Henrywood and Agarwal, Equation (12)

Percentage Accurate: 67.1% → 79.5%

Time: 26.4s

Alternatives: 22

Speedup: 1.0×

• Rival profile vector overflowed, profile may not be complete

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: 67.1% 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: 79.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := 1 + \frac{h}{\ell} \cdot \left({\left(D \cdot \frac{\frac{M\_m}{2}}{d}\right)}^{2} \cdot -0.5\right)\\ t_1 := \sqrt{-d}\\ \mathbf{if}\;h \leq -1 \cdot 10^{-311}:\\ \;\;\;\;\frac{t\_1}{\sqrt{-\ell}} \cdot \left(\frac{t\_1}{\sqrt{-h}} \cdot t\_0\right)\\ \mathbf{elif}\;h \leq 6 \cdot 10^{+50}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(t\_0 \cdot \sqrt{\frac{d}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + {\left(\frac{D}{d} \cdot \left(M\_m \cdot 0.5\right)\right)}^{2} \cdot \frac{h \cdot -0.5}{\ell}\right)\right)\\ \end{array} \end{array} \]

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 (+ 1.0 (* (/ h l) (* (pow (* D (/ (/ M_m 2.0) d)) 2.0) -0.5))))
        (t_1 (sqrt (- d))))
   (if (<= h -1e-311)
     (* (/ t_1 (sqrt (- l))) (* (/ t_1 (sqrt (- h))) t_0))
     (if (<= h 6e+50)
       (* (/ (sqrt d) (sqrt l)) (* t_0 (sqrt (/ d h))))
       (*
        (/ (sqrt d) (sqrt h))
        (*
         (sqrt (/ d l))
         (+ 1.0 (* (pow (* (/ D d) (* M_m 0.5)) 2.0) (/ (* h -0.5) 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 = 1.0 + ((h / l) * (pow((D * ((M_m / 2.0) / d)), 2.0) * -0.5));
	double t_1 = sqrt(-d);
	double tmp;
	if (h <= -1e-311) {
		tmp = (t_1 / sqrt(-l)) * ((t_1 / sqrt(-h)) * t_0);
	} else if (h <= 6e+50) {
		tmp = (sqrt(d) / sqrt(l)) * (t_0 * sqrt((d / h)));
	} else {
		tmp = (sqrt(d) / sqrt(h)) * (sqrt((d / l)) * (1.0 + (pow(((D / d) * (M_m * 0.5)), 2.0) * ((h * -0.5) / 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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 + ((h / l) * (((d_1 * ((m_m / 2.0d0) / d)) ** 2.0d0) * (-0.5d0)))
    t_1 = sqrt(-d)
    if (h <= (-1d-311)) then
        tmp = (t_1 / sqrt(-l)) * ((t_1 / sqrt(-h)) * t_0)
    else if (h <= 6d+50) then
        tmp = (sqrt(d) / sqrt(l)) * (t_0 * sqrt((d / h)))
    else
        tmp = (sqrt(d) / sqrt(h)) * (sqrt((d / l)) * (1.0d0 + ((((d_1 / d) * (m_m * 0.5d0)) ** 2.0d0) * ((h * (-0.5d0)) / 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 t_0 = 1.0 + ((h / l) * (Math.pow((D * ((M_m / 2.0) / d)), 2.0) * -0.5));
	double t_1 = Math.sqrt(-d);
	double tmp;
	if (h <= -1e-311) {
		tmp = (t_1 / Math.sqrt(-l)) * ((t_1 / Math.sqrt(-h)) * t_0);
	} else if (h <= 6e+50) {
		tmp = (Math.sqrt(d) / Math.sqrt(l)) * (t_0 * Math.sqrt((d / h)));
	} else {
		tmp = (Math.sqrt(d) / Math.sqrt(h)) * (Math.sqrt((d / l)) * (1.0 + (Math.pow(((D / d) * (M_m * 0.5)), 2.0) * ((h * -0.5) / 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):
	t_0 = 1.0 + ((h / l) * (math.pow((D * ((M_m / 2.0) / d)), 2.0) * -0.5))
	t_1 = math.sqrt(-d)
	tmp = 0
	if h <= -1e-311:
		tmp = (t_1 / math.sqrt(-l)) * ((t_1 / math.sqrt(-h)) * t_0)
	elif h <= 6e+50:
		tmp = (math.sqrt(d) / math.sqrt(l)) * (t_0 * math.sqrt((d / h)))
	else:
		tmp = (math.sqrt(d) / math.sqrt(h)) * (math.sqrt((d / l)) * (1.0 + (math.pow(((D / d) * (M_m * 0.5)), 2.0) * ((h * -0.5) / 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(1.0 + Float64(Float64(h / l) * Float64((Float64(D * Float64(Float64(M_m / 2.0) / d)) ^ 2.0) * -0.5)))
	t_1 = sqrt(Float64(-d))
	tmp = 0.0
	if (h <= -1e-311)
		tmp = Float64(Float64(t_1 / sqrt(Float64(-l))) * Float64(Float64(t_1 / sqrt(Float64(-h))) * t_0));
	elseif (h <= 6e+50)
		tmp = Float64(Float64(sqrt(d) / sqrt(l)) * Float64(t_0 * sqrt(Float64(d / h))));
	else
		tmp = Float64(Float64(sqrt(d) / sqrt(h)) * Float64(sqrt(Float64(d / l)) * Float64(1.0 + Float64((Float64(Float64(D / d) * Float64(M_m * 0.5)) ^ 2.0) * Float64(Float64(h * -0.5) / 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)
	t_0 = 1.0 + ((h / l) * (((D * ((M_m / 2.0) / d)) ^ 2.0) * -0.5));
	t_1 = sqrt(-d);
	tmp = 0.0;
	if (h <= -1e-311)
		tmp = (t_1 / sqrt(-l)) * ((t_1 / sqrt(-h)) * t_0);
	elseif (h <= 6e+50)
		tmp = (sqrt(d) / sqrt(l)) * (t_0 * sqrt((d / h)));
	else
		tmp = (sqrt(d) / sqrt(h)) * (sqrt((d / l)) * (1.0 + ((((D / d) * (M_m * 0.5)) ^ 2.0) * ((h * -0.5) / 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_] := Block[{t$95$0 = 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]}, Block[{t$95$1 = N[Sqrt[(-d)], $MachinePrecision]}, If[LessEqual[h, -1e-311], N[(N[(t$95$1 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$1 / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, 6e+50], N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(N[Power[N[(N[(D / d), $MachinePrecision] * N[(M$95$m * 0.5), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(h * -0.5), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if h < -9.99999999999948e-312

