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

       5. associate-/r* 53.4%

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

   15. Simplified 53.4%

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

   if -2.3999999999999998e103 < l < 7.8e-292

   1. Initial program 80.6%

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

   3. Simplified 80.5%

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

      1. associate-*r/ 85.0%

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

      2. frac-times 86.2%

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

      3. associate-/l* 85.0%

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

      4. *-un-lft-identity 85.0%

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

      5. times-frac 85.0%

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

      6. metadata-eval 85.0%

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

   6. Applied egg-rr 85.0%

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

      1. *-commutative 85.0%

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

      2. sqrt-unprod 74.2%

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

   8. Applied egg-rr 74.2%

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

   if 7.8e-292 < l < 2.3999999999999998e192

   1. Initial program 68.5%

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

   3. Simplified 68.4%

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

      1. associate-*r/ 73.5%

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

      2. frac-times 73.5%

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

      3. associate-/l* 73.5%

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

      4. *-un-lft-identity 73.5%

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

      5. times-frac 73.5%

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

      6. metadata-eval 73.5%

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

   6. Applied egg-rr 73.5%

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

      1. clear-num 73.6%

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

      2. sqrt-div 74.3%

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

      3. metadata-eval 74.3%

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

   8. Applied egg-rr 74.3%

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

   9. Taylor expanded in d around 0 78.4%

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

       1. unpow-1 78.4%

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

       2. metadata-eval 78.4%

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

       3. pow-sqr 78.4%

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

       4. rem-sqrt-square 79.0%

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

       5. metadata-eval 79.0%

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

       6. pow-sqr 78.9%

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

       7. fabs-sqr 78.9%

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

       8. pow-sqr 79.0%

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

       9. metadata-eval 79.0%

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

   11. Simplified 79.0%

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

   if 2.3999999999999998e192 < l

   1. Initial program 62.9%

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

   3. Simplified 58.4%

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

   5. Taylor expanded in d around inf 58.4%

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

      1. sqrt-div 58.3%

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

      2. metadata-eval 58.3%

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

      3. *-commutative 58.3%

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

      4. sqrt-unprod 89.9%

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

      5. div-inv 89.9%

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

      6. associate-/r* 90.2%

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

   7. Applied egg-rr 90.2%

      \[\leadsto \color{blue}{\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.4 \cdot 10^{+103}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{h}}{\ell}}\right)\\ \mathbf{elif}\;\ell \leq 7.8 \cdot 10^{-292}:\\ \;\;\;\;\left(1 - 0.5 \cdot \frac{h \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell}\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{elif}\;\ell \leq 2.4 \cdot 10^{+192}:\\ \;\;\;\;\left(1 - 0.5 \cdot \frac{h \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell}\right) \cdot \left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 66.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -7.2 \cdot 10^{+95}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{h}}{\ell}}\right)\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D\_m}{d} \cdot \frac{M\_m}{2}\right)}^{2}\right)\right)\\ \mathbf{elif}\;\ell \leq 2.4 \cdot 10^{+192}:\\ \;\;\;\;\left(1 - 0.5 \cdot \frac{h \cdot {\left(M\_m \cdot \left(0.5 \cdot \frac{D\_m}{d}\right)\right)}^{2}}{\ell}\right) \cdot \left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= l -7.2e+95)
   (* d (- (sqrt (/ (/ 1.0 h) l))))
   (if (<= l -5e-310)
     (*
      (sqrt (* (/ d h) (/ d l)))
      (- 1.0 (* 0.5 (* (/ h l) (pow (* (/ D_m d) (/ M_m 2.0)) 2.0)))))
     (if (<= l 2.4e+192)
       (*
        (- 1.0 (* 0.5 (/ (* h (pow (* M_m (* 0.5 (/ D_m d))) 2.0)) l)))
        (* d (pow (* l h) -0.5)))
       (/ (/ d (sqrt l)) (sqrt h))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= -7.2e+95) {
		tmp = d * -sqrt(((1.0 / h) / l));
	} else if (l <= -5e-310) {
		tmp = sqrt(((d / h) * (d / l))) * (1.0 - (0.5 * ((h / l) * pow(((D_m / d) * (M_m / 2.0)), 2.0))));
	} else if (l <= 2.4e+192) {
		tmp = (1.0 - (0.5 * ((h * pow((M_m * (0.5 * (D_m / d))), 2.0)) / l))) * (d * pow((l * h), -0.5));
	} else {
		tmp = (d / sqrt(l)) / sqrt(h);
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (l <= (-7.2d+95)) then
        tmp = d * -sqrt(((1.0d0 / h) / l))
    else if (l <= (-5d-310)) then
        tmp = sqrt(((d / h) * (d / l))) * (1.0d0 - (0.5d0 * ((h / l) * (((d_m / d) * (m_m / 2.0d0)) ** 2.0d0))))
    else if (l <= 2.4d+192) then
        tmp = (1.0d0 - (0.5d0 * ((h * ((m_m * (0.5d0 * (d_m / d))) ** 2.0d0)) / l))) * (d * ((l * h) ** (-0.5d0)))
    else
        tmp = (d / sqrt(l)) / sqrt(h)
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= -7.2e+95) {
		tmp = d * -Math.sqrt(((1.0 / h) / l));
	} else if (l <= -5e-310) {
		tmp = Math.sqrt(((d / h) * (d / l))) * (1.0 - (0.5 * ((h / l) * Math.pow(((D_m / d) * (M_m / 2.0)), 2.0))));
	} else if (l <= 2.4e+192) {
		tmp = (1.0 - (0.5 * ((h * Math.pow((M_m * (0.5 * (D_m / d))), 2.0)) / l))) * (d * Math.pow((l * h), -0.5));
	} else {
		tmp = (d / Math.sqrt(l)) / Math.sqrt(h);
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if l <= -7.2e+95:
		tmp = d * -math.sqrt(((1.0 / h) / l))
	elif l <= -5e-310:
		tmp = math.sqrt(((d / h) * (d / l))) * (1.0 - (0.5 * ((h / l) * math.pow(((D_m / d) * (M_m / 2.0)), 2.0))))
	elif l <= 2.4e+192:
		tmp = (1.0 - (0.5 * ((h * math.pow((M_m * (0.5 * (D_m / d))), 2.0)) / l))) * (d * math.pow((l * h), -0.5))
	else:
		tmp = (d / math.sqrt(l)) / math.sqrt(h)
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (l <= -7.2e+95)
		tmp = Float64(d * Float64(-sqrt(Float64(Float64(1.0 / h) / l))));
	elseif (l <= -5e-310)
		tmp = Float64(sqrt(Float64(Float64(d / h) * Float64(d / l))) * Float64(1.0 - Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(D_m / d) * Float64(M_m / 2.0)) ^ 2.0)))));
	elseif (l <= 2.4e+192)
		tmp = Float64(Float64(1.0 - Float64(0.5 * Float64(Float64(h * (Float64(M_m * Float64(0.5 * Float64(D_m / d))) ^ 2.0)) / l))) * Float64(d * (Float64(l * h) ^ -0.5)));
	else
		tmp = Float64(Float64(d / sqrt(l)) / sqrt(h));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (l <= -7.2e+95)
		tmp = d * -sqrt(((1.0 / h) / l));
	elseif (l <= -5e-310)
		tmp = sqrt(((d / h) * (d / l))) * (1.0 - (0.5 * ((h / l) * (((D_m / d) * (M_m / 2.0)) ^ 2.0))));
	elseif (l <= 2.4e+192)
		tmp = (1.0 - (0.5 * ((h * ((M_m * (0.5 * (D_m / d))) ^ 2.0)) / l))) * (d * ((l * h) ^ -0.5));
	else
		tmp = (d / sqrt(l)) / sqrt(h);
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[l, -7.2e+95], N[(d * (-N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], If[LessEqual[l, -5e-310], N[(N[Sqrt[N[(N[(d / h), $MachinePrecision] * N[(d / l), $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 / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 2.4e+192], N[(N[(1.0 - N[(0.5 * N[(N[(h * N[Power[N[(M$95$m * N[(0.5 * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(d * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(d / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if l < -7.19999999999999955e95

   1. Initial program 42.1%

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

   3. Simplified 44.2%

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

   5. Taylor expanded in d around inf 2.9%

      \[\leadsto \color{blue}{d \cdot \left(\sqrt{\frac{1}{h \cdot \ell}} + -0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative 2.9%

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

      2. fma-define 2.9%

         \[\leadsto d \cdot \color{blue}{\mathsf{fma}\left(-0.125, \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}} \cdot \sqrt{\frac{h}{{\ell}^{3}}}, \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      3. *-commutative 2.9%

         \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}}}, \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      4. *-commutative 2.9%

         \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot \frac{\color{blue}{{M}^{2} \cdot {D}^{2}}}{{d}^{2}}, \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      5. associate-/l* 2.9%

         \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot \color{blue}{\left({M}^{2} \cdot \frac{{D}^{2}}{{d}^{2}}\right)}, \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      6. unpow2 2.9%

         \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\color{blue}{\left(M \cdot M\right)} \cdot \frac{{D}^{2}}{{d}^{2}}\right), \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      7. unpow2 2.9%

         \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\left(M \cdot M\right) \cdot \frac{\color{blue}{D \cdot D}}{{d}^{2}}\right), \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      8. unpow2 2.9%

         \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\left(M \cdot M\right) \cdot \frac{D \cdot D}{\color{blue}{d \cdot d}}\right), \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      9. times-frac 3.3%

         \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\left(M \cdot M\right) \cdot \color{blue}{\left(\frac{D}{d} \cdot \frac{D}{d}\right)}\right), \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      10. swap-sqr 3.3%

          \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot \color{blue}{\left(\left(M \cdot \frac{D}{d}\right) \cdot \left(M \cdot \frac{D}{d}\right)\right)}, \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      11. unpow2 3.3%

          \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot \color{blue}{{\left(M \cdot \frac{D}{d}\right)}^{2}}, \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      12. associate-*r/ 3.3%

          \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot {\color{blue}{\left(\frac{M \cdot D}{d}\right)}}^{2}, \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      13. *-commutative 3.3%

          \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot {\left(\frac{\color{blue}{D \cdot M}}{d}\right)}^{2}, \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      14. associate-/l* 3.3%

          \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot {\color{blue}{\left(D \cdot \frac{M}{d}\right)}}^{2}, \sqrt{\frac{1}{h \cdot \ell}}\right) \]

   7. Simplified 3.3%

      \[\leadsto \color{blue}{d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}, \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

   8. Taylor expanded in h around 0 4.8%

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

      1. associate-/r* 4.7%

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

      2. unpow1/2 4.7%

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

      3. associate-/r* 4.8%

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

      4. rem-exp-log 4.8%

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

      5. exp-neg 4.8%

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

      6. exp-prod 4.8%

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

      7. distribute-lft-neg-out 4.8%

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

      8. distribute-rgt-neg-in 4.8%

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

      9. metadata-eval 4.8%

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

      10. exp-to-pow 4.8%

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

   10. Simplified 4.8%

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

       1. unpow-prod-down 0.0%

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

       2. metadata-eval 0.0%

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

       3. sqrt-pow1 0.0%

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

       4. inv-pow 0.0%

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

       5. metadata-eval 0.0%

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

       6. sqrt-pow1 0.0%

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

       7. inv-pow 0.0%

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

       8. sqrt-prod 4.7%

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

       9. div-inv 4.7%

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

       10. sqrt-div 0.0%

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

       11. inv-pow 0.0%

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

       12. sqrt-pow1 0.0%

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

       13. metadata-eval 0.0%

           \[\leadsto d \cdot \frac{{h}^{\color{blue}{-0.5}}}{\sqrt{\ell}} \]

   12. Applied egg-rr 0.0%

       \[\leadsto d \cdot \color{blue}{\frac{{h}^{-0.5}}{\sqrt{\ell}}} \]

   13. Taylor expanded in h around -inf 0.0%

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

       1. *-commutative 0.0%

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

       2. unpow2 0.0%

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

       3. rem-square-sqrt 51.6%

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

       4. mul-1-neg 51.6%

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

       5. associate-/r* 53.4%

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

   15. Simplified 53.4%

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

   if -7.19999999999999955e95 < l < -4.999999999999985e-310

   1. Initial program 81.6%

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

   3. Simplified 81.4%

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

      1. *-commutative 84.0%

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

      2. sqrt-unprod 72.3%

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

   6. Applied egg-rr 69.8%

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

   if -4.999999999999985e-310 < l < 2.3999999999999998e192

   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 68.4%

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

      1. associate-*r/ 75.0%

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

      2. frac-times 75.0%

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

      3. associate-/l* 75.0%

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

      4. *-un-lft-identity 75.0%

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

      5. times-frac 75.0%

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

      6. metadata-eval 75.0%

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

   6. Applied egg-rr 75.0%

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

      1. clear-num 75.0%

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

      2. sqrt-div 75.8%

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

      3. metadata-eval 75.8%

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

   8. Applied egg-rr 75.8%

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

   9. Taylor expanded in d around 0 79.6%

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

       1. unpow-1 79.6%

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

       2. metadata-eval 79.6%

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

       3. pow-sqr 79.6%

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

       4. rem-sqrt-square 80.2%

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

       5. metadata-eval 80.2%

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

       6. pow-sqr 80.1%

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

       7. fabs-sqr 80.1%

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

       8. pow-sqr 80.2%

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

       9. metadata-eval 80.2%

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

   11. Simplified 80.2%

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

   if 2.3999999999999998e192 < l

   1. Initial program 62.9%

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

   3. Simplified 58.4%

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

   5. Taylor expanded in d around inf 58.4%

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

      1. sqrt-div 58.3%

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

      2. metadata-eval 58.3%

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

      3. *-commutative 58.3%

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

      4. sqrt-unprod 89.9%

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

      5. div-inv 89.9%

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

      6. associate-/r* 90.2%

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

   7. Applied egg-rr 90.2%

      \[\leadsto \color{blue}{\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 72.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -7.2 \cdot 10^{+95}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{h}}{\ell}}\right)\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right)\\ \mathbf{elif}\;\ell \leq 2.4 \cdot 10^{+192}:\\ \;\;\;\;\left(1 - 0.5 \cdot \frac{h \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell}\right) \cdot \left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 62.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.55 \cdot 10^{-92}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{h}}{\ell}}\right)\\ \mathbf{elif}\;\ell \leq -4.8 \cdot 10^{-307}:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{0}}\\ \mathbf{elif}\;\ell \leq 1.06 \cdot 10^{+195}:\\ \;\;\;\;\left(1 - 0.5 \cdot \frac{h \cdot {\left(M\_m \cdot \left(0.5 \cdot \frac{D\_m}{d}\right)\right)}^{2}}{\ell}\right) \cdot \left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= l -1.55e-92)
   (* d (- (sqrt (/ (/ 1.0 h) l))))
   (if (<= l -4.8e-307)
     (* d (sqrt (/ 1.0 0.0)))
     (if (<= l 1.06e+195)
       (*
        (- 1.0 (* 0.5 (/ (* h (pow (* M_m (* 0.5 (/ D_m d))) 2.0)) l)))
        (* d (pow (* l h) -0.5)))
       (/ (/ d (sqrt l)) (sqrt h))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= -1.55e-92) {
		tmp = d * -sqrt(((1.0 / h) / l));
	} else if (l <= -4.8e-307) {
		tmp = d * sqrt((1.0 / 0.0));
	} else if (l <= 1.06e+195) {
		tmp = (1.0 - (0.5 * ((h * pow((M_m * (0.5 * (D_m / d))), 2.0)) / l))) * (d * pow((l * h), -0.5));
	} else {
		tmp = (d / sqrt(l)) / sqrt(h);
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (l <= (-1.55d-92)) then
        tmp = d * -sqrt(((1.0d0 / h) / l))
    else if (l <= (-4.8d-307)) then
        tmp = d * sqrt((1.0d0 / 0.0d0))
    else if (l <= 1.06d+195) then
        tmp = (1.0d0 - (0.5d0 * ((h * ((m_m * (0.5d0 * (d_m / d))) ** 2.0d0)) / l))) * (d * ((l * h) ** (-0.5d0)))
    else
        tmp = (d / sqrt(l)) / sqrt(h)
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= -1.55e-92) {
		tmp = d * -Math.sqrt(((1.0 / h) / l));
	} else if (l <= -4.8e-307) {
		tmp = d * Math.sqrt((1.0 / 0.0));
	} else if (l <= 1.06e+195) {
		tmp = (1.0 - (0.5 * ((h * Math.pow((M_m * (0.5 * (D_m / d))), 2.0)) / l))) * (d * Math.pow((l * h), -0.5));
	} else {
		tmp = (d / Math.sqrt(l)) / Math.sqrt(h);
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if l <= -1.55e-92:
		tmp = d * -math.sqrt(((1.0 / h) / l))
	elif l <= -4.8e-307:
		tmp = d * math.sqrt((1.0 / 0.0))
	elif l <= 1.06e+195:
		tmp = (1.0 - (0.5 * ((h * math.pow((M_m * (0.5 * (D_m / d))), 2.0)) / l))) * (d * math.pow((l * h), -0.5))
	else:
		tmp = (d / math.sqrt(l)) / math.sqrt(h)
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (l <= -1.55e-92)
		tmp = Float64(d * Float64(-sqrt(Float64(Float64(1.0 / h) / l))));
	elseif (l <= -4.8e-307)
		tmp = Float64(d * sqrt(Float64(1.0 / 0.0)));
	elseif (l <= 1.06e+195)
		tmp = Float64(Float64(1.0 - Float64(0.5 * Float64(Float64(h * (Float64(M_m * Float64(0.5 * Float64(D_m / d))) ^ 2.0)) / l))) * Float64(d * (Float64(l * h) ^ -0.5)));
	else
		tmp = Float64(Float64(d / sqrt(l)) / sqrt(h));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (l <= -1.55e-92)
		tmp = d * -sqrt(((1.0 / h) / l));
	elseif (l <= -4.8e-307)
		tmp = d * sqrt((1.0 / 0.0));
	elseif (l <= 1.06e+195)
		tmp = (1.0 - (0.5 * ((h * ((M_m * (0.5 * (D_m / d))) ^ 2.0)) / l))) * (d * ((l * h) ^ -0.5));
	else
		tmp = (d / sqrt(l)) / sqrt(h);
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[l, -1.55e-92], N[(d * (-N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], If[LessEqual[l, -4.8e-307], N[(d * N[Sqrt[N[(1.0 / 0.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.06e+195], N[(N[(1.0 - N[(0.5 * N[(N[(h * N[Power[N[(M$95$m * N[(0.5 * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(d * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(d / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if l < -1.55e-92

   1. Initial program 60.2%

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

   3. Simplified 61.3%

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

   5. Taylor expanded in d around inf 1.9%

      \[\leadsto \color{blue}{d \cdot \left(\sqrt{\frac{1}{h \cdot \ell}} + -0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative 1.9%

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

      2. fma-define 1.9%

         \[\leadsto d \cdot \color{blue}{\mathsf{fma}\left(-0.125, \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}} \cdot \sqrt{\frac{h}{{\ell}^{3}}}, \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      3. *-commutative 1.9%

         \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}}}, \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      4. *-commutative 1.9%

         \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot \frac{\color{blue}{{M}^{2} \cdot {D}^{2}}}{{d}^{2}}, \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      5. associate-/l* 1.9%

         \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot \color{blue}{\left({M}^{2} \cdot \frac{{D}^{2}}{{d}^{2}}\right)}, \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      6. unpow2 1.9%

         \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\color{blue}{\left(M \cdot M\right)} \cdot \frac{{D}^{2}}{{d}^{2}}\right), \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      7. unpow2 1.9%

         \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\left(M \cdot M\right) \cdot \frac{\color{blue}{D \cdot D}}{{d}^{2}}\right), \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      8. unpow2 1.9%

         \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\left(M \cdot M\right) \cdot \frac{D \cdot D}{\color{blue}{d \cdot d}}\right), \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      9. times-frac 2.1%

