Henrywood and Agarwal, Equation (12)

Percentage Accurate: 66.8% → 80.1%

Time: 23.9s

Alternatives: 17

Speedup: 1.0×

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: 66.8% 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: 80.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;h \leq -4.9 \cdot 10^{+104}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(D \cdot \frac{\frac{M\_m}{2}}{d}\right)}^{2} \cdot -0.5\right)\right)\right)\\ \mathbf{elif}\;h \leq 1.38 \cdot 10^{-151}:\\ \;\;\;\;\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - 0.5 \cdot {\left(0.5 \cdot \left(\frac{D \cdot M\_m}{d} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}} \cdot \left(1 - 0.5 \cdot \left(h \cdot \left(0.25 \cdot \frac{{\left(M\_m \cdot \frac{D}{d}\right)}^{2}}{\ell}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (if (<= h -4.9e+104)
   (*
    (sqrt (/ d l))
    (*
     (/ (sqrt (- d)) (sqrt (- h)))
     (+ 1.0 (* (/ h l) (* (pow (* D (/ (/ M_m 2.0) d)) 2.0) -0.5)))))
   (if (<= h 1.38e-151)
     (*
      (/ (fabs d) (sqrt (* h l)))
      (- 1.0 (* 0.5 (pow (* 0.5 (* (/ (* D M_m) d) (sqrt (/ h l)))) 2.0))))
     (*
      (/ (/ d (sqrt h)) (sqrt l))
      (- 1.0 (* 0.5 (* h (* 0.25 (/ (pow (* M_m (/ D d)) 2.0) l)))))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (h <= -4.9e+104) {
		tmp = sqrt((d / l)) * ((sqrt(-d) / sqrt(-h)) * (1.0 + ((h / l) * (pow((D * ((M_m / 2.0) / d)), 2.0) * -0.5))));
	} else if (h <= 1.38e-151) {
		tmp = (fabs(d) / sqrt((h * l))) * (1.0 - (0.5 * pow((0.5 * (((D * M_m) / d) * sqrt((h / l)))), 2.0)));
	} else {
		tmp = ((d / sqrt(h)) / sqrt(l)) * (1.0 - (0.5 * (h * (0.25 * (pow((M_m * (D / d)), 2.0) / l)))));
	}
	return tmp;
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (h <= (-4.9d+104)) then
        tmp = sqrt((d / l)) * ((sqrt(-d) / sqrt(-h)) * (1.0d0 + ((h / l) * (((d_1 * ((m_m / 2.0d0) / d)) ** 2.0d0) * (-0.5d0)))))
    else if (h <= 1.38d-151) then
        tmp = (abs(d) / sqrt((h * l))) * (1.0d0 - (0.5d0 * ((0.5d0 * (((d_1 * m_m) / d) * sqrt((h / l)))) ** 2.0d0)))
    else
        tmp = ((d / sqrt(h)) / sqrt(l)) * (1.0d0 - (0.5d0 * (h * (0.25d0 * (((m_m * (d_1 / d)) ** 2.0d0) / l)))))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (h <= -4.9e+104) {
		tmp = Math.sqrt((d / l)) * ((Math.sqrt(-d) / Math.sqrt(-h)) * (1.0 + ((h / l) * (Math.pow((D * ((M_m / 2.0) / d)), 2.0) * -0.5))));
	} else if (h <= 1.38e-151) {
		tmp = (Math.abs(d) / Math.sqrt((h * l))) * (1.0 - (0.5 * Math.pow((0.5 * (((D * M_m) / d) * Math.sqrt((h / l)))), 2.0)));
	} else {
		tmp = ((d / Math.sqrt(h)) / Math.sqrt(l)) * (1.0 - (0.5 * (h * (0.25 * (Math.pow((M_m * (D / d)), 2.0) / l)))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	tmp = 0
	if h <= -4.9e+104:
		tmp = math.sqrt((d / l)) * ((math.sqrt(-d) / math.sqrt(-h)) * (1.0 + ((h / l) * (math.pow((D * ((M_m / 2.0) / d)), 2.0) * -0.5))))
	elif h <= 1.38e-151:
		tmp = (math.fabs(d) / math.sqrt((h * l))) * (1.0 - (0.5 * math.pow((0.5 * (((D * M_m) / d) * math.sqrt((h / l)))), 2.0)))
	else:
		tmp = ((d / math.sqrt(h)) / math.sqrt(l)) * (1.0 - (0.5 * (h * (0.25 * (math.pow((M_m * (D / d)), 2.0) / l)))))
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	tmp = 0.0
	if (h <= -4.9e+104)
		tmp = Float64(sqrt(Float64(d / l)) * Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-h))) * Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(D * Float64(Float64(M_m / 2.0) / d)) ^ 2.0) * -0.5)))));
	elseif (h <= 1.38e-151)
		tmp = Float64(Float64(abs(d) / sqrt(Float64(h * l))) * Float64(1.0 - Float64(0.5 * (Float64(0.5 * Float64(Float64(Float64(D * M_m) / d) * sqrt(Float64(h / l)))) ^ 2.0))));
	else
		tmp = Float64(Float64(Float64(d / sqrt(h)) / sqrt(l)) * Float64(1.0 - Float64(0.5 * Float64(h * Float64(0.25 * Float64((Float64(M_m * Float64(D / d)) ^ 2.0) / l))))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp_2 = code(d, h, l, M_m, D)
	tmp = 0.0;
	if (h <= -4.9e+104)
		tmp = sqrt((d / l)) * ((sqrt(-d) / sqrt(-h)) * (1.0 + ((h / l) * (((D * ((M_m / 2.0) / d)) ^ 2.0) * -0.5))));
	elseif (h <= 1.38e-151)
		tmp = (abs(d) / sqrt((h * l))) * (1.0 - (0.5 * ((0.5 * (((D * M_m) / d) * sqrt((h / l)))) ^ 2.0)));
	else
		tmp = ((d / sqrt(h)) / sqrt(l)) * (1.0 - (0.5 * (h * (0.25 * (((M_m * (D / d)) ^ 2.0) / l)))));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := If[LessEqual[h, -4.9e+104], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(D * N[(N[(M$95$m / 2.0), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, 1.38e-151], N[(N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[Power[N[(0.5 * N[(N[(N[(D * M$95$m), $MachinePrecision] / d), $MachinePrecision] * N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(d / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(h * N[(0.25 * N[(N[Power[N[(M$95$m * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if h < -4.89999999999999985e104

   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 60.7%

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

      1. frac-2neg 60.7%

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

      2. sqrt-div 72.9%

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

   6. Applied egg-rr 72.9%

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

   if -4.89999999999999985e104 < h < 1.38000000000000008e-151

   1. Initial program 72.1%

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

   3. Simplified 72.1%

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

      1. add-sqr-sqrt 72.1%

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

      2. pow2 72.1%

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

      3. sqrt-prod 72.1%

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

      4. sqrt-pow1 77.3%

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

      5. metadata-eval 77.3%

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

      6. associate-*l/ 77.3%

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

      7. div-inv 77.3%

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

      8. metadata-eval 77.3%

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

      9. *-commutative 77.3%

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

      10. pow1 77.3%

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

      11. clear-num 77.3%

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

      12. un-div-inv 77.3%

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

   6. Applied egg-rr 77.3%

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

   7. Taylor expanded in M around 0 77.4%

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

      1. sqrt-unprod 64.5%

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

      2. pow1/2 64.5%

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

      3. frac-times 55.9%

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

      4. pow2 55.9%

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

   9. Applied egg-rr 55.9%

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

       1. unpow1/2 55.9%

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

   11. Simplified 55.9%

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

       1. sqrt-div 59.0%

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

   13. Applied egg-rr 59.0%

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

       1. unpow2 59.0%

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

       2. rem-sqrt-square 87.5%

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

   15. Simplified 87.5%

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

   if 1.38000000000000008e-151 < h

   1. Initial program 71.3%

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

   3. Simplified 70.2%

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

      1. sqrt-div 73.1%

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

      2. sqrt-div 75.8%

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

      3. frac-times 75.8%

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

      4. add-sqr-sqrt 75.9%

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

   6. Applied egg-rr 75.9%

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

      1. associate-/r* 75.9%

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

   8. Simplified 75.9%

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

   9. Taylor expanded in M around 0 58.6%

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

       1. associate-*r* 58.6%

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

       2. times-frac 54.6%

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

       3. associate-/l* 55.6%

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

       4. *-commutative 55.6%

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

       5. unpow2 55.6%

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

       6. unpow2 55.6%

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

       7. times-frac 64.9%

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

       8. unpow2 64.9%

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

       9. swap-sqr 77.0%

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

       10. associate-/r/ 74.9%

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

       11. associate-/r/ 75.9%

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

       12. unpow2 75.9%

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

       13. associate-*l* 75.9%

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

       14. *-commutative 75.9%

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

       15. associate-*l/ 83.6%

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

       16. associate-/l* 83.6%

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

       17. associate-/l* 83.6%

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

   11. Simplified 83.5%

       \[\leadsto \frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}} \cdot \left(1 - 0.5 \cdot \color{blue}{\left(h \cdot \left(0.25 \cdot \frac{{\left(M \cdot \frac{D}{d}\right)}^{2}}{\ell}\right)\right)}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 78.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := {\left(M\_m \cdot \frac{D}{d}\right)}^{2}\\ \mathbf{if}\;h \leq -4.1 \cdot 10^{+103}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(h, t\_0 \cdot \frac{-0.125}{\ell}, 1\right)\right)\\ \mathbf{elif}\;h \leq 9.5 \cdot 10^{-152}:\\ \;\;\;\;\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - 0.5 \cdot {\left(0.5 \cdot \left(\frac{D \cdot M\_m}{d} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}} \cdot \left(1 - 0.5 \cdot \left(h \cdot \left(0.25 \cdot \frac{t\_0}{\ell}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (let* ((t_0 (pow (* M_m (/ D d)) 2.0)))
   (if (<= h -4.1e+103)
     (* (sqrt (/ d h)) (* (sqrt (/ d l)) (fma h (* t_0 (/ -0.125 l)) 1.0)))
     (if (<= h 9.5e-152)
       (*
        (/ (fabs d) (sqrt (* h l)))
        (- 1.0 (* 0.5 (pow (* 0.5 (* (/ (* D M_m) d) (sqrt (/ h l)))) 2.0))))
       (*
        (/ (/ d (sqrt h)) (sqrt l))
        (- 1.0 (* 0.5 (* h (* 0.25 (/ t_0 l))))))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double t_0 = pow((M_m * (D / d)), 2.0);
	double tmp;
	if (h <= -4.1e+103) {
		tmp = sqrt((d / h)) * (sqrt((d / l)) * fma(h, (t_0 * (-0.125 / l)), 1.0));
	} else if (h <= 9.5e-152) {
		tmp = (fabs(d) / sqrt((h * l))) * (1.0 - (0.5 * pow((0.5 * (((D * M_m) / d) * sqrt((h / l)))), 2.0)));
	} else {
		tmp = ((d / sqrt(h)) / sqrt(l)) * (1.0 - (0.5 * (h * (0.25 * (t_0 / l)))));
	}
	return tmp;
}

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	t_0 = Float64(M_m * Float64(D / d)) ^ 2.0
	tmp = 0.0
	if (h <= -4.1e+103)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(sqrt(Float64(d / l)) * fma(h, Float64(t_0 * Float64(-0.125 / l)), 1.0)));
	elseif (h <= 9.5e-152)
		tmp = Float64(Float64(abs(d) / sqrt(Float64(h * l))) * Float64(1.0 - Float64(0.5 * (Float64(0.5 * Float64(Float64(Float64(D * M_m) / d) * sqrt(Float64(h / l)))) ^ 2.0))));
	else
		tmp = Float64(Float64(Float64(d / sqrt(h)) / sqrt(l)) * Float64(1.0 - Float64(0.5 * Float64(h * Float64(0.25 * Float64(t_0 / l))))));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := Block[{t$95$0 = N[Power[N[(M$95$m * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[h, -4.1e+103], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(h * N[(t$95$0 * N[(-0.125 / l), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, 9.5e-152], N[(N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[Power[N[(0.5 * N[(N[(N[(D * M$95$m), $MachinePrecision] / d), $MachinePrecision] * N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(d / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(h * N[(0.25 * N[(t$95$0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if h < -4.1000000000000002e103