   1. Initial program 64.2%

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

   3. Simplified 62.7%

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

      1. frac-2neg 62.7%

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

      2. sqrt-div 68.6%

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

   6. Applied egg-rr 68.6%

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

      1. frac-2neg 68.6%

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

      2. sqrt-div 82.2%

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

   8. Applied egg-rr 82.2%

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

   if -9.99999999999948e-312 < h < 5.9999999999999996e50

   1. Initial program 71.8%

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

   3. Simplified 71.8%

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

      1. sqrt-div 85.0%

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

   6. Applied egg-rr 85.0%

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

   if 5.9999999999999996e50 < h

   1. Initial program 59.8%

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

   3. Simplified 59.6%

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

      1. fma-undefine 59.6%

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

      2. associate-*r* 59.6%

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

      3. associate-*r/ 59.6%

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

   6. Applied egg-rr 59.6%

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

      1. sqrt-div 76.5%

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

   8. Applied egg-rr 76.5%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -1 \cdot 10^{-311}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(D \cdot \frac{\frac{M}{2}}{d}\right)}^{2} \cdot -0.5\right)\right)\right)\\ \mathbf{elif}\;h \leq 6 \cdot 10^{+50}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(\left(1 + \frac{h}{\ell} \cdot \left({\left(D \cdot \frac{\frac{M}{2}}{d}\right)}^{2} \cdot -0.5\right)\right) \cdot \sqrt{\frac{d}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + {\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2} \cdot \frac{h \cdot -0.5}{\ell}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 70.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{D \cdot M\_m}{d \cdot 2}\right)}^{2}\right)\right)\\ \mathbf{if}\;t\_0 \leq 2 \cdot 10^{+102}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left(h \cdot \frac{{\left(D \cdot \left(0.5 \cdot \frac{M\_m}{d}\right)\right)}^{2}}{\ell}\right)\right) \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;-1 + \frac{\ell + d \cdot \sqrt{\frac{\ell}{h}}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left({\left(\sqrt[3]{D \cdot M\_m}\right)}^{6} \cdot \frac{\sqrt{\frac{h}{{\ell}^{3}}}}{d}\right)\\ \end{array} \end{array} \]

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 (/ d h) 0.5) (pow (/ d l) 0.5))
          (- 1.0 (* (/ h l) (* 0.5 (pow (/ (* D M_m) (* d 2.0)) 2.0)))))))
   (if (<= t_0 2e+102)
     (*
      (- 1.0 (* 0.5 (* h (/ (pow (* D (* 0.5 (/ M_m d))) 2.0) l))))
      (* (sqrt (/ d h)) (sqrt (/ d l))))
     (if (<= t_0 INFINITY)
       (+ -1.0 (/ (+ l (* d (sqrt (/ l h)))) l))
       (*
        -0.125
        (* (pow (cbrt (* D M_m)) 6.0) (/ (sqrt (/ h (pow l 3.0))) d)))))))

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((d / h), 0.5) * pow((d / l), 0.5)) * (1.0 - ((h / l) * (0.5 * pow(((D * M_m) / (d * 2.0)), 2.0))));
	double tmp;
	if (t_0 <= 2e+102) {
		tmp = (1.0 - (0.5 * (h * (pow((D * (0.5 * (M_m / d))), 2.0) / l)))) * (sqrt((d / h)) * sqrt((d / l)));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = -1.0 + ((l + (d * sqrt((l / h)))) / l);
	} else {
		tmp = -0.125 * (pow(cbrt((D * M_m)), 6.0) * (sqrt((h / pow(l, 3.0))) / d));
	}
	return tmp;
}

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((d / h), 0.5) * Math.pow((d / l), 0.5)) * (1.0 - ((h / l) * (0.5 * Math.pow(((D * M_m) / (d * 2.0)), 2.0))));
	double tmp;
	if (t_0 <= 2e+102) {
		tmp = (1.0 - (0.5 * (h * (Math.pow((D * (0.5 * (M_m / d))), 2.0) / l)))) * (Math.sqrt((d / h)) * Math.sqrt((d / l)));
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = -1.0 + ((l + (d * Math.sqrt((l / h)))) / l);
	} else {
		tmp = -0.125 * (Math.pow(Math.cbrt((D * M_m)), 6.0) * (Math.sqrt((h / Math.pow(l, 3.0))) / d));
	}
	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((Float64(d / h) ^ 0.5) * (Float64(d / l) ^ 0.5)) * Float64(1.0 - Float64(Float64(h / l) * Float64(0.5 * (Float64(Float64(D * M_m) / Float64(d * 2.0)) ^ 2.0)))))
	tmp = 0.0
	if (t_0 <= 2e+102)
		tmp = Float64(Float64(1.0 - Float64(0.5 * Float64(h * Float64((Float64(D * Float64(0.5 * Float64(M_m / d))) ^ 2.0) / l)))) * Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l))));
	elseif (t_0 <= Inf)
		tmp = Float64(-1.0 + Float64(Float64(l + Float64(d * sqrt(Float64(l / h)))) / l));
	else
		tmp = Float64(-0.125 * Float64((cbrt(Float64(D * M_m)) ^ 6.0) * Float64(sqrt(Float64(h / (l ^ 3.0))) / d)));
	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[(N[(N[Power[N[(d / h), $MachinePrecision], 0.5], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(0.5 * N[Power[N[(N[(D * M$95$m), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 2e+102], N[(N[(1.0 - N[(0.5 * N[(h * N[(N[Power[N[(D * N[(0.5 * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(-1.0 + N[(N[(l + N[(d * N[Sqrt[N[(l / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(-0.125 * N[(N[Power[N[Power[N[(D * M$95$m), $MachinePrecision], 1/3], $MachinePrecision], 6.0], $MachinePrecision] * N[(N[Sqrt[N[(h / N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 1.99999999999999995e102

   1. Initial program 81.9%

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

   3. Simplified 81.2%

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

      1. associate-*r/ 81.8%

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

      2. frac-times 82.1%

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

      3. associate-/l* 81.8%

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

      4. *-commutative 81.8%

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

   6. Applied egg-rr 81.8%

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

      1. *-commutative 81.8%

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

      2. associate-/l* 82.4%

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

      3. associate-*r/ 82.7%

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

      4. *-rgt-identity 82.7%

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

      5. times-frac 82.7%

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

      6. metadata-eval 82.7%

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

      7. *-commutative 82.7%

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

      8. associate-/l* 81.8%

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

      9. associate-*l* 81.8%

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

   8. Simplified 81.8%

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

   if 1.99999999999999995e102 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < +inf.0