         \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\left(M \cdot M\right) \cdot \color{blue}{\left(\frac{D}{d} \cdot \frac{D}{d}\right)}\right), \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      10. swap-sqr 2.3%

          \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot \color{blue}{\left(\left(M \cdot \frac{D}{d}\right) \cdot \left(M \cdot \frac{D}{d}\right)\right)}, \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      11. unpow2 2.3%

          \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot \color{blue}{{\left(M \cdot \frac{D}{d}\right)}^{2}}, \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      12. associate-*r/ 2.2%

          \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot {\color{blue}{\left(\frac{M \cdot D}{d}\right)}}^{2}, \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      13. *-commutative 2.2%

          \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot {\left(\frac{\color{blue}{D \cdot M}}{d}\right)}^{2}, \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      14. associate-/l* 2.3%

          \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot {\color{blue}{\left(D \cdot \frac{M}{d}\right)}}^{2}, \sqrt{\frac{1}{h \cdot \ell}}\right) \]

   7. Simplified 2.3%

      \[\leadsto \color{blue}{d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}, \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

   8. Taylor expanded in h around 0 4.8%

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

      1. associate-/r* 4.9%

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

      2. unpow1/2 4.9%

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

      3. associate-/r* 4.8%

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

      4. rem-exp-log 4.8%

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

      5. exp-neg 4.8%

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

      6. exp-prod 4.8%

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

      7. distribute-lft-neg-out 4.8%

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

      8. distribute-rgt-neg-in 4.8%

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

      9. metadata-eval 4.8%

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

      10. exp-to-pow 4.8%

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

   10. Simplified 4.8%

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

       1. unpow-prod-down 0.0%

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

       2. metadata-eval 0.0%

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

       3. sqrt-pow1 0.0%

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

       4. inv-pow 0.0%

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

       5. metadata-eval 0.0%

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

       6. sqrt-pow1 0.0%

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

       7. inv-pow 0.0%

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

       8. sqrt-prod 4.9%

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

       9. div-inv 4.9%

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

       10. sqrt-div 0.0%

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

       11. inv-pow 0.0%

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

       12. sqrt-pow1 0.0%

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

       13. metadata-eval 0.0%

           \[\leadsto d \cdot \frac{{h}^{\color{blue}{-0.5}}}{\sqrt{\ell}} \]

   12. Applied egg-rr 0.0%

       \[\leadsto d \cdot \color{blue}{\frac{{h}^{-0.5}}{\sqrt{\ell}}} \]

   13. Taylor expanded in h around -inf 0.0%

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

       1. *-commutative 0.0%

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

       2. unpow2 0.0%

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

       3. rem-square-sqrt 49.4%

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

       4. mul-1-neg 49.4%

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

       5. associate-/r* 50.5%

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

   15. Simplified 50.5%

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

   if -1.55e-92 < l < -4.80000000000000036e-307

   1. Initial program 78.1%

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

   3. Simplified 78.0%

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

   5. Taylor expanded in d around inf 18.3%

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

      1. add-log-exp 41.4%

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

      2. *-commutative 41.4%

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

      3. exp-prod 57.1%

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

   7. Applied egg-rr 57.1%

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

   8. Taylor expanded in l around 0 57.1%

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

   if -4.80000000000000036e-307 < l < 1.06000000000000001e195

   1. Initial program 68.7%

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

   3. Simplified 68.7%

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

      1. associate-*r/ 75.2%

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

      2. frac-times 75.3%

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

      3. associate-/l* 75.2%

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

      4. *-un-lft-identity 75.2%

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

      5. times-frac 75.2%

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

      6. metadata-eval 75.2%

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

   6. Applied egg-rr 75.2%

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

      1. clear-num 75.3%

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

      2. sqrt-div 76.0%

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

      3. metadata-eval 76.0%

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

   8. Applied egg-rr 76.0%

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

   9. Taylor expanded in d around 0 78.8%

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

       1. unpow-1 78.8%

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

       2. metadata-eval 78.8%

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

       3. pow-sqr 78.9%

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

       4. rem-sqrt-square 79.5%

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

       5. metadata-eval 79.5%

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

       6. pow-sqr 79.4%

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

       7. fabs-sqr 79.4%

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

       8. pow-sqr 79.5%

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

       9. metadata-eval 79.5%

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

   11. Simplified 79.5%

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

   if 1.06000000000000001e195 < l

   1. Initial program 62.9%

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

   3. Simplified 58.4%

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

   5. Taylor expanded in d around inf 58.4%

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

      1. sqrt-div 58.3%

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

      2. metadata-eval 58.3%

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

      3. *-commutative 58.3%

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

      4. sqrt-unprod 89.9%

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

      5. div-inv 89.9%

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

      6. associate-/r* 90.2%

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

   7. Applied egg-rr 90.2%

      \[\leadsto \color{blue}{\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.55 \cdot 10^{-92}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{h}}{\ell}}\right)\\ \mathbf{elif}\;\ell \leq -4.8 \cdot 10^{-307}:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{0}}\\ \mathbf{elif}\;\ell \leq 1.06 \cdot 10^{+195}:\\ \;\;\;\;\left(1 - 0.5 \cdot \frac{h \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell}\right) \cdot \left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 60.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -5.2 \cdot 10^{-93}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{h}}{\ell}}\right)\\ \mathbf{elif}\;\ell \leq 2.7 \cdot 10^{-304}:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{0}}\\ \mathbf{elif}\;\ell \leq 3.4 \cdot 10^{+192}:\\ \;\;\;\;\frac{d \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D\_m}{d} \cdot \left(0.5 \cdot M\_m\right)\right)}^{2}\right)\right)}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= l -5.2e-93)
   (* d (- (sqrt (/ (/ 1.0 h) l))))
   (if (<= l 2.7e-304)
     (* d (sqrt (/ 1.0 0.0)))
     (if (<= l 3.4e+192)
       (/
        (* d (+ 1.0 (* -0.5 (* (/ h l) (pow (* (/ D_m d) (* 0.5 M_m)) 2.0)))))
        (sqrt (* l h)))
       (/ (/ d (sqrt l)) (sqrt h))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= -5.2e-93) {
		tmp = d * -sqrt(((1.0 / h) / l));
	} else if (l <= 2.7e-304) {
		tmp = d * sqrt((1.0 / 0.0));
	} else if (l <= 3.4e+192) {
		tmp = (d * (1.0 + (-0.5 * ((h / l) * pow(((D_m / d) * (0.5 * M_m)), 2.0))))) / sqrt((l * h));
	} else {
		tmp = (d / sqrt(l)) / sqrt(h);
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (l <= (-5.2d-93)) then
        tmp = d * -sqrt(((1.0d0 / h) / l))
    else if (l <= 2.7d-304) then
        tmp = d * sqrt((1.0d0 / 0.0d0))
    else if (l <= 3.4d+192) then
        tmp = (d * (1.0d0 + ((-0.5d0) * ((h / l) * (((d_m / d) * (0.5d0 * m_m)) ** 2.0d0))))) / sqrt((l * h))
    else
        tmp = (d / sqrt(l)) / sqrt(h)
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= -5.2e-93) {
		tmp = d * -Math.sqrt(((1.0 / h) / l));
	} else if (l <= 2.7e-304) {
		tmp = d * Math.sqrt((1.0 / 0.0));
	} else if (l <= 3.4e+192) {
		tmp = (d * (1.0 + (-0.5 * ((h / l) * Math.pow(((D_m / d) * (0.5 * M_m)), 2.0))))) / Math.sqrt((l * h));
	} else {
		tmp = (d / Math.sqrt(l)) / Math.sqrt(h);
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if l <= -5.2e-93:
		tmp = d * -math.sqrt(((1.0 / h) / l))
	elif l <= 2.7e-304:
		tmp = d * math.sqrt((1.0 / 0.0))
	elif l <= 3.4e+192:
		tmp = (d * (1.0 + (-0.5 * ((h / l) * math.pow(((D_m / d) * (0.5 * M_m)), 2.0))))) / math.sqrt((l * h))
	else:
		tmp = (d / math.sqrt(l)) / math.sqrt(h)
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (l <= -5.2e-93)
		tmp = Float64(d * Float64(-sqrt(Float64(Float64(1.0 / h) / l))));
	elseif (l <= 2.7e-304)
		tmp = Float64(d * sqrt(Float64(1.0 / 0.0)));
	elseif (l <= 3.4e+192)
		tmp = Float64(Float64(d * Float64(1.0 + Float64(-0.5 * Float64(Float64(h / l) * (Float64(Float64(D_m / d) * Float64(0.5 * M_m)) ^ 2.0))))) / sqrt(Float64(l * h)));
	else
		tmp = Float64(Float64(d / sqrt(l)) / sqrt(h));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (l <= -5.2e-93)
		tmp = d * -sqrt(((1.0 / h) / l));
	elseif (l <= 2.7e-304)
		tmp = d * sqrt((1.0 / 0.0));
	elseif (l <= 3.4e+192)
		tmp = (d * (1.0 + (-0.5 * ((h / l) * (((D_m / d) * (0.5 * M_m)) ^ 2.0))))) / sqrt((l * h));
	else
		tmp = (d / sqrt(l)) / sqrt(h);
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[l, -5.2e-93], N[(d * (-N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], If[LessEqual[l, 2.7e-304], N[(d * N[Sqrt[N[(1.0 / 0.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 3.4e+192], N[(N[(d * N[(1.0 + N[(-0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D$95$m / d), $MachinePrecision] * N[(0.5 * M$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(d / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if l < -5.1999999999999997e-93

   1. Initial program 60.2%

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

   3. Simplified 61.3%

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

   5. Taylor expanded in d around inf 1.9%

      \[\leadsto \color{blue}{d \cdot \left(\sqrt{\frac{1}{h \cdot \ell}} + -0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative 1.9%

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

      2. fma-define 1.9%

         \[\leadsto d \cdot \color{blue}{\mathsf{fma}\left(-0.125, \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}} \cdot \sqrt{\frac{h}{{\ell}^{3}}}, \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      3. *-commutative 1.9%

         \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}}}, \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      4. *-commutative 1.9%

         \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot \frac{\color{blue}{{M}^{2} \cdot {D}^{2}}}{{d}^{2}}, \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      5. associate-/l* 1.9%

         \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot \color{blue}{\left({M}^{2} \cdot \frac{{D}^{2}}{{d}^{2}}\right)}, \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      6. unpow2 1.9%

         \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\color{blue}{\left(M \cdot M\right)} \cdot \frac{{D}^{2}}{{d}^{2}}\right), \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      7. unpow2 1.9%

         \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\left(M \cdot M\right) \cdot \frac{\color{blue}{D \cdot D}}{{d}^{2}}\right), \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      8. unpow2 1.9%

         \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\left(M \cdot M\right) \cdot \frac{D \cdot D}{\color{blue}{d \cdot d}}\right), \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      9. times-frac 2.1%

         \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\left(M \cdot M\right) \cdot \color{blue}{\left(\frac{D}{d} \cdot \frac{D}{d}\right)}\right), \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      10. swap-sqr 2.3%

          \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot \color{blue}{\left(\left(M \cdot \frac{D}{d}\right) \cdot \left(M \cdot \frac{D}{d}\right)\right)}, \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      11. unpow2 2.3%

          \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot \color{blue}{{\left(M \cdot \frac{D}{d}\right)}^{2}}, \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      12. associate-*r/ 2.2%

          \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot {\color{blue}{\left(\frac{M \cdot D}{d}\right)}}^{2}, \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      13. *-commutative 2.2%

          \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot {\left(\frac{\color{blue}{D \cdot M}}{d}\right)}^{2}, \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      14. associate-/l* 2.3%

          \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot {\color{blue}{\left(D \cdot \frac{M}{d}\right)}}^{2}, \sqrt{\frac{1}{h \cdot \ell}}\right) \]

   7. Simplified 2.3%

      \[\leadsto \color{blue}{d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}, \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

   8. Taylor expanded in h around 0 4.8%

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

      1. associate-/r* 4.9%

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

      2. unpow1/2 4.9%

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

      3. associate-/r* 4.8%

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

      4. rem-exp-log 4.8%

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

      5. exp-neg 4.8%

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

      6. exp-prod 4.8%

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

      7. distribute-lft-neg-out 4.8%

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

      8. distribute-rgt-neg-in 4.8%

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

      9. metadata-eval 4.8%

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

      10. exp-to-pow 4.8%

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

   10. Simplified 4.8%

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

       1. unpow-prod-down 0.0%

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

       2. metadata-eval 0.0%

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

       3. sqrt-pow1 0.0%

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

       4. inv-pow 0.0%

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

       5. metadata-eval 0.0%

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

       6. sqrt-pow1 0.0%

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

       7. inv-pow 0.0%

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

       8. sqrt-prod 4.9%

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

       9. div-inv 4.9%

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

       10. sqrt-div 0.0%

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

       11. inv-pow 0.0%

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

       12. sqrt-pow1 0.0%

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

       13. metadata-eval 0.0%

           \[\leadsto d \cdot \frac{{h}^{\color{blue}{-0.5}}}{\sqrt{\ell}} \]

   12. Applied egg-rr 0.0%

       \[\leadsto d \cdot \color{blue}{\frac{{h}^{-0.5}}{\sqrt{\ell}}} \]

   13. Taylor expanded in h around -inf 0.0%

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

       1. *-commutative 0.0%

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

       2. unpow2 0.0%

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

       3. rem-square-sqrt 49.4%

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

       4. mul-1-neg 49.4%

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

       5. associate-/r* 50.5%

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

   15. Simplified 50.5%

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

   if -5.1999999999999997e-93 < l < 2.7000000000000001e-304

   1. Initial program 77.4%

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

   3. Simplified 77.3%

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

   5. Taylor expanded in d around inf 21.4%

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

      1. add-log-exp 42.9%

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

      2. *-commutative 42.9%

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

      3. exp-prod 57.7%

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

   7. Applied egg-rr 57.7%

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

   8. Taylor expanded in l around 0 57.7%

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

   if 2.7000000000000001e-304 < l < 3.39999999999999996e192

   1. Initial program 68.8%

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

   3. Simplified 68.7%

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

      1. *-commutative 74.5%

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

      2. sqrt-div 81.6%

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

      3. sqrt-div 87.2%

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

      4. frac-times 87.1%

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

      5. add-sqr-sqrt 87.4%

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

   6. Applied egg-rr 79.8%

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

      1. associate-*l/ 82.6%

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

      2. cancel-sign-sub-inv 82.6%

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

      3. metadata-eval 82.6%

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

      4. *-commutative 82.6%

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

      5. div-inv 82.6%

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

      6. metadata-eval 82.6%

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

      7. sqrt-unprod 73.2%

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

      8. *-commutative 73.2%

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

   8. Applied egg-rr 73.2%

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

   if 3.39999999999999996e192 < l

   1. Initial program 62.9%

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

   3. Simplified 58.4%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 58.4%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. sqrt-div 58.3%

         \[\leadsto d \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{h \cdot \ell}}} \]

      2. metadata-eval 58.3%

         \[\leadsto d \cdot \frac{\color{blue}{1}}{\sqrt{h \cdot \ell}} \]

      3. *-commutative 58.3%

         \[\leadsto d \cdot \frac{1}{\sqrt{\color{blue}{\ell \cdot h}}} \]

      4. sqrt-unprod 89.9%

         \[\leadsto d \cdot \frac{1}{\color{blue}{\sqrt{\ell} \cdot \sqrt{h}}} \]

      5. div-inv 89.9%

         \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}} \]

      6. associate-/r* 90.2%

         \[\leadsto \color{blue}{\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}} \]

   7. Applied egg-rr 90.2%

      \[\leadsto \color{blue}{\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 64.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5.2 \cdot 10^{-93}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{h}}{\ell}}\right)\\ \mathbf{elif}\;\ell \leq 2.7 \cdot 10^{-304}:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{0}}\\ \mathbf{elif}\;\ell \leq 3.4 \cdot 10^{+192}:\\ \;\;\;\;\frac{d \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \left(0.5 \cdot M\right)\right)}^{2}\right)\right)}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 60.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.35 \cdot 10^{-92}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{h}}{\ell}}\right)\\ \mathbf{elif}\;\ell \leq 2.7 \cdot 10^{-304}:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{0}}\\ \mathbf{elif}\;\ell \leq 2.4 \cdot 10^{+192}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D\_m}{d} \cdot \frac{M\_m}{2}\right)}^{2}\right)\right) \cdot \frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= l -1.35e-92)
   (* d (- (sqrt (/ (/ 1.0 h) l))))
   (if (<= l 2.7e-304)
     (* d (sqrt (/ 1.0 0.0)))
     (if (<= l 2.4e+192)
       (*
        (- 1.0 (* 0.5 (* (/ h l) (pow (* (/ D_m d) (/ M_m 2.0)) 2.0))))
        (/ d (sqrt (* l h))))
       (/ (/ d (sqrt l)) (sqrt h))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= -1.35e-92) {
		tmp = d * -sqrt(((1.0 / h) / l));
	} else if (l <= 2.7e-304) {
		tmp = d * sqrt((1.0 / 0.0));
	} else if (l <= 2.4e+192) {
		tmp = (1.0 - (0.5 * ((h / l) * pow(((D_m / d) * (M_m / 2.0)), 2.0)))) * (d / sqrt((l * h)));
	} else {
		tmp = (d / sqrt(l)) / sqrt(h);
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (l <= (-1.35d-92)) then
        tmp = d * -sqrt(((1.0d0 / h) / l))
    else if (l <= 2.7d-304) then
        tmp = d * sqrt((1.0d0 / 0.0d0))
    else if (l <= 2.4d+192) then
        tmp = (1.0d0 - (0.5d0 * ((h / l) * (((d_m / d) * (m_m / 2.0d0)) ** 2.0d0)))) * (d / sqrt((l * h)))
    else
        tmp = (d / sqrt(l)) / sqrt(h)
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= -1.35e-92) {
		tmp = d * -Math.sqrt(((1.0 / h) / l));
	} else if (l <= 2.7e-304) {
		tmp = d * Math.sqrt((1.0 / 0.0));
	} else if (l <= 2.4e+192) {
		tmp = (1.0 - (0.5 * ((h / l) * Math.pow(((D_m / d) * (M_m / 2.0)), 2.0)))) * (d / Math.sqrt((l * h)));
	} else {
		tmp = (d / Math.sqrt(l)) / Math.sqrt(h);
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if l <= -1.35e-92:
		tmp = d * -math.sqrt(((1.0 / h) / l))
	elif l <= 2.7e-304:
		tmp = d * math.sqrt((1.0 / 0.0))
	elif l <= 2.4e+192:
		tmp = (1.0 - (0.5 * ((h / l) * math.pow(((D_m / d) * (M_m / 2.0)), 2.0)))) * (d / math.sqrt((l * h)))
	else:
		tmp = (d / math.sqrt(l)) / math.sqrt(h)
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (l <= -1.35e-92)
		tmp = Float64(d * Float64(-sqrt(Float64(Float64(1.0 / h) / l))));
	elseif (l <= 2.7e-304)
		tmp = Float64(d * sqrt(Float64(1.0 / 0.0)));
	elseif (l <= 2.4e+192)
		tmp = Float64(Float64(1.0 - Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(D_m / d) * Float64(M_m / 2.0)) ^ 2.0)))) * Float64(d / sqrt(Float64(l * h))));
	else
		tmp = Float64(Float64(d / sqrt(l)) / sqrt(h));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (l <= -1.35e-92)
		tmp = d * -sqrt(((1.0 / h) / l));
	elseif (l <= 2.7e-304)
		tmp = d * sqrt((1.0 / 0.0));
	elseif (l <= 2.4e+192)
		tmp = (1.0 - (0.5 * ((h / l) * (((D_m / d) * (M_m / 2.0)) ^ 2.0)))) * (d / sqrt((l * h)));
	else
		tmp = (d / sqrt(l)) / sqrt(h);
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[l, -1.35e-92], N[(d * (-N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], If[LessEqual[l, 2.7e-304], N[(d * N[Sqrt[N[(1.0 / 0.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 2.4e+192], N[(N[(1.0 - N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D$95$m / d), $MachinePrecision] * N[(M$95$m / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(d / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if l < -1.34999999999999998e-92