   1. Initial program 59.5%

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

   3. Simplified 59.6%

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

   5. Taylor expanded in M around 0 27.5%

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

      1. associate-*r/ 27.5%

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

      2. associate-*r* 35.5%

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

      3. associate-*r* 35.5%

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

      4. associate-*l/ 39.7%

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

      5. associate-*r/ 39.7%

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

      6. *-commutative 39.7%

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

      7. +-commutative 39.7%

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

      8. fma-undefine 39.7%

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

   7. Simplified 64.1%

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

   if -4.1000000000000002e103 < h < 9.49999999999999925e-152

   1. Initial program 72.8%

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

   3. Simplified 72.8%

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

      1. add-sqr-sqrt 72.7%

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

      2. pow2 72.7%

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

      3. sqrt-prod 72.7%

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

      4. sqrt-pow1 78.0%

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

      5. metadata-eval 78.0%

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

      6. associate-*l/ 78.0%

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

      7. div-inv 78.0%

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

      8. metadata-eval 78.0%

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

      9. *-commutative 78.0%

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

      10. pow1 78.0%

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

      11. clear-num 78.0%

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

      12. un-div-inv 78.0%

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

   6. Applied egg-rr 78.0%

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

   7. Taylor expanded in M around 0 78.1%

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

      1. sqrt-unprod 65.1%

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

      2. pow1/2 65.1%

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

      3. frac-times 56.4%

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

      4. pow2 56.4%

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

   9. Applied egg-rr 56.4%

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

       1. unpow1/2 56.4%

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

   11. Simplified 56.4%

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

       1. sqrt-div 59.6%

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

   13. Applied egg-rr 59.6%

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

       1. unpow2 59.6%

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

       2. rem-sqrt-square 88.3%

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

   15. Simplified 88.3%

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

   if 9.49999999999999925e-152 < h

   1. Initial program 71.3%

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

   3. Simplified 70.2%

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

      1. sqrt-div 73.1%

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

      2. sqrt-div 75.8%

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

      3. frac-times 75.8%

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

      4. add-sqr-sqrt 75.9%

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

   6. Applied egg-rr 75.9%

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

      1. associate-/r* 75.9%

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

   8. Simplified 75.9%

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

   9. Taylor expanded in M around 0 58.6%

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

       1. associate-*r* 58.6%

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

       2. times-frac 54.6%

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

       3. associate-/l* 55.6%

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

       4. *-commutative 55.6%

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

       5. unpow2 55.6%

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

       6. unpow2 55.6%

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

       7. times-frac 64.9%

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

       8. unpow2 64.9%

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

       9. swap-sqr 77.0%

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

       10. associate-/r/ 74.9%

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

       11. associate-/r/ 75.9%

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

       12. unpow2 75.9%

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

       13. associate-*l* 75.9%

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

       14. *-commutative 75.9%

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

       15. associate-*l/ 83.6%

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

       16. associate-/l* 83.6%

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

       17. associate-/l* 83.6%

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

   11. Simplified 83.5%

       \[\leadsto \frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}} \cdot \left(1 - 0.5 \cdot \color{blue}{\left(h \cdot \left(0.25 \cdot \frac{{\left(M \cdot \frac{D}{d}\right)}^{2}}{\ell}\right)\right)}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -4.1 \cdot 10^{+103}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(h, {\left(M \cdot \frac{D}{d}\right)}^{2} \cdot \frac{-0.125}{\ell}, 1\right)\right)\\ \mathbf{elif}\;h \leq 9.5 \cdot 10^{-152}:\\ \;\;\;\;\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - 0.5 \cdot {\left(0.5 \cdot \left(\frac{D \cdot M}{d} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}} \cdot \left(1 - 0.5 \cdot \left(h \cdot \left(0.25 \cdot \frac{{\left(M \cdot \frac{D}{d}\right)}^{2}}{\ell}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := {\left(M\_m \cdot \frac{D}{d}\right)}^{2}\\ \mathbf{if}\;h \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(h, t\_0 \cdot \frac{-0.125}{\ell}, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}} \cdot \left(1 - 0.5 \cdot \left(h \cdot \left(0.25 \cdot \frac{t\_0}{\ell}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (let* ((t_0 (pow (* M_m (/ D d)) 2.0)))
   (if (<= h -1e-309)
     (* (sqrt (/ d h)) (* (sqrt (/ d l)) (fma h (* t_0 (/ -0.125 l)) 1.0)))
     (*
      (/ (/ d (sqrt h)) (sqrt l))
      (- 1.0 (* 0.5 (* h (* 0.25 (/ t_0 l)))))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double t_0 = pow((M_m * (D / d)), 2.0);
	double tmp;
	if (h <= -1e-309) {
		tmp = sqrt((d / h)) * (sqrt((d / l)) * fma(h, (t_0 * (-0.125 / l)), 1.0));
	} else {
		tmp = ((d / sqrt(h)) / sqrt(l)) * (1.0 - (0.5 * (h * (0.25 * (t_0 / l)))));
	}
	return tmp;
}

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	t_0 = Float64(M_m * Float64(D / d)) ^ 2.0
	tmp = 0.0
	if (h <= -1e-309)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(sqrt(Float64(d / l)) * fma(h, Float64(t_0 * Float64(-0.125 / l)), 1.0)));
	else
		tmp = Float64(Float64(Float64(d / sqrt(h)) / sqrt(l)) * Float64(1.0 - Float64(0.5 * Float64(h * Float64(0.25 * Float64(t_0 / l))))));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := Block[{t$95$0 = N[Power[N[(M$95$m * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[h, -1e-309], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(h * N[(t$95$0 * N[(-0.125 / l), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(d / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(h * N[(0.25 * N[(t$95$0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if h < -1.000000000000002e-309

   1. Initial program 69.9%

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

   3. Simplified 70.0%

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

   5. Taylor expanded in M around 0 38.8%

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

      1. associate-*r/ 38.8%

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

      2. associate-*r* 41.9%

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

      3. associate-*r* 41.9%

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

      4. associate-*l/ 43.6%

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

      5. associate-*r/ 43.6%

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

      6. *-commutative 43.6%

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

      7. +-commutative 43.6%

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

      8. fma-undefine 43.6%

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

   7. Simplified 71.7%

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

   if -1.000000000000002e-309 < h

   1. Initial program 69.5%

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

   3. Simplified 68.6%

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

      1. sqrt-div 73.7%

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

      2. sqrt-div 76.4%

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

      3. frac-times 76.4%

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

      4. add-sqr-sqrt 76.5%

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

   6. Applied egg-rr 76.5%

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

      1. associate-/r* 75.8%

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

   8. Simplified 75.8%

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

   9. Taylor expanded in M around 0 59.6%

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

       1. associate-*r* 58.9%

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

       2. times-frac 53.6%

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

       3. associate-/l* 53.6%

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

       4. *-commutative 53.6%

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

       5. unpow2 53.6%

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

       6. unpow2 53.6%

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

       7. times-frac 65.2%

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

       8. unpow2 65.2%

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

       9. swap-sqr 75.8%

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

       10. associate-/r/ 74.3%

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

       11. associate-/r/ 75.8%

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

       12. unpow2 75.8%

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

       13. associate-*l* 75.8%

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

       14. *-commutative 75.8%

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

       15. associate-*l/ 81.6%

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

       16. associate-/l* 81.6%

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

       17. associate-/l* 81.6%

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

   11. Simplified 81.6%

       \[\leadsto \frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}} \cdot \left(1 - 0.5 \cdot \color{blue}{\left(h \cdot \left(0.25 \cdot \frac{{\left(M \cdot \frac{D}{d}\right)}^{2}}{\ell}\right)\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(h, {\left(M \cdot \frac{D}{d}\right)}^{2} \cdot \frac{-0.125}{\ell}, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}} \cdot \left(1 - 0.5 \cdot \left(h \cdot \left(0.25 \cdot \frac{{\left(M \cdot \frac{D}{d}\right)}^{2}}{\ell}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := \frac{{\left(M\_m \cdot \frac{D}{d}\right)}^{2}}{\ell}\\ \mathbf{if}\;h \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 + h \cdot \left(t\_0 \cdot -0.125\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}} \cdot \left(1 - 0.5 \cdot \left(h \cdot \left(0.25 \cdot t\_0\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (let* ((t_0 (/ (pow (* M_m (/ D d)) 2.0) l)))
   (if (<= h -1e-309)
     (* (* (sqrt (/ d l)) (sqrt (/ d h))) (+ 1.0 (* h (* t_0 -0.125))))
     (* (/ (/ d (sqrt h)) (sqrt l)) (- 1.0 (* 0.5 (* h (* 0.25 t_0))))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double t_0 = pow((M_m * (D / d)), 2.0) / l;
	double tmp;
	if (h <= -1e-309) {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 + (h * (t_0 * -0.125)));
	} else {
		tmp = ((d / sqrt(h)) / sqrt(l)) * (1.0 - (0.5 * (h * (0.25 * t_0))));
	}
	return tmp;
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((m_m * (d_1 / d)) ** 2.0d0) / l
    if (h <= (-1d-309)) then
        tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0d0 + (h * (t_0 * (-0.125d0))))
    else
        tmp = ((d / sqrt(h)) / sqrt(l)) * (1.0d0 - (0.5d0 * (h * (0.25d0 * t_0))))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double t_0 = Math.pow((M_m * (D / d)), 2.0) / l;
	double tmp;
	if (h <= -1e-309) {
		tmp = (Math.sqrt((d / l)) * Math.sqrt((d / h))) * (1.0 + (h * (t_0 * -0.125)));
	} else {
		tmp = ((d / Math.sqrt(h)) / Math.sqrt(l)) * (1.0 - (0.5 * (h * (0.25 * t_0))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	t_0 = math.pow((M_m * (D / d)), 2.0) / l
	tmp = 0
	if h <= -1e-309:
		tmp = (math.sqrt((d / l)) * math.sqrt((d / h))) * (1.0 + (h * (t_0 * -0.125)))
	else:
		tmp = ((d / math.sqrt(h)) / math.sqrt(l)) * (1.0 - (0.5 * (h * (0.25 * t_0))))
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	t_0 = Float64((Float64(M_m * Float64(D / d)) ^ 2.0) / l)
	tmp = 0.0
	if (h <= -1e-309)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * Float64(1.0 + Float64(h * Float64(t_0 * -0.125))));
	else
		tmp = Float64(Float64(Float64(d / sqrt(h)) / sqrt(l)) * Float64(1.0 - Float64(0.5 * Float64(h * Float64(0.25 * t_0)))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp_2 = code(d, h, l, M_m, D)
	t_0 = ((M_m * (D / d)) ^ 2.0) / l;
	tmp = 0.0;
	if (h <= -1e-309)
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 + (h * (t_0 * -0.125)));
	else
		tmp = ((d / sqrt(h)) / sqrt(l)) * (1.0 - (0.5 * (h * (0.25 * t_0))));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := Block[{t$95$0 = N[(N[Power[N[(M$95$m * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]}, If[LessEqual[h, -1e-309], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(h * N[(t$95$0 * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(d / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(h * N[(0.25 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if h < -1.000000000000002e-309