   1. Initial program 78.4%

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

   3. Simplified 78.3%

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

   5. Taylor expanded in d around inf 54.9%

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

      1. expm1-log1p-u 52.1%

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

      2. expm1-undefine 52.1%

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

      3. pow1/2 52.1%

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

      4. inv-pow 52.1%

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

      5. pow-pow 52.1%

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

      6. metadata-eval 52.1%

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

   7. Applied egg-rr 52.1%

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

      1. sub-neg 52.1%

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

      2. metadata-eval 52.1%

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

      3. +-commutative 52.1%

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

      4. log1p-undefine 52.1%

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

      5. rem-exp-log 55.0%

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

      6. +-commutative 55.0%

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

      7. fma-define 55.0%

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

   9. Simplified 55.0%

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

   10. Taylor expanded in l around 0 93.0%

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

   if +inf.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 0.0%

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

   3. Simplified 0.1%

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

   5. Taylor expanded in M around inf 0.2%

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

      1. associate-*r* 0.2%

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

      2. times-frac 0.0%

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

      3. *-commutative 0.0%

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

      4. associate-/l* 0.0%

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

      5. unpow2 0.0%

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

      6. unpow2 0.0%

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

      7. unpow2 0.0%

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

      8. times-frac 0.0%

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

      9. swap-sqr 0.0%

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

      10. unpow2 0.0%

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

      11. associate-*r/ 0.0%

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

      12. *-commutative 0.0%

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

      13. associate-/l* 0.0%

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

   7. Simplified 0.0%

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

      1. frac-2neg 0.1%

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

      2. sqrt-div 4.1%

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

   9. Applied egg-rr 4.0%

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

   10. Taylor expanded in d around 0 20.6%

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

       1. associate-*l/ 20.6%

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

       2. associate-/l* 20.6%

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

       3. unpow2 20.6%

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

       4. unpow2 20.6%

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

       5. swap-sqr 23.2%

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

       6. rem-cube-cbrt 23.2%

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

       7. rem-cube-cbrt 23.2%

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

       8. pow-sqr 23.2%

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

       9. metadata-eval 23.2%

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

   12. Simplified 23.2%

       \[\leadsto \color{blue}{-0.125 \cdot \left({\left(\sqrt[3]{D \cdot M}\right)}^{6} \cdot \frac{\sqrt{\frac{h}{{\ell}^{3}}}}{d}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}\right)\right) \leq 2 \cdot 10^{+102}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left(h \cdot \frac{{\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2}}{\ell}\right)\right) \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}\right)\right) \leq \infty:\\ \;\;\;\;-1 + \frac{\ell + d \cdot \sqrt{\frac{\ell}{h}}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left({\left(\sqrt[3]{D \cdot M}\right)}^{6} \cdot \frac{\sqrt{\frac{h}{{\ell}^{3}}}}{d}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 71.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{D \cdot M\_m}{d \cdot 2}\right)}^{2}\right)\right)\\ \mathbf{if}\;t\_0 \leq 2 \cdot 10^{+102}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left(h \cdot \frac{{\left(D \cdot \left(0.5 \cdot \frac{M\_m}{d}\right)\right)}^{2}}{\ell}\right)\right) \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;-1 + \frac{\ell + d \cdot \sqrt{\frac{\ell}{h}}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;d \cdot {0}^{-0.5}\\ \end{array} \end{array} \]

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 (/ d h) 0.5) (pow (/ d l) 0.5))
          (- 1.0 (* (/ h l) (* 0.5 (pow (/ (* D M_m) (* d 2.0)) 2.0)))))))
   (if (<= t_0 2e+102)
     (*
      (- 1.0 (* 0.5 (* h (/ (pow (* D (* 0.5 (/ M_m d))) 2.0) l))))
      (* (sqrt (/ d h)) (sqrt (/ d l))))
     (if (<= t_0 INFINITY)
       (+ -1.0 (/ (+ l (* d (sqrt (/ l h)))) l))
       (* d (pow 0.0 -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 = (pow((d / h), 0.5) * pow((d / l), 0.5)) * (1.0 - ((h / l) * (0.5 * pow(((D * M_m) / (d * 2.0)), 2.0))));
	double tmp;
	if (t_0 <= 2e+102) {
		tmp = (1.0 - (0.5 * (h * (pow((D * (0.5 * (M_m / d))), 2.0) / l)))) * (sqrt((d / h)) * sqrt((d / l)));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = -1.0 + ((l + (d * sqrt((l / h)))) / l);
	} else {
		tmp = d * pow(0.0, -0.5);
	}
	return tmp;
}

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((d / h), 0.5) * Math.pow((d / l), 0.5)) * (1.0 - ((h / l) * (0.5 * Math.pow(((D * M_m) / (d * 2.0)), 2.0))));
	double tmp;
	if (t_0 <= 2e+102) {
		tmp = (1.0 - (0.5 * (h * (Math.pow((D * (0.5 * (M_m / d))), 2.0) / l)))) * (Math.sqrt((d / h)) * Math.sqrt((d / l)));
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = -1.0 + ((l + (d * Math.sqrt((l / h)))) / l);
	} else {
		tmp = d * Math.pow(0.0, -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):
	t_0 = (math.pow((d / h), 0.5) * math.pow((d / l), 0.5)) * (1.0 - ((h / l) * (0.5 * math.pow(((D * M_m) / (d * 2.0)), 2.0))))
	tmp = 0
	if t_0 <= 2e+102:
		tmp = (1.0 - (0.5 * (h * (math.pow((D * (0.5 * (M_m / d))), 2.0) / l)))) * (math.sqrt((d / h)) * math.sqrt((d / l)))
	elif t_0 <= math.inf:
		tmp = -1.0 + ((l + (d * math.sqrt((l / h)))) / l)
	else:
		tmp = d * math.pow(0.0, -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((Float64(d / h) ^ 0.5) * (Float64(d / l) ^ 0.5)) * Float64(1.0 - Float64(Float64(h / l) * Float64(0.5 * (Float64(Float64(D * M_m) / Float64(d * 2.0)) ^ 2.0)))))
	tmp = 0.0
	if (t_0 <= 2e+102)
		tmp = Float64(Float64(1.0 - Float64(0.5 * Float64(h * Float64((Float64(D * Float64(0.5 * Float64(M_m / d))) ^ 2.0) / l)))) * Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l))));
	elseif (t_0 <= Inf)
		tmp = Float64(-1.0 + Float64(Float64(l + Float64(d * sqrt(Float64(l / h)))) / l));
	else
		tmp = Float64(d * (0.0 ^ -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)
	t_0 = (((d / h) ^ 0.5) * ((d / l) ^ 0.5)) * (1.0 - ((h / l) * (0.5 * (((D * M_m) / (d * 2.0)) ^ 2.0))));
	tmp = 0.0;
	if (t_0 <= 2e+102)
		tmp = (1.0 - (0.5 * (h * (((D * (0.5 * (M_m / d))) ^ 2.0) / l)))) * (sqrt((d / h)) * sqrt((d / l)));
	elseif (t_0 <= Inf)
		tmp = -1.0 + ((l + (d * sqrt((l / h)))) / l);
	else
		tmp = d * (0.0 ^ -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_] := Block[{t$95$0 = N[(N[(N[Power[N[(d / h), $MachinePrecision], 0.5], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(0.5 * N[Power[N[(N[(D * M$95$m), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 2e+102], N[(N[(1.0 - N[(0.5 * N[(h * N[(N[Power[N[(D * N[(0.5 * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(-1.0 + N[(N[(l + N[(d * N[Sqrt[N[(l / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(d * N[Power[0.0, -0.5], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 1.99999999999999995e102

   1. Initial program 81.9%

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

   3. Simplified 81.2%

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

      1. associate-*r/ 81.8%

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

      2. frac-times 82.1%

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

      3. associate-/l* 81.8%

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

      4. *-commutative 81.8%

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

   6. Applied egg-rr 81.8%

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

      1. *-commutative 81.8%

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

      2. associate-/l* 82.4%

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

      3. associate-*r/ 82.7%

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

      4. *-rgt-identity 82.7%

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

      5. times-frac 82.7%

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

      6. metadata-eval 82.7%

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

      7. *-commutative 82.7%

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

      8. associate-/l* 81.8%

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

      9. associate-*l* 81.8%

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

   8. Simplified 81.8%

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

   if 1.99999999999999995e102 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < +inf.0

   1. Initial program 78.4%

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

   3. Simplified 78.3%

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

   5. Taylor expanded in d around inf 54.9%

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

      1. expm1-log1p-u 52.1%

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

      2. expm1-undefine 52.1%

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

      3. pow1/2 52.1%

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

      4. inv-pow 52.1%

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

      5. pow-pow 52.1%

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

      6. metadata-eval 52.1%

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

   7. Applied egg-rr 52.1%

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

      1. sub-neg 52.1%

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

      2. metadata-eval 52.1%

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

      3. +-commutative 52.1%

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

      4. log1p-undefine 52.1%

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

      5. rem-exp-log 55.0%

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

      6. +-commutative 55.0%

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

      7. fma-define 55.0%

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

   9. Simplified 55.0%

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

   10. Taylor expanded in l around 0 93.0%

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

   if +inf.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 0.0%

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

   3. Simplified 0.1%

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

      1. pow1 0.1%

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

      2. associate-*r* 0.1%

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

      3. sqrt-unprod 0.1%

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

      4. associate-*r* 0.1%

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

      5. associate-*r/ 0.1%

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

   6. Applied egg-rr 0.1%

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

   8. Simplified 0.1%

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

   9. Taylor expanded in d around inf 11.4%

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

       1. unpow-1 11.4%

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

       2. metadata-eval 11.4%

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

       3. pow-sqr 11.4%

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

       4. rem-sqrt-square 11.4%

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

       5. rem-square-sqrt 11.4%

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

       6. fabs-sqr 11.4%

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

       7. rem-square-sqrt 11.4%

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

   11. Simplified 11.4%

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

       1. expm1-log1p-u 11.2%

          \[\leadsto d \cdot {\color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(h \cdot \ell\right)\right)\right)}}^{-0.5} \]