   1. Initial program 60.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 61.3%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 1.9%

      \[\leadsto \color{blue}{d \cdot \left(\sqrt{\frac{1}{h \cdot \ell}} + -0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative 1.9%

         \[\leadsto d \cdot \color{blue}{\left(-0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) + \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. fma-define 1.9%

         \[\leadsto d \cdot \color{blue}{\mathsf{fma}\left(-0.125, \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}} \cdot \sqrt{\frac{h}{{\ell}^{3}}}, \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      3. *-commutative 1.9%

         \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}}}, \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      4. *-commutative 1.9%

         \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot \frac{\color{blue}{{M}^{2} \cdot {D}^{2}}}{{d}^{2}}, \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      5. associate-/l* 1.9%

         \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot \color{blue}{\left({M}^{2} \cdot \frac{{D}^{2}}{{d}^{2}}\right)}, \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      6. unpow2 1.9%

         \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\color{blue}{\left(M \cdot M\right)} \cdot \frac{{D}^{2}}{{d}^{2}}\right), \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      7. unpow2 1.9%

         \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\left(M \cdot M\right) \cdot \frac{\color{blue}{D \cdot D}}{{d}^{2}}\right), \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      8. unpow2 1.9%

         \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\left(M \cdot M\right) \cdot \frac{D \cdot D}{\color{blue}{d \cdot d}}\right), \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      9. times-frac 2.1%

         \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\left(M \cdot M\right) \cdot \color{blue}{\left(\frac{D}{d} \cdot \frac{D}{d}\right)}\right), \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      10. swap-sqr 2.3%

          \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot \color{blue}{\left(\left(M \cdot \frac{D}{d}\right) \cdot \left(M \cdot \frac{D}{d}\right)\right)}, \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      11. unpow2 2.3%

          \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot \color{blue}{{\left(M \cdot \frac{D}{d}\right)}^{2}}, \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      12. associate-*r/ 2.2%

          \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot {\color{blue}{\left(\frac{M \cdot D}{d}\right)}}^{2}, \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      13. *-commutative 2.2%

          \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot {\left(\frac{\color{blue}{D \cdot M}}{d}\right)}^{2}, \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      14. associate-/l* 2.3%

          \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot {\color{blue}{\left(D \cdot \frac{M}{d}\right)}}^{2}, \sqrt{\frac{1}{h \cdot \ell}}\right) \]

   7. Simplified 2.3%

      \[\leadsto \color{blue}{d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}, \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

   8. Taylor expanded in h around 0 4.8%

      \[\leadsto d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]
   9. Step-by-step derivation

      1. associate-/r* 4.9%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \]

      2. unpow1/2 4.9%

         \[\leadsto d \cdot \color{blue}{{\left(\frac{\frac{1}{h}}{\ell}\right)}^{0.5}} \]

      3. associate-/r* 4.8%

         \[\leadsto d \cdot {\color{blue}{\left(\frac{1}{h \cdot \ell}\right)}}^{0.5} \]

      4. rem-exp-log 4.8%

         \[\leadsto d \cdot {\left(\frac{1}{\color{blue}{e^{\log \left(h \cdot \ell\right)}}}\right)}^{0.5} \]

      5. exp-neg 4.8%

         \[\leadsto d \cdot {\color{blue}{\left(e^{-\log \left(h \cdot \ell\right)}\right)}}^{0.5} \]

      6. exp-prod 4.8%

         \[\leadsto d \cdot \color{blue}{e^{\left(-\log \left(h \cdot \ell\right)\right) \cdot 0.5}} \]

      7. distribute-lft-neg-out 4.8%

         \[\leadsto d \cdot e^{\color{blue}{-\log \left(h \cdot \ell\right) \cdot 0.5}} \]

      8. distribute-rgt-neg-in 4.8%

         \[\leadsto d \cdot e^{\color{blue}{\log \left(h \cdot \ell\right) \cdot \left(-0.5\right)}} \]

      9. metadata-eval 4.8%

         \[\leadsto d \cdot e^{\log \left(h \cdot \ell\right) \cdot \color{blue}{-0.5}} \]

      10. exp-to-pow 4.8%

          \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

   10. Simplified 4.8%

       \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]
   11. Step-by-step derivation

       1. unpow-prod-down 0.0%

          \[\leadsto d \cdot \color{blue}{\left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)} \]

       2. metadata-eval 0.0%

          \[\leadsto d \cdot \left({h}^{\color{blue}{\left(\frac{-1}{2}\right)}} \cdot {\ell}^{-0.5}\right) \]

       3. sqrt-pow1 0.0%

          \[\leadsto d \cdot \left(\color{blue}{\sqrt{{h}^{-1}}} \cdot {\ell}^{-0.5}\right) \]

       4. inv-pow 0.0%

          \[\leadsto d \cdot \left(\sqrt{\color{blue}{\frac{1}{h}}} \cdot {\ell}^{-0.5}\right) \]

       5. metadata-eval 0.0%

          \[\leadsto d \cdot \left(\sqrt{\frac{1}{h}} \cdot {\ell}^{\color{blue}{\left(\frac{-1}{2}\right)}}\right) \]

       6. sqrt-pow1 0.0%

          \[\leadsto d \cdot \left(\sqrt{\frac{1}{h}} \cdot \color{blue}{\sqrt{{\ell}^{-1}}}\right) \]

       7. inv-pow 0.0%

          \[\leadsto d \cdot \left(\sqrt{\frac{1}{h}} \cdot \sqrt{\color{blue}{\frac{1}{\ell}}}\right) \]

       8. sqrt-prod 4.9%

          \[\leadsto d \cdot \color{blue}{\sqrt{\frac{1}{h} \cdot \frac{1}{\ell}}} \]

       9. div-inv 4.9%

          \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \]

       10. sqrt-div 0.0%

           \[\leadsto d \cdot \color{blue}{\frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}} \]

       11. inv-pow 0.0%

           \[\leadsto d \cdot \frac{\sqrt{\color{blue}{{h}^{-1}}}}{\sqrt{\ell}} \]

       12. sqrt-pow1 0.0%

           \[\leadsto d \cdot \frac{\color{blue}{{h}^{\left(\frac{-1}{2}\right)}}}{\sqrt{\ell}} \]

       13. metadata-eval 0.0%

           \[\leadsto d \cdot \frac{{h}^{\color{blue}{-0.5}}}{\sqrt{\ell}} \]

   12. Applied egg-rr 0.0%

       \[\leadsto d \cdot \color{blue}{\frac{{h}^{-0.5}}{\sqrt{\ell}}} \]

   13. Taylor expanded in h around -inf 0.0%

       \[\leadsto d \cdot \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]
   14. Step-by-step derivation

       1. *-commutative 0.0%

          \[\leadsto d \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

       2. unpow2 0.0%

          \[\leadsto d \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

       3. rem-square-sqrt 49.4%

          \[\leadsto d \cdot \left(\color{blue}{-1} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

       4. mul-1-neg 49.4%

          \[\leadsto d \cdot \color{blue}{\left(-\sqrt{\frac{1}{h \cdot \ell}}\right)} \]

       5. associate-/r* 50.5%

          \[\leadsto d \cdot \left(-\sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}}\right) \]

   15. Simplified 50.5%

       \[\leadsto d \cdot \color{blue}{\left(-\sqrt{\frac{\frac{1}{h}}{\ell}}\right)} \]

   if -1.34999999999999998e-92 < l < 2.7000000000000001e-304

   1. Initial program 77.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 77.3%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 21.4%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. add-log-exp 42.9%

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\log \left(e^{h \cdot \ell}\right)}}} \]

      2. *-commutative 42.9%

         \[\leadsto d \cdot \sqrt{\frac{1}{\log \left(e^{\color{blue}{\ell \cdot h}}\right)}} \]

      3. exp-prod 57.7%

         \[\leadsto d \cdot \sqrt{\frac{1}{\log \color{blue}{\left({\left(e^{\ell}\right)}^{h}\right)}}} \]

   7. Applied egg-rr 57.7%

      \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\log \left({\left(e^{\ell}\right)}^{h}\right)}}} \]

   8. Taylor expanded in l around 0 57.7%

      \[\leadsto d \cdot \sqrt{\frac{1}{\log \color{blue}{1}}} \]

   if 2.7000000000000001e-304 < l < 2.3999999999999998e192

   1. Initial program 68.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 68.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 74.5%

         \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - 0.5 \cdot \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot h}{\ell}\right) \]

      2. sqrt-div 81.6%

         \[\leadsto \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot h}{\ell}\right) \]

      3. sqrt-div 87.2%

         \[\leadsto \left(\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{h}}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot h}{\ell}\right) \]

      4. frac-times 87.1%

         \[\leadsto \color{blue}{\frac{\sqrt{d} \cdot \sqrt{d}}{\sqrt{\ell} \cdot \sqrt{h}}} \cdot \left(1 - 0.5 \cdot \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot h}{\ell}\right) \]

      5. add-sqr-sqrt 87.4%

         \[\leadsto \frac{\color{blue}{d}}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 - 0.5 \cdot \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot h}{\ell}\right) \]

   6. Applied egg-rr 79.8%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   7. Taylor expanded in d around 0 71.6%

      \[\leadsto \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]
   8. Step-by-step derivation

      1. *-commutative 71.6%

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot d\right)} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      2. associate-/r* 71.6%

         \[\leadsto \left(\sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \cdot d\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      3. unpow1/2 71.6%

         \[\leadsto \left(\color{blue}{{\left(\frac{\frac{1}{h}}{\ell}\right)}^{0.5}} \cdot d\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      4. associate-/r* 71.6%

         \[\leadsto \left({\color{blue}{\left(\frac{1}{h \cdot \ell}\right)}}^{0.5} \cdot d\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      5. rem-exp-log 69.8%

         \[\leadsto \left({\left(\frac{1}{\color{blue}{e^{\log \left(h \cdot \ell\right)}}}\right)}^{0.5} \cdot d\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      6. exp-neg 69.8%

         \[\leadsto \left({\color{blue}{\left(e^{-\log \left(h \cdot \ell\right)}\right)}}^{0.5} \cdot d\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      7. exp-prod 70.4%

         \[\leadsto \left(\color{blue}{e^{\left(-\log \left(h \cdot \ell\right)\right) \cdot 0.5}} \cdot d\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. distribute-lft-neg-out 70.4%

         \[\leadsto \left(e^{\color{blue}{-\log \left(h \cdot \ell\right) \cdot 0.5}} \cdot d\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      9. exp-neg 70.4%

         \[\leadsto \left(\color{blue}{\frac{1}{e^{\log \left(h \cdot \ell\right) \cdot 0.5}}} \cdot d\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      10. exp-to-pow 72.2%

          \[\leadsto \left(\frac{1}{\color{blue}{{\left(h \cdot \ell\right)}^{0.5}}} \cdot d\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      11. unpow1/2 72.2%

          \[\leadsto \left(\frac{1}{\color{blue}{\sqrt{h \cdot \ell}}} \cdot d\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      12. unpow-1 72.2%

          \[\leadsto \left(\color{blue}{{\left(\sqrt{h \cdot \ell}\right)}^{-1}} \cdot d\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      13. unpow-1 72.2%

          \[\leadsto \left(\color{blue}{\frac{1}{\sqrt{h \cdot \ell}}} \cdot d\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      14. associate-*l/ 72.2%

          \[\leadsto \color{blue}{\frac{1 \cdot d}{\sqrt{h \cdot \ell}}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      15. *-lft-identity 72.2%

          \[\leadsto \frac{\color{blue}{d}}{\sqrt{h \cdot \ell}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   9. Simplified 72.2%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   if 2.3999999999999998e192 < l

   1. Initial program 62.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 58.4%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 58.4%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. sqrt-div 58.3%

         \[\leadsto d \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{h \cdot \ell}}} \]

      2. metadata-eval 58.3%

         \[\leadsto d \cdot \frac{\color{blue}{1}}{\sqrt{h \cdot \ell}} \]

      3. *-commutative 58.3%

         \[\leadsto d \cdot \frac{1}{\sqrt{\color{blue}{\ell \cdot h}}} \]

      4. sqrt-unprod 89.9%

         \[\leadsto d \cdot \frac{1}{\color{blue}{\sqrt{\ell} \cdot \sqrt{h}}} \]

      5. div-inv 89.9%

         \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}} \]

      6. associate-/r* 90.2%

         \[\leadsto \color{blue}{\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}} \]

   7. Applied egg-rr 90.2%

      \[\leadsto \color{blue}{\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 63.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.35 \cdot 10^{-92}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{h}}{\ell}}\right)\\ \mathbf{elif}\;\ell \leq 2.7 \cdot 10^{-304}:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{0}}\\ \mathbf{elif}\;\ell \leq 2.4 \cdot 10^{+192}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right) \cdot \frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 74.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := 0.5 \cdot \frac{h \cdot {\left(M\_m \cdot \left(0.5 \cdot \frac{D\_m}{d}\right)\right)}^{2}}{\ell}\\ t_1 := d \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \mathbf{if}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;t\_1 \cdot \left(-1 + t\_0\right)\\ \mathbf{elif}\;\ell \leq 1.3 \cdot 10^{+194}:\\ \;\;\;\;\left(1 - t\_0\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (* 0.5 (/ (* h (pow (* M_m (* 0.5 (/ D_m d))) 2.0)) l)))
        (t_1 (* d (pow (* l h) -0.5))))
   (if (<= l -5e-310)
     (* t_1 (+ -1.0 t_0))
     (if (<= l 1.3e+194) (* (- 1.0 t_0) t_1) (/ (/ d (sqrt l)) (sqrt h))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = 0.5 * ((h * pow((M_m * (0.5 * (D_m / d))), 2.0)) / l);
	double t_1 = d * pow((l * h), -0.5);
	double tmp;
	if (l <= -5e-310) {
		tmp = t_1 * (-1.0 + t_0);
	} else if (l <= 1.3e+194) {
		tmp = (1.0 - t_0) * t_1;
	} else {
		tmp = (d / sqrt(l)) / sqrt(h);
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 0.5d0 * ((h * ((m_m * (0.5d0 * (d_m / d))) ** 2.0d0)) / l)
    t_1 = d * ((l * h) ** (-0.5d0))
    if (l <= (-5d-310)) then
        tmp = t_1 * ((-1.0d0) + t_0)
    else if (l <= 1.3d+194) then
        tmp = (1.0d0 - t_0) * t_1
    else
        tmp = (d / sqrt(l)) / sqrt(h)
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = 0.5 * ((h * Math.pow((M_m * (0.5 * (D_m / d))), 2.0)) / l);
	double t_1 = d * Math.pow((l * h), -0.5);
	double tmp;
	if (l <= -5e-310) {
		tmp = t_1 * (-1.0 + t_0);
	} else if (l <= 1.3e+194) {
		tmp = (1.0 - t_0) * t_1;
	} else {
		tmp = (d / Math.sqrt(l)) / Math.sqrt(h);
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = 0.5 * ((h * math.pow((M_m * (0.5 * (D_m / d))), 2.0)) / l)
	t_1 = d * math.pow((l * h), -0.5)
	tmp = 0
	if l <= -5e-310:
		tmp = t_1 * (-1.0 + t_0)
	elif l <= 1.3e+194:
		tmp = (1.0 - t_0) * t_1
	else:
		tmp = (d / math.sqrt(l)) / math.sqrt(h)
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(0.5 * Float64(Float64(h * (Float64(M_m * Float64(0.5 * Float64(D_m / d))) ^ 2.0)) / l))
	t_1 = Float64(d * (Float64(l * h) ^ -0.5))
	tmp = 0.0
	if (l <= -5e-310)
		tmp = Float64(t_1 * Float64(-1.0 + t_0));
	elseif (l <= 1.3e+194)
		tmp = Float64(Float64(1.0 - t_0) * t_1);
	else
		tmp = Float64(Float64(d / sqrt(l)) / sqrt(h));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = 0.5 * ((h * ((M_m * (0.5 * (D_m / d))) ^ 2.0)) / l);
	t_1 = d * ((l * h) ^ -0.5);
	tmp = 0.0;
	if (l <= -5e-310)
		tmp = t_1 * (-1.0 + t_0);
	elseif (l <= 1.3e+194)
		tmp = (1.0 - t_0) * t_1;
	else
		tmp = (d / sqrt(l)) / sqrt(h);
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(0.5 * N[(N[(h * N[Power[N[(M$95$m * N[(0.5 * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(d * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -5e-310], N[(t$95$1 * N[(-1.0 + t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.3e+194], N[(N[(1.0 - t$95$0), $MachinePrecision] * t$95$1), $MachinePrecision], N[(N[(d / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -4.999999999999985e-310

   1. Initial program 66.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 67.4%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r/ 70.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot h}{\ell}}\right) \]

      2. frac-times 70.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot h}{\ell}\right) \]

      3. associate-/l* 70.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)}}^{2} \cdot h}{\ell}\right) \]

      4. *-un-lft-identity 70.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(M \cdot \frac{\color{blue}{1 \cdot D}}{2 \cdot d}\right)}^{2} \cdot h}{\ell}\right) \]

      5. times-frac 70.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(M \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{D}{d}\right)}\right)}^{2} \cdot h}{\ell}\right) \]

      6. metadata-eval 70.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(M \cdot \left(\color{blue}{0.5} \cdot \frac{D}{d}\right)\right)}^{2} \cdot h}{\ell}\right) \]

   6. Applied egg-rr 70.6%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot h}{\ell}}\right) \]
   7. Step-by-step derivation

      1. clear-num 70.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\color{blue}{\frac{1}{\frac{\ell}{d}}}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot h}{\ell}\right) \]

      2. sqrt-div 70.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot h}{\ell}\right) \]

      3. metadata-eval 70.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \frac{\color{blue}{1}}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot h}{\ell}\right) \]

   8. Applied egg-rr 70.5%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\frac{1}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot h}{\ell}\right) \]

   9. Taylor expanded in d around -inf 67.1%

      \[\leadsto \color{blue}{\left(-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)\right)} \cdot \left(1 - 0.5 \cdot \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot h}{\ell}\right) \]
   10. Step-by-step derivation

       1. mul-1-neg 67.1%

          \[\leadsto \color{blue}{\left(-d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot \left(1 - 0.5 \cdot \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot h}{\ell}\right) \]

       2. unpow-1 67.1%

          \[\leadsto \left(-d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot h}{\ell}\right) \]

       3. metadata-eval 67.1%

          \[\leadsto \left(-d \cdot \sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot h}{\ell}\right) \]

       4. pow-sqr 67.1%

          \[\leadsto \left(-d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot h}{\ell}\right) \]

       5. rem-sqrt-square 67.6%

          \[\leadsto \left(-d \cdot \color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot h}{\ell}\right) \]

       6. metadata-eval 67.6%

          \[\leadsto \left(-d \cdot \left|{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.25\right)}}\right|\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot h}{\ell}\right) \]

       7. pow-sqr 67.4%

          \[\leadsto \left(-d \cdot \left|\color{blue}{{\left(h \cdot \ell\right)}^{-0.25} \cdot {\left(h \cdot \ell\right)}^{-0.25}}\right|\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot h}{\ell}\right) \]

       8. fabs-sqr 67.4%

          \[\leadsto \left(-d \cdot \color{blue}{\left({\left(h \cdot \ell\right)}^{-0.25} \cdot {\left(h \cdot \ell\right)}^{-0.25}\right)}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot h}{\ell}\right) \]

       9. pow-sqr 67.6%

          \[\leadsto \left(-d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{\left(2 \cdot -0.25\right)}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot h}{\ell}\right) \]