   1. Initial program 69.9%

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

   3. Simplified 69.9%

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

      1. add-sqr-sqrt 69.9%

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

      2. pow2 69.9%

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

      3. sqrt-prod 69.9%

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

      4. sqrt-pow1 71.4%

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

      5. metadata-eval 71.4%

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

      6. associate-*l/ 71.4%

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

      7. div-inv 71.4%

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

      8. metadata-eval 71.4%

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

      9. *-commutative 71.4%

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

      10. pow1 71.4%

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

      11. clear-num 71.4%

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

      12. un-div-inv 71.4%

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

   6. Applied egg-rr 71.4%

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

   7. Taylor expanded in M around 0 72.2%

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

   8. Taylor expanded in h around inf 43.5%

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

      1. sub-neg 43.5%

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

      2. distribute-lft-in 43.5%

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

      3. rgt-mult-inverse 43.6%

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

      4. distribute-lft-neg-in 43.6%

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

      5. metadata-eval 43.6%

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

      6. associate-/l* 42.8%

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

      7. associate-/r* 42.8%

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

      8. associate-/l* 45.8%

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

      9. unpow2 45.8%

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

      10. unpow2 45.8%

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

      11. unpow2 45.8%

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

      12. times-frac 60.2%

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

      13. swap-sqr 70.2%

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

      14. unpow2 70.2%

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

      15. associate-*r/ 71.7%

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

      16. *-commutative 71.7%

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

      17. associate-/l* 71.7%

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

   10. Simplified 71.7%

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

   if -1.000000000000002e-309 < h

   1. Initial program 69.5%

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

   3. Simplified 68.6%

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

      1. sqrt-div 73.7%

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

      2. sqrt-div 76.4%

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

      3. frac-times 76.4%

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

      4. add-sqr-sqrt 76.5%

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

   6. Applied egg-rr 76.5%

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

      1. associate-/r* 75.8%

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

   8. Simplified 75.8%

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

   9. Taylor expanded in M around 0 59.6%

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

       1. associate-*r* 58.9%

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

       2. times-frac 53.6%

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

       3. associate-/l* 53.6%

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

       4. *-commutative 53.6%

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

       5. unpow2 53.6%

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

       6. unpow2 53.6%

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

       7. times-frac 65.2%

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

       8. unpow2 65.2%

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

       9. swap-sqr 75.8%

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

       10. associate-/r/ 74.3%

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

       11. associate-/r/ 75.8%

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

       12. unpow2 75.8%

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

       13. associate-*l* 75.8%

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

       14. *-commutative 75.8%

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

       15. associate-*l/ 81.6%

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

       16. associate-/l* 81.6%

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

       17. associate-/l* 81.6%

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

   11. Simplified 81.6%

       \[\leadsto \frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}} \cdot \left(1 - 0.5 \cdot \color{blue}{\left(h \cdot \left(0.25 \cdot \frac{{\left(M \cdot \frac{D}{d}\right)}^{2}}{\ell}\right)\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 + h \cdot \left(\frac{{\left(M \cdot \frac{D}{d}\right)}^{2}}{\ell} \cdot -0.125\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}} \cdot \left(1 - 0.5 \cdot \left(h \cdot \left(0.25 \cdot \frac{{\left(M \cdot \frac{D}{d}\right)}^{2}}{\ell}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 68.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 1.15 \cdot 10^{+171}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 + h \cdot \left(\frac{{\left(M\_m \cdot \frac{D}{d}\right)}^{2}}{\ell} \cdot -0.125\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right) \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M\_m}{2} \cdot \frac{D}{d}\right)}^{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (if (<= d 1.15e+171)
   (*
    (* (sqrt (/ d l)) (sqrt (/ d h)))
    (+ 1.0 (* h (* (/ (pow (* M_m (/ D d)) 2.0) l) -0.125))))
   (*
    (* d (sqrt (/ (/ 1.0 l) h)))
    (- 1.0 (* 0.5 (* (/ h l) (pow (* (/ M_m 2.0) (/ D d)) 2.0)))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (d <= 1.15e+171) {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 + (h * ((pow((M_m * (D / d)), 2.0) / l) * -0.125)));
	} else {
		tmp = (d * sqrt(((1.0 / l) / h))) * (1.0 - (0.5 * ((h / l) * pow(((M_m / 2.0) * (D / d)), 2.0))));
	}
	return tmp;
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (d <= 1.15d+171) then
        tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0d0 + (h * ((((m_m * (d_1 / d)) ** 2.0d0) / l) * (-0.125d0))))
    else
        tmp = (d * sqrt(((1.0d0 / l) / h))) * (1.0d0 - (0.5d0 * ((h / l) * (((m_m / 2.0d0) * (d_1 / d)) ** 2.0d0))))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (d <= 1.15e+171) {
		tmp = (Math.sqrt((d / l)) * Math.sqrt((d / h))) * (1.0 + (h * ((Math.pow((M_m * (D / d)), 2.0) / l) * -0.125)));
	} else {
		tmp = (d * Math.sqrt(((1.0 / l) / h))) * (1.0 - (0.5 * ((h / l) * Math.pow(((M_m / 2.0) * (D / d)), 2.0))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	tmp = 0
	if d <= 1.15e+171:
		tmp = (math.sqrt((d / l)) * math.sqrt((d / h))) * (1.0 + (h * ((math.pow((M_m * (D / d)), 2.0) / l) * -0.125)))
	else:
		tmp = (d * math.sqrt(((1.0 / l) / h))) * (1.0 - (0.5 * ((h / l) * math.pow(((M_m / 2.0) * (D / d)), 2.0))))
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	tmp = 0.0
	if (d <= 1.15e+171)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * Float64(1.0 + Float64(h * Float64(Float64((Float64(M_m * Float64(D / d)) ^ 2.0) / l) * -0.125))));
	else
		tmp = Float64(Float64(d * sqrt(Float64(Float64(1.0 / l) / h))) * Float64(1.0 - Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(M_m / 2.0) * Float64(D / d)) ^ 2.0)))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp_2 = code(d, h, l, M_m, D)
	tmp = 0.0;
	if (d <= 1.15e+171)
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 + (h * ((((M_m * (D / d)) ^ 2.0) / l) * -0.125)));
	else
		tmp = (d * sqrt(((1.0 / l) / h))) * (1.0 - (0.5 * ((h / l) * (((M_m / 2.0) * (D / d)) ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := If[LessEqual[d, 1.15e+171], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(h * N[(N[(N[Power[N[(M$95$m * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(d * N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M$95$m / 2.0), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 1.15000000000000009e171

   1. Initial program 69.1%

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

   3. Simplified 68.6%

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

      1. add-sqr-sqrt 68.6%

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

      2. pow2 68.6%

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

      3. sqrt-prod 68.6%

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

      4. sqrt-pow1 73.3%

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

      5. metadata-eval 73.3%

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

      6. associate-*l/ 73.3%

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

      7. div-inv 73.3%

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

      8. metadata-eval 73.3%

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

      9. *-commutative 73.3%

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

      10. pow1 73.3%

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

      11. clear-num 73.3%

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

      12. un-div-inv 73.3%

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

   6. Applied egg-rr 73.3%

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

   7. Taylor expanded in M around 0 73.4%

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

   8. Taylor expanded in h around inf 50.5%

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

      1. sub-neg 50.5%

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

      2. distribute-lft-in 50.5%

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

      3. rgt-mult-inverse 50.6%

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

      4. distribute-lft-neg-in 50.6%

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

      5. metadata-eval 50.6%

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

      6. associate-/l* 48.9%

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

      7. associate-/r* 48.1%

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

      8. associate-/l* 49.8%

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

      9. unpow2 49.8%

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

      10. unpow2 49.8%

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

      11. unpow2 49.8%

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

      12. times-frac 60.3%

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

      13. swap-sqr 70.3%

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

      14. unpow2 70.3%

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

      15. associate-*r/ 71.6%

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

      16. *-commutative 71.6%

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

      17. associate-/l* 71.2%

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

   10. Simplified 71.2%

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

   if 1.15000000000000009e171 < d

   1. Initial program 73.9%

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

   3. Simplified 73.7%

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

      1. sqrt-div 85.3%

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

      2. sqrt-div 93.3%

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

      3. frac-times 93.3%

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

      4. add-sqr-sqrt 93.3%

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

   6. Applied egg-rr 93.3%

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

      1. associate-/r* 90.5%

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

   8. Simplified 90.5%

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

   9. Taylor expanded in d around 0 84.4%

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

       1. associate-/r* 84.4%

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

       2. associate-/l/ 84.4%

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

       3. associate-/r* 84.4%

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

   11. Simplified 84.4%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 1.15 \cdot 10^{+171}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 + h \cdot \left(\frac{{\left(M \cdot \frac{D}{d}\right)}^{2}}{\ell} \cdot -0.125\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right) \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 64.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -4 \cdot 10^{-116}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M\_m \cdot 0.5}{\frac{d}{D}}\right)}^{2}\right)\right)\\ \mathbf{elif}\;d \leq 2.6 \cdot 10^{-304}:\\ \;\;\;\;-0.125 \cdot \frac{{\left(D \cdot M\_m\right)}^{2} \cdot \left(-\sqrt{\frac{h}{{\ell}^{3}}}\right)}{d}\\ \mathbf{else}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right) \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M\_m}{2} \cdot \frac{D}{d}\right)}^{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (if (<= d -4e-116)
   (*
    (sqrt (* (/ d l) (/ d h)))
    (+ 1.0 (* -0.5 (* (/ h l) (pow (/ (* M_m 0.5) (/ d D)) 2.0)))))
   (if (<= d 2.6e-304)
     (* -0.125 (/ (* (pow (* D M_m) 2.0) (- (sqrt (/ h (pow l 3.0))))) d))
     (*
      (* d (sqrt (/ (/ 1.0 l) h)))
      (- 1.0 (* 0.5 (* (/ h l) (pow (* (/ M_m 2.0) (/ D d)) 2.0))))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (d <= -4e-116) {
		tmp = sqrt(((d / l) * (d / h))) * (1.0 + (-0.5 * ((h / l) * pow(((M_m * 0.5) / (d / D)), 2.0))));
	} else if (d <= 2.6e-304) {
		tmp = -0.125 * ((pow((D * M_m), 2.0) * -sqrt((h / pow(l, 3.0)))) / d);
	} else {
		tmp = (d * sqrt(((1.0 / l) / h))) * (1.0 - (0.5 * ((h / l) * pow(((M_m / 2.0) * (D / d)), 2.0))));
	}
	return tmp;
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (d <= (-4d-116)) then
        tmp = sqrt(((d / l) * (d / h))) * (1.0d0 + ((-0.5d0) * ((h / l) * (((m_m * 0.5d0) / (d / d_1)) ** 2.0d0))))
    else if (d <= 2.6d-304) then
        tmp = (-0.125d0) * ((((d_1 * m_m) ** 2.0d0) * -sqrt((h / (l ** 3.0d0)))) / d)
    else
        tmp = (d * sqrt(((1.0d0 / l) / h))) * (1.0d0 - (0.5d0 * ((h / l) * (((m_m / 2.0d0) * (d_1 / d)) ** 2.0d0))))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (d <= -4e-116) {
		tmp = Math.sqrt(((d / l) * (d / h))) * (1.0 + (-0.5 * ((h / l) * Math.pow(((M_m * 0.5) / (d / D)), 2.0))));
	} else if (d <= 2.6e-304) {
		tmp = -0.125 * ((Math.pow((D * M_m), 2.0) * -Math.sqrt((h / Math.pow(l, 3.0)))) / d);
	} else {
		tmp = (d * Math.sqrt(((1.0 / l) / h))) * (1.0 - (0.5 * ((h / l) * Math.pow(((M_m / 2.0) * (D / d)), 2.0))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	tmp = 0
	if d <= -4e-116:
		tmp = math.sqrt(((d / l) * (d / h))) * (1.0 + (-0.5 * ((h / l) * math.pow(((M_m * 0.5) / (d / D)), 2.0))))
	elif d <= 2.6e-304:
		tmp = -0.125 * ((math.pow((D * M_m), 2.0) * -math.sqrt((h / math.pow(l, 3.0)))) / d)
	else:
		tmp = (d * math.sqrt(((1.0 / l) / h))) * (1.0 - (0.5 * ((h / l) * math.pow(((M_m / 2.0) * (D / d)), 2.0))))
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	tmp = 0.0
	if (d <= -4e-116)
		tmp = Float64(sqrt(Float64(Float64(d / l) * Float64(d / h))) * Float64(1.0 + Float64(-0.5 * Float64(Float64(h / l) * (Float64(Float64(M_m * 0.5) / Float64(d / D)) ^ 2.0)))));
	elseif (d <= 2.6e-304)
		tmp = Float64(-0.125 * Float64(Float64((Float64(D * M_m) ^ 2.0) * Float64(-sqrt(Float64(h / (l ^ 3.0))))) / d));
	else
		tmp = Float64(Float64(d * sqrt(Float64(Float64(1.0 / l) / h))) * Float64(1.0 - Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(M_m / 2.0) * Float64(D / d)) ^ 2.0)))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp_2 = code(d, h, l, M_m, D)
	tmp = 0.0;
	if (d <= -4e-116)
		tmp = sqrt(((d / l) * (d / h))) * (1.0 + (-0.5 * ((h / l) * (((M_m * 0.5) / (d / D)) ^ 2.0))));
	elseif (d <= 2.6e-304)
		tmp = -0.125 * ((((D * M_m) ^ 2.0) * -sqrt((h / (l ^ 3.0)))) / d);
	else
		tmp = (d * sqrt(((1.0 / l) / h))) * (1.0 - (0.5 * ((h / l) * (((M_m / 2.0) * (D / d)) ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := If[LessEqual[d, -4e-116], N[(N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(-0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M$95$m * 0.5), $MachinePrecision] / N[(d / D), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 2.6e-304], N[(-0.125 * N[(N[(N[Power[N[(D * M$95$m), $MachinePrecision], 2.0], $MachinePrecision] * (-N[Sqrt[N[(h / N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], N[(N[(d * N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M$95$m / 2.0), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -4e-116