       2. expm1-undefine 12.3%

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

   13. Applied egg-rr 12.3%

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

       1. sub-neg 12.3%

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

       2. metadata-eval 12.3%

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

       3. +-commutative 12.3%

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

       4. log1p-undefine 12.3%

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

       5. rem-exp-log 12.5%

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

       6. +-commutative 12.5%

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

       7. fma-define 12.5%

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

   15. Simplified 12.5%

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

   16. Taylor expanded in h around 0 23.0%

       \[\leadsto d \cdot {\left(-1 + \color{blue}{1}\right)}^{-0.5} \]
3. Recombined 3 regimes into one program.

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}\right)\right) \leq 2 \cdot 10^{+102}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left(h \cdot \frac{{\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2}}{\ell}\right)\right) \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}\right)\right) \leq \infty:\\ \;\;\;\;-1 + \frac{\ell + d \cdot \sqrt{\frac{\ell}{h}}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;d \cdot {0}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 77.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := \sqrt{\frac{d}{h}}\\ t_2 := \sqrt{-d}\\ \mathbf{if}\;h \leq -1.3 \cdot 10^{+80}:\\ \;\;\;\;\frac{t\_2}{\sqrt{-h}} \cdot \left(t\_0 \cdot \left(1 + h \cdot \left(\frac{-0.125}{\ell} \cdot {\left(\frac{D \cdot M\_m}{d}\right)}^{2}\right)\right)\right)\\ \mathbf{elif}\;h \leq -1 \cdot 10^{-311}:\\ \;\;\;\;\left(\frac{t\_2}{\sqrt{-\ell}} \cdot t\_1\right) \cdot \left(1 - 0.5 \cdot {\left(\left(M\_m \cdot \frac{D}{d \cdot 2}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right)\\ \mathbf{elif}\;h \leq 6.2 \cdot 10^{+51}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(\left(1 + \frac{h}{\ell} \cdot \left({\left(D \cdot \frac{\frac{M\_m}{2}}{d}\right)}^{2} \cdot -0.5\right)\right) \cdot t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(t\_0 \cdot \left(1 + {\left(\frac{D}{d} \cdot \left(M\_m \cdot 0.5\right)\right)}^{2} \cdot \frac{h \cdot -0.5}{\ell}\right)\right)\\ \end{array} \end{array} \]

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 (sqrt (/ d l))) (t_1 (sqrt (/ d h))) (t_2 (sqrt (- d))))
   (if (<= h -1.3e+80)
     (*
      (/ t_2 (sqrt (- h)))
      (* t_0 (+ 1.0 (* h (* (/ -0.125 l) (pow (/ (* D M_m) d) 2.0))))))
     (if (<= h -1e-311)
       (*
        (* (/ t_2 (sqrt (- l))) t_1)
        (- 1.0 (* 0.5 (pow (* (* M_m (/ D (* d 2.0))) (sqrt (/ h l))) 2.0))))
       (if (<= h 6.2e+51)
         (*
          (/ (sqrt d) (sqrt l))
          (*
           (+ 1.0 (* (/ h l) (* (pow (* D (/ (/ M_m 2.0) d)) 2.0) -0.5)))
           t_1))
         (*
          (/ (sqrt d) (sqrt h))
          (*
           t_0
           (+
            1.0
            (* (pow (* (/ D d) (* M_m 0.5)) 2.0) (/ (* h -0.5) 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 = sqrt((d / l));
	double t_1 = sqrt((d / h));
	double t_2 = sqrt(-d);
	double tmp;
	if (h <= -1.3e+80) {
		tmp = (t_2 / sqrt(-h)) * (t_0 * (1.0 + (h * ((-0.125 / l) * pow(((D * M_m) / d), 2.0)))));
	} else if (h <= -1e-311) {
		tmp = ((t_2 / sqrt(-l)) * t_1) * (1.0 - (0.5 * pow(((M_m * (D / (d * 2.0))) * sqrt((h / l))), 2.0)));
	} else if (h <= 6.2e+51) {
		tmp = (sqrt(d) / sqrt(l)) * ((1.0 + ((h / l) * (pow((D * ((M_m / 2.0) / d)), 2.0) * -0.5))) * t_1);
	} else {
		tmp = (sqrt(d) / sqrt(h)) * (t_0 * (1.0 + (pow(((D / d) * (M_m * 0.5)), 2.0) * ((h * -0.5) / 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) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = sqrt((d / l))
    t_1 = sqrt((d / h))
    t_2 = sqrt(-d)
    if (h <= (-1.3d+80)) then
        tmp = (t_2 / sqrt(-h)) * (t_0 * (1.0d0 + (h * (((-0.125d0) / l) * (((d_1 * m_m) / d) ** 2.0d0)))))
    else if (h <= (-1d-311)) then
        tmp = ((t_2 / sqrt(-l)) * t_1) * (1.0d0 - (0.5d0 * (((m_m * (d_1 / (d * 2.0d0))) * sqrt((h / l))) ** 2.0d0)))
    else if (h <= 6.2d+51) then
        tmp = (sqrt(d) / sqrt(l)) * ((1.0d0 + ((h / l) * (((d_1 * ((m_m / 2.0d0) / d)) ** 2.0d0) * (-0.5d0)))) * t_1)
    else
        tmp = (sqrt(d) / sqrt(h)) * (t_0 * (1.0d0 + ((((d_1 / d) * (m_m * 0.5d0)) ** 2.0d0) * ((h * (-0.5d0)) / 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 t_0 = Math.sqrt((d / l));
	double t_1 = Math.sqrt((d / h));
	double t_2 = Math.sqrt(-d);
	double tmp;
	if (h <= -1.3e+80) {
		tmp = (t_2 / Math.sqrt(-h)) * (t_0 * (1.0 + (h * ((-0.125 / l) * Math.pow(((D * M_m) / d), 2.0)))));
	} else if (h <= -1e-311) {
		tmp = ((t_2 / Math.sqrt(-l)) * t_1) * (1.0 - (0.5 * Math.pow(((M_m * (D / (d * 2.0))) * Math.sqrt((h / l))), 2.0)));
	} else if (h <= 6.2e+51) {
		tmp = (Math.sqrt(d) / Math.sqrt(l)) * ((1.0 + ((h / l) * (Math.pow((D * ((M_m / 2.0) / d)), 2.0) * -0.5))) * t_1);
	} else {
		tmp = (Math.sqrt(d) / Math.sqrt(h)) * (t_0 * (1.0 + (Math.pow(((D / d) * (M_m * 0.5)), 2.0) * ((h * -0.5) / 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):
	t_0 = math.sqrt((d / l))
	t_1 = math.sqrt((d / h))
	t_2 = math.sqrt(-d)
	tmp = 0
	if h <= -1.3e+80:
		tmp = (t_2 / math.sqrt(-h)) * (t_0 * (1.0 + (h * ((-0.125 / l) * math.pow(((D * M_m) / d), 2.0)))))
	elif h <= -1e-311:
		tmp = ((t_2 / math.sqrt(-l)) * t_1) * (1.0 - (0.5 * math.pow(((M_m * (D / (d * 2.0))) * math.sqrt((h / l))), 2.0)))
	elif h <= 6.2e+51:
		tmp = (math.sqrt(d) / math.sqrt(l)) * ((1.0 + ((h / l) * (math.pow((D * ((M_m / 2.0) / d)), 2.0) * -0.5))) * t_1)
	else:
		tmp = (math.sqrt(d) / math.sqrt(h)) * (t_0 * (1.0 + (math.pow(((D / d) * (M_m * 0.5)), 2.0) * ((h * -0.5) / 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 = sqrt(Float64(d / l))
	t_1 = sqrt(Float64(d / h))
	t_2 = sqrt(Float64(-d))
	tmp = 0.0
	if (h <= -1.3e+80)
		tmp = Float64(Float64(t_2 / sqrt(Float64(-h))) * Float64(t_0 * Float64(1.0 + Float64(h * Float64(Float64(-0.125 / l) * (Float64(Float64(D * M_m) / d) ^ 2.0))))));
	elseif (h <= -1e-311)
		tmp = Float64(Float64(Float64(t_2 / sqrt(Float64(-l))) * t_1) * Float64(1.0 - Float64(0.5 * (Float64(Float64(M_m * Float64(D / Float64(d * 2.0))) * sqrt(Float64(h / l))) ^ 2.0))));
	elseif (h <= 6.2e+51)
		tmp = Float64(Float64(sqrt(d) / sqrt(l)) * Float64(Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(D * Float64(Float64(M_m / 2.0) / d)) ^ 2.0) * -0.5))) * t_1));
	else
		tmp = Float64(Float64(sqrt(d) / sqrt(h)) * Float64(t_0 * Float64(1.0 + Float64((Float64(Float64(D / d) * Float64(M_m * 0.5)) ^ 2.0) * Float64(Float64(h * -0.5) / 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)
	t_0 = sqrt((d / l));
	t_1 = sqrt((d / h));
	t_2 = sqrt(-d);
	tmp = 0.0;
	if (h <= -1.3e+80)
		tmp = (t_2 / sqrt(-h)) * (t_0 * (1.0 + (h * ((-0.125 / l) * (((D * M_m) / d) ^ 2.0)))));
	elseif (h <= -1e-311)
		tmp = ((t_2 / sqrt(-l)) * t_1) * (1.0 - (0.5 * (((M_m * (D / (d * 2.0))) * sqrt((h / l))) ^ 2.0)));
	elseif (h <= 6.2e+51)
		tmp = (sqrt(d) / sqrt(l)) * ((1.0 + ((h / l) * (((D * ((M_m / 2.0) / d)) ^ 2.0) * -0.5))) * t_1);
	else
		tmp = (sqrt(d) / sqrt(h)) * (t_0 * (1.0 + ((((D / d) * (M_m * 0.5)) ^ 2.0) * ((h * -0.5) / 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_] := Block[{t$95$0 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[(-d)], $MachinePrecision]}, If[LessEqual[h, -1.3e+80], N[(N[(t$95$2 / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * N[(1.0 + N[(h * N[(N[(-0.125 / l), $MachinePrecision] * N[Power[N[(N[(D * M$95$m), $MachinePrecision] / d), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, -1e-311], N[(N[(N[(t$95$2 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * N[(1.0 - N[(0.5 * N[Power[N[(N[(M$95$m * N[(D / N[(d * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, 6.2e+51], N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * N[(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] * t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * N[(1.0 + N[(N[Power[N[(N[(D / d), $MachinePrecision] * N[(M$95$m * 0.5), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(h * -0.5), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if h < -1.29999999999999991e80