       10. metadata-eval 67.6%

           \[\leadsto \left(-d \cdot {\left(h \cdot \ell\right)}^{\color{blue}{-0.5}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot h}{\ell}\right) \]

       11. distribute-rgt-neg-in 67.6%

           \[\leadsto \color{blue}{\left(d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\right)} \cdot \left(1 - 0.5 \cdot \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot h}{\ell}\right) \]

   11. Simplified 67.6%

       \[\leadsto \color{blue}{\left(d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\right)} \cdot \left(1 - 0.5 \cdot \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot h}{\ell}\right) \]

   if -4.999999999999985e-310 < l < 1.2999999999999999e194

   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 68.4%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r/ 75.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot h}{\ell}}\right) \]

      2. frac-times 75.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot h}{\ell}\right) \]

      3. associate-/l* 75.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)}}^{2} \cdot h}{\ell}\right) \]

      4. *-un-lft-identity 75.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(M \cdot \frac{\color{blue}{1 \cdot D}}{2 \cdot d}\right)}^{2} \cdot h}{\ell}\right) \]

      5. times-frac 75.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(M \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{D}{d}\right)}\right)}^{2} \cdot h}{\ell}\right) \]

      6. metadata-eval 75.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(M \cdot \left(\color{blue}{0.5} \cdot \frac{D}{d}\right)\right)}^{2} \cdot h}{\ell}\right) \]

   6. Applied egg-rr 75.0%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot h}{\ell}}\right) \]
   7. Step-by-step derivation

      1. clear-num 75.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\color{blue}{\frac{1}{\frac{\ell}{d}}}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot h}{\ell}\right) \]

      2. sqrt-div 75.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot h}{\ell}\right) \]

      3. metadata-eval 75.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \frac{\color{blue}{1}}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot h}{\ell}\right) \]

   8. Applied egg-rr 75.8%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\frac{1}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot h}{\ell}\right) \]

   9. Taylor expanded in d around 0 79.6%

      \[\leadsto \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot \left(1 - 0.5 \cdot \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot h}{\ell}\right) \]
   10. Step-by-step derivation

       1. unpow-1 79.6%

          \[\leadsto \left(d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot h}{\ell}\right) \]

       2. metadata-eval 79.6%

          \[\leadsto \left(d \cdot \sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot h}{\ell}\right) \]

       3. pow-sqr 79.6%

          \[\leadsto \left(d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot h}{\ell}\right) \]

       4. rem-sqrt-square 80.2%

          \[\leadsto \left(d \cdot \color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot h}{\ell}\right) \]

       5. metadata-eval 80.2%

          \[\leadsto \left(d \cdot \left|{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.25\right)}}\right|\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot h}{\ell}\right) \]

       6. pow-sqr 80.1%

          \[\leadsto \left(d \cdot \left|\color{blue}{{\left(h \cdot \ell\right)}^{-0.25} \cdot {\left(h \cdot \ell\right)}^{-0.25}}\right|\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot h}{\ell}\right) \]

       7. fabs-sqr 80.1%

          \[\leadsto \left(d \cdot \color{blue}{\left({\left(h \cdot \ell\right)}^{-0.25} \cdot {\left(h \cdot \ell\right)}^{-0.25}\right)}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot h}{\ell}\right) \]

       8. pow-sqr 80.2%

          \[\leadsto \left(d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{\left(2 \cdot -0.25\right)}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot h}{\ell}\right) \]

       9. metadata-eval 80.2%

          \[\leadsto \left(d \cdot {\left(h \cdot \ell\right)}^{\color{blue}{-0.5}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot h}{\ell}\right) \]

   11. Simplified 80.2%

       \[\leadsto \color{blue}{\left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right)} \cdot \left(1 - 0.5 \cdot \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot h}{\ell}\right) \]

   if 1.2999999999999999e194 < l

   1. Initial program 62.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 58.4%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 58.4%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. sqrt-div 58.3%

         \[\leadsto d \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{h \cdot \ell}}} \]

      2. metadata-eval 58.3%

         \[\leadsto d \cdot \frac{\color{blue}{1}}{\sqrt{h \cdot \ell}} \]

      3. *-commutative 58.3%

         \[\leadsto d \cdot \frac{1}{\sqrt{\color{blue}{\ell \cdot h}}} \]

      4. sqrt-unprod 89.9%

         \[\leadsto d \cdot \frac{1}{\color{blue}{\sqrt{\ell} \cdot \sqrt{h}}} \]

      5. div-inv 89.9%

         \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}} \]

      6. associate-/r* 90.2%

         \[\leadsto \color{blue}{\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}} \]

   7. Applied egg-rr 90.2%

      \[\leadsto \color{blue}{\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right) \cdot \left(-1 + 0.5 \cdot \frac{h \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell}\right)\\ \mathbf{elif}\;\ell \leq 1.3 \cdot 10^{+194}:\\ \;\;\;\;\left(1 - 0.5 \cdot \frac{h \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell}\right) \cdot \left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 75.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := d \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \mathbf{if}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;t\_0 \cdot \left(-1 + 0.5 \cdot \left(h \cdot \frac{0.25 \cdot {\left(D\_m \cdot \frac{M\_m}{d}\right)}^{2}}{\ell}\right)\right)\\ \mathbf{elif}\;\ell \leq 1.55 \cdot 10^{+194}:\\ \;\;\;\;\left(1 - 0.5 \cdot \frac{h \cdot {\left(M\_m \cdot \left(0.5 \cdot \frac{D\_m}{d}\right)\right)}^{2}}{\ell}\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (* d (pow (* l h) -0.5))))
   (if (<= l -5e-310)
     (* t_0 (+ -1.0 (* 0.5 (* h (/ (* 0.25 (pow (* D_m (/ M_m d)) 2.0)) l)))))
     (if (<= l 1.55e+194)
       (* (- 1.0 (* 0.5 (/ (* h (pow (* M_m (* 0.5 (/ D_m d))) 2.0)) l))) t_0)
       (/ (/ d (sqrt l)) (sqrt h))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = d * pow((l * h), -0.5);
	double tmp;
	if (l <= -5e-310) {
		tmp = t_0 * (-1.0 + (0.5 * (h * ((0.25 * pow((D_m * (M_m / d)), 2.0)) / l))));
	} else if (l <= 1.55e+194) {
		tmp = (1.0 - (0.5 * ((h * pow((M_m * (0.5 * (D_m / d))), 2.0)) / l))) * t_0;
	} else {
		tmp = (d / sqrt(l)) / sqrt(h);
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = d * ((l * h) ** (-0.5d0))
    if (l <= (-5d-310)) then
        tmp = t_0 * ((-1.0d0) + (0.5d0 * (h * ((0.25d0 * ((d_m * (m_m / d)) ** 2.0d0)) / l))))
    else if (l <= 1.55d+194) then
        tmp = (1.0d0 - (0.5d0 * ((h * ((m_m * (0.5d0 * (d_m / d))) ** 2.0d0)) / l))) * t_0
    else
        tmp = (d / sqrt(l)) / sqrt(h)
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = d * Math.pow((l * h), -0.5);
	double tmp;
	if (l <= -5e-310) {
		tmp = t_0 * (-1.0 + (0.5 * (h * ((0.25 * Math.pow((D_m * (M_m / d)), 2.0)) / l))));
	} else if (l <= 1.55e+194) {
		tmp = (1.0 - (0.5 * ((h * Math.pow((M_m * (0.5 * (D_m / d))), 2.0)) / l))) * t_0;
	} else {
		tmp = (d / Math.sqrt(l)) / Math.sqrt(h);
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = d * math.pow((l * h), -0.5)
	tmp = 0
	if l <= -5e-310:
		tmp = t_0 * (-1.0 + (0.5 * (h * ((0.25 * math.pow((D_m * (M_m / d)), 2.0)) / l))))
	elif l <= 1.55e+194:
		tmp = (1.0 - (0.5 * ((h * math.pow((M_m * (0.5 * (D_m / d))), 2.0)) / l))) * t_0
	else:
		tmp = (d / math.sqrt(l)) / math.sqrt(h)
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(d * (Float64(l * h) ^ -0.5))
	tmp = 0.0
	if (l <= -5e-310)
		tmp = Float64(t_0 * Float64(-1.0 + Float64(0.5 * Float64(h * Float64(Float64(0.25 * (Float64(D_m * Float64(M_m / d)) ^ 2.0)) / l)))));
	elseif (l <= 1.55e+194)
		tmp = Float64(Float64(1.0 - Float64(0.5 * Float64(Float64(h * (Float64(M_m * Float64(0.5 * Float64(D_m / d))) ^ 2.0)) / l))) * t_0);
	else
		tmp = Float64(Float64(d / sqrt(l)) / sqrt(h));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = d * ((l * h) ^ -0.5);
	tmp = 0.0;
	if (l <= -5e-310)
		tmp = t_0 * (-1.0 + (0.5 * (h * ((0.25 * ((D_m * (M_m / d)) ^ 2.0)) / l))));
	elseif (l <= 1.55e+194)
		tmp = (1.0 - (0.5 * ((h * ((M_m * (0.5 * (D_m / d))) ^ 2.0)) / l))) * t_0;
	else
		tmp = (d / sqrt(l)) / sqrt(h);
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(d * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -5e-310], N[(t$95$0 * N[(-1.0 + N[(0.5 * N[(h * N[(N[(0.25 * N[Power[N[(D$95$m * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.55e+194], N[(N[(1.0 - N[(0.5 * N[(N[(h * N[Power[N[(M$95$m * N[(0.5 * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[(d / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < -4.999999999999985e-310

   1. Initial program 66.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 67.4%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r/ 70.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot h}{\ell}}\right) \]

      2. frac-times 70.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot h}{\ell}\right) \]

      3. associate-/l* 70.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)}}^{2} \cdot h}{\ell}\right) \]

      4. *-un-lft-identity 70.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(M \cdot \frac{\color{blue}{1 \cdot D}}{2 \cdot d}\right)}^{2} \cdot h}{\ell}\right) \]

      5. times-frac 70.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(M \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{D}{d}\right)}\right)}^{2} \cdot h}{\ell}\right) \]

      6. metadata-eval 70.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(M \cdot \left(\color{blue}{0.5} \cdot \frac{D}{d}\right)\right)}^{2} \cdot h}{\ell}\right) \]

   6. Applied egg-rr 70.6%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot h}{\ell}}\right) \]
   7. Step-by-step derivation

      1. clear-num 70.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\color{blue}{\frac{1}{\frac{\ell}{d}}}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot h}{\ell}\right) \]

      2. sqrt-div 70.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot h}{\ell}\right) \]

      3. metadata-eval 70.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \frac{\color{blue}{1}}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot h}{\ell}\right) \]

   8. Applied egg-rr 70.5%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\frac{1}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot h}{\ell}\right) \]

   9. Taylor expanded in d around -inf 67.1%

      \[\leadsto \color{blue}{\left(-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)\right)} \cdot \left(1 - 0.5 \cdot \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot h}{\ell}\right) \]
   10. Step-by-step derivation

       1. mul-1-neg 67.1%

          \[\leadsto \color{blue}{\left(-d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot \left(1 - 0.5 \cdot \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot h}{\ell}\right) \]

       2. unpow-1 67.1%

          \[\leadsto \left(-d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot h}{\ell}\right) \]

       3. metadata-eval 67.1%

          \[\leadsto \left(-d \cdot \sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot h}{\ell}\right) \]

       4. pow-sqr 67.1%

          \[\leadsto \left(-d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot h}{\ell}\right) \]

       5. rem-sqrt-square 67.6%

          \[\leadsto \left(-d \cdot \color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot h}{\ell}\right) \]

       6. metadata-eval 67.6%

          \[\leadsto \left(-d \cdot \left|{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.25\right)}}\right|\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot h}{\ell}\right) \]

       7. pow-sqr 67.4%

          \[\leadsto \left(-d \cdot \left|\color{blue}{{\left(h \cdot \ell\right)}^{-0.25} \cdot {\left(h \cdot \ell\right)}^{-0.25}}\right|\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot h}{\ell}\right) \]

       8. fabs-sqr 67.4%

          \[\leadsto \left(-d \cdot \color{blue}{\left({\left(h \cdot \ell\right)}^{-0.25} \cdot {\left(h \cdot \ell\right)}^{-0.25}\right)}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot h}{\ell}\right) \]

       9. pow-sqr 67.6%

          \[\leadsto \left(-d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{\left(2 \cdot -0.25\right)}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot h}{\ell}\right) \]

       10. metadata-eval 67.6%

           \[\leadsto \left(-d \cdot {\left(h \cdot \ell\right)}^{\color{blue}{-0.5}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot h}{\ell}\right) \]

       11. distribute-rgt-neg-in 67.6%

           \[\leadsto \color{blue}{\left(d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\right)} \cdot \left(1 - 0.5 \cdot \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot h}{\ell}\right) \]

   11. Simplified 67.6%

       \[\leadsto \color{blue}{\left(d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\right)} \cdot \left(1 - 0.5 \cdot \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot h}{\ell}\right) \]
   12. Step-by-step derivation

       1. *-un-lft-identity 70.6%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(1 \cdot \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot h}{\ell}\right)}\right) \]

       2. associate-/l* 67.4%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(1 \cdot \color{blue}{\left({\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}\right)}\right)\right) \]

       3. associate-*r* 67.4%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(1 \cdot \left({\color{blue}{\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

       4. metadata-eval 67.4%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(1 \cdot \left({\left(\left(M \cdot \color{blue}{\frac{1}{2}}\right) \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

       5. div-inv 67.4%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(1 \cdot \left({\left(\color{blue}{\frac{M}{2}} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

       6. *-commutative 67.4%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(1 \cdot \left({\color{blue}{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

       7. div-inv 67.4%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(1 \cdot \left({\left(\frac{D}{d} \cdot \color{blue}{\left(M \cdot \frac{1}{2}\right)}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

       8. metadata-eval 67.4%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(1 \cdot \left({\left(\frac{D}{d} \cdot \left(M \cdot \color{blue}{0.5}\right)\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

   13. Applied egg-rr 62.8%

       \[\leadsto \left(d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(1 \cdot \left({\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2} \cdot \frac{h}{\ell}\right)\right)}\right) \]
   14. Step-by-step derivation

       1. *-lft-identity 67.4%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left({\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2} \cdot \frac{h}{\ell}\right)}\right) \]

       2. *-commutative 67.4%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2}\right)}\right) \]

       3. associate-*l/ 70.6%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{h \cdot {\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2}}{\ell}}\right) \]

       4. associate-/l* 69.2%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(h \cdot \frac{{\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2}}{\ell}\right)}\right) \]

       5. unpow2 69.2%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{\color{blue}{\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right) \cdot \left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}}{\ell}\right)\right) \]

       6. associate-*l/ 68.5%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{\color{blue}{\frac{D \cdot \left(M \cdot 0.5\right)}{d}} \cdot \left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}{\ell}\right)\right) \]

       7. associate-*r* 68.5%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{\frac{\color{blue}{\left(D \cdot M\right) \cdot 0.5}}{d} \cdot \left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}{\ell}\right)\right) \]

       8. associate-*l/ 68.5%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{\color{blue}{\left(\frac{D \cdot M}{d} \cdot 0.5\right)} \cdot \left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}{\ell}\right)\right) \]

       9. *-commutative 68.5%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{\color{blue}{\left(0.5 \cdot \frac{D \cdot M}{d}\right)} \cdot \left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}{\ell}\right)\right) \]

       10. associate-*l/ 69.2%

           \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{\left(0.5 \cdot \frac{D \cdot M}{d}\right) \cdot \color{blue}{\frac{D \cdot \left(M \cdot 0.5\right)}{d}}}{\ell}\right)\right) \]

       11. associate-*r* 69.2%

           \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{\left(0.5 \cdot \frac{D \cdot M}{d}\right) \cdot \frac{\color{blue}{\left(D \cdot M\right) \cdot 0.5}}{d}}{\ell}\right)\right) \]

       12. associate-*l/ 69.2%

           \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{\left(0.5 \cdot \frac{D \cdot M}{d}\right) \cdot \color{blue}{\left(\frac{D \cdot M}{d} \cdot 0.5\right)}}{\ell}\right)\right) \]

       13. *-commutative 69.2%

           \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{\left(0.5 \cdot \frac{D \cdot M}{d}\right) \cdot \color{blue}{\left(0.5 \cdot \frac{D \cdot M}{d}\right)}}{\ell}\right)\right) \]

       14. swap-sqr 69.2%

           \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{\color{blue}{\left(0.5 \cdot 0.5\right) \cdot \left(\frac{D \cdot M}{d} \cdot \frac{D \cdot M}{d}\right)}}{\ell}\right)\right) \]

       15. metadata-eval 69.2%

           \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{\color{blue}{0.25} \cdot \left(\frac{D \cdot M}{d} \cdot \frac{D \cdot M}{d}\right)}{\ell}\right)\right) \]

       16. unpow2 69.2%

           \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{0.25 \cdot \color{blue}{{\left(\frac{D \cdot M}{d}\right)}^{2}}}{\ell}\right)\right) \]

       17. associate-/l* 70.0%

           \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{0.25 \cdot {\color{blue}{\left(D \cdot \frac{M}{d}\right)}}^{2}}{\ell}\right)\right) \]

   15. Simplified 68.3%

       \[\leadsto \left(d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(h \cdot \frac{0.25 \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}}{\ell}\right)}\right) \]

   if -4.999999999999985e-310 < l < 1.55e194

   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 68.4%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r/ 75.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot h}{\ell}}\right) \]

      2. frac-times 75.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot h}{\ell}\right) \]

      3. associate-/l* 75.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)}}^{2} \cdot h}{\ell}\right) \]

      4. *-un-lft-identity 75.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(M \cdot \frac{\color{blue}{1 \cdot D}}{2 \cdot d}\right)}^{2} \cdot h}{\ell}\right) \]

      5. times-frac 75.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(M \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{D}{d}\right)}\right)}^{2} \cdot h}{\ell}\right) \]

      6. metadata-eval 75.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(M \cdot \left(\color{blue}{0.5} \cdot \frac{D}{d}\right)\right)}^{2} \cdot h}{\ell}\right) \]

   6. Applied egg-rr 75.0%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot h}{\ell}}\right) \]
   7. Step-by-step derivation

      1. clear-num 75.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\color{blue}{\frac{1}{\frac{\ell}{d}}}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot h}{\ell}\right) \]

      2. sqrt-div 75.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot h}{\ell}\right) \]

      3. metadata-eval 75.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \frac{\color{blue}{1}}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot h}{\ell}\right) \]

   8. Applied egg-rr 75.8%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\frac{1}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot h}{\ell}\right) \]

   9. Taylor expanded in d around 0 79.6%

      \[\leadsto \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot \left(1 - 0.5 \cdot \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot h}{\ell}\right) \]
   10. Step-by-step derivation

       1. unpow-1 79.6%

          \[\leadsto \left(d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot h}{\ell}\right) \]

       2. metadata-eval 79.6%

          \[\leadsto \left(d \cdot \sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot h}{\ell}\right) \]

       3. pow-sqr 79.6%

          \[\leadsto \left(d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot h}{\ell}\right) \]

       4. rem-sqrt-square 80.2%

          \[\leadsto \left(d \cdot \color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot h}{\ell}\right) \]

       5. metadata-eval 80.2%

          \[\leadsto \left(d \cdot \left|{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.25\right)}}\right|\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot h}{\ell}\right) \]

       6. pow-sqr 80.1%

          \[\leadsto \left(d \cdot \left|\color{blue}{{\left(h \cdot \ell\right)}^{-0.25} \cdot {\left(h \cdot \ell\right)}^{-0.25}}\right|\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot h}{\ell}\right) \]

       7. fabs-sqr 80.1%

          \[\leadsto \left(d \cdot \color{blue}{\left({\left(h \cdot \ell\right)}^{-0.25} \cdot {\left(h \cdot \ell\right)}^{-0.25}\right)}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot h}{\ell}\right) \]