   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 78.1%

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

      1. add-sqr-sqrt 78.1%

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

      2. pow2 78.1%

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

      3. sqrt-prod 78.1%

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

      4. sqrt-pow1 80.4%

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

      5. metadata-eval 80.4%

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

      6. associate-*l/ 80.4%

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

      7. div-inv 80.4%

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

      8. metadata-eval 80.4%

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

      9. *-commutative 80.4%

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

      10. pow1 80.4%

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

      11. clear-num 80.4%

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

      12. un-div-inv 80.4%

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

   6. Applied egg-rr 80.4%

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

      1. *-commutative 80.4%

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

      2. sqrt-unprod 72.4%

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

   8. Applied egg-rr 72.4%

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

      1. cancel-sign-sub-inv 72.4%

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

      2. metadata-eval 72.4%

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

      3. *-commutative 72.4%

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

      4. unpow-prod-down 70.1%

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

      5. pow2 70.1%

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

      6. add-sqr-sqrt 70.1%

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

      7. associate-*r/ 70.1%

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

   10. Applied egg-rr 70.1%

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

   if -4e-116 < d < 2.59999999999999997e-304

   1. Initial program 53.4%

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

   3. Simplified 53.5%

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

   5. Taylor expanded in M around inf 25.5%

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

   6. Taylor expanded in h around -inf 0.0%

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

      1. *-commutative 0.0%

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

      2. associate-*l/ 0.0%

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

      3. associate-*r* 0.0%

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

      4. unpow2 0.0%

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

      5. rem-square-sqrt 40.8%

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

      6. unpow2 40.8%

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

      7. unpow2 40.8%

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

      8. swap-sqr 49.1%

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

      9. unpow2 49.1%

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

      10. associate-*l* 49.1%

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

      11. mul-1-neg 49.1%

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

   8. Simplified 49.1%

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

   if 2.59999999999999997e-304 < d

   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 69.7%

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

      1. sqrt-div 74.8%

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

      2. sqrt-div 76.8%

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

      3. frac-times 76.8%

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

      4. add-sqr-sqrt 76.9%

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

   6. Applied egg-rr 76.9%

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

      1. associate-/r* 76.2%

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

   8. Simplified 76.2%

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

   9. Taylor expanded in d around 0 70.1%

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

       1. associate-/r* 70.9%

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

       2. associate-/l/ 70.1%

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

       3. associate-/r* 70.9%

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

   11. Simplified 70.9%

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

4. Final simplification 66.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -4 \cdot 10^{-116}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M \cdot 0.5}{\frac{d}{D}}\right)}^{2}\right)\right)\\ \mathbf{elif}\;d \leq 2.6 \cdot 10^{-304}:\\ \;\;\;\;-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2} \cdot \left(-\sqrt{\frac{h}{{\ell}^{3}}}\right)}{d}\\ \mathbf{else}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right) \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 63.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -1.6 \cdot 10^{-165}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M\_m \cdot 0.5}{\frac{d}{D}}\right)}^{2}\right)\right)\\ \mathbf{elif}\;d \leq 2.6 \cdot 10^{-304}:\\ \;\;\;\;d \cdot \sqrt{\frac{\log \left(e^{\frac{1}{\ell}}\right)}{h}}\\ \mathbf{else}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right) \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M\_m}{2} \cdot \frac{D}{d}\right)}^{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (if (<= d -1.6e-165)
   (*
    (sqrt (* (/ d l) (/ d h)))
    (+ 1.0 (* -0.5 (* (/ h l) (pow (/ (* M_m 0.5) (/ d D)) 2.0)))))
   (if (<= d 2.6e-304)
     (* d (sqrt (/ (log (exp (/ 1.0 l))) h)))
     (*
      (* d (sqrt (/ (/ 1.0 l) h)))
      (- 1.0 (* 0.5 (* (/ h l) (pow (* (/ M_m 2.0) (/ D d)) 2.0))))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (d <= -1.6e-165) {
		tmp = sqrt(((d / l) * (d / h))) * (1.0 + (-0.5 * ((h / l) * pow(((M_m * 0.5) / (d / D)), 2.0))));
	} else if (d <= 2.6e-304) {
		tmp = d * sqrt((log(exp((1.0 / l))) / h));
	} else {
		tmp = (d * sqrt(((1.0 / l) / h))) * (1.0 - (0.5 * ((h / l) * pow(((M_m / 2.0) * (D / d)), 2.0))));
	}
	return tmp;
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (d <= (-1.6d-165)) then
        tmp = sqrt(((d / l) * (d / h))) * (1.0d0 + ((-0.5d0) * ((h / l) * (((m_m * 0.5d0) / (d / d_1)) ** 2.0d0))))
    else if (d <= 2.6d-304) then
        tmp = d * sqrt((log(exp((1.0d0 / l))) / h))
    else
        tmp = (d * sqrt(((1.0d0 / l) / h))) * (1.0d0 - (0.5d0 * ((h / l) * (((m_m / 2.0d0) * (d_1 / d)) ** 2.0d0))))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (d <= -1.6e-165) {
		tmp = Math.sqrt(((d / l) * (d / h))) * (1.0 + (-0.5 * ((h / l) * Math.pow(((M_m * 0.5) / (d / D)), 2.0))));
	} else if (d <= 2.6e-304) {
		tmp = d * Math.sqrt((Math.log(Math.exp((1.0 / l))) / h));
	} else {
		tmp = (d * Math.sqrt(((1.0 / l) / h))) * (1.0 - (0.5 * ((h / l) * Math.pow(((M_m / 2.0) * (D / d)), 2.0))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	tmp = 0
	if d <= -1.6e-165:
		tmp = math.sqrt(((d / l) * (d / h))) * (1.0 + (-0.5 * ((h / l) * math.pow(((M_m * 0.5) / (d / D)), 2.0))))
	elif d <= 2.6e-304:
		tmp = d * math.sqrt((math.log(math.exp((1.0 / l))) / h))
	else:
		tmp = (d * math.sqrt(((1.0 / l) / h))) * (1.0 - (0.5 * ((h / l) * math.pow(((M_m / 2.0) * (D / d)), 2.0))))
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	tmp = 0.0
	if (d <= -1.6e-165)
		tmp = Float64(sqrt(Float64(Float64(d / l) * Float64(d / h))) * Float64(1.0 + Float64(-0.5 * Float64(Float64(h / l) * (Float64(Float64(M_m * 0.5) / Float64(d / D)) ^ 2.0)))));
	elseif (d <= 2.6e-304)
		tmp = Float64(d * sqrt(Float64(log(exp(Float64(1.0 / l))) / h)));
	else
		tmp = Float64(Float64(d * sqrt(Float64(Float64(1.0 / l) / h))) * Float64(1.0 - Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(M_m / 2.0) * Float64(D / d)) ^ 2.0)))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp_2 = code(d, h, l, M_m, D)
	tmp = 0.0;
	if (d <= -1.6e-165)
		tmp = sqrt(((d / l) * (d / h))) * (1.0 + (-0.5 * ((h / l) * (((M_m * 0.5) / (d / D)) ^ 2.0))));
	elseif (d <= 2.6e-304)
		tmp = d * sqrt((log(exp((1.0 / l))) / h));
	else
		tmp = (d * sqrt(((1.0 / l) / h))) * (1.0 - (0.5 * ((h / l) * (((M_m / 2.0) * (D / d)) ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := If[LessEqual[d, -1.6e-165], N[(N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(-0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M$95$m * 0.5), $MachinePrecision] / N[(d / D), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 2.6e-304], N[(d * N[Sqrt[N[(N[Log[N[Exp[N[(1.0 / l), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(d * N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M$95$m / 2.0), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.60000000000000006e-165

   1. Initial program 78.0%

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

   3. Simplified 78.1%

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

      1. add-sqr-sqrt 78.0%

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

      2. pow2 78.0%

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

      3. sqrt-prod 78.0%

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

      4. sqrt-pow1 80.1%

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

      5. metadata-eval 80.1%

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

      6. associate-*l/ 80.1%

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

      7. div-inv 80.1%

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

      8. metadata-eval 80.1%

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

      9. *-commutative 80.1%

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

      10. pow1 80.1%

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

      11. clear-num 80.2%

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

      12. un-div-inv 80.1%

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

   6. Applied egg-rr 80.1%

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

      1. *-commutative 80.1%

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

      2. sqrt-unprod 68.6%

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

   8. Applied egg-rr 68.6%

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

      1. cancel-sign-sub-inv 68.6%

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

      2. metadata-eval 68.6%

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

      3. *-commutative 68.6%

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

      4. unpow-prod-down 66.5%

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

      5. pow2 66.5%

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

      6. add-sqr-sqrt 66.5%

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

      7. associate-*r/ 66.5%

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

   10. Applied egg-rr 66.5%

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

   if -1.60000000000000006e-165 < d < 2.59999999999999997e-304

   1. Initial program 47.1%

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

   3. Simplified 47.1%

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

   5. Taylor expanded in d around inf 17.5%

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

      1. *-un-lft-identity 17.5%

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

      2. associate-/r* 17.5%

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

   7. Applied egg-rr 17.5%

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

      1. *-lft-identity 17.5%

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

      2. associate-/l/ 17.5%

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

      3. associate-/r* 17.5%

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

   9. Simplified 17.5%

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

       1. add-log-exp 43.6%

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

   11. Applied egg-rr 43.6%

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

   if 2.59999999999999997e-304 < d

   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 69.7%

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

      1. sqrt-div 74.8%

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

      2. sqrt-div 76.8%

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

      3. frac-times 76.8%

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

      4. add-sqr-sqrt 76.9%

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

   6. Applied egg-rr 76.9%

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

      1. associate-/r* 76.2%

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

   8. Simplified 76.2%

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

   9. Taylor expanded in d around 0 70.1%

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

       1. associate-/r* 70.9%

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

       2. associate-/l/ 70.1%

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

       3. associate-/r* 70.9%

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

   11. Simplified 70.9%

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

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.6 \cdot 10^{-165}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M \cdot 0.5}{\frac{d}{D}}\right)}^{2}\right)\right)\\ \mathbf{elif}\;d \leq 2.6 \cdot 10^{-304}:\\ \;\;\;\;d \cdot \sqrt{\frac{\log \left(e^{\frac{1}{\ell}}\right)}{h}}\\ \mathbf{else}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right) \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 59.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -7 \cdot 10^{-94}:\\ \;\;\;\;\left(-d\right) \cdot {\left(h \cdot \ell\right)}^{-0.5}\\ \mathbf{elif}\;\ell \leq -1.05 \cdot 10^{-235}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(-0.125 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D \cdot M\_m}{d}\right)}^{2}\right)\right)\\ \mathbf{elif}\;\ell \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;\ell \leq 9 \cdot 10^{+199}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M\_m}{2} \cdot \frac{D}{d}\right)}^{2}\right)\right) \cdot \frac{d}{\sqrt{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (if (<= l -7e-94)
   (* (- d) (pow (* h l) -0.5))
   (if (<= l -1.05e-235)
     (*
      (sqrt (* (/ d l) (/ d h)))
      (* -0.125 (* (/ h l) (pow (/ (* D M_m) d) 2.0))))
     (if (<= l -4e-310)
       (* (sqrt (/ d l)) (sqrt (/ d h)))
       (if (<= l 9e+199)
         (*
          (- 1.0 (* 0.5 (* (/ h l) (pow (* (/ M_m 2.0) (/ D d)) 2.0))))
          (/ d (sqrt (* h l))))
         (* d (* (pow l -0.5) (pow h -0.5))))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (l <= -7e-94) {
		tmp = -d * pow((h * l), -0.5);
	} else if (l <= -1.05e-235) {
		tmp = sqrt(((d / l) * (d / h))) * (-0.125 * ((h / l) * pow(((D * M_m) / d), 2.0)));
	} else if (l <= -4e-310) {
		tmp = sqrt((d / l)) * sqrt((d / h));
	} else if (l <= 9e+199) {
		tmp = (1.0 - (0.5 * ((h / l) * pow(((M_m / 2.0) * (D / d)), 2.0)))) * (d / sqrt((h * l)));
	} else {
		tmp = d * (pow(l, -0.5) * pow(h, -0.5));
	}
	return tmp;
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (l <= (-7d-94)) then
        tmp = -d * ((h * l) ** (-0.5d0))
    else if (l <= (-1.05d-235)) then
        tmp = sqrt(((d / l) * (d / h))) * ((-0.125d0) * ((h / l) * (((d_1 * m_m) / d) ** 2.0d0)))
    else if (l <= (-4d-310)) then
        tmp = sqrt((d / l)) * sqrt((d / h))
    else if (l <= 9d+199) then
        tmp = (1.0d0 - (0.5d0 * ((h / l) * (((m_m / 2.0d0) * (d_1 / d)) ** 2.0d0)))) * (d / sqrt((h * l)))
    else
        tmp = d * ((l ** (-0.5d0)) * (h ** (-0.5d0)))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (l <= -7e-94) {
		tmp = -d * Math.pow((h * l), -0.5);
	} else if (l <= -1.05e-235) {
		tmp = Math.sqrt(((d / l) * (d / h))) * (-0.125 * ((h / l) * Math.pow(((D * M_m) / d), 2.0)));
	} else if (l <= -4e-310) {
		tmp = Math.sqrt((d / l)) * Math.sqrt((d / h));
	} else if (l <= 9e+199) {
		tmp = (1.0 - (0.5 * ((h / l) * Math.pow(((M_m / 2.0) * (D / d)), 2.0)))) * (d / Math.sqrt((h * l)));
	} else {
		tmp = d * (Math.pow(l, -0.5) * Math.pow(h, -0.5));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	tmp = 0
	if l <= -7e-94:
		tmp = -d * math.pow((h * l), -0.5)
	elif l <= -1.05e-235:
		tmp = math.sqrt(((d / l) * (d / h))) * (-0.125 * ((h / l) * math.pow(((D * M_m) / d), 2.0)))
	elif l <= -4e-310:
		tmp = math.sqrt((d / l)) * math.sqrt((d / h))
	elif l <= 9e+199:
		tmp = (1.0 - (0.5 * ((h / l) * math.pow(((M_m / 2.0) * (D / d)), 2.0)))) * (d / math.sqrt((h * l)))
	else:
		tmp = d * (math.pow(l, -0.5) * math.pow(h, -0.5))
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	tmp = 0.0
	if (l <= -7e-94)
		tmp = Float64(Float64(-d) * (Float64(h * l) ^ -0.5));
	elseif (l <= -1.05e-235)
		tmp = Float64(sqrt(Float64(Float64(d / l) * Float64(d / h))) * Float64(-0.125 * Float64(Float64(h / l) * (Float64(Float64(D * M_m) / d) ^ 2.0))));
	elseif (l <= -4e-310)
		tmp = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)));
	elseif (l <= 9e+199)
		tmp = Float64(Float64(1.0 - Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(M_m / 2.0) * Float64(D / d)) ^ 2.0)))) * Float64(d / sqrt(Float64(h * l))));
	else
		tmp = Float64(d * Float64((l ^ -0.5) * (h ^ -0.5)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp_2 = code(d, h, l, M_m, D)
	tmp = 0.0;
	if (l <= -7e-94)
		tmp = -d * ((h * l) ^ -0.5);
	elseif (l <= -1.05e-235)
		tmp = sqrt(((d / l) * (d / h))) * (-0.125 * ((h / l) * (((D * M_m) / d) ^ 2.0)));
	elseif (l <= -4e-310)
		tmp = sqrt((d / l)) * sqrt((d / h));
	elseif (l <= 9e+199)
		tmp = (1.0 - (0.5 * ((h / l) * (((M_m / 2.0) * (D / d)) ^ 2.0)))) * (d / sqrt((h * l)));
	else
		tmp = d * ((l ^ -0.5) * (h ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := If[LessEqual[l, -7e-94], N[((-d) * N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -1.05e-235], N[(N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-0.125 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D * M$95$m), $MachinePrecision] / d), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -4e-310], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 9e+199], N[(N[(1.0 - N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M$95$m / 2.0), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[l, -0.5], $MachinePrecision] * N[Power[h, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if l < -6.99999999999999996e-94