   1. Initial program 45.3%

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

   3. Simplified 45.3%

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

      1. fma-undefine 45.3%

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

      2. associate-*r* 45.3%

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

      3. associate-*r/ 45.3%

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

   6. Applied egg-rr 45.3%

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

   7. Taylor expanded in M around 0 34.9%

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

      1. associate-*r/ 34.9%

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

      2. associate-*r* 38.3%

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

      3. associate-*r* 38.3%

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

      4. associate-*l/ 40.6%

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

      5. associate-*r/ 40.6%

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

      6. *-commutative 40.6%

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

      7. associate-*r/ 40.6%

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

      8. *-commutative 40.6%

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

      9. times-frac 43.0%

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

      10. *-commutative 43.0%

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

      11. associate-/l* 43.0%

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

      12. unpow2 43.0%

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

      13. unpow2 43.0%

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

      14. unpow2 43.0%

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

      15. times-frac 45.9%

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

      16. swap-sqr 50.7%

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

      17. unpow2 50.7%

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

   9. Simplified 50.7%

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

       1. frac-2neg 47.4%

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

       2. sqrt-div 78.6%

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

   11. Applied egg-rr 77.0%

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

   if -1.29999999999999991e80 < h < -9.99999999999948e-312

   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 74.0%

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

      1. add-sqr-sqrt 74.0%

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

      2. pow2 74.0%

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

      3. sqrt-prod 74.0%

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

      4. sqrt-pow1 78.1%

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

      5. frac-times 78.0%

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

      6. metadata-eval 78.0%

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

      7. pow1 78.0%

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

      8. associate-/l* 78.1%

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

      9. *-commutative 78.1%

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

   6. Applied egg-rr 78.1%

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

      1. frac-2neg 71.6%

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

      2. sqrt-div 79.5%

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

   8. Applied egg-rr 86.5%

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

   if -9.99999999999948e-312 < h < 6.20000000000000022e51

   1. Initial program 71.8%

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

   3. Simplified 71.8%

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

      1. sqrt-div 85.0%

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

   6. Applied egg-rr 85.0%

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

   if 6.20000000000000022e51 < h

   1. Initial program 59.8%

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

   3. Simplified 59.6%

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

      1. fma-undefine 59.6%

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

      2. associate-*r* 59.6%

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

      3. associate-*r/ 59.6%

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

   6. Applied egg-rr 59.6%

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

      1. sqrt-div 76.5%

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

   8. Applied egg-rr 76.5%

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

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -1.3 \cdot 10^{+80}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \left(\frac{-0.125}{\ell} \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)\right)\right)\\ \mathbf{elif}\;h \leq -1 \cdot 10^{-311}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(M \cdot \frac{D}{d \cdot 2}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right)\\ \mathbf{elif}\;h \leq 6.2 \cdot 10^{+51}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(\left(1 + \frac{h}{\ell} \cdot \left({\left(D \cdot \frac{\frac{M}{2}}{d}\right)}^{2} \cdot -0.5\right)\right) \cdot \sqrt{\frac{d}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + {\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2} \cdot \frac{h \cdot -0.5}{\ell}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 70.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ t_1 := \sqrt{\frac{d}{\ell}}\\ t_2 := D \cdot \frac{M\_m}{d}\\ \mathbf{if}\;d \leq -6.6 \cdot 10^{+268}:\\ \;\;\;\;d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{elif}\;d \leq -9.2 \cdot 10^{-70}:\\ \;\;\;\;t\_0 \cdot \left(t\_1 \cdot \left(1 + h \cdot {\left(t\_2 \cdot \sqrt{\frac{-0.125}{\ell}}\right)}^{2}\right)\right)\\ \mathbf{elif}\;d \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(t\_1 \cdot \left(-0.125 \cdot \left(\frac{h}{\ell} \cdot {t\_2}^{2}\right)\right)\right)\\ \mathbf{elif}\;d \leq 3.5 \cdot 10^{-247}:\\ \;\;\;\;-0.125 \cdot \left({\left(\sqrt[3]{D \cdot M\_m}\right)}^{6} \cdot \frac{\sqrt{\frac{h}{{\ell}^{3}}}}{d}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(\left(1 + \frac{h}{\ell} \cdot \left({\left(D \cdot \frac{\frac{M\_m}{2}}{d}\right)}^{2} \cdot -0.5\right)\right) \cdot t\_0\right)\\ \end{array} \end{array} \]