       8. pow-sqr 80.2%

          \[\leadsto \left(d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{\left(2 \cdot -0.25\right)}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot h}{\ell}\right) \]

       9. metadata-eval 80.2%

          \[\leadsto \left(d \cdot {\left(h \cdot \ell\right)}^{\color{blue}{-0.5}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot h}{\ell}\right) \]

   11. Simplified 80.2%

       \[\leadsto \color{blue}{\left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right)} \cdot \left(1 - 0.5 \cdot \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot h}{\ell}\right) \]

   if 1.55e194 < l

   1. Initial program 62.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 58.4%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 58.4%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. sqrt-div 58.3%

         \[\leadsto d \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{h \cdot \ell}}} \]

      2. metadata-eval 58.3%

         \[\leadsto d \cdot \frac{\color{blue}{1}}{\sqrt{h \cdot \ell}} \]

      3. *-commutative 58.3%

         \[\leadsto d \cdot \frac{1}{\sqrt{\color{blue}{\ell \cdot h}}} \]

      4. sqrt-unprod 89.9%

         \[\leadsto d \cdot \frac{1}{\color{blue}{\sqrt{\ell} \cdot \sqrt{h}}} \]

      5. div-inv 89.9%

         \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}} \]

      6. associate-/r* 90.2%

         \[\leadsto \color{blue}{\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}} \]

   7. Applied egg-rr 90.2%

      \[\leadsto \color{blue}{\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right) \cdot \left(-1 + 0.5 \cdot \left(h \cdot \frac{0.25 \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}}{\ell}\right)\right)\\ \mathbf{elif}\;\ell \leq 1.55 \cdot 10^{+194}:\\ \;\;\;\;\left(1 - 0.5 \cdot \frac{h \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell}\right) \cdot \left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 49.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -2.1 \cdot 10^{-48}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{h}}{\ell}}\right)\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{0}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{h}^{-0.5}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

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 (<= d -2.1e-48)
   (* d (- (sqrt (/ (/ 1.0 h) l))))
   (if (<= d -5e-310)
     (* d (sqrt (/ 1.0 0.0)))
     (* d (/ (pow h -0.5) (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 (d <= -2.1e-48) {
		tmp = d * -sqrt(((1.0 / h) / l));
	} else if (d <= -5e-310) {
		tmp = d * sqrt((1.0 / 0.0));
	} else {
		tmp = d * (pow(h, -0.5) / 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 (d <= (-2.1d-48)) then
        tmp = d * -sqrt(((1.0d0 / h) / l))
    else if (d <= (-5d-310)) then
        tmp = d * sqrt((1.0d0 / 0.0d0))
    else
        tmp = d * ((h ** (-0.5d0)) / 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 (d <= -2.1e-48) {
		tmp = d * -Math.sqrt(((1.0 / h) / l));
	} else if (d <= -5e-310) {
		tmp = d * Math.sqrt((1.0 / 0.0));
	} else {
		tmp = d * (Math.pow(h, -0.5) / 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 d <= -2.1e-48:
		tmp = d * -math.sqrt(((1.0 / h) / l))
	elif d <= -5e-310:
		tmp = d * math.sqrt((1.0 / 0.0))
	else:
		tmp = d * (math.pow(h, -0.5) / 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 (d <= -2.1e-48)
		tmp = Float64(d * Float64(-sqrt(Float64(Float64(1.0 / h) / l))));
	elseif (d <= -5e-310)
		tmp = Float64(d * sqrt(Float64(1.0 / 0.0)));
	else
		tmp = Float64(d * Float64((h ^ -0.5) / 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 (d <= -2.1e-48)
		tmp = d * -sqrt(((1.0 / h) / l));
	elseif (d <= -5e-310)
		tmp = d * sqrt((1.0 / 0.0));
	else
		tmp = d * ((h ^ -0.5) / 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[d, -2.1e-48], N[(d * (-N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], If[LessEqual[d, -5e-310], N[(d * N[Sqrt[N[(1.0 / 0.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[h, -0.5], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -2.09999999999999989e-48

   1. Initial program 79.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 81.0%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 0.7%

      \[\leadsto \color{blue}{d \cdot \left(\sqrt{\frac{1}{h \cdot \ell}} + -0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative 0.7%

         \[\leadsto d \cdot \color{blue}{\left(-0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) + \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. fma-define 0.7%

         \[\leadsto d \cdot \color{blue}{\mathsf{fma}\left(-0.125, \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}} \cdot \sqrt{\frac{h}{{\ell}^{3}}}, \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      3. *-commutative 0.7%

         \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}}}, \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      4. *-commutative 0.7%

         \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot \frac{\color{blue}{{M}^{2} \cdot {D}^{2}}}{{d}^{2}}, \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      5. associate-/l* 0.8%

         \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot \color{blue}{\left({M}^{2} \cdot \frac{{D}^{2}}{{d}^{2}}\right)}, \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      6. unpow2 0.8%

         \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\color{blue}{\left(M \cdot M\right)} \cdot \frac{{D}^{2}}{{d}^{2}}\right), \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      7. unpow2 0.8%

         \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\left(M \cdot M\right) \cdot \frac{\color{blue}{D \cdot D}}{{d}^{2}}\right), \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      8. unpow2 0.8%

         \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\left(M \cdot M\right) \cdot \frac{D \cdot D}{\color{blue}{d \cdot d}}\right), \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      9. times-frac 0.9%

         \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\left(M \cdot M\right) \cdot \color{blue}{\left(\frac{D}{d} \cdot \frac{D}{d}\right)}\right), \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      10. swap-sqr 1.0%

          \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot \color{blue}{\left(\left(M \cdot \frac{D}{d}\right) \cdot \left(M \cdot \frac{D}{d}\right)\right)}, \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      11. unpow2 1.0%

          \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot \color{blue}{{\left(M \cdot \frac{D}{d}\right)}^{2}}, \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      12. associate-*r/ 1.0%

          \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot {\color{blue}{\left(\frac{M \cdot D}{d}\right)}}^{2}, \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      13. *-commutative 1.0%

          \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot {\left(\frac{\color{blue}{D \cdot M}}{d}\right)}^{2}, \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      14. associate-/l* 1.0%

          \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot {\color{blue}{\left(D \cdot \frac{M}{d}\right)}}^{2}, \sqrt{\frac{1}{h \cdot \ell}}\right) \]

   7. Simplified 1.0%

      \[\leadsto \color{blue}{d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}, \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

   8. Taylor expanded in h around 0 3.6%

      \[\leadsto d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]
   9. Step-by-step derivation

      1. associate-/r* 3.6%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \]

      2. unpow1/2 3.6%

         \[\leadsto d \cdot \color{blue}{{\left(\frac{\frac{1}{h}}{\ell}\right)}^{0.5}} \]

      3. associate-/r* 3.6%

         \[\leadsto d \cdot {\color{blue}{\left(\frac{1}{h \cdot \ell}\right)}}^{0.5} \]

      4. rem-exp-log 3.6%

         \[\leadsto d \cdot {\left(\frac{1}{\color{blue}{e^{\log \left(h \cdot \ell\right)}}}\right)}^{0.5} \]

      5. exp-neg 3.6%

         \[\leadsto d \cdot {\color{blue}{\left(e^{-\log \left(h \cdot \ell\right)}\right)}}^{0.5} \]

      6. exp-prod 3.6%

         \[\leadsto d \cdot \color{blue}{e^{\left(-\log \left(h \cdot \ell\right)\right) \cdot 0.5}} \]

      7. distribute-lft-neg-out 3.6%

         \[\leadsto d \cdot e^{\color{blue}{-\log \left(h \cdot \ell\right) \cdot 0.5}} \]

      8. distribute-rgt-neg-in 3.6%

         \[\leadsto d \cdot e^{\color{blue}{\log \left(h \cdot \ell\right) \cdot \left(-0.5\right)}} \]

      9. metadata-eval 3.6%

         \[\leadsto d \cdot e^{\log \left(h \cdot \ell\right) \cdot \color{blue}{-0.5}} \]

      10. exp-to-pow 3.6%

          \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

   10. Simplified 3.6%

       \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]
   11. Step-by-step derivation

       1. unpow-prod-down 0.0%

          \[\leadsto d \cdot \color{blue}{\left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)} \]

       2. metadata-eval 0.0%

          \[\leadsto d \cdot \left({h}^{\color{blue}{\left(\frac{-1}{2}\right)}} \cdot {\ell}^{-0.5}\right) \]

       3. sqrt-pow1 0.0%

          \[\leadsto d \cdot \left(\color{blue}{\sqrt{{h}^{-1}}} \cdot {\ell}^{-0.5}\right) \]

       4. inv-pow 0.0%

          \[\leadsto d \cdot \left(\sqrt{\color{blue}{\frac{1}{h}}} \cdot {\ell}^{-0.5}\right) \]

       5. metadata-eval 0.0%

          \[\leadsto d \cdot \left(\sqrt{\frac{1}{h}} \cdot {\ell}^{\color{blue}{\left(\frac{-1}{2}\right)}}\right) \]

       6. sqrt-pow1 0.0%

          \[\leadsto d \cdot \left(\sqrt{\frac{1}{h}} \cdot \color{blue}{\sqrt{{\ell}^{-1}}}\right) \]

       7. inv-pow 0.0%

          \[\leadsto d \cdot \left(\sqrt{\frac{1}{h}} \cdot \sqrt{\color{blue}{\frac{1}{\ell}}}\right) \]

       8. sqrt-prod 3.6%

          \[\leadsto d \cdot \color{blue}{\sqrt{\frac{1}{h} \cdot \frac{1}{\ell}}} \]

       9. div-inv 3.6%

          \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \]

       10. sqrt-div 0.0%

           \[\leadsto d \cdot \color{blue}{\frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}} \]

       11. inv-pow 0.0%

           \[\leadsto d \cdot \frac{\sqrt{\color{blue}{{h}^{-1}}}}{\sqrt{\ell}} \]

       12. sqrt-pow1 0.0%

           \[\leadsto d \cdot \frac{\color{blue}{{h}^{\left(\frac{-1}{2}\right)}}}{\sqrt{\ell}} \]

       13. metadata-eval 0.0%

           \[\leadsto d \cdot \frac{{h}^{\color{blue}{-0.5}}}{\sqrt{\ell}} \]

   12. Applied egg-rr 0.0%

       \[\leadsto d \cdot \color{blue}{\frac{{h}^{-0.5}}{\sqrt{\ell}}} \]

   13. Taylor expanded in h around -inf 0.0%

       \[\leadsto d \cdot \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]
   14. Step-by-step derivation

       1. *-commutative 0.0%

          \[\leadsto d \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

       2. unpow2 0.0%

          \[\leadsto d \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

       3. rem-square-sqrt 56.4%

          \[\leadsto d \cdot \left(\color{blue}{-1} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

       4. mul-1-neg 56.4%

          \[\leadsto d \cdot \color{blue}{\left(-\sqrt{\frac{1}{h \cdot \ell}}\right)} \]

       5. associate-/r* 57.5%

          \[\leadsto d \cdot \left(-\sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}}\right) \]

   15. Simplified 57.5%

       \[\leadsto d \cdot \color{blue}{\left(-\sqrt{\frac{\frac{1}{h}}{\ell}}\right)} \]

   if -2.09999999999999989e-48 < d < -4.999999999999985e-310

   1. Initial program 47.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 47.7%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 18.0%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. add-log-exp 24.9%

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\log \left(e^{h \cdot \ell}\right)}}} \]

      2. *-commutative 24.9%

         \[\leadsto d \cdot \sqrt{\frac{1}{\log \left(e^{\color{blue}{\ell \cdot h}}\right)}} \]

      3. exp-prod 31.0%

         \[\leadsto d \cdot \sqrt{\frac{1}{\log \color{blue}{\left({\left(e^{\ell}\right)}^{h}\right)}}} \]

   7. Applied egg-rr 31.0%

      \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\log \left({\left(e^{\ell}\right)}^{h}\right)}}} \]

   8. Taylor expanded in l around 0 49.1%

      \[\leadsto d \cdot \sqrt{\frac{1}{\log \color{blue}{1}}} \]

   if -4.999999999999985e-310 < d

   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 66.7%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 26.2%

      \[\leadsto \color{blue}{d \cdot \left(\sqrt{\frac{1}{h \cdot \ell}} + -0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative 26.2%

         \[\leadsto d \cdot \color{blue}{\left(-0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) + \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. fma-define 26.2%

         \[\leadsto d \cdot \color{blue}{\mathsf{fma}\left(-0.125, \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}} \cdot \sqrt{\frac{h}{{\ell}^{3}}}, \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      3. *-commutative 26.2%

         \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}}}, \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      4. *-commutative 26.2%

         \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot \frac{\color{blue}{{M}^{2} \cdot {D}^{2}}}{{d}^{2}}, \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      5. associate-/l* 25.7%

         \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot \color{blue}{\left({M}^{2} \cdot \frac{{D}^{2}}{{d}^{2}}\right)}, \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      6. unpow2 25.7%

         \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\color{blue}{\left(M \cdot M\right)} \cdot \frac{{D}^{2}}{{d}^{2}}\right), \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      7. unpow2 25.7%

         \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\left(M \cdot M\right) \cdot \frac{\color{blue}{D \cdot D}}{{d}^{2}}\right), \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      8. unpow2 25.7%

         \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\left(M \cdot M\right) \cdot \frac{D \cdot D}{\color{blue}{d \cdot d}}\right), \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      9. times-frac 39.2%

         \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\left(M \cdot M\right) \cdot \color{blue}{\left(\frac{D}{d} \cdot \frac{D}{d}\right)}\right), \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      10. swap-sqr 49.6%

          \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot \color{blue}{\left(\left(M \cdot \frac{D}{d}\right) \cdot \left(M \cdot \frac{D}{d}\right)\right)}, \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      11. unpow2 49.6%

          \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot \color{blue}{{\left(M \cdot \frac{D}{d}\right)}^{2}}, \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      12. associate-*r/ 49.6%

          \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot {\color{blue}{\left(\frac{M \cdot D}{d}\right)}}^{2}, \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      13. *-commutative 49.6%

          \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot {\left(\frac{\color{blue}{D \cdot M}}{d}\right)}^{2}, \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      14. associate-/l* 49.6%

          \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot {\color{blue}{\left(D \cdot \frac{M}{d}\right)}}^{2}, \sqrt{\frac{1}{h \cdot \ell}}\right) \]

   7. Simplified 49.6%

      \[\leadsto \color{blue}{d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}, \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

   8. Taylor expanded in h around 0 43.5%

      \[\leadsto d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]
   9. Step-by-step derivation

      1. associate-/r* 44.9%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \]

      2. unpow1/2 44.9%

         \[\leadsto d \cdot \color{blue}{{\left(\frac{\frac{1}{h}}{\ell}\right)}^{0.5}} \]

      3. associate-/r* 43.5%

         \[\leadsto d \cdot {\color{blue}{\left(\frac{1}{h \cdot \ell}\right)}}^{0.5} \]

      4. rem-exp-log 41.4%

         \[\leadsto d \cdot {\left(\frac{1}{\color{blue}{e^{\log \left(h \cdot \ell\right)}}}\right)}^{0.5} \]

      5. exp-neg 41.4%

         \[\leadsto d \cdot {\color{blue}{\left(e^{-\log \left(h \cdot \ell\right)}\right)}}^{0.5} \]

      6. exp-prod 41.9%

         \[\leadsto d \cdot \color{blue}{e^{\left(-\log \left(h \cdot \ell\right)\right) \cdot 0.5}} \]

      7. distribute-lft-neg-out 41.9%

         \[\leadsto d \cdot e^{\color{blue}{-\log \left(h \cdot \ell\right) \cdot 0.5}} \]

      8. distribute-rgt-neg-in 41.9%

         \[\leadsto d \cdot e^{\color{blue}{\log \left(h \cdot \ell\right) \cdot \left(-0.5\right)}} \]

      9. metadata-eval 41.9%

         \[\leadsto d \cdot e^{\log \left(h \cdot \ell\right) \cdot \color{blue}{-0.5}} \]

      10. exp-to-pow 44.0%

          \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

   10. Simplified 44.0%

       \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]
   11. Step-by-step derivation

       1. unpow-prod-down 52.6%

          \[\leadsto d \cdot \color{blue}{\left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)} \]

       2. metadata-eval 52.6%

          \[\leadsto d \cdot \left({h}^{\color{blue}{\left(\frac{-1}{2}\right)}} \cdot {\ell}^{-0.5}\right) \]

       3. sqrt-pow1 52.5%

          \[\leadsto d \cdot \left(\color{blue}{\sqrt{{h}^{-1}}} \cdot {\ell}^{-0.5}\right) \]

       4. inv-pow 52.5%

          \[\leadsto d \cdot \left(\sqrt{\color{blue}{\frac{1}{h}}} \cdot {\ell}^{-0.5}\right) \]

       5. metadata-eval 52.5%

          \[\leadsto d \cdot \left(\sqrt{\frac{1}{h}} \cdot {\ell}^{\color{blue}{\left(\frac{-1}{2}\right)}}\right) \]

       6. sqrt-pow1 52.6%

          \[\leadsto d \cdot \left(\sqrt{\frac{1}{h}} \cdot \color{blue}{\sqrt{{\ell}^{-1}}}\right) \]

       7. inv-pow 52.6%

          \[\leadsto d \cdot \left(\sqrt{\frac{1}{h}} \cdot \sqrt{\color{blue}{\frac{1}{\ell}}}\right) \]

       8. sqrt-prod 44.9%

          \[\leadsto d \cdot \color{blue}{\sqrt{\frac{1}{h} \cdot \frac{1}{\ell}}} \]

       9. div-inv 44.9%

          \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \]

       10. sqrt-div 52.5%

           \[\leadsto d \cdot \color{blue}{\frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}} \]

       11. inv-pow 52.5%

           \[\leadsto d \cdot \frac{\sqrt{\color{blue}{{h}^{-1}}}}{\sqrt{\ell}} \]

       12. sqrt-pow1 52.5%

           \[\leadsto d \cdot \frac{\color{blue}{{h}^{\left(\frac{-1}{2}\right)}}}{\sqrt{\ell}} \]

       13. metadata-eval 52.5%

           \[\leadsto d \cdot \frac{{h}^{\color{blue}{-0.5}}}{\sqrt{\ell}} \]

   12. Applied egg-rr 52.5%

       \[\leadsto d \cdot \color{blue}{\frac{{h}^{-0.5}}{\sqrt{\ell}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 53.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.1 \cdot 10^{-48}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{h}}{\ell}}\right)\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{0}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{h}^{-0.5}}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 49.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -1.65 \cdot 10^{-53}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{h}}{\ell}}\right)\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
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 -1.65e-53)
   (* d (- (sqrt (/ (/ 1.0 h) l))))
   (if (<= d -5e-310) (* d (sqrt (/ 1.0 0.0))) (/ d (* (sqrt l) (sqrt h))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= -1.65e-53) {
		tmp = d * -sqrt(((1.0 / h) / l));
	} else if (d <= -5e-310) {
		tmp = d * sqrt((1.0 / 0.0));
	} else {
		tmp = d / (sqrt(l) * sqrt(h));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (d <= (-1.65d-53)) then
        tmp = d * -sqrt(((1.0d0 / h) / l))
    else if (d <= (-5d-310)) then
        tmp = d * sqrt((1.0d0 / 0.0d0))
    else
        tmp = d / (sqrt(l) * sqrt(h))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
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 <= -1.65e-53) {
		tmp = d * -Math.sqrt(((1.0 / h) / l));
	} else if (d <= -5e-310) {
		tmp = d * Math.sqrt((1.0 / 0.0));
	} else {
		tmp = d / (Math.sqrt(l) * Math.sqrt(h));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_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 <= -1.65e-53:
		tmp = d * -math.sqrt(((1.0 / h) / l))
	elif d <= -5e-310:
		tmp = d * math.sqrt((1.0 / 0.0))
	else:
		tmp = d / (math.sqrt(l) * math.sqrt(h))
	return tmp