   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 66.8%

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

   5. Taylor expanded in d around inf 7.2%

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

   6. Taylor expanded in h around -inf 0.0%

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

      1. *-commutative 0.0%

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

      2. associate-/r* 0.0%

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

      3. unpow1/2 0.0%

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

      4. associate-/r* 0.0%

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

      5. rem-exp-log 0.0%

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

      6. exp-neg 0.0%

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

      7. exp-prod 0.0%

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

      8. distribute-lft-neg-out 0.0%

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

      9. distribute-rgt-neg-in 0.0%

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

      10. metadata-eval 0.0%

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

      11. exp-to-pow 0.0%

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

      12. *-commutative 0.0%

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

      13. unpow2 0.0%

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

      14. rem-square-sqrt 46.3%

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

      15. mul-1-neg 46.3%

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

   8. Simplified 46.3%

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

   if -6.99999999999999996e-94 < l < -1.05e-235

   1. Initial program 87.3%

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

   3. Simplified 87.3%

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

      1. add-sqr-sqrt 87.2%

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

      2. pow2 87.2%

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

      3. sqrt-prod 87.2%

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

      4. sqrt-pow1 87.2%

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

      5. metadata-eval 87.2%

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

      6. associate-*l/ 87.2%

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

      7. div-inv 87.2%

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

      8. metadata-eval 87.2%

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

      9. *-commutative 87.2%

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

      10. pow1 87.2%

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

      11. clear-num 87.2%

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

      12. un-div-inv 87.2%

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

   6. Applied egg-rr 87.2%

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

   7. Taylor expanded in M around 0 87.2%

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

   8. Taylor expanded in D around inf 52.2%

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

      1. associate-/l* 52.2%

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

      2. times-frac 52.2%

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

      3. associate-*r* 52.2%

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

      4. unpow2 52.2%

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

      5. unpow2 52.2%

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

      6. unpow2 52.2%

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

      7. times-frac 69.6%

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

      8. swap-sqr 78.3%

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

      9. unpow2 78.3%

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

      10. associate-*r/ 78.3%

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

      11. *-commutative 78.3%

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

      12. associate-/l* 78.3%

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

   10. Simplified 78.3%

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

       1. pow1 78.3%

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

       2. sqrt-unprod 74.3%

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

       3. associate-*r* 74.3%

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

       4. associate-*r/ 74.3%

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

       5. *-commutative 74.3%

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

       6. associate-/l* 74.3%

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

   12. Applied egg-rr 74.3%

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

       1. unpow1 74.3%

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

       2. *-commutative 74.3%

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

       3. associate-*l* 74.3%

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

       4. *-commutative 74.3%

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

       5. associate-*r/ 74.3%

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

   14. Simplified 74.3%

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

   if -1.05e-235 < l < -3.999999999999988e-310

   1. Initial program 62.3%

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

   3. Simplified 62.4%

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

   5. Taylor expanded in d around inf 57.2%

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

   if -3.999999999999988e-310 < l < 8.9999999999999994e199

   1. Initial program 74.9%

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

   3. Simplified 73.9%

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

      1. sqrt-div 77.4%

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

      2. sqrt-div 80.5%

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

      3. frac-times 80.5%

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

      4. add-sqr-sqrt 80.6%

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

   6. Applied egg-rr 80.6%

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

      1. associate-/r* 80.6%

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

   8. Simplified 80.6%

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

      1. *-un-lft-identity 80.6%

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

      2. associate-/l/ 80.6%

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

      3. pow1/2 80.6%

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

      4. pow1/2 80.6%

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

      5. pow-prod-down 77.0%

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

      6. *-commutative 77.0%

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

      7. pow1/2 77.0%

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

   10. Applied egg-rr 77.0%

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

       1. *-lft-identity 77.0%

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

   12. Simplified 77.0%

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

   if 8.9999999999999994e199 < l

   1. Initial program 44.2%

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

   3. Simplified 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 40.4%

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

      1. *-un-lft-identity 40.4%

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

      2. pow1/2 40.4%

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

      3. inv-pow 40.4%

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

      4. pow-pow 40.4%

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

      5. metadata-eval 40.4%

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

   7. Applied egg-rr 40.4%

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

      1. *-lft-identity 40.4%

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

   9. Simplified 40.4%

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

       1. *-commutative 40.4%

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

       2. unpow-prod-down 56.9%

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

   11. Applied egg-rr 56.9%

       \[\leadsto d \cdot \color{blue}{\left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -7 \cdot 10^{-94}:\\ \;\;\;\;\left(-d\right) \cdot {\left(h \cdot \ell\right)}^{-0.5}\\ \mathbf{elif}\;\ell \leq -1.05 \cdot 10^{-235}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(-0.125 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)\right)\\ \mathbf{elif}\;\ell \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;\ell \leq 9 \cdot 10^{+199}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}\right)\right) \cdot \frac{d}{\sqrt{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 60.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M\_m \cdot 0.5}{\frac{d}{D}}\right)}^{2}\right)\right)\\ \mathbf{if}\;h \leq -1.95 \cdot 10^{-49}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;h \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\left(-d\right) \cdot {\left(h \cdot \ell\right)}^{-0.5}\\ \mathbf{elif}\;h \leq 2.75 \cdot 10^{+137}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right) \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M\_m}{2} \cdot \frac{D}{d}\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (let* ((t_0
         (*
          (sqrt (* (/ d l) (/ d h)))
          (+ 1.0 (* -0.5 (* (/ h l) (pow (/ (* M_m 0.5) (/ d D)) 2.0)))))))
   (if (<= h -1.95e-49)
     t_0
     (if (<= h -1e-309)
       (* (- d) (pow (* h l) -0.5))
       (if (<= h 2.75e+137)
         (*
          (* d (sqrt (/ (/ 1.0 l) h)))
          (- 1.0 (* 0.5 (* (/ h l) (pow (* (/ M_m 2.0) (/ D d)) 2.0)))))
         t_0)))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double t_0 = sqrt(((d / l) * (d / h))) * (1.0 + (-0.5 * ((h / l) * pow(((M_m * 0.5) / (d / D)), 2.0))));
	double tmp;
	if (h <= -1.95e-49) {
		tmp = t_0;
	} else if (h <= -1e-309) {
		tmp = -d * pow((h * l), -0.5);
	} else if (h <= 2.75e+137) {
		tmp = (d * sqrt(((1.0 / l) / h))) * (1.0 - (0.5 * ((h / l) * pow(((M_m / 2.0) * (D / d)), 2.0))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(((d / l) * (d / h))) * (1.0d0 + ((-0.5d0) * ((h / l) * (((m_m * 0.5d0) / (d / d_1)) ** 2.0d0))))
    if (h <= (-1.95d-49)) then
        tmp = t_0
    else if (h <= (-1d-309)) then
        tmp = -d * ((h * l) ** (-0.5d0))
    else if (h <= 2.75d+137) then
        tmp = (d * sqrt(((1.0d0 / l) / h))) * (1.0d0 - (0.5d0 * ((h / l) * (((m_m / 2.0d0) * (d_1 / d)) ** 2.0d0))))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double t_0 = Math.sqrt(((d / l) * (d / h))) * (1.0 + (-0.5 * ((h / l) * Math.pow(((M_m * 0.5) / (d / D)), 2.0))));
	double tmp;
	if (h <= -1.95e-49) {
		tmp = t_0;
	} else if (h <= -1e-309) {
		tmp = -d * Math.pow((h * l), -0.5);
	} else if (h <= 2.75e+137) {
		tmp = (d * Math.sqrt(((1.0 / l) / h))) * (1.0 - (0.5 * ((h / l) * Math.pow(((M_m / 2.0) * (D / d)), 2.0))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	t_0 = math.sqrt(((d / l) * (d / h))) * (1.0 + (-0.5 * ((h / l) * math.pow(((M_m * 0.5) / (d / D)), 2.0))))
	tmp = 0
	if h <= -1.95e-49:
		tmp = t_0
	elif h <= -1e-309:
		tmp = -d * math.pow((h * l), -0.5)
	elif h <= 2.75e+137:
		tmp = (d * math.sqrt(((1.0 / l) / h))) * (1.0 - (0.5 * ((h / l) * math.pow(((M_m / 2.0) * (D / d)), 2.0))))
	else:
		tmp = t_0
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	t_0 = Float64(sqrt(Float64(Float64(d / l) * Float64(d / h))) * Float64(1.0 + Float64(-0.5 * Float64(Float64(h / l) * (Float64(Float64(M_m * 0.5) / Float64(d / D)) ^ 2.0)))))
	tmp = 0.0
	if (h <= -1.95e-49)
		tmp = t_0;
	elseif (h <= -1e-309)
		tmp = Float64(Float64(-d) * (Float64(h * l) ^ -0.5));
	elseif (h <= 2.75e+137)
		tmp = Float64(Float64(d * sqrt(Float64(Float64(1.0 / l) / h))) * Float64(1.0 - Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(M_m / 2.0) * Float64(D / d)) ^ 2.0)))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp_2 = code(d, h, l, M_m, D)
	t_0 = sqrt(((d / l) * (d / h))) * (1.0 + (-0.5 * ((h / l) * (((M_m * 0.5) / (d / D)) ^ 2.0))));
	tmp = 0.0;
	if (h <= -1.95e-49)
		tmp = t_0;
	elseif (h <= -1e-309)
		tmp = -d * ((h * l) ^ -0.5);
	elseif (h <= 2.75e+137)
		tmp = (d * sqrt(((1.0 / l) / h))) * (1.0 - (0.5 * ((h / l) * (((M_m / 2.0) * (D / d)) ^ 2.0))));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := Block[{t$95$0 = N[(N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(-0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M$95$m * 0.5), $MachinePrecision] / N[(d / D), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[h, -1.95e-49], t$95$0, If[LessEqual[h, -1e-309], N[((-d) * N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], If[LessEqual[h, 2.75e+137], N[(N[(d * N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M$95$m / 2.0), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if h < -1.95000000000000006e-49 or 2.7500000000000001e137 < h