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 (sqrt (/ d h))) (t_1 (sqrt (/ d l))) (t_2 (* D (/ M_m d))))
   (if (<= d -6.6e+268)
     (* d (- (pow (* h l) -0.5)))
     (if (<= d -9.2e-70)
       (* t_0 (* t_1 (+ 1.0 (* h (pow (* t_2 (sqrt (/ -0.125 l))) 2.0)))))
       (if (<= d -1e-309)
         (*
          (/ (sqrt (- d)) (sqrt (- h)))
          (* t_1 (* -0.125 (* (/ h l) (pow t_2 2.0)))))
         (if (<= d 3.5e-247)
           (*
            -0.125
            (* (pow (cbrt (* D M_m)) 6.0) (/ (sqrt (/ h (pow l 3.0))) d)))
           (*
            (/ (sqrt d) (sqrt l))
            (*
             (+ 1.0 (* (/ h l) (* (pow (* D (/ (/ M_m 2.0) d)) 2.0) -0.5)))
             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 / h));
	double t_1 = sqrt((d / l));
	double t_2 = D * (M_m / d);
	double tmp;
	if (d <= -6.6e+268) {
		tmp = d * -pow((h * l), -0.5);
	} else if (d <= -9.2e-70) {
		tmp = t_0 * (t_1 * (1.0 + (h * pow((t_2 * sqrt((-0.125 / l))), 2.0))));
	} else if (d <= -1e-309) {
		tmp = (sqrt(-d) / sqrt(-h)) * (t_1 * (-0.125 * ((h / l) * pow(t_2, 2.0))));
	} else if (d <= 3.5e-247) {
		tmp = -0.125 * (pow(cbrt((D * M_m)), 6.0) * (sqrt((h / pow(l, 3.0))) / d));
	} else {
		tmp = (sqrt(d) / sqrt(l)) * ((1.0 + ((h / l) * (pow((D * ((M_m / 2.0) / d)), 2.0) * -0.5))) * t_0);
	}
	return tmp;
}

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 / h));
	double t_1 = Math.sqrt((d / l));
	double t_2 = D * (M_m / d);
	double tmp;
	if (d <= -6.6e+268) {
		tmp = d * -Math.pow((h * l), -0.5);
	} else if (d <= -9.2e-70) {
		tmp = t_0 * (t_1 * (1.0 + (h * Math.pow((t_2 * Math.sqrt((-0.125 / l))), 2.0))));
	} else if (d <= -1e-309) {
		tmp = (Math.sqrt(-d) / Math.sqrt(-h)) * (t_1 * (-0.125 * ((h / l) * Math.pow(t_2, 2.0))));
	} else if (d <= 3.5e-247) {
		tmp = -0.125 * (Math.pow(Math.cbrt((D * M_m)), 6.0) * (Math.sqrt((h / Math.pow(l, 3.0))) / d));
	} else {
		tmp = (Math.sqrt(d) / Math.sqrt(l)) * ((1.0 + ((h / l) * (Math.pow((D * ((M_m / 2.0) / d)), 2.0) * -0.5))) * 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 = sqrt(Float64(d / h))
	t_1 = sqrt(Float64(d / l))
	t_2 = Float64(D * Float64(M_m / d))
	tmp = 0.0
	if (d <= -6.6e+268)
		tmp = Float64(d * Float64(-(Float64(h * l) ^ -0.5)));
	elseif (d <= -9.2e-70)
		tmp = Float64(t_0 * Float64(t_1 * Float64(1.0 + Float64(h * (Float64(t_2 * sqrt(Float64(-0.125 / l))) ^ 2.0)))));
	elseif (d <= -1e-309)
		tmp = Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-h))) * Float64(t_1 * Float64(-0.125 * Float64(Float64(h / l) * (t_2 ^ 2.0)))));
	elseif (d <= 3.5e-247)
		tmp = Float64(-0.125 * Float64((cbrt(Float64(D * M_m)) ^ 6.0) * Float64(sqrt(Float64(h / (l ^ 3.0))) / d)));
	else
		tmp = Float64(Float64(sqrt(d) / sqrt(l)) * Float64(Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(D * Float64(Float64(M_m / 2.0) / d)) ^ 2.0) * -0.5))) * t_0));
	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[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(D * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -6.6e+268], N[(d * (-N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision])), $MachinePrecision], If[LessEqual[d, -9.2e-70], N[(t$95$0 * N[(t$95$1 * N[(1.0 + N[(h * N[Power[N[(t$95$2 * N[Sqrt[N[(-0.125 / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -1e-309], N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * N[(-0.125 * N[(N[(h / l), $MachinePrecision] * N[Power[t$95$2, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 3.5e-247], N[(-0.125 * N[(N[Power[N[Power[N[(D * M$95$m), $MachinePrecision], 1/3], $MachinePrecision], 6.0], $MachinePrecision] * N[(N[Sqrt[N[(h / N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * N[(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] * t$95$0), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if d < -6.6000000000000002e268