Julia

M_m = abs(M)
D_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 <= -1.65e-53)
		tmp = Float64(d * Float64(-sqrt(Float64(Float64(1.0 / h) / l))));
	elseif (d <= -5e-310)
		tmp = Float64(d * sqrt(Float64(1.0 / 0.0)));
	else
		tmp = Float64(d / Float64(sqrt(l) * sqrt(h)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_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 <= -1.65e-53)
		tmp = d * -sqrt(((1.0 / h) / l));
	elseif (d <= -5e-310)
		tmp = d * sqrt((1.0 / 0.0));
	else
		tmp = d / (sqrt(l) * sqrt(h));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
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, -1.65e-53], N[(d * (-N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], If[LessEqual[d, -5e-310], N[(d * N[Sqrt[N[(1.0 / 0.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.65000000000000002e-53

   1. Initial program 79.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 81.0%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 0.7%

      \[\leadsto \color{blue}{d \cdot \left(\sqrt{\frac{1}{h \cdot \ell}} + -0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative 0.7%

         \[\leadsto d \cdot \color{blue}{\left(-0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) + \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. fma-define 0.7%

         \[\leadsto d \cdot \color{blue}{\mathsf{fma}\left(-0.125, \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}} \cdot \sqrt{\frac{h}{{\ell}^{3}}}, \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      3. *-commutative 0.7%

         \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}}}, \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      4. *-commutative 0.7%

         \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot \frac{\color{blue}{{M}^{2} \cdot {D}^{2}}}{{d}^{2}}, \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      5. associate-/l* 0.8%

         \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot \color{blue}{\left({M}^{2} \cdot \frac{{D}^{2}}{{d}^{2}}\right)}, \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      6. unpow2 0.8%

         \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\color{blue}{\left(M \cdot M\right)} \cdot \frac{{D}^{2}}{{d}^{2}}\right), \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      7. unpow2 0.8%

         \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\left(M \cdot M\right) \cdot \frac{\color{blue}{D \cdot D}}{{d}^{2}}\right), \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      8. unpow2 0.8%

         \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\left(M \cdot M\right) \cdot \frac{D \cdot D}{\color{blue}{d \cdot d}}\right), \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      9. times-frac 0.9%

         \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\left(M \cdot M\right) \cdot \color{blue}{\left(\frac{D}{d} \cdot \frac{D}{d}\right)}\right), \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      10. swap-sqr 1.0%

          \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot \color{blue}{\left(\left(M \cdot \frac{D}{d}\right) \cdot \left(M \cdot \frac{D}{d}\right)\right)}, \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      11. unpow2 1.0%

          \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot \color{blue}{{\left(M \cdot \frac{D}{d}\right)}^{2}}, \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      12. associate-*r/ 1.0%

          \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot {\color{blue}{\left(\frac{M \cdot D}{d}\right)}}^{2}, \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      13. *-commutative 1.0%

          \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot {\left(\frac{\color{blue}{D \cdot M}}{d}\right)}^{2}, \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      14. associate-/l* 1.0%

          \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot {\color{blue}{\left(D \cdot \frac{M}{d}\right)}}^{2}, \sqrt{\frac{1}{h \cdot \ell}}\right) \]

   7. Simplified 1.0%

      \[\leadsto \color{blue}{d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}, \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

   8. Taylor expanded in h around 0 3.6%

      \[\leadsto d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]
   9. Step-by-step derivation

      1. associate-/r* 3.6%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \]

      2. unpow1/2 3.6%

         \[\leadsto d \cdot \color{blue}{{\left(\frac{\frac{1}{h}}{\ell}\right)}^{0.5}} \]

      3. associate-/r* 3.6%

         \[\leadsto d \cdot {\color{blue}{\left(\frac{1}{h \cdot \ell}\right)}}^{0.5} \]

      4. rem-exp-log 3.6%

         \[\leadsto d \cdot {\left(\frac{1}{\color{blue}{e^{\log \left(h \cdot \ell\right)}}}\right)}^{0.5} \]

      5. exp-neg 3.6%

         \[\leadsto d \cdot {\color{blue}{\left(e^{-\log \left(h \cdot \ell\right)}\right)}}^{0.5} \]

      6. exp-prod 3.6%

         \[\leadsto d \cdot \color{blue}{e^{\left(-\log \left(h \cdot \ell\right)\right) \cdot 0.5}} \]

      7. distribute-lft-neg-out 3.6%

         \[\leadsto d \cdot e^{\color{blue}{-\log \left(h \cdot \ell\right) \cdot 0.5}} \]

      8. distribute-rgt-neg-in 3.6%

         \[\leadsto d \cdot e^{\color{blue}{\log \left(h \cdot \ell\right) \cdot \left(-0.5\right)}} \]

      9. metadata-eval 3.6%

         \[\leadsto d \cdot e^{\log \left(h \cdot \ell\right) \cdot \color{blue}{-0.5}} \]

      10. exp-to-pow 3.6%

          \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

   10. Simplified 3.6%

       \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]
   11. Step-by-step derivation

       1. unpow-prod-down 0.0%

          \[\leadsto d \cdot \color{blue}{\left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)} \]

       2. metadata-eval 0.0%

          \[\leadsto d \cdot \left({h}^{\color{blue}{\left(\frac{-1}{2}\right)}} \cdot {\ell}^{-0.5}\right) \]

       3. sqrt-pow1 0.0%

          \[\leadsto d \cdot \left(\color{blue}{\sqrt{{h}^{-1}}} \cdot {\ell}^{-0.5}\right) \]

       4. inv-pow 0.0%

          \[\leadsto d \cdot \left(\sqrt{\color{blue}{\frac{1}{h}}} \cdot {\ell}^{-0.5}\right) \]

       5. metadata-eval 0.0%

          \[\leadsto d \cdot \left(\sqrt{\frac{1}{h}} \cdot {\ell}^{\color{blue}{\left(\frac{-1}{2}\right)}}\right) \]

       6. sqrt-pow1 0.0%

          \[\leadsto d \cdot \left(\sqrt{\frac{1}{h}} \cdot \color{blue}{\sqrt{{\ell}^{-1}}}\right) \]

       7. inv-pow 0.0%

          \[\leadsto d \cdot \left(\sqrt{\frac{1}{h}} \cdot \sqrt{\color{blue}{\frac{1}{\ell}}}\right) \]

       8. sqrt-prod 3.6%

          \[\leadsto d \cdot \color{blue}{\sqrt{\frac{1}{h} \cdot \frac{1}{\ell}}} \]

       9. div-inv 3.6%

          \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \]

       10. sqrt-div 0.0%

           \[\leadsto d \cdot \color{blue}{\frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}} \]

       11. inv-pow 0.0%

           \[\leadsto d \cdot \frac{\sqrt{\color{blue}{{h}^{-1}}}}{\sqrt{\ell}} \]

       12. sqrt-pow1 0.0%

           \[\leadsto d \cdot \frac{\color{blue}{{h}^{\left(\frac{-1}{2}\right)}}}{\sqrt{\ell}} \]

       13. metadata-eval 0.0%

           \[\leadsto d \cdot \frac{{h}^{\color{blue}{-0.5}}}{\sqrt{\ell}} \]

   12. Applied egg-rr 0.0%

       \[\leadsto d \cdot \color{blue}{\frac{{h}^{-0.5}}{\sqrt{\ell}}} \]

   13. Taylor expanded in h around -inf 0.0%

       \[\leadsto d \cdot \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]
   14. Step-by-step derivation

       1. *-commutative 0.0%

          \[\leadsto d \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

       2. unpow2 0.0%

          \[\leadsto d \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

       3. rem-square-sqrt 56.4%

          \[\leadsto d \cdot \left(\color{blue}{-1} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

       4. mul-1-neg 56.4%

          \[\leadsto d \cdot \color{blue}{\left(-\sqrt{\frac{1}{h \cdot \ell}}\right)} \]

       5. associate-/r* 57.5%

          \[\leadsto d \cdot \left(-\sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}}\right) \]

   15. Simplified 57.5%

       \[\leadsto d \cdot \color{blue}{\left(-\sqrt{\frac{\frac{1}{h}}{\ell}}\right)} \]

   if -1.65000000000000002e-53 < d < -4.999999999999985e-310

   1. Initial program 47.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 47.7%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 18.0%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. add-log-exp 24.9%

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\log \left(e^{h \cdot \ell}\right)}}} \]

      2. *-commutative 24.9%

         \[\leadsto d \cdot \sqrt{\frac{1}{\log \left(e^{\color{blue}{\ell \cdot h}}\right)}} \]

      3. exp-prod 31.0%

         \[\leadsto d \cdot \sqrt{\frac{1}{\log \color{blue}{\left({\left(e^{\ell}\right)}^{h}\right)}}} \]

   7. Applied egg-rr 31.0%

      \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\log \left({\left(e^{\ell}\right)}^{h}\right)}}} \]

   8. Taylor expanded in l around 0 49.1%

      \[\leadsto d \cdot \sqrt{\frac{1}{\log \color{blue}{1}}} \]

   if -4.999999999999985e-310 < d

   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 66.7%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 43.5%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. sqrt-div 44.0%

         \[\leadsto d \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{h \cdot \ell}}} \]

      2. metadata-eval 44.0%

         \[\leadsto d \cdot \frac{\color{blue}{1}}{\sqrt{h \cdot \ell}} \]

      3. *-commutative 44.0%

         \[\leadsto d \cdot \frac{1}{\sqrt{\color{blue}{\ell \cdot h}}} \]

      4. sqrt-unprod 52.5%

         \[\leadsto d \cdot \frac{1}{\color{blue}{\sqrt{\ell} \cdot \sqrt{h}}} \]

      5. div-inv 52.5%

         \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}} \]

      6. associate-/r* 50.5%

         \[\leadsto \color{blue}{\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}} \]

   7. Applied egg-rr 50.5%

      \[\leadsto \color{blue}{\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}} \]
   8. Step-by-step derivation

      1. associate-/l/ 52.5%

         \[\leadsto \color{blue}{\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}} \]

      2. *-commutative 52.5%

         \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell} \cdot \sqrt{h}}} \]

   9. Simplified 52.5%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 53.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.65 \cdot 10^{-53}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{h}}{\ell}}\right)\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 46.2% accurate, 2.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{if}\;d \leq -2.2 \cdot 10^{-51}:\\ \;\;\;\;d \cdot \left(-t\_0\right)\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{0}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot t\_0\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (sqrt (/ (/ 1.0 h) l))))
   (if (<= d -2.2e-51)
     (* d (- t_0))
     (if (<= d -5e-310) (* d (sqrt (/ 1.0 0.0))) (* d t_0)))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = sqrt(((1.0 / h) / l));
	double tmp;
	if (d <= -2.2e-51) {
		tmp = d * -t_0;
	} else if (d <= -5e-310) {
		tmp = d * sqrt((1.0 / 0.0));
	} else {
		tmp = d * t_0;
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(((1.0d0 / h) / l))
    if (d <= (-2.2d-51)) then
        tmp = d * -t_0
    else if (d <= (-5d-310)) then
        tmp = d * sqrt((1.0d0 / 0.0d0))
    else
        tmp = d * t_0
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = Math.sqrt(((1.0 / h) / l));
	double tmp;
	if (d <= -2.2e-51) {
		tmp = d * -t_0;
	} else if (d <= -5e-310) {
		tmp = d * Math.sqrt((1.0 / 0.0));
	} else {
		tmp = d * t_0;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = math.sqrt(((1.0 / h) / l))
	tmp = 0
	if d <= -2.2e-51:
		tmp = d * -t_0
	elif d <= -5e-310:
		tmp = d * math.sqrt((1.0 / 0.0))
	else:
		tmp = d * t_0
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = sqrt(Float64(Float64(1.0 / h) / l))
	tmp = 0.0
	if (d <= -2.2e-51)
		tmp = Float64(d * Float64(-t_0));
	elseif (d <= -5e-310)
		tmp = Float64(d * sqrt(Float64(1.0 / 0.0)));
	else
		tmp = Float64(d * t_0);
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = sqrt(((1.0 / h) / l));
	tmp = 0.0;
	if (d <= -2.2e-51)
		tmp = d * -t_0;
	elseif (d <= -5e-310)
		tmp = d * sqrt((1.0 / 0.0));
	else
		tmp = d * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[d, -2.2e-51], N[(d * (-t$95$0)), $MachinePrecision], If[LessEqual[d, -5e-310], N[(d * N[Sqrt[N[(1.0 / 0.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d * t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if d < -2.2e-51

   1. Initial program 79.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 81.0%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 0.7%

      \[\leadsto \color{blue}{d \cdot \left(\sqrt{\frac{1}{h \cdot \ell}} + -0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative 0.7%

         \[\leadsto d \cdot \color{blue}{\left(-0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) + \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. fma-define 0.7%

         \[\leadsto d \cdot \color{blue}{\mathsf{fma}\left(-0.125, \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}} \cdot \sqrt{\frac{h}{{\ell}^{3}}}, \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      3. *-commutative 0.7%

         \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}}}, \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      4. *-commutative 0.7%

         \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot \frac{\color{blue}{{M}^{2} \cdot {D}^{2}}}{{d}^{2}}, \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      5. associate-/l* 0.8%

         \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot \color{blue}{\left({M}^{2} \cdot \frac{{D}^{2}}{{d}^{2}}\right)}, \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      6. unpow2 0.8%

         \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\color{blue}{\left(M \cdot M\right)} \cdot \frac{{D}^{2}}{{d}^{2}}\right), \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      7. unpow2 0.8%

         \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\left(M \cdot M\right) \cdot \frac{\color{blue}{D \cdot D}}{{d}^{2}}\right), \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      8. unpow2 0.8%

         \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\left(M \cdot M\right) \cdot \frac{D \cdot D}{\color{blue}{d \cdot d}}\right), \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      9. times-frac 0.9%

         \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\left(M \cdot M\right) \cdot \color{blue}{\left(\frac{D}{d} \cdot \frac{D}{d}\right)}\right), \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      10. swap-sqr 1.0%

          \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot \color{blue}{\left(\left(M \cdot \frac{D}{d}\right) \cdot \left(M \cdot \frac{D}{d}\right)\right)}, \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      11. unpow2 1.0%

          \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot \color{blue}{{\left(M \cdot \frac{D}{d}\right)}^{2}}, \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      12. associate-*r/ 1.0%

          \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot {\color{blue}{\left(\frac{M \cdot D}{d}\right)}}^{2}, \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      13. *-commutative 1.0%

          \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot {\left(\frac{\color{blue}{D \cdot M}}{d}\right)}^{2}, \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      14. associate-/l* 1.0%

          \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot {\color{blue}{\left(D \cdot \frac{M}{d}\right)}}^{2}, \sqrt{\frac{1}{h \cdot \ell}}\right) \]

   7. Simplified 1.0%

      \[\leadsto \color{blue}{d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}, \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

   8. Taylor expanded in h around 0 3.6%

      \[\leadsto d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]
   9. Step-by-step derivation

      1. associate-/r* 3.6%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \]

      2. unpow1/2 3.6%

         \[\leadsto d \cdot \color{blue}{{\left(\frac{\frac{1}{h}}{\ell}\right)}^{0.5}} \]

      3. associate-/r* 3.6%

         \[\leadsto d \cdot {\color{blue}{\left(\frac{1}{h \cdot \ell}\right)}}^{0.5} \]

      4. rem-exp-log 3.6%

         \[\leadsto d \cdot {\left(\frac{1}{\color{blue}{e^{\log \left(h \cdot \ell\right)}}}\right)}^{0.5} \]

      5. exp-neg 3.6%

         \[\leadsto d \cdot {\color{blue}{\left(e^{-\log \left(h \cdot \ell\right)}\right)}}^{0.5} \]

      6. exp-prod 3.6%

         \[\leadsto d \cdot \color{blue}{e^{\left(-\log \left(h \cdot \ell\right)\right) \cdot 0.5}} \]

      7. distribute-lft-neg-out 3.6%

         \[\leadsto d \cdot e^{\color{blue}{-\log \left(h \cdot \ell\right) \cdot 0.5}} \]

      8. distribute-rgt-neg-in 3.6%

         \[\leadsto d \cdot e^{\color{blue}{\log \left(h \cdot \ell\right) \cdot \left(-0.5\right)}} \]

      9. metadata-eval 3.6%

         \[\leadsto d \cdot e^{\log \left(h \cdot \ell\right) \cdot \color{blue}{-0.5}} \]

      10. exp-to-pow 3.6%

          \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

   10. Simplified 3.6%

       \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]
   11. Step-by-step derivation

       1. unpow-prod-down 0.0%

          \[\leadsto d \cdot \color{blue}{\left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)} \]

       2. metadata-eval 0.0%

          \[\leadsto d \cdot \left({h}^{\color{blue}{\left(\frac{-1}{2}\right)}} \cdot {\ell}^{-0.5}\right) \]

       3. sqrt-pow1 0.0%

          \[\leadsto d \cdot \left(\color{blue}{\sqrt{{h}^{-1}}} \cdot {\ell}^{-0.5}\right) \]

       4. inv-pow 0.0%

          \[\leadsto d \cdot \left(\sqrt{\color{blue}{\frac{1}{h}}} \cdot {\ell}^{-0.5}\right) \]

       5. metadata-eval 0.0%

          \[\leadsto d \cdot \left(\sqrt{\frac{1}{h}} \cdot {\ell}^{\color{blue}{\left(\frac{-1}{2}\right)}}\right) \]

       6. sqrt-pow1 0.0%

          \[\leadsto d \cdot \left(\sqrt{\frac{1}{h}} \cdot \color{blue}{\sqrt{{\ell}^{-1}}}\right) \]

       7. inv-pow 0.0%

          \[\leadsto d \cdot \left(\sqrt{\frac{1}{h}} \cdot \sqrt{\color{blue}{\frac{1}{\ell}}}\right) \]

       8. sqrt-prod 3.6%

          \[\leadsto d \cdot \color{blue}{\sqrt{\frac{1}{h} \cdot \frac{1}{\ell}}} \]

       9. div-inv 3.6%

          \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \]

       10. sqrt-div 0.0%

           \[\leadsto d \cdot \color{blue}{\frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}} \]

       11. inv-pow 0.0%

           \[\leadsto d \cdot \frac{\sqrt{\color{blue}{{h}^{-1}}}}{\sqrt{\ell}} \]

       12. sqrt-pow1 0.0%

           \[\leadsto d \cdot \frac{\color{blue}{{h}^{\left(\frac{-1}{2}\right)}}}{\sqrt{\ell}} \]

       13. metadata-eval 0.0%

           \[\leadsto d \cdot \frac{{h}^{\color{blue}{-0.5}}}{\sqrt{\ell}} \]

   12. Applied egg-rr 0.0%

       \[\leadsto d \cdot \color{blue}{\frac{{h}^{-0.5}}{\sqrt{\ell}}} \]

   13. Taylor expanded in h around -inf 0.0%

       \[\leadsto d \cdot \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]
   14. Step-by-step derivation

       1. *-commutative 0.0%

          \[\leadsto d \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

       2. unpow2 0.0%

          \[\leadsto d \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

       3. rem-square-sqrt 56.4%

          \[\leadsto d \cdot \left(\color{blue}{-1} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

       4. mul-1-neg 56.4%

          \[\leadsto d \cdot \color{blue}{\left(-\sqrt{\frac{1}{h \cdot \ell}}\right)} \]

       5. associate-/r* 57.5%

          \[\leadsto d \cdot \left(-\sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}}\right) \]

   15. Simplified 57.5%

       \[\leadsto d \cdot \color{blue}{\left(-\sqrt{\frac{\frac{1}{h}}{\ell}}\right)} \]

   if -2.2e-51 < d < -4.999999999999985e-310

   1. Initial program 47.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 47.7%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 18.0%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. add-log-exp 24.9%