   1. Initial program 69.1%

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

   3. Simplified 68.2%

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

      1. add-sqr-sqrt 68.2%

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

      2. pow2 68.2%

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

      3. sqrt-prod 68.1%

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

      4. sqrt-pow1 70.8%

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

      5. metadata-eval 70.8%

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

      6. associate-*l/ 70.8%

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

      7. div-inv 70.8%

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

      8. metadata-eval 70.8%

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

      9. *-commutative 70.8%

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

      10. pow1 70.8%

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

      11. clear-num 70.8%

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

      12. un-div-inv 70.8%

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

   6. Applied egg-rr 70.8%

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

      1. *-commutative 70.8%

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

      2. sqrt-unprod 59.6%

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

   8. Applied egg-rr 59.6%

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

      1. cancel-sign-sub-inv 59.6%

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

      2. metadata-eval 59.6%

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

      3. *-commutative 59.6%

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

      4. unpow-prod-down 57.8%

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

      5. pow2 57.8%

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

      6. add-sqr-sqrt 57.8%

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

      7. associate-*r/ 57.8%

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

   10. Applied egg-rr 57.8%

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

   if -1.95000000000000006e-49 < h < -1.000000000000002e-309

   1. Initial program 71.5%

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

   3. Simplified 71.6%

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

   5. Taylor expanded in d around inf 16.0%

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

   6. Taylor expanded in h around -inf 0.0%

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

      1. *-commutative 0.0%

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

      2. associate-/r* 0.0%

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

      3. unpow1/2 0.0%

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

      4. associate-/r* 0.0%

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

      5. rem-exp-log 0.0%

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

      6. exp-neg 0.0%

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

      7. exp-prod 0.0%

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

      8. distribute-lft-neg-out 0.0%

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

      9. distribute-rgt-neg-in 0.0%

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

      10. metadata-eval 0.0%

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

      11. exp-to-pow 0.0%

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

      12. *-commutative 0.0%

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

      13. unpow2 0.0%

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

      14. rem-square-sqrt 64.2%

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

      15. mul-1-neg 64.2%

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

   8. Simplified 64.2%

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

   if -1.000000000000002e-309 < h < 2.7500000000000001e137

   1. Initial program 69.5%

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

   3. Simplified 69.4%

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

      1. sqrt-div 75.8%

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

      2. sqrt-div 79.1%

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

      3. frac-times 79.2%

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

      4. add-sqr-sqrt 79.3%

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

   6. Applied egg-rr 79.3%

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

      1. associate-/r* 78.4%

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

   8. Simplified 78.4%

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

   9. Taylor expanded in d around 0 76.6%

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

       1. associate-/r* 77.5%

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

       2. associate-/l/ 76.6%

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

       3. associate-/r* 77.5%

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

   11. Simplified 77.5%

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

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -1.95 \cdot 10^{-49}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M \cdot 0.5}{\frac{d}{D}}\right)}^{2}\right)\right)\\ \mathbf{elif}\;h \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\left(-d\right) \cdot {\left(h \cdot \ell\right)}^{-0.5}\\ \mathbf{elif}\;h \leq 2.75 \cdot 10^{+137}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right) \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M \cdot 0.5}{\frac{d}{D}}\right)}^{2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 60.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M\_m \cdot 0.5}{\frac{d}{D}}\right)}^{2}\right)\right)\\ \mathbf{if}\;h \leq -1.75 \cdot 10^{-50}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;h \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\left(-d\right) \cdot {\left(h \cdot \ell\right)}^{-0.5}\\ \mathbf{elif}\;h \leq 6.5 \cdot 10^{+138}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M\_m}{2} \cdot \frac{D}{d}\right)}^{2}\right)\right) \cdot \frac{d}{\sqrt{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (let* ((t_0
         (*
          (sqrt (* (/ d l) (/ d h)))
          (+ 1.0 (* -0.5 (* (/ h l) (pow (/ (* M_m 0.5) (/ d D)) 2.0)))))))
   (if (<= h -1.75e-50)
     t_0
     (if (<= h -1e-309)
       (* (- d) (pow (* h l) -0.5))
       (if (<= h 6.5e+138)
         (*
          (- 1.0 (* 0.5 (* (/ h l) (pow (* (/ M_m 2.0) (/ D d)) 2.0))))
          (/ d (sqrt (* h l))))
         t_0)))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double t_0 = sqrt(((d / l) * (d / h))) * (1.0 + (-0.5 * ((h / l) * pow(((M_m * 0.5) / (d / D)), 2.0))));
	double tmp;
	if (h <= -1.75e-50) {
		tmp = t_0;
	} else if (h <= -1e-309) {
		tmp = -d * pow((h * l), -0.5);
	} else if (h <= 6.5e+138) {
		tmp = (1.0 - (0.5 * ((h / l) * pow(((M_m / 2.0) * (D / d)), 2.0)))) * (d / sqrt((h * l)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(((d / l) * (d / h))) * (1.0d0 + ((-0.5d0) * ((h / l) * (((m_m * 0.5d0) / (d / d_1)) ** 2.0d0))))
    if (h <= (-1.75d-50)) then
        tmp = t_0
    else if (h <= (-1d-309)) then
        tmp = -d * ((h * l) ** (-0.5d0))
    else if (h <= 6.5d+138) then
        tmp = (1.0d0 - (0.5d0 * ((h / l) * (((m_m / 2.0d0) * (d_1 / d)) ** 2.0d0)))) * (d / sqrt((h * l)))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double t_0 = Math.sqrt(((d / l) * (d / h))) * (1.0 + (-0.5 * ((h / l) * Math.pow(((M_m * 0.5) / (d / D)), 2.0))));
	double tmp;
	if (h <= -1.75e-50) {
		tmp = t_0;
	} else if (h <= -1e-309) {
		tmp = -d * Math.pow((h * l), -0.5);
	} else if (h <= 6.5e+138) {
		tmp = (1.0 - (0.5 * ((h / l) * Math.pow(((M_m / 2.0) * (D / d)), 2.0)))) * (d / Math.sqrt((h * l)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	t_0 = math.sqrt(((d / l) * (d / h))) * (1.0 + (-0.5 * ((h / l) * math.pow(((M_m * 0.5) / (d / D)), 2.0))))
	tmp = 0
	if h <= -1.75e-50:
		tmp = t_0
	elif h <= -1e-309:
		tmp = -d * math.pow((h * l), -0.5)
	elif h <= 6.5e+138:
		tmp = (1.0 - (0.5 * ((h / l) * math.pow(((M_m / 2.0) * (D / d)), 2.0)))) * (d / math.sqrt((h * l)))
	else:
		tmp = t_0
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	t_0 = Float64(sqrt(Float64(Float64(d / l) * Float64(d / h))) * Float64(1.0 + Float64(-0.5 * Float64(Float64(h / l) * (Float64(Float64(M_m * 0.5) / Float64(d / D)) ^ 2.0)))))
	tmp = 0.0
	if (h <= -1.75e-50)
		tmp = t_0;
	elseif (h <= -1e-309)
		tmp = Float64(Float64(-d) * (Float64(h * l) ^ -0.5));
	elseif (h <= 6.5e+138)
		tmp = Float64(Float64(1.0 - Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(M_m / 2.0) * Float64(D / d)) ^ 2.0)))) * Float64(d / sqrt(Float64(h * l))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp_2 = code(d, h, l, M_m, D)
	t_0 = sqrt(((d / l) * (d / h))) * (1.0 + (-0.5 * ((h / l) * (((M_m * 0.5) / (d / D)) ^ 2.0))));
	tmp = 0.0;
	if (h <= -1.75e-50)
		tmp = t_0;
	elseif (h <= -1e-309)
		tmp = -d * ((h * l) ^ -0.5);
	elseif (h <= 6.5e+138)
		tmp = (1.0 - (0.5 * ((h / l) * (((M_m / 2.0) * (D / d)) ^ 2.0)))) * (d / sqrt((h * l)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := Block[{t$95$0 = N[(N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(-0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M$95$m * 0.5), $MachinePrecision] / N[(d / D), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[h, -1.75e-50], t$95$0, If[LessEqual[h, -1e-309], N[((-d) * N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], If[LessEqual[h, 6.5e+138], N[(N[(1.0 - N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M$95$m / 2.0), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if h < -1.74999999999999998e-50 or 6.50000000000000054e138 < h

   1. Initial program 69.1%

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

   3. Simplified 68.2%

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

      1. add-sqr-sqrt 68.2%

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

      2. pow2 68.2%

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

      3. sqrt-prod 68.1%

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

      4. sqrt-pow1 70.8%

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

      5. metadata-eval 70.8%

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

      6. associate-*l/ 70.8%

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

      7. div-inv 70.8%

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

      8. metadata-eval 70.8%

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

      9. *-commutative 70.8%

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

      10. pow1 70.8%

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

      11. clear-num 70.8%

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

      12. un-div-inv 70.8%

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

   6. Applied egg-rr 70.8%

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

      1. *-commutative 70.8%

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

      2. sqrt-unprod 59.6%

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

   8. Applied egg-rr 59.6%

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

      1. cancel-sign-sub-inv 59.6%

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

      2. metadata-eval 59.6%

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

      3. *-commutative 59.6%

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

      4. unpow-prod-down 57.8%

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

      5. pow2 57.8%

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

      6. add-sqr-sqrt 57.8%

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

      7. associate-*r/ 57.8%

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

   10. Applied egg-rr 57.8%

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

   if -1.74999999999999998e-50 < h < -1.000000000000002e-309

   1. Initial program 71.5%

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

   3. Simplified 71.6%

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

   5. Taylor expanded in d around inf 16.0%

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

   6. Taylor expanded in h around -inf 0.0%

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

      1. *-commutative 0.0%

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

      2. associate-/r* 0.0%

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

      3. unpow1/2 0.0%

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

      4. associate-/r* 0.0%

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

      5. rem-exp-log 0.0%

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

      6. exp-neg 0.0%

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

      7. exp-prod 0.0%

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

      8. distribute-lft-neg-out 0.0%

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

      9. distribute-rgt-neg-in 0.0%

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

      10. metadata-eval 0.0%

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

      11. exp-to-pow 0.0%

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

      12. *-commutative 0.0%

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

      13. unpow2 0.0%

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

      14. rem-square-sqrt 64.2%

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

      15. mul-1-neg 64.2%

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

   8. Simplified 64.2%

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

   if -1.000000000000002e-309 < h < 6.50000000000000054e138

   1. Initial program 69.5%

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

   3. Simplified 69.4%

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

      1. sqrt-div 75.8%

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

      2. sqrt-div 79.1%

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

      3. frac-times 79.2%

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

      4. add-sqr-sqrt 79.3%

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

   6. Applied egg-rr 79.3%

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

      1. associate-/r* 78.4%

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

   8. Simplified 78.4%

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

      1. *-un-lft-identity 78.4%

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

      2. associate-/l/ 79.3%

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

      3. pow1/2 79.3%

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

      4. pow1/2 79.3%

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

      5. pow-prod-down 76.5%

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

      6. *-commutative 76.5%

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

      7. pow1/2 76.5%

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

   10. Applied egg-rr 76.5%

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

       1. *-lft-identity 76.5%

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

   12. Simplified 76.5%

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

4. Final simplification 66.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -1.75 \cdot 10^{-50}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M \cdot 0.5}{\frac{d}{D}}\right)}^{2}\right)\right)\\ \mathbf{elif}\;h \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\left(-d\right) \cdot {\left(h \cdot \ell\right)}^{-0.5}\\ \mathbf{elif}\;h \leq 6.5 \cdot 10^{+138}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}\right)\right) \cdot \frac{d}{\sqrt{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M \cdot 0.5}{\frac{d}{D}}\right)}^{2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 52.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -6.8 \cdot 10^{+122}:\\ \;\;\;\;\left(-d\right) \cdot {\left(h \cdot \ell\right)}^{-0.5}\\ \mathbf{elif}\;d \leq -4.6 \cdot 10^{-154}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;d \leq -1.25 \cdot 10^{-294}:\\ \;\;\;\;d \cdot {\left(-1 + \mathsf{fma}\left(h, \ell, 1\right)\right)}^{-0.5}\\ \mathbf{elif}\;d \leq 1.35 \cdot 10^{-77}:\\ \;\;\;\;-0.125 \cdot \frac{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\left(D \cdot M\_m\right) \cdot \left(D \cdot M\_m\right)\right)}{d}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (if (<= d -6.8e+122)
   (* (- d) (pow (* h l) -0.5))
   (if (<= d -4.6e-154)
     (* (sqrt (/ d l)) (sqrt (/ d h)))
     (if (<= d -1.25e-294)
       (* d (pow (+ -1.0 (fma h l 1.0)) -0.5))
       (if (<= d 1.35e-77)
         (* -0.125 (/ (* (sqrt (/ h (pow l 3.0))) (* (* D M_m) (* D M_m))) d))
         (* d (* (pow l -0.5) (pow h -0.5))))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (d <= -6.8e+122) {
		tmp = -d * pow((h * l), -0.5);
	} else if (d <= -4.6e-154) {
		tmp = sqrt((d / l)) * sqrt((d / h));
	} else if (d <= -1.25e-294) {
		tmp = d * pow((-1.0 + fma(h, l, 1.0)), -0.5);
	} else if (d <= 1.35e-77) {
		tmp = -0.125 * ((sqrt((h / pow(l, 3.0))) * ((D * M_m) * (D * M_m))) / d);
	} else {
		tmp = d * (pow(l, -0.5) * pow(h, -0.5));
	}
	return tmp;
}