   1. Initial program 40.6%

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

   3. Simplified 41.3%

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

   5. Taylor expanded in d around inf 10.2%

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

   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. associate-*l* 0.0%

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

      2. unpow2 0.0%

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

      3. rem-square-sqrt 89.7%

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

      4. neg-mul-1 89.7%

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

      5. unpow-1 89.7%

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

      6. metadata-eval 89.7%

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

      7. pow-sqr 89.8%

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

      8. rem-sqrt-square 89.8%

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

      9. rem-square-sqrt 89.1%

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

      10. fabs-sqr 89.1%

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

      11. rem-square-sqrt 89.8%

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

   8. Simplified 89.8%

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

   if -6.6000000000000002e268 < d < -9.20000000000000002e-70

   1. Initial program 81.6%

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

   3. Simplified 80.3%

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

      1. fma-undefine 80.3%

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

      2. associate-*r* 80.3%

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

      3. associate-*r/ 80.3%

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

   6. Applied egg-rr 80.3%

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

   7. Taylor expanded in M around 0 64.7%

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

      1. associate-*r/ 64.7%

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

      2. associate-*r* 65.3%

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

      3. associate-*r* 65.3%

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

      4. associate-*l/ 66.7%

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

      5. associate-*r/ 66.7%

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

      6. *-commutative 66.7%

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

      7. associate-*r/ 66.7%

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

      8. *-commutative 66.7%

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

      9. times-frac 71.3%

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

      10. *-commutative 71.3%

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

      11. associate-/l* 71.3%

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

      12. unpow2 71.3%

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

      13. unpow2 71.3%

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

      14. unpow2 71.3%

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

      15. times-frac 75.7%

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

      16. swap-sqr 81.9%

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

      17. unpow2 81.9%

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

   9. Simplified 81.9%

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

       1. add-sqr-sqrt 81.9%

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

       2. pow2 81.9%

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

       3. *-commutative 81.9%

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

       4. associate-*r/ 81.9%

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

       5. sqrt-prod 81.9%

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

       6. sqrt-pow1 84.7%

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

       7. metadata-eval 84.7%

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

       8. pow1 84.7%

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

   11. Applied egg-rr 84.7%

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

   if -9.20000000000000002e-70 < d < -1.000000000000002e-309

   1. Initial program 44.5%

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

   3. Simplified 44.4%

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

   5. Taylor expanded in M around inf 26.9%

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

      1. associate-*r* 29.1%

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

      2. times-frac 27.3%

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

      3. *-commutative 27.3%

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

      4. associate-/l* 25.1%

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

      5. unpow2 25.1%

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

      6. unpow2 25.1%

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

      7. unpow2 25.1%

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

      8. times-frac 31.9%

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

      9. swap-sqr 40.3%

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

      10. unpow2 40.3%

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

      11. associate-*r/ 40.3%

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

      12. *-commutative 40.3%

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

      13. associate-/l* 38.3%

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

   7. Simplified 38.3%

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

      1. frac-2neg 50.6%

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

      2. sqrt-div 77.0%

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

   9. Applied egg-rr 62.4%

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

   if -1.000000000000002e-309 < d < 3.4999999999999999e-247

   1. Initial program 7.2%

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

   3. Simplified 7.2%

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

   5. Taylor expanded in M around inf 6.7%

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

      1. associate-*r* 6.7%

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

      2. times-frac 6.7%

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

      3. *-commutative 6.7%

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

      4. associate-/l* 6.7%

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

      5. unpow2 6.7%

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

      6. unpow2 6.7%

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

      7. unpow2 6.7%

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

      8. times-frac 7.2%

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

      9. swap-sqr 7.2%

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

      10. unpow2 7.2%

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

      11. associate-*r/ 7.2%

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

      12. *-commutative 7.2%

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

      13. associate-/l* 7.2%

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

   7. Simplified 7.2%

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

      1. frac-2neg 7.2%

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

      2. sqrt-div 0.0%

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

   9. Applied egg-rr 0.0%

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

   10. Taylor expanded in d around 0 54.7%

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

       1. associate-*l/ 54.7%

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

       2. associate-/l* 54.7%

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

       3. unpow2 54.7%

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

       4. unpow2 54.7%

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

       5. swap-sqr 61.8%

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

       6. rem-cube-cbrt 61.8%

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

       7. rem-cube-cbrt 61.8%

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

       8. pow-sqr 61.8%

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

       9. metadata-eval 61.8%

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

   12. Simplified 61.8%

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

   if 3.4999999999999999e-247 < 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 73.1%

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

      1. sqrt-div 81.2%

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

   6. Applied egg-rr 81.2%

      \[\leadsto \color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(D \cdot \frac{\frac{M}{2}}{d}\right)}^{2} \cdot -0.5\right)\right)\right) \]
3. Recombined 5 regimes into one program.

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -6.6 \cdot 10^{+268}:\\ \;\;\;\;d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{elif}\;d \leq -9.2 \cdot 10^{-70}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot {\left(\left(D \cdot \frac{M}{d}\right) \cdot \sqrt{\frac{-0.125}{\ell}}\right)}^{2}\right)\right)\\ \mathbf{elif}\;d \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(-0.125 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}\right)\right)\right)\\ \mathbf{elif}\;d \leq 3.5 \cdot 10^{-247}:\\ \;\;\;\;-0.125 \cdot \left({\left(\sqrt[3]{D \cdot M}\right)}^{6} \cdot \frac{\sqrt{\frac{h}{{\ell}^{3}}}}{d}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(\left(1 + \frac{h}{\ell} \cdot \left({\left(D \cdot \frac{\frac{M}{2}}{d}\right)}^{2} \cdot -0.5\right)\right) \cdot \sqrt{\frac{d}{h}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := 1 + \frac{h}{\ell} \cdot \left({\left(D \cdot \frac{\frac{M\_m}{2}}{d}\right)}^{2} \cdot -0.5\right)\\ t_1 := \sqrt{\frac{d}{\ell}}\\ t_2 := 1 - 0.5 \cdot \left(h \cdot \frac{{\left(D \cdot \left(0.5 \cdot \frac{M\_m}{d}\right)\right)}^{2}}{\ell}\right)\\ t_3 := \sqrt{\frac{d}{h}}\\ t_4 := \sqrt{-d}\\ \mathbf{if}\;\ell \leq -1.5 \cdot 10^{-201}:\\ \;\;\;\;\left(\frac{t\_4}{\sqrt{-h}} \cdot t\_1\right) \cdot t\_2\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(\frac{t\_4}{\sqrt{-\ell}} \cdot t\_3\right) \cdot t\_2\\ \mathbf{elif}\;\ell \leq 1.5 \cdot 10^{+97}:\\ \;\;\;\;t\_1 \cdot \left(t\_0 \cdot \frac{\sqrt{d}}{\sqrt{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(t\_0 \cdot t\_3\right)\\ \end{array} \end{array} \]

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 (+ 1.0 (* (/ h l) (* (pow (* D (/ (/ M_m 2.0) d)) 2.0) -0.5))))
        (t_1 (sqrt (/ d l)))
        (t_2 (- 1.0 (* 0.5 (* h (/ (pow (* D (* 0.5 (/ M_m d))) 2.0) l)))))
        (t_3 (sqrt (/ d h)))
        (t_4 (sqrt (- d))))
   (if (<= l -1.5e-201)
     (* (* (/ t_4 (sqrt (- h))) t_1) t_2)
     (if (<= l -5e-310)
       (* (* (/ t_4 (sqrt (- l))) t_3) t_2)
       (if (<= l 1.5e+97)
         (* t_1 (* t_0 (/ (sqrt d) (sqrt h))))
         (* (/ (sqrt d) (sqrt l)) (* t_0 t_3)))))))