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\log \left(e^{h \cdot \ell}\right)}}} \]

      2. *-commutative 24.9%

         \[\leadsto d \cdot \sqrt{\frac{1}{\log \left(e^{\color{blue}{\ell \cdot h}}\right)}} \]

      3. exp-prod 31.0%

         \[\leadsto d \cdot \sqrt{\frac{1}{\log \color{blue}{\left({\left(e^{\ell}\right)}^{h}\right)}}} \]

   7. Applied egg-rr 31.0%

      \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\log \left({\left(e^{\ell}\right)}^{h}\right)}}} \]

   8. Taylor expanded in l around 0 49.1%

      \[\leadsto d \cdot \sqrt{\frac{1}{\log \color{blue}{1}}} \]

   if -4.999999999999985e-310 < d

   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 66.7%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 26.2%

      \[\leadsto \color{blue}{d \cdot \left(\sqrt{\frac{1}{h \cdot \ell}} + -0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative 26.2%

         \[\leadsto d \cdot \color{blue}{\left(-0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) + \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. fma-define 26.2%

         \[\leadsto d \cdot \color{blue}{\mathsf{fma}\left(-0.125, \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}} \cdot \sqrt{\frac{h}{{\ell}^{3}}}, \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      3. *-commutative 26.2%

         \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}}}, \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      4. *-commutative 26.2%

         \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot \frac{\color{blue}{{M}^{2} \cdot {D}^{2}}}{{d}^{2}}, \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      5. associate-/l* 25.7%

         \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot \color{blue}{\left({M}^{2} \cdot \frac{{D}^{2}}{{d}^{2}}\right)}, \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      6. unpow2 25.7%

         \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\color{blue}{\left(M \cdot M\right)} \cdot \frac{{D}^{2}}{{d}^{2}}\right), \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      7. unpow2 25.7%

         \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\left(M \cdot M\right) \cdot \frac{\color{blue}{D \cdot D}}{{d}^{2}}\right), \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      8. unpow2 25.7%

         \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\left(M \cdot M\right) \cdot \frac{D \cdot D}{\color{blue}{d \cdot d}}\right), \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      9. times-frac 39.2%

         \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\left(M \cdot M\right) \cdot \color{blue}{\left(\frac{D}{d} \cdot \frac{D}{d}\right)}\right), \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      10. swap-sqr 49.6%

          \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot \color{blue}{\left(\left(M \cdot \frac{D}{d}\right) \cdot \left(M \cdot \frac{D}{d}\right)\right)}, \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      11. unpow2 49.6%

          \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot \color{blue}{{\left(M \cdot \frac{D}{d}\right)}^{2}}, \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      12. associate-*r/ 49.6%

          \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot {\color{blue}{\left(\frac{M \cdot D}{d}\right)}}^{2}, \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      13. *-commutative 49.6%

          \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot {\left(\frac{\color{blue}{D \cdot M}}{d}\right)}^{2}, \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      14. associate-/l* 49.6%

          \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot {\color{blue}{\left(D \cdot \frac{M}{d}\right)}}^{2}, \sqrt{\frac{1}{h \cdot \ell}}\right) \]

   7. Simplified 49.6%

      \[\leadsto \color{blue}{d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}, \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

   8. Taylor expanded in h around 0 43.5%

      \[\leadsto d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]
   9. Step-by-step derivation

      1. associate-/r* 44.9%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \]

   10. Simplified 44.9%

       \[\leadsto d \cdot \color{blue}{\sqrt{\frac{\frac{1}{h}}{\ell}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 49.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.2 \cdot 10^{-51}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{h}}{\ell}}\right)\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{0}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 42.7% accurate, 2.9× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{if}\;\ell \leq -5.6 \cdot 10^{-241}:\\ \;\;\;\;d \cdot \left(-t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot t\_0\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (sqrt (/ (/ 1.0 h) l))))
   (if (<= l -5.6e-241) (* d (- t_0)) (* d t_0))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = sqrt(((1.0 / h) / l));
	double tmp;
	if (l <= -5.6e-241) {
		tmp = d * -t_0;
	} else {
		tmp = d * t_0;
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(((1.0d0 / h) / l))
    if (l <= (-5.6d-241)) then
        tmp = d * -t_0
    else
        tmp = d * t_0
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = Math.sqrt(((1.0 / h) / l));
	double tmp;
	if (l <= -5.6e-241) {
		tmp = d * -t_0;
	} else {
		tmp = d * t_0;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = math.sqrt(((1.0 / h) / l))
	tmp = 0
	if l <= -5.6e-241:
		tmp = d * -t_0
	else:
		tmp = d * t_0
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = sqrt(Float64(Float64(1.0 / h) / l))
	tmp = 0.0
	if (l <= -5.6e-241)
		tmp = Float64(d * Float64(-t_0));
	else
		tmp = Float64(d * t_0);
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = sqrt(((1.0 / h) / l));
	tmp = 0.0;
	if (l <= -5.6e-241)
		tmp = d * -t_0;
	else
		tmp = d * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -5.6e-241], N[(d * (-t$95$0)), $MachinePrecision], N[(d * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < -5.5999999999999998e-241

   1. Initial program 63.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 63.9%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 1.4%

      \[\leadsto \color{blue}{d \cdot \left(\sqrt{\frac{1}{h \cdot \ell}} + -0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative 1.4%

         \[\leadsto d \cdot \color{blue}{\left(-0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) + \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. fma-define 1.4%

         \[\leadsto d \cdot \color{blue}{\mathsf{fma}\left(-0.125, \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}} \cdot \sqrt{\frac{h}{{\ell}^{3}}}, \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      3. *-commutative 1.4%

         \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}}}, \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      4. *-commutative 1.4%

         \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot \frac{\color{blue}{{M}^{2} \cdot {D}^{2}}}{{d}^{2}}, \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      5. associate-/l* 1.5%

         \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot \color{blue}{\left({M}^{2} \cdot \frac{{D}^{2}}{{d}^{2}}\right)}, \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      6. unpow2 1.5%

         \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\color{blue}{\left(M \cdot M\right)} \cdot \frac{{D}^{2}}{{d}^{2}}\right), \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      7. unpow2 1.5%

         \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\left(M \cdot M\right) \cdot \frac{\color{blue}{D \cdot D}}{{d}^{2}}\right), \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      8. unpow2 1.5%

         \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\left(M \cdot M\right) \cdot \frac{D \cdot D}{\color{blue}{d \cdot d}}\right), \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      9. times-frac 1.7%

         \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\left(M \cdot M\right) \cdot \color{blue}{\left(\frac{D}{d} \cdot \frac{D}{d}\right)}\right), \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      10. swap-sqr 1.8%

          \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot \color{blue}{\left(\left(M \cdot \frac{D}{d}\right) \cdot \left(M \cdot \frac{D}{d}\right)\right)}, \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      11. unpow2 1.8%

          \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot \color{blue}{{\left(M \cdot \frac{D}{d}\right)}^{2}}, \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      12. associate-*r/ 1.7%

          \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot {\color{blue}{\left(\frac{M \cdot D}{d}\right)}}^{2}, \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      13. *-commutative 1.7%

          \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot {\left(\frac{\color{blue}{D \cdot M}}{d}\right)}^{2}, \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      14. associate-/l* 1.8%

          \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot {\color{blue}{\left(D \cdot \frac{M}{d}\right)}}^{2}, \sqrt{\frac{1}{h \cdot \ell}}\right) \]

   7. Simplified 1.8%

      \[\leadsto \color{blue}{d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}, \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

   8. Taylor expanded in h around 0 6.9%

      \[\leadsto d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]
   9. Step-by-step derivation

      1. associate-/r* 6.9%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \]

      2. unpow1/2 6.9%

         \[\leadsto d \cdot \color{blue}{{\left(\frac{\frac{1}{h}}{\ell}\right)}^{0.5}} \]

      3. associate-/r* 6.9%

         \[\leadsto d \cdot {\color{blue}{\left(\frac{1}{h \cdot \ell}\right)}}^{0.5} \]

      4. rem-exp-log 6.9%

         \[\leadsto d \cdot {\left(\frac{1}{\color{blue}{e^{\log \left(h \cdot \ell\right)}}}\right)}^{0.5} \]

      5. exp-neg 6.9%

         \[\leadsto d \cdot {\color{blue}{\left(e^{-\log \left(h \cdot \ell\right)}\right)}}^{0.5} \]

      6. exp-prod 6.9%

         \[\leadsto d \cdot \color{blue}{e^{\left(-\log \left(h \cdot \ell\right)\right) \cdot 0.5}} \]

      7. distribute-lft-neg-out 6.9%

         \[\leadsto d \cdot e^{\color{blue}{-\log \left(h \cdot \ell\right) \cdot 0.5}} \]

      8. distribute-rgt-neg-in 6.9%

         \[\leadsto d \cdot e^{\color{blue}{\log \left(h \cdot \ell\right) \cdot \left(-0.5\right)}} \]

      9. metadata-eval 6.9%

         \[\leadsto d \cdot e^{\log \left(h \cdot \ell\right) \cdot \color{blue}{-0.5}} \]

      10. exp-to-pow 6.9%

          \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

   10. Simplified 6.9%

       \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]
   11. Step-by-step derivation

       1. unpow-prod-down 0.0%

          \[\leadsto d \cdot \color{blue}{\left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)} \]

       2. metadata-eval 0.0%

          \[\leadsto d \cdot \left({h}^{\color{blue}{\left(\frac{-1}{2}\right)}} \cdot {\ell}^{-0.5}\right) \]

       3. sqrt-pow1 0.0%

          \[\leadsto d \cdot \left(\color{blue}{\sqrt{{h}^{-1}}} \cdot {\ell}^{-0.5}\right) \]

       4. inv-pow 0.0%

          \[\leadsto d \cdot \left(\sqrt{\color{blue}{\frac{1}{h}}} \cdot {\ell}^{-0.5}\right) \]

       5. metadata-eval 0.0%

          \[\leadsto d \cdot \left(\sqrt{\frac{1}{h}} \cdot {\ell}^{\color{blue}{\left(\frac{-1}{2}\right)}}\right) \]

       6. sqrt-pow1 0.0%

          \[\leadsto d \cdot \left(\sqrt{\frac{1}{h}} \cdot \color{blue}{\sqrt{{\ell}^{-1}}}\right) \]

       7. inv-pow 0.0%

          \[\leadsto d \cdot \left(\sqrt{\frac{1}{h}} \cdot \sqrt{\color{blue}{\frac{1}{\ell}}}\right) \]

       8. sqrt-prod 6.9%

          \[\leadsto d \cdot \color{blue}{\sqrt{\frac{1}{h} \cdot \frac{1}{\ell}}} \]

       9. div-inv 6.9%

          \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \]

       10. sqrt-div 0.0%

           \[\leadsto d \cdot \color{blue}{\frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}} \]

       11. inv-pow 0.0%

           \[\leadsto d \cdot \frac{\sqrt{\color{blue}{{h}^{-1}}}}{\sqrt{\ell}} \]

       12. sqrt-pow1 0.0%

           \[\leadsto d \cdot \frac{\color{blue}{{h}^{\left(\frac{-1}{2}\right)}}}{\sqrt{\ell}} \]

       13. metadata-eval 0.0%

           \[\leadsto d \cdot \frac{{h}^{\color{blue}{-0.5}}}{\sqrt{\ell}} \]

   12. Applied egg-rr 0.0%

       \[\leadsto d \cdot \color{blue}{\frac{{h}^{-0.5}}{\sqrt{\ell}}} \]

   13. Taylor expanded in h around -inf 0.0%

       \[\leadsto d \cdot \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]
   14. Step-by-step derivation

       1. *-commutative 0.0%

          \[\leadsto d \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

       2. unpow2 0.0%

          \[\leadsto d \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

       3. rem-square-sqrt 44.3%

          \[\leadsto d \cdot \left(\color{blue}{-1} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

       4. mul-1-neg 44.3%

          \[\leadsto d \cdot \color{blue}{\left(-\sqrt{\frac{1}{h \cdot \ell}}\right)} \]

       5. associate-/r* 45.1%

          \[\leadsto d \cdot \left(-\sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}}\right) \]

   15. Simplified 45.1%

       \[\leadsto d \cdot \color{blue}{\left(-\sqrt{\frac{\frac{1}{h}}{\ell}}\right)} \]

   if -5.5999999999999998e-241 < l

   1. Initial program 70.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 69.5%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 23.0%

      \[\leadsto \color{blue}{d \cdot \left(\sqrt{\frac{1}{h \cdot \ell}} + -0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative 23.0%

         \[\leadsto d \cdot \color{blue}{\left(-0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) + \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. fma-define 23.0%

         \[\leadsto d \cdot \color{blue}{\mathsf{fma}\left(-0.125, \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}} \cdot \sqrt{\frac{h}{{\ell}^{3}}}, \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      3. *-commutative 23.0%

         \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}}}, \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      4. *-commutative 23.0%

         \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot \frac{\color{blue}{{M}^{2} \cdot {D}^{2}}}{{d}^{2}}, \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      5. associate-/l* 22.5%

         \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot \color{blue}{\left({M}^{2} \cdot \frac{{D}^{2}}{{d}^{2}}\right)}, \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      6. unpow2 22.5%

         \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\color{blue}{\left(M \cdot M\right)} \cdot \frac{{D}^{2}}{{d}^{2}}\right), \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      7. unpow2 22.5%

         \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\left(M \cdot M\right) \cdot \frac{\color{blue}{D \cdot D}}{{d}^{2}}\right), \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      8. unpow2 22.5%

         \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\left(M \cdot M\right) \cdot \frac{D \cdot D}{\color{blue}{d \cdot d}}\right), \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      9. times-frac 34.3%

         \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\left(M \cdot M\right) \cdot \color{blue}{\left(\frac{D}{d} \cdot \frac{D}{d}\right)}\right), \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      10. swap-sqr 43.4%

          \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot \color{blue}{\left(\left(M \cdot \frac{D}{d}\right) \cdot \left(M \cdot \frac{D}{d}\right)\right)}, \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      11. unpow2 43.4%

          \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot \color{blue}{{\left(M \cdot \frac{D}{d}\right)}^{2}}, \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      12. associate-*r/ 43.4%

          \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot {\color{blue}{\left(\frac{M \cdot D}{d}\right)}}^{2}, \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      13. *-commutative 43.4%

          \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot {\left(\frac{\color{blue}{D \cdot M}}{d}\right)}^{2}, \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      14. associate-/l* 43.4%

          \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot {\color{blue}{\left(D \cdot \frac{M}{d}\right)}}^{2}, \sqrt{\frac{1}{h \cdot \ell}}\right) \]

   7. Simplified 43.4%

      \[\leadsto \color{blue}{d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}, \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

   8. Taylor expanded in h around 0 41.3%

      \[\leadsto d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]
   9. Step-by-step derivation

      1. associate-/r* 42.5%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \]

   10. Simplified 42.5%

       \[\leadsto d \cdot \color{blue}{\sqrt{\frac{\frac{1}{h}}{\ell}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 42.6% accurate, 3.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -2.9 \cdot 10^{-239}:\\ \;\;\;\;\frac{d}{-\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= l -2.9e-239) (/ d (- (sqrt (* l h)))) (* d (sqrt (/ (/ 1.0 h) l)))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= -2.9e-239) {
		tmp = d / -sqrt((l * h));
	} else {
		tmp = d * sqrt(((1.0 / h) / l));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (l <= (-2.9d-239)) then
        tmp = d / -sqrt((l * h))
    else
        tmp = d * sqrt(((1.0d0 / h) / l))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= -2.9e-239) {
		tmp = d / -Math.sqrt((l * h));
	} else {
		tmp = d * Math.sqrt(((1.0 / h) / l));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if l <= -2.9e-239:
		tmp = d / -math.sqrt((l * h))
	else:
		tmp = d * math.sqrt(((1.0 / h) / l))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (l <= -2.9e-239)
		tmp = Float64(d / Float64(-sqrt(Float64(l * h))));
	else
		tmp = Float64(d * sqrt(Float64(Float64(1.0 / h) / l)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (l <= -2.9e-239)
		tmp = d / -sqrt((l * h));
	else
		tmp = d * sqrt(((1.0 / h) / l));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[l, -2.9e-239], N[(d / (-N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(d * N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -2.9000000000000002e-239

   1. Initial program 63.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 63.9%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 6.9%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

   6. Taylor expanded in h around -inf 0.0%

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   7. Step-by-step derivation

      1. associate-*l* 0.0%

         \[\leadsto \color{blue}{d \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. unpow2 0.0%

         \[\leadsto d \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      3. rem-square-sqrt 44.3%

         \[\leadsto d \cdot \left(\color{blue}{-1} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      4. mul-1-neg 44.3%

         \[\leadsto d \cdot \color{blue}{\left(-\sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      5. distribute-rgt-neg-out 44.3%

         \[\leadsto \color{blue}{-d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      6. *-commutative 44.3%

         \[\leadsto -\color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      7. associate-/r* 45.1%

         \[\leadsto -\sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \cdot d \]

      8. unpow1/2 45.1%

         \[\leadsto -\color{blue}{{\left(\frac{\frac{1}{h}}{\ell}\right)}^{0.5}} \cdot d \]

      9. associate-/r* 44.3%

         \[\leadsto -{\color{blue}{\left(\frac{1}{h \cdot \ell}\right)}}^{0.5} \cdot d \]

      10. rem-exp-log 42.1%

          \[\leadsto -{\left(\frac{1}{\color{blue}{e^{\log \left(h \cdot \ell\right)}}}\right)}^{0.5} \cdot d \]

      11. exp-neg 42.1%

          \[\leadsto -{\color{blue}{\left(e^{-\log \left(h \cdot \ell\right)}\right)}}^{0.5} \cdot d \]

      12. exp-prod 42.1%

          \[\leadsto -\color{blue}{e^{\left(-\log \left(h \cdot \ell\right)\right) \cdot 0.5}} \cdot d \]

      13. distribute-lft-neg-out 42.1%

          \[\leadsto -e^{\color{blue}{-\log \left(h \cdot \ell\right) \cdot 0.5}} \cdot d \]

      14. exp-neg 42.1%

          \[\leadsto -\color{blue}{\frac{1}{e^{\log \left(h \cdot \ell\right) \cdot 0.5}}} \cdot d \]

      15. exp-to-pow 44.2%

          \[\leadsto -\frac{1}{\color{blue}{{\left(h \cdot \ell\right)}^{0.5}}} \cdot d \]

      16. unpow1/2 44.2%

          \[\leadsto -\frac{1}{\color{blue}{\sqrt{h \cdot \ell}}} \cdot d \]

      17. unpow-1 44.2%

          \[\leadsto -\color{blue}{{\left(\sqrt{h \cdot \ell}\right)}^{-1}} \cdot d \]

      18. unpow-1 44.2%

          \[\leadsto -\color{blue}{\frac{1}{\sqrt{h \cdot \ell}}} \cdot d \]

      19. associate-*l/ 44.3%

          \[\leadsto -\color{blue}{\frac{1 \cdot d}{\sqrt{h \cdot \ell}}} \]

      20. *-lft-identity 44.3%

          \[\leadsto -\frac{\color{blue}{d}}{\sqrt{h \cdot \ell}} \]

   8. Simplified 44.3%

      \[\leadsto \color{blue}{\frac{d}{-\sqrt{h \cdot \ell}}} \]

   if -2.9000000000000002e-239 < l

   1. Initial program 70.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 69.5%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 23.0%

      \[\leadsto \color{blue}{d \cdot \left(\sqrt{\frac{1}{h \cdot \ell}} + -0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative 23.0%

         \[\leadsto d \cdot \color{blue}{\left(-0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) + \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. fma-define 23.0%

         \[\leadsto d \cdot \color{blue}{\mathsf{fma}\left(-0.125, \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}} \cdot \sqrt{\frac{h}{{\ell}^{3}}}, \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      3. *-commutative 23.0%