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	tmp = 0.0
	if (d <= -6.8e+122)
		tmp = Float64(Float64(-d) * (Float64(h * l) ^ -0.5));
	elseif (d <= -4.6e-154)
		tmp = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)));
	elseif (d <= -1.25e-294)
		tmp = Float64(d * (Float64(-1.0 + fma(h, l, 1.0)) ^ -0.5));
	elseif (d <= 1.35e-77)
		tmp = Float64(-0.125 * Float64(Float64(sqrt(Float64(h / (l ^ 3.0))) * Float64(Float64(D * M_m) * Float64(D * M_m))) / d));
	else
		tmp = Float64(d * Float64((l ^ -0.5) * (h ^ -0.5)));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := If[LessEqual[d, -6.8e+122], N[((-d) * N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -4.6e-154], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -1.25e-294], N[(d * N[Power[N[(-1.0 + N[(h * l + 1.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.35e-77], N[(-0.125 * N[(N[(N[Sqrt[N[(h / N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(D * M$95$m), $MachinePrecision] * N[(D * M$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[l, -0.5], $MachinePrecision] * N[Power[h, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if d < -6.8e122

   1. Initial program 72.2%

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

   3. Simplified 72.3%

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

   5. Taylor expanded in d around inf 4.2%

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

   6. Taylor expanded in h around -inf 0.0%

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

      1. *-commutative 0.0%

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

      2. associate-/r* 0.0%

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

      3. unpow1/2 0.0%

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

      4. associate-/r* 0.0%

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

      5. rem-exp-log 0.0%

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

      6. exp-neg 0.0%

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

      7. exp-prod 0.0%

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

      8. distribute-lft-neg-out 0.0%

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

      9. distribute-rgt-neg-in 0.0%

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

      10. metadata-eval 0.0%

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

      11. exp-to-pow 0.0%

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

      12. *-commutative 0.0%

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

      13. unpow2 0.0%

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

      14. rem-square-sqrt 66.3%

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

      15. mul-1-neg 66.3%

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

   8. Simplified 66.3%

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

   if -6.8e122 < d < -4.5999999999999999e-154

   1. Initial program 82.5%

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

   3. Simplified 82.5%

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

   5. Taylor expanded in d around inf 44.0%

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

   if -4.5999999999999999e-154 < d < -1.2500000000000001e-294

   1. Initial program 51.8%

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

   3. Simplified 51.9%

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

   5. Taylor expanded in d around inf 13.2%

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

      1. expm1-log1p-u 13.2%

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

      2. expm1-undefine 39.1%

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

   7. Applied egg-rr 39.1%

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

      1. sub-neg 39.1%

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

      2. metadata-eval 39.1%

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

      3. +-commutative 39.1%

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

      4. log1p-undefine 39.1%

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

      5. rem-exp-log 39.1%

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

      6. +-commutative 39.1%

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

      7. fma-define 39.1%

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

   9. Simplified 39.1%

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

       1. pow1/2 39.1%

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

       2. inv-pow 39.1%

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

       3. pow-pow 39.1%

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

       4. metadata-eval 39.1%

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

   11. Applied egg-rr 39.1%

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

   if -1.2500000000000001e-294 < d < 1.35e-77

   1. Initial program 58.5%

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

   3. Simplified 58.5%

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

   5. Taylor expanded in d around 0 45.7%

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

      1. associate-*l/ 47.8%

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

      2. pow-prod-down 54.6%

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

   7. Applied egg-rr 54.6%

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

      1. unpow2 54.6%

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

   9. Applied egg-rr 54.6%

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

   if 1.35e-77 < d

   1. Initial program 74.4%

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

   3. Simplified 73.2%

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

   5. Taylor expanded in d around inf 60.4%

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

      1. *-un-lft-identity 60.4%

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

      2. pow1/2 60.4%

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

      3. inv-pow 60.4%

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

      4. pow-pow 60.4%

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

      5. metadata-eval 60.4%

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

   7. Applied egg-rr 60.4%

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

      1. *-lft-identity 60.4%

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

   9. Simplified 60.4%

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

       1. *-commutative 60.4%

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

       2. unpow-prod-down 66.5%

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

   11. Applied egg-rr 66.5%

       \[\leadsto d \cdot \color{blue}{\left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 56.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -6.8 \cdot 10^{+122}:\\ \;\;\;\;\left(-d\right) \cdot {\left(h \cdot \ell\right)}^{-0.5}\\ \mathbf{elif}\;d \leq -4.6 \cdot 10^{-154}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;d \leq -1.25 \cdot 10^{-294}:\\ \;\;\;\;d \cdot {\left(-1 + \mathsf{fma}\left(h, \ell, 1\right)\right)}^{-0.5}\\ \mathbf{elif}\;d \leq 1.35 \cdot 10^{-77}:\\ \;\;\;\;-0.125 \cdot \frac{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right)}{d}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 48.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;D \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(-0.125 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D \cdot M\_m}{d}\right)}^{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (if (<= D 1.9e+154)
   (* (sqrt (/ d l)) (sqrt (/ d h)))
   (*
    (sqrt (* (/ d l) (/ d h)))
    (* -0.125 (* (/ h l) (pow (/ (* D M_m) d) 2.0))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (D <= 1.9e+154) {
		tmp = sqrt((d / l)) * sqrt((d / h));
	} else {
		tmp = sqrt(((d / l) * (d / h))) * (-0.125 * ((h / l) * pow(((D * M_m) / d), 2.0)));
	}
	return tmp;
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (d_1 <= 1.9d+154) then
        tmp = sqrt((d / l)) * sqrt((d / h))
    else
        tmp = sqrt(((d / l) * (d / h))) * ((-0.125d0) * ((h / l) * (((d_1 * m_m) / d) ** 2.0d0)))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (D <= 1.9e+154) {
		tmp = Math.sqrt((d / l)) * Math.sqrt((d / h));
	} else {
		tmp = Math.sqrt(((d / l) * (d / h))) * (-0.125 * ((h / l) * Math.pow(((D * M_m) / d), 2.0)));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	tmp = 0
	if D <= 1.9e+154:
		tmp = math.sqrt((d / l)) * math.sqrt((d / h))
	else:
		tmp = math.sqrt(((d / l) * (d / h))) * (-0.125 * ((h / l) * math.pow(((D * M_m) / d), 2.0)))
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	tmp = 0.0
	if (D <= 1.9e+154)
		tmp = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)));
	else
		tmp = Float64(sqrt(Float64(Float64(d / l) * Float64(d / h))) * Float64(-0.125 * Float64(Float64(h / l) * (Float64(Float64(D * M_m) / d) ^ 2.0))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp_2 = code(d, h, l, M_m, D)
	tmp = 0.0;
	if (D <= 1.9e+154)
		tmp = sqrt((d / l)) * sqrt((d / h));
	else
		tmp = sqrt(((d / l) * (d / h))) * (-0.125 * ((h / l) * (((D * M_m) / d) ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := If[LessEqual[D, 1.9e+154], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-0.125 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D * M$95$m), $MachinePrecision] / d), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if D < 1.8999999999999999e154

   1. Initial program 70.3%

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

   3. Simplified 69.9%

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

   5. Taylor expanded in d around inf 45.4%

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

   if 1.8999999999999999e154 < D

   1. Initial program 65.4%

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

   3. Simplified 65.4%

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

      1. add-sqr-sqrt 65.4%

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

      2. pow2 65.4%

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

      3. sqrt-prod 65.4%

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

      4. sqrt-pow1 68.1%

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

      5. metadata-eval 68.1%

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

      6. associate-*l/ 68.1%

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

      7. div-inv 68.1%

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

      8. metadata-eval 68.1%

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

      9. *-commutative 68.1%

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

      10. pow1 68.1%

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

      11. clear-num 68.1%

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

      12. un-div-inv 68.1%

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

   6. Applied egg-rr 68.1%

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

   7. Taylor expanded in M around 0 65.4%

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

   8. Taylor expanded in D around inf 38.6%

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

      1. associate-/l* 38.6%

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

      2. times-frac 35.7%

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

      3. associate-*r* 35.7%

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

      4. unpow2 35.7%

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

      5. unpow2 35.7%

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

      6. unpow2 35.7%

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

      7. times-frac 50.6%

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

      8. swap-sqr 59.4%

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

      9. unpow2 59.4%

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

      10. associate-*r/ 62.3%

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

      11. *-commutative 62.3%

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

      12. associate-/l* 62.3%

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

   10. Simplified 62.3%

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

       1. pow1 62.3%

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

       2. sqrt-unprod 53.4%

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

       3. associate-*r* 53.4%

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

       4. associate-*r/ 53.4%

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

       5. *-commutative 53.4%

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

       6. associate-/l* 50.5%

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

   12. Applied egg-rr 50.5%

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

       1. unpow1 50.5%

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

       2. *-commutative 50.5%

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

       3. associate-*l* 50.5%

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

       4. *-commutative 50.5%

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

       5. associate-*r/ 53.4%

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

   14. Simplified 53.4%

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

4. Final simplification 46.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(-0.125 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 45.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := \left(-d\right) \cdot {\left(h \cdot \ell\right)}^{-0.5}\\ \mathbf{if}\;\ell \leq -9.5 \cdot 10^{-99}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq -1.05 \cdot 10^{-235}:\\ \;\;\;\;d \cdot {\left(-1 + \mathsf{fma}\left(h, \ell, 1\right)\right)}^{-0.5}\\ \mathbf{elif}\;\ell \leq 2.8 \cdot 10^{-238}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (let* ((t_0 (* (- d) (pow (* h l) -0.5))))
   (if (<= l -9.5e-99)
     t_0
     (if (<= l -1.05e-235)
       (* d (pow (+ -1.0 (fma h l 1.0)) -0.5))
       (if (<= l 2.8e-238) t_0 (* d (* (pow l -0.5) (pow h -0.5))))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double t_0 = -d * pow((h * l), -0.5);
	double tmp;
	if (l <= -9.5e-99) {
		tmp = t_0;
	} else if (l <= -1.05e-235) {
		tmp = d * pow((-1.0 + fma(h, l, 1.0)), -0.5);
	} else if (l <= 2.8e-238) {
		tmp = t_0;
	} else {
		tmp = d * (pow(l, -0.5) * pow(h, -0.5));
	}
	return tmp;
}

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	t_0 = Float64(Float64(-d) * (Float64(h * l) ^ -0.5))
	tmp = 0.0
	if (l <= -9.5e-99)
		tmp = t_0;
	elseif (l <= -1.05e-235)
		tmp = Float64(d * (Float64(-1.0 + fma(h, l, 1.0)) ^ -0.5));
	elseif (l <= 2.8e-238)
		tmp = t_0;
	else
		tmp = Float64(d * Float64((l ^ -0.5) * (h ^ -0.5)));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := Block[{t$95$0 = N[((-d) * N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -9.5e-99], t$95$0, If[LessEqual[l, -1.05e-235], N[(d * N[Power[N[(-1.0 + N[(h * l + 1.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 2.8e-238], t$95$0, N[(d * N[(N[Power[l, -0.5], $MachinePrecision] * N[Power[h, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -9.5000000000000008e-99 or -1.05e-235 < l < 2.80000000000000004e-238