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 = 1.0 + ((h / l) * (pow((D * ((M_m / 2.0) / d)), 2.0) * -0.5));
	double t_1 = sqrt((d / l));
	double t_2 = 1.0 - (0.5 * (h * (pow((D * (0.5 * (M_m / d))), 2.0) / l)));
	double t_3 = sqrt((d / h));
	double t_4 = sqrt(-d);
	double tmp;
	if (l <= -1.5e-201) {
		tmp = ((t_4 / sqrt(-h)) * t_1) * t_2;
	} else if (l <= -5e-310) {
		tmp = ((t_4 / sqrt(-l)) * t_3) * t_2;
	} else if (l <= 1.5e+97) {
		tmp = t_1 * (t_0 * (sqrt(d) / sqrt(h)));
	} else {
		tmp = (sqrt(d) / sqrt(l)) * (t_0 * t_3);
	}
	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) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_0 = 1.0d0 + ((h / l) * (((d_1 * ((m_m / 2.0d0) / d)) ** 2.0d0) * (-0.5d0)))
    t_1 = sqrt((d / l))
    t_2 = 1.0d0 - (0.5d0 * (h * (((d_1 * (0.5d0 * (m_m / d))) ** 2.0d0) / l)))
    t_3 = sqrt((d / h))
    t_4 = sqrt(-d)
    if (l <= (-1.5d-201)) then
        tmp = ((t_4 / sqrt(-h)) * t_1) * t_2
    else if (l <= (-5d-310)) then
        tmp = ((t_4 / sqrt(-l)) * t_3) * t_2
    else if (l <= 1.5d+97) then
        tmp = t_1 * (t_0 * (sqrt(d) / sqrt(h)))
    else
        tmp = (sqrt(d) / sqrt(l)) * (t_0 * t_3)
    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 = 1.0 + ((h / l) * (Math.pow((D * ((M_m / 2.0) / d)), 2.0) * -0.5));
	double t_1 = Math.sqrt((d / l));
	double t_2 = 1.0 - (0.5 * (h * (Math.pow((D * (0.5 * (M_m / d))), 2.0) / l)));
	double t_3 = Math.sqrt((d / h));
	double t_4 = Math.sqrt(-d);
	double tmp;
	if (l <= -1.5e-201) {
		tmp = ((t_4 / Math.sqrt(-h)) * t_1) * t_2;
	} else if (l <= -5e-310) {
		tmp = ((t_4 / Math.sqrt(-l)) * t_3) * t_2;
	} else if (l <= 1.5e+97) {
		tmp = t_1 * (t_0 * (Math.sqrt(d) / Math.sqrt(h)));
	} else {
		tmp = (Math.sqrt(d) / Math.sqrt(l)) * (t_0 * t_3);
	}
	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 = 1.0 + ((h / l) * (math.pow((D * ((M_m / 2.0) / d)), 2.0) * -0.5))
	t_1 = math.sqrt((d / l))
	t_2 = 1.0 - (0.5 * (h * (math.pow((D * (0.5 * (M_m / d))), 2.0) / l)))
	t_3 = math.sqrt((d / h))
	t_4 = math.sqrt(-d)
	tmp = 0
	if l <= -1.5e-201:
		tmp = ((t_4 / math.sqrt(-h)) * t_1) * t_2
	elif l <= -5e-310:
		tmp = ((t_4 / math.sqrt(-l)) * t_3) * t_2
	elif l <= 1.5e+97:
		tmp = t_1 * (t_0 * (math.sqrt(d) / math.sqrt(h)))
	else:
		tmp = (math.sqrt(d) / math.sqrt(l)) * (t_0 * t_3)
	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(1.0 + Float64(Float64(h / l) * Float64((Float64(D * Float64(Float64(M_m / 2.0) / d)) ^ 2.0) * -0.5)))
	t_1 = sqrt(Float64(d / l))
	t_2 = Float64(1.0 - Float64(0.5 * Float64(h * Float64((Float64(D * Float64(0.5 * Float64(M_m / d))) ^ 2.0) / l))))
	t_3 = sqrt(Float64(d / h))
	t_4 = sqrt(Float64(-d))
	tmp = 0.0
	if (l <= -1.5e-201)
		tmp = Float64(Float64(Float64(t_4 / sqrt(Float64(-h))) * t_1) * t_2);
	elseif (l <= -5e-310)
		tmp = Float64(Float64(Float64(t_4 / sqrt(Float64(-l))) * t_3) * t_2);
	elseif (l <= 1.5e+97)
		tmp = Float64(t_1 * Float64(t_0 * Float64(sqrt(d) / sqrt(h))));
	else
		tmp = Float64(Float64(sqrt(d) / sqrt(l)) * Float64(t_0 * t_3));
	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 = 1.0 + ((h / l) * (((D * ((M_m / 2.0) / d)) ^ 2.0) * -0.5));
	t_1 = sqrt((d / l));
	t_2 = 1.0 - (0.5 * (h * (((D * (0.5 * (M_m / d))) ^ 2.0) / l)));
	t_3 = sqrt((d / h));
	t_4 = sqrt(-d);
	tmp = 0.0;
	if (l <= -1.5e-201)
		tmp = ((t_4 / sqrt(-h)) * t_1) * t_2;
	elseif (l <= -5e-310)
		tmp = ((t_4 / sqrt(-l)) * t_3) * t_2;
	elseif (l <= 1.5e+97)
		tmp = t_1 * (t_0 * (sqrt(d) / sqrt(h)));
	else
		tmp = (sqrt(d) / sqrt(l)) * (t_0 * t_3);
	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[(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]}, Block[{t$95$1 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(1.0 - N[(0.5 * N[(h * N[(N[Power[N[(D * N[(0.5 * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[(-d)], $MachinePrecision]}, If[LessEqual[l, -1.5e-201], N[(N[(N[(t$95$4 / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[l, -5e-310], N[(N[(N[(t$95$4 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[l, 1.5e+97], N[(t$95$1 * N[(t$95$0 * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * t$95$3), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -1.50000000000000001e-201

   1. Initial program 62.0%

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

   3. Simplified 62.1%

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

      1. associate-*r/ 63.3%

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

      2. frac-times 63.2%

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

      3. associate-/l* 63.3%

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

      4. *-commutative 63.3%

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

   6. Applied egg-rr 63.3%

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

      1. *-commutative 63.3%

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

      2. associate-/l* 63.3%

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

      3. associate-*r/ 63.2%

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

      4. *-rgt-identity 63.2%

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

      5. times-frac 63.2%

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

      6. metadata-eval 63.2%

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

      7. *-commutative 63.2%

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

      8. associate-/l* 63.2%

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

      9. associate-*l* 63.2%

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

   8. Simplified 63.2%

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

      1. frac-2neg 66.8%

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

      2. sqrt-div 82.7%

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

   10. Applied egg-rr 77.0%

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

   if -1.50000000000000001e-201 < l < -4.999999999999985e-310

   1. Initial program 72.9%

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

   3. Simplified 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. associate-*r/ 73.8%

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

      2. frac-times 73.8%

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

      3. associate-/l* 73.8%

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

      4. *-commutative 73.8%

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

   6. Applied egg-rr 73.8%

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

      1. *-commutative 73.8%

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

      2. associate-/l* 73.8%

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

      3. associate-*r/ 73.8%

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

      4. *-rgt-identity 73.8%

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

      5. times-frac 73.8%

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

      6. metadata-eval 73.8%

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

      7. *-commutative 73.8%

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

      8. associate-/l* 69.8%

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

      9. associate-*l* 69.8%

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

   8. Simplified 69.8%

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

      1. frac-2neg 68.9%

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

      2. sqrt-div 76.0%

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

   10. Applied egg-rr 91.7%

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

   if -4.999999999999985e-310 < l < 1.4999999999999999e97

   1. Initial program 75.0%

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

   3. Simplified 72.8%

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

      1. sqrt-div 85.2%

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

   6. Applied egg-rr 85.3%

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

   if 1.4999999999999999e97 < l

   1. Initial program 49.1%

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

   3. Simplified 51.1%

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

      1. sqrt-div 72.7%

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

   6. Applied egg-rr 72.7%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.5 \cdot 10^{-201}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{{\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2}}{\ell}\right)\right)\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{{\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2}}{\ell}\right)\right)\\ \mathbf{elif}\;\ell \leq 1.5 \cdot 10^{+97}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\left(1 + \frac{h}{\ell} \cdot \left({\left(D \cdot \frac{\frac{M}{2}}{d}\right)}^{2} \cdot -0.5\right)\right) \cdot \frac{\sqrt{d}}{\sqrt{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(\left(1 + \frac{h}{\ell} \cdot \left({\left(D \cdot \frac{\frac{M}{2}}{d}\right)}^{2} \cdot -0.5\right)\right) \cdot \sqrt{\frac{d}{h}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := 1 + \frac{h}{\ell} \cdot \left({\left(D \cdot \frac{\frac{M\_m}{2}}{d}\right)}^{2} \cdot -0.5\right)\\ t_1 := \sqrt{\frac{d}{h}}\\ t_2 := \sqrt{-d}\\ t_3 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;\ell \leq -5 \cdot 10^{-190}:\\ \;\;\;\;\left(\frac{t\_2}{\sqrt{-h}} \cdot t\_0\right) \cdot t\_3\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(\frac{t\_2}{\sqrt{-\ell}} \cdot t\_1\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{{\left(D \cdot \left(0.5 \cdot \frac{M\_m}{d}\right)\right)}^{2}}{\ell}\right)\right)\\ \mathbf{elif}\;\ell \leq 1.05 \cdot 