         \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}}}, \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      4. *-commutative 23.0%

         \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot \frac{\color{blue}{{M}^{2} \cdot {D}^{2}}}{{d}^{2}}, \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      5. associate-/l* 22.5%

         \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot \color{blue}{\left({M}^{2} \cdot \frac{{D}^{2}}{{d}^{2}}\right)}, \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      6. unpow2 22.5%

         \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\color{blue}{\left(M \cdot M\right)} \cdot \frac{{D}^{2}}{{d}^{2}}\right), \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      7. unpow2 22.5%

         \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\left(M \cdot M\right) \cdot \frac{\color{blue}{D \cdot D}}{{d}^{2}}\right), \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      8. unpow2 22.5%

         \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\left(M \cdot M\right) \cdot \frac{D \cdot D}{\color{blue}{d \cdot d}}\right), \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      9. times-frac 34.3%

         \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\left(M \cdot M\right) \cdot \color{blue}{\left(\frac{D}{d} \cdot \frac{D}{d}\right)}\right), \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      10. swap-sqr 43.4%

          \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot \color{blue}{\left(\left(M \cdot \frac{D}{d}\right) \cdot \left(M \cdot \frac{D}{d}\right)\right)}, \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      11. unpow2 43.4%

          \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot \color{blue}{{\left(M \cdot \frac{D}{d}\right)}^{2}}, \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      12. associate-*r/ 43.4%

          \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot {\color{blue}{\left(\frac{M \cdot D}{d}\right)}}^{2}, \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      13. *-commutative 43.4%

          \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot {\left(\frac{\color{blue}{D \cdot M}}{d}\right)}^{2}, \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      14. associate-/l* 43.4%

          \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot {\color{blue}{\left(D \cdot \frac{M}{d}\right)}}^{2}, \sqrt{\frac{1}{h \cdot \ell}}\right) \]

   7. Simplified 43.4%

      \[\leadsto \color{blue}{d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}, \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

   8. Taylor expanded in h around 0 41.3%

      \[\leadsto d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]
   9. Step-by-step derivation

      1. associate-/r* 42.5%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \]

   10. Simplified 42.5%

       \[\leadsto d \cdot \color{blue}{\sqrt{\frac{\frac{1}{h}}{\ell}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 43.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.9 \cdot 10^{-239}:\\ \;\;\;\;\frac{d}{-\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 42.5% accurate, 3.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\ell \cdot h}\\ \mathbf{if}\;\ell \leq -5.4 \cdot 10^{-244}:\\ \;\;\;\;\frac{d}{-t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{t\_0}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (sqrt (* l h)))) (if (<= l -5.4e-244) (/ d (- t_0)) (/ d t_0))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = sqrt((l * h));
	double tmp;
	if (l <= -5.4e-244) {
		tmp = d / -t_0;
	} else {
		tmp = d / t_0;
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((l * h))
    if (l <= (-5.4d-244)) then
        tmp = d / -t_0
    else
        tmp = d / t_0
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = Math.sqrt((l * h));
	double tmp;
	if (l <= -5.4e-244) {
		tmp = d / -t_0;
	} else {
		tmp = d / t_0;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = math.sqrt((l * h))
	tmp = 0
	if l <= -5.4e-244:
		tmp = d / -t_0
	else:
		tmp = d / t_0
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = sqrt(Float64(l * h))
	tmp = 0.0
	if (l <= -5.4e-244)
		tmp = Float64(d / Float64(-t_0));
	else
		tmp = Float64(d / t_0);
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = sqrt((l * h));
	tmp = 0.0;
	if (l <= -5.4e-244)
		tmp = d / -t_0;
	else
		tmp = d / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -5.4e-244], N[(d / (-t$95$0)), $MachinePrecision], N[(d / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < -5.3999999999999999e-244

   1. Initial program 63.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 63.9%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 6.9%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

   6. Taylor expanded in h around -inf 0.0%

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   7. Step-by-step derivation

      1. associate-*l* 0.0%

         \[\leadsto \color{blue}{d \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. unpow2 0.0%

         \[\leadsto d \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      3. rem-square-sqrt 44.3%

         \[\leadsto d \cdot \left(\color{blue}{-1} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      4. mul-1-neg 44.3%

         \[\leadsto d \cdot \color{blue}{\left(-\sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      5. distribute-rgt-neg-out 44.3%

         \[\leadsto \color{blue}{-d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      6. *-commutative 44.3%

         \[\leadsto -\color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      7. associate-/r* 45.1%

         \[\leadsto -\sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \cdot d \]

      8. unpow1/2 45.1%

         \[\leadsto -\color{blue}{{\left(\frac{\frac{1}{h}}{\ell}\right)}^{0.5}} \cdot d \]

      9. associate-/r* 44.3%

         \[\leadsto -{\color{blue}{\left(\frac{1}{h \cdot \ell}\right)}}^{0.5} \cdot d \]

      10. rem-exp-log 42.1%

          \[\leadsto -{\left(\frac{1}{\color{blue}{e^{\log \left(h \cdot \ell\right)}}}\right)}^{0.5} \cdot d \]

      11. exp-neg 42.1%

          \[\leadsto -{\color{blue}{\left(e^{-\log \left(h \cdot \ell\right)}\right)}}^{0.5} \cdot d \]

      12. exp-prod 42.1%

          \[\leadsto -\color{blue}{e^{\left(-\log \left(h \cdot \ell\right)\right) \cdot 0.5}} \cdot d \]

      13. distribute-lft-neg-out 42.1%

          \[\leadsto -e^{\color{blue}{-\log \left(h \cdot \ell\right) \cdot 0.5}} \cdot d \]

      14. exp-neg 42.1%

          \[\leadsto -\color{blue}{\frac{1}{e^{\log \left(h \cdot \ell\right) \cdot 0.5}}} \cdot d \]

      15. exp-to-pow 44.2%

          \[\leadsto -\frac{1}{\color{blue}{{\left(h \cdot \ell\right)}^{0.5}}} \cdot d \]

      16. unpow1/2 44.2%

          \[\leadsto -\frac{1}{\color{blue}{\sqrt{h \cdot \ell}}} \cdot d \]

      17. unpow-1 44.2%

          \[\leadsto -\color{blue}{{\left(\sqrt{h \cdot \ell}\right)}^{-1}} \cdot d \]

      18. unpow-1 44.2%

          \[\leadsto -\color{blue}{\frac{1}{\sqrt{h \cdot \ell}}} \cdot d \]

      19. associate-*l/ 44.3%

          \[\leadsto -\color{blue}{\frac{1 \cdot d}{\sqrt{h \cdot \ell}}} \]

      20. *-lft-identity 44.3%

          \[\leadsto -\frac{\color{blue}{d}}{\sqrt{h \cdot \ell}} \]

   8. Simplified 44.3%

      \[\leadsto \color{blue}{\frac{d}{-\sqrt{h \cdot \ell}}} \]

   if -5.3999999999999999e-244 < l

   1. Initial program 70.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 69.5%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 23.0%

      \[\leadsto \color{blue}{d \cdot \left(\sqrt{\frac{1}{h \cdot \ell}} + -0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative 23.0%

         \[\leadsto d \cdot \color{blue}{\left(-0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) + \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. fma-define 23.0%

         \[\leadsto d \cdot \color{blue}{\mathsf{fma}\left(-0.125, \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}} \cdot \sqrt{\frac{h}{{\ell}^{3}}}, \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      3. *-commutative 23.0%

         \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}}}, \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      4. *-commutative 23.0%

         \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot \frac{\color{blue}{{M}^{2} \cdot {D}^{2}}}{{d}^{2}}, \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      5. associate-/l* 22.5%

         \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot \color{blue}{\left({M}^{2} \cdot \frac{{D}^{2}}{{d}^{2}}\right)}, \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      6. unpow2 22.5%

         \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\color{blue}{\left(M \cdot M\right)} \cdot \frac{{D}^{2}}{{d}^{2}}\right), \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      7. unpow2 22.5%

         \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\left(M \cdot M\right) \cdot \frac{\color{blue}{D \cdot D}}{{d}^{2}}\right), \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      8. unpow2 22.5%

         \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\left(M \cdot M\right) \cdot \frac{D \cdot D}{\color{blue}{d \cdot d}}\right), \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      9. times-frac 34.3%

         \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\left(M \cdot M\right) \cdot \color{blue}{\left(\frac{D}{d} \cdot \frac{D}{d}\right)}\right), \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      10. swap-sqr 43.4%

          \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot \color{blue}{\left(\left(M \cdot \frac{D}{d}\right) \cdot \left(M \cdot \frac{D}{d}\right)\right)}, \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      11. unpow2 43.4%

          \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot \color{blue}{{\left(M \cdot \frac{D}{d}\right)}^{2}}, \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      12. associate-*r/ 43.4%

          \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot {\color{blue}{\left(\frac{M \cdot D}{d}\right)}}^{2}, \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      13. *-commutative 43.4%

          \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot {\left(\frac{\color{blue}{D \cdot M}}{d}\right)}^{2}, \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      14. associate-/l* 43.4%

          \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot {\color{blue}{\left(D \cdot \frac{M}{d}\right)}}^{2}, \sqrt{\frac{1}{h \cdot \ell}}\right) \]

   7. Simplified 43.4%

      \[\leadsto \color{blue}{d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}, \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

   8. Taylor expanded in h around 0 41.3%

      \[\leadsto d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]
   9. Step-by-step derivation

      1. associate-/r* 42.5%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \]

      2. unpow1/2 42.5%

         \[\leadsto d \cdot \color{blue}{{\left(\frac{\frac{1}{h}}{\ell}\right)}^{0.5}} \]

      3. associate-/r* 41.3%

         \[\leadsto d \cdot {\color{blue}{\left(\frac{1}{h \cdot \ell}\right)}}^{0.5} \]

      4. rem-exp-log 39.4%

         \[\leadsto d \cdot {\left(\frac{1}{\color{blue}{e^{\log \left(h \cdot \ell\right)}}}\right)}^{0.5} \]

      5. exp-neg 39.4%

         \[\leadsto d \cdot {\color{blue}{\left(e^{-\log \left(h \cdot \ell\right)}\right)}}^{0.5} \]

      6. exp-prod 39.8%

         \[\leadsto d \cdot \color{blue}{e^{\left(-\log \left(h \cdot \ell\right)\right) \cdot 0.5}} \]

      7. distribute-lft-neg-out 39.8%

         \[\leadsto d \cdot e^{\color{blue}{-\log \left(h \cdot \ell\right) \cdot 0.5}} \]

      8. distribute-rgt-neg-in 39.8%

         \[\leadsto d \cdot e^{\color{blue}{\log \left(h \cdot \ell\right) \cdot \left(-0.5\right)}} \]

      9. metadata-eval 39.8%

         \[\leadsto d \cdot e^{\log \left(h \cdot \ell\right) \cdot \color{blue}{-0.5}} \]

      10. exp-to-pow 41.7%

          \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

   10. Simplified 41.7%

       \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]
   11. Step-by-step derivation

       1. metadata-eval 41.7%

          \[\leadsto d \cdot {\left(h \cdot \ell\right)}^{\color{blue}{\left(\frac{-1}{2}\right)}} \]

       2. sqrt-pow1 41.3%

          \[\leadsto d \cdot \color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-1}}} \]

       3. add-log-exp 10.1%

          \[\leadsto d \cdot \sqrt{{\left(h \cdot \color{blue}{\log \left(e^{\ell}\right)}\right)}^{-1}} \]

       4. log-pow 17.3%

          \[\leadsto d \cdot \sqrt{{\color{blue}{\log \left({\left(e^{\ell}\right)}^{h}\right)}}^{-1}} \]

       5. inv-pow 17.3%

          \[\leadsto d \cdot \sqrt{\color{blue}{\frac{1}{\log \left({\left(e^{\ell}\right)}^{h}\right)}}} \]

       6. sqrt-div 17.3%

          \[\leadsto d \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\log \left({\left(e^{\ell}\right)}^{h}\right)}}} \]

       7. metadata-eval 17.3%

          \[\leadsto d \cdot \frac{\color{blue}{1}}{\sqrt{\log \left({\left(e^{\ell}\right)}^{h}\right)}} \]

       8. un-div-inv 17.3%

          \[\leadsto \color{blue}{\frac{d}{\sqrt{\log \left({\left(e^{\ell}\right)}^{h}\right)}}} \]

       9. log-pow 11.5%

          \[\leadsto \frac{d}{\sqrt{\color{blue}{h \cdot \log \left(e^{\ell}\right)}}} \]

       10. add-log-exp 41.7%

           \[\leadsto \frac{d}{\sqrt{h \cdot \color{blue}{\ell}}} \]

   12. Applied egg-rr 41.7%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 42.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5.4 \cdot 10^{-244}:\\ \;\;\;\;\frac{d}{-\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 26.3% accurate, 3.2× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \frac{d}{\sqrt{\ell \cdot h}} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m) :precision binary64 (/ d (sqrt (* l h))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	return d / sqrt((l * h));
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    code = d / sqrt((l * h))
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	return d / Math.sqrt((l * h));
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	return d / math.sqrt((l * h))

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	return Float64(d / sqrt(Float64(l * h)))
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp = code(d, h, l, M_m, D_m)
	tmp = d / sqrt((l * h));
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 67.1%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

3. Simplified 67.1%

   \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in d around inf 13.5%

   \[\leadsto \color{blue}{d \cdot \left(\sqrt{\frac{1}{h \cdot \ell}} + -0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)\right)} \]
6. Step-by-step derivation

   1. +-commutative 13.5%

      \[\leadsto d \cdot \color{blue}{\left(-0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) + \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

   2. fma-define 13.5%

      \[\leadsto d \cdot \color{blue}{\mathsf{fma}\left(-0.125, \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}} \cdot \sqrt{\frac{h}{{\ell}^{3}}}, \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

   3. *-commutative 13.5%

      \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}}}, \sqrt{\frac{1}{h \cdot \ell}}\right) \]

   4. *-commutative 13.5%

      \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot \frac{\color{blue}{{M}^{2} \cdot {D}^{2}}}{{d}^{2}}, \sqrt{\frac{1}{h \cdot \ell}}\right) \]

   5. associate-/l* 13.3%

      \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot \color{blue}{\left({M}^{2} \cdot \frac{{D}^{2}}{{d}^{2}}\right)}, \sqrt{\frac{1}{h \cdot \ell}}\right) \]

   6. unpow2 13.3%

      \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\color{blue}{\left(M \cdot M\right)} \cdot \frac{{D}^{2}}{{d}^{2}}\right), \sqrt{\frac{1}{h \cdot \ell}}\right) \]

   7. unpow2 13.3%

      \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\left(M \cdot M\right) \cdot \frac{\color{blue}{D \cdot D}}{{d}^{2}}\right), \sqrt{\frac{1}{h \cdot \ell}}\right) \]

   8. unpow2 13.3%

      \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\left(M \cdot M\right) \cdot \frac{D \cdot D}{\color{blue}{d \cdot d}}\right), \sqrt{\frac{1}{h \cdot \ell}}\right) \]

   9. times-frac 20.0%

      \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\left(M \cdot M\right) \cdot \color{blue}{\left(\frac{D}{d} \cdot \frac{D}{d}\right)}\right), \sqrt{\frac{1}{h \cdot \ell}}\right) \]

   10. swap-sqr 25.2%

       \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot \color{blue}{\left(\left(M \cdot \frac{D}{d}\right) \cdot \left(M \cdot \frac{D}{d}\right)\right)}, \sqrt{\frac{1}{h \cdot \ell}}\right) \]

   11. unpow2 25.2%

       \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot \color{blue}{{\left(M \cdot \frac{D}{d}\right)}^{2}}, \sqrt{\frac{1}{h \cdot \ell}}\right) \]

   12. associate-*r/ 25.2%

       \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot {\color{blue}{\left(\frac{M \cdot D}{d}\right)}}^{2}, \sqrt{\frac{1}{h \cdot \ell}}\right) \]

   13. *-commutative 25.2%

       \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot {\left(\frac{\color{blue}{D \cdot M}}{d}\right)}^{2}, \sqrt{\frac{1}{h \cdot \ell}}\right) \]

   14. associate-/l* 25.2%

       \[\leadsto d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot {\color{blue}{\left(D \cdot \frac{M}{d}\right)}}^{2}, \sqrt{\frac{1}{h \cdot \ell}}\right) \]

7. Simplified 25.2%

   \[\leadsto \color{blue}{d \cdot \mathsf{fma}\left(-0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}, \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

8. Taylor expanded in h around 0 26.2%

   \[\leadsto d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]
9. Step-by-step derivation

   1. associate-/r* 26.9%

      \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \]

   2. unpow1/2 26.9%

      \[\leadsto d \cdot \color{blue}{{\left(\frac{\frac{1}{h}}{\ell}\right)}^{0.5}} \]

   3. associate-/r* 26.2%

      \[\leadsto d \cdot {\color{blue}{\left(\frac{1}{h \cdot \ell}\right)}}^{0.5} \]

   4. rem-exp-log 25.2%

      \[\leadsto d \cdot {\left(\frac{1}{\color{blue}{e^{\log \left(h \cdot \ell\right)}}}\right)}^{0.5} \]

   5. exp-neg 25.2%

      \[\leadsto d \cdot {\color{blue}{\left(e^{-\log \left(h \cdot \ell\right)}\right)}}^{0.5} \]

   6. exp-prod 25.4%

      \[\leadsto d \cdot \color{blue}{e^{\left(-\log \left(h \cdot \ell\right)\right) \cdot 0.5}} \]

   7. distribute-lft-neg-out 25.4%

      \[\leadsto d \cdot e^{\color{blue}{-\log \left(h \cdot \ell\right) \cdot 0.5}} \]

   8. distribute-rgt-neg-in 25.4%

      \[\leadsto d \cdot e^{\color{blue}{\log \left(h \cdot \ell\right) \cdot \left(-0.5\right)}} \]

   9. metadata-eval 25.4%

      \[\leadsto d \cdot e^{\log \left(h \cdot \ell\right) \cdot \color{blue}{-0.5}} \]

   10. exp-to-pow 26.5%

       \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

10. Simplified 26.5%

    \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]
11. Step-by-step derivation

    1. metadata-eval 26.5%

       \[\leadsto d \cdot {\left(h \cdot \ell\right)}^{\color{blue}{\left(\frac{-1}{2}\right)}} \]

    2. sqrt-pow1 26.2%

       \[\leadsto d \cdot \color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-1}}} \]

    3. add-log-exp 6.9%

       \[\leadsto d \cdot \sqrt{{\left(h \cdot \color{blue}{\log \left(e^{\ell}\right)}\right)}^{-1}} \]

    4. log-pow 17.4%

       \[\leadsto d \cdot \sqrt{{\color{blue}{\log \left({\left(e^{\ell}\right)}^{h}\right)}}^{-1}} \]

    5. inv-pow 17.4%

       \[\leadsto d \cdot \sqrt{\color{blue}{\frac{1}{\log \left({\left(e^{\ell}\right)}^{h}\right)}}} \]

    6. sqrt-div 17.4%

       \[\leadsto d \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\log \left({\left(e^{\ell}\right)}^{h}\right)}}} \]

    7. metadata-eval 17.4%

       \[\leadsto d \cdot \frac{\color{blue}{1}}{\sqrt{\log \left({\left(e^{\ell}\right)}^{h}\right)}} \]

    8. un-div-inv 17.4%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{\log \left({\left(e^{\ell}\right)}^{h}\right)}}} \]

    9. log-pow 9.8%

       \[\leadsto \frac{d}{\sqrt{\color{blue}{h \cdot \log \left(e^{\ell}\right)}}} \]

    10. add-log-exp 26.5%

        \[\leadsto \frac{d}{\sqrt{h \cdot \color{blue}{\ell}}} \]

12. Applied egg-rr 26.5%

    \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]

13. Final simplification 26.5%

    \[\leadsto \frac{d}{\sqrt{\ell \cdot h}} \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024182 
(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)))))