   1. Initial program 69.2%

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

   3. Simplified 69.2%

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

   5. Taylor expanded in d around inf 8.1%

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

   6. Taylor expanded in h around -inf 0.0%

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

      1. *-commutative 0.0%

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

      2. associate-/r* 0.0%

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

      3. unpow1/2 0.0%

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

      4. associate-/r* 0.0%

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

      5. rem-exp-log 0.0%

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

      6. exp-neg 0.0%

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

      7. exp-prod 0.0%

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

      8. distribute-lft-neg-out 0.0%

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

      9. distribute-rgt-neg-in 0.0%

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

      10. metadata-eval 0.0%

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

      11. exp-to-pow 0.0%

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

      12. *-commutative 0.0%

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

      13. unpow2 0.0%

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

      14. rem-square-sqrt 47.2%

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

      15. mul-1-neg 47.2%

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

   8. Simplified 47.2%

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

   if -9.5000000000000008e-99 < l < -1.05e-235

   1. Initial program 87.3%

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

   3. Simplified 87.3%

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

   5. Taylor expanded in d around inf 20.1%

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

      1. expm1-log1p-u 20.1%

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

      2. expm1-undefine 61.9%

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

   7. Applied egg-rr 61.9%

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

      1. sub-neg 61.9%

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

      2. metadata-eval 61.9%

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

      3. +-commutative 61.9%

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

      4. log1p-undefine 61.9%

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

      5. rem-exp-log 61.9%

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

      6. +-commutative 61.9%

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

      7. fma-define 61.9%

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

   9. Simplified 61.9%

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

       1. pow1/2 61.9%

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

       2. inv-pow 61.9%

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

       3. pow-pow 61.9%

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

       4. metadata-eval 61.9%

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

   11. Applied egg-rr 61.9%

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

   if 2.80000000000000004e-238 < l

   1. Initial program 66.6%

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

   3. Simplified 65.7%

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

   5. Taylor expanded in d around inf 52.8%

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

      1. *-un-lft-identity 52.8%

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

      2. pow1/2 52.8%

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

      3. inv-pow 52.8%

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

      4. pow-pow 52.8%

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

      5. metadata-eval 52.8%

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

   7. Applied egg-rr 52.8%

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

      1. *-lft-identity 52.8%

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

   9. Simplified 52.8%

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

       1. *-commutative 52.8%

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

       2. unpow-prod-down 57.7%

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

   11. Applied egg-rr 57.7%

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

4. Final simplification 53.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -9.5 \cdot 10^{-99}:\\ \;\;\;\;\left(-d\right) \cdot {\left(h \cdot \ell\right)}^{-0.5}\\ \mathbf{elif}\;\ell \leq -1.05 \cdot 10^{-235}:\\ \;\;\;\;d \cdot {\left(-1 + \mathsf{fma}\left(h, \ell, 1\right)\right)}^{-0.5}\\ \mathbf{elif}\;\ell \leq 2.8 \cdot 10^{-238}:\\ \;\;\;\;\left(-d\right) \cdot {\left(h \cdot \ell\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 45.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 5 \cdot 10^{-238}:\\ \;\;\;\;\left(-d\right) \cdot {\left(h \cdot \ell\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (if (<= l 5e-238)
   (* (- d) (pow (* h l) -0.5))
   (* d (* (pow l -0.5) (pow h -0.5)))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (l <= 5e-238) {
		tmp = -d * pow((h * l), -0.5);
	} else {
		tmp = d * (pow(l, -0.5) * pow(h, -0.5));
	}
	return tmp;
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (l <= 5d-238) then
        tmp = -d * ((h * l) ** (-0.5d0))
    else
        tmp = d * ((l ** (-0.5d0)) * (h ** (-0.5d0)))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (l <= 5e-238) {
		tmp = -d * Math.pow((h * l), -0.5);
	} else {
		tmp = d * (Math.pow(l, -0.5) * Math.pow(h, -0.5));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	tmp = 0
	if l <= 5e-238:
		tmp = -d * math.pow((h * l), -0.5)
	else:
		tmp = d * (math.pow(l, -0.5) * math.pow(h, -0.5))
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	tmp = 0.0
	if (l <= 5e-238)
		tmp = Float64(Float64(-d) * (Float64(h * l) ^ -0.5));
	else
		tmp = Float64(d * Float64((l ^ -0.5) * (h ^ -0.5)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp_2 = code(d, h, l, M_m, D)
	tmp = 0.0;
	if (l <= 5e-238)
		tmp = -d * ((h * l) ^ -0.5);
	else
		tmp = d * ((l ^ -0.5) * (h ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := If[LessEqual[l, 5e-238], N[((-d) * N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[l, -0.5], $MachinePrecision] * N[Power[h, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 5e-238

   1. Initial program 72.1%

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

   3. Simplified 72.1%

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

   5. Taylor expanded in d around inf 10.0%

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

   6. Taylor expanded in h around -inf 0.0%

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

      1. *-commutative 0.0%

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

      2. associate-/r* 0.0%

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

      3. unpow1/2 0.0%

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

      4. associate-/r* 0.0%

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

      5. rem-exp-log 0.0%

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

      6. exp-neg 0.0%

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

      7. exp-prod 0.0%

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

      8. distribute-lft-neg-out 0.0%

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

      9. distribute-rgt-neg-in 0.0%

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

      10. metadata-eval 0.0%

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

      11. exp-to-pow 0.0%

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

      12. *-commutative 0.0%

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

      13. unpow2 0.0%

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

      14. rem-square-sqrt 42.4%

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

      15. mul-1-neg 42.4%

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

   8. Simplified 42.4%

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

   if 5e-238 < l

   1. Initial program 66.6%

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

   3. Simplified 65.7%

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

   5. Taylor expanded in d around inf 52.8%

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

      1. *-un-lft-identity 52.8%

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

      2. pow1/2 52.8%

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

      3. inv-pow 52.8%

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

      4. pow-pow 52.8%

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

      5. metadata-eval 52.8%

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

   7. Applied egg-rr 52.8%

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

      1. *-lft-identity 52.8%

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

   9. Simplified 52.8%

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

       1. *-commutative 52.8%

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

       2. unpow-prod-down 57.7%

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

   11. Applied egg-rr 57.7%

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

4. Final simplification 49.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 5 \cdot 10^{-238}:\\ \;\;\;\;\left(-d\right) \cdot {\left(h \cdot \ell\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 42.4% accurate, 3.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 2.6 \cdot 10^{-238}:\\ \;\;\;\;\left(-d\right) \cdot {\left(h \cdot \ell\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (if (<= l 2.6e-238)
   (* (- d) (pow (* h l) -0.5))
   (* d (sqrt (/ 1.0 (* h l))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (l <= 2.6e-238) {
		tmp = -d * pow((h * l), -0.5);
	} else {
		tmp = d * sqrt((1.0 / (h * l)));
	}
	return tmp;
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (l <= 2.6d-238) then
        tmp = -d * ((h * l) ** (-0.5d0))
    else
        tmp = d * sqrt((1.0d0 / (h * l)))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (l <= 2.6e-238) {
		tmp = -d * Math.pow((h * l), -0.5);
	} else {
		tmp = d * Math.sqrt((1.0 / (h * l)));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	tmp = 0
	if l <= 2.6e-238:
		tmp = -d * math.pow((h * l), -0.5)
	else:
		tmp = d * math.sqrt((1.0 / (h * l)))
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	tmp = 0.0
	if (l <= 2.6e-238)
		tmp = Float64(Float64(-d) * (Float64(h * l) ^ -0.5));
	else
		tmp = Float64(d * sqrt(Float64(1.0 / Float64(h * l))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp_2 = code(d, h, l, M_m, D)
	tmp = 0.0;
	if (l <= 2.6e-238)
		tmp = -d * ((h * l) ^ -0.5);
	else
		tmp = d * sqrt((1.0 / (h * l)));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := If[LessEqual[l, 2.6e-238], N[((-d) * N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], N[(d * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 2.6000000000000001e-238

   1. Initial program 72.1%

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

   3. Simplified 72.1%

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

   5. Taylor expanded in d around inf 10.0%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

   6. Taylor expanded in h around -inf 0.0%

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   7. Step-by-step derivation

      1. *-commutative 0.0%

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]

      2. associate-/r* 0.0%

         \[\leadsto \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      3. unpow1/2 0.0%

         \[\leadsto \color{blue}{{\left(\frac{\frac{1}{h}}{\ell}\right)}^{0.5}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      4. associate-/r* 0.0%

         \[\leadsto {\color{blue}{\left(\frac{1}{h \cdot \ell}\right)}}^{0.5} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      5. rem-exp-log 0.0%

         \[\leadsto {\left(\frac{1}{\color{blue}{e^{\log \left(h \cdot \ell\right)}}}\right)}^{0.5} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      6. exp-neg 0.0%

         \[\leadsto {\color{blue}{\left(e^{-\log \left(h \cdot \ell\right)}\right)}}^{0.5} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      7. exp-prod 0.0%

         \[\leadsto \color{blue}{e^{\left(-\log \left(h \cdot \ell\right)\right) \cdot 0.5}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      8. distribute-lft-neg-out 0.0%

         \[\leadsto e^{\color{blue}{-\log \left(h \cdot \ell\right) \cdot 0.5}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      9. distribute-rgt-neg-in 0.0%

         \[\leadsto e^{\color{blue}{\log \left(h \cdot \ell\right) \cdot \left(-0.5\right)}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      10. metadata-eval 0.0%

          \[\leadsto e^{\log \left(h \cdot \ell\right) \cdot \color{blue}{-0.5}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      11. exp-to-pow 0.0%

          \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      12. *-commutative 0.0%

          \[\leadsto {\left(h \cdot \ell\right)}^{-0.5} \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot d\right)} \]

      13. unpow2 0.0%

          \[\leadsto {\left(h \cdot \ell\right)}^{-0.5} \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right) \]

      14. rem-square-sqrt 42.4%

          \[\leadsto {\left(h \cdot \ell\right)}^{-0.5} \cdot \left(\color{blue}{-1} \cdot d\right) \]

      15. mul-1-neg 42.4%

          \[\leadsto {\left(h \cdot \ell\right)}^{-0.5} \cdot \color{blue}{\left(-d\right)} \]

   8. Simplified 42.4%

      \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot \left(-d\right)} \]

   if 2.6000000000000001e-238 < l

   1. Initial program 66.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 65.7%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 52.8%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 47.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2.6 \cdot 10^{-238}:\\ \;\;\;\;\left(-d\right) \cdot {\left(h \cdot \ell\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 26.9% accurate, 3.1× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ d \cdot \sqrt{\frac{1}{h \cdot \ell}} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D) :precision binary64 (* d (sqrt (/ 1.0 (* h l)))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	return d * sqrt((1.0 / (h * l)));
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_1
    code = d * sqrt((1.0d0 / (h * l)))
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	return d * Math.sqrt((1.0 / (h * l)));
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	return d * math.sqrt((1.0 / (h * l)))

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	return Float64(d * sqrt(Float64(1.0 / Float64(h * l))))
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp = code(d, h, l, M_m, D)
	tmp = d * sqrt((1.0 / (h * l)));
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := N[(d * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 69.7%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

3. Simplified 69.3%

   \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in d around inf 28.9%

   \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
6. Add Preprocessing

Alternative 17: 26.8% accurate, 3.1× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ d \cdot {\left(h \cdot \ell\right)}^{-0.5} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D) :precision binary64 (* d (pow (* h l) -0.5)))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	return d * pow((h * l), -0.5);
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_1
    code = d * ((h * l) ** (-0.5d0))
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	return d * Math.pow((h * l), -0.5);
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	return d * math.pow((h * l), -0.5)

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	return Float64(d * (Float64(h * l) ^ -0.5))
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp = code(d, h, l, M_m, D)
	tmp = d * ((h * l) ^ -0.5);
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := N[(d * N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 69.7%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

3. Simplified 69.3%

   \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in d around inf 28.9%

   \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
6. Step-by-step derivation

   1. *-un-lft-identity 28.9%

      \[\leadsto d \cdot \color{blue}{\left(1 \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

   2. pow1/2 28.9%

      \[\leadsto d \cdot \left(1 \cdot \color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{0.5}}\right) \]

   3. inv-pow 28.9%

      \[\leadsto d \cdot \left(1 \cdot {\color{blue}{\left({\left(h \cdot \ell\right)}^{-1}\right)}}^{0.5}\right) \]

   4. pow-pow 28.5%

      \[\leadsto d \cdot \left(1 \cdot \color{blue}{{\left(h \cdot \ell\right)}^{\left(-1 \cdot 0.5\right)}}\right) \]

   5. metadata-eval 28.5%

      \[\leadsto d \cdot \left(1 \cdot {\left(h \cdot \ell\right)}^{\color{blue}{-0.5}}\right) \]

7. Applied egg-rr 28.5%

   \[\leadsto d \cdot \color{blue}{\left(1 \cdot {\left(h \cdot \ell\right)}^{-0.5}\right)} \]
8. Step-by-step derivation

   1. *-lft-identity 28.5%

      \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

9. Simplified 28.5%

   \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024146 
(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)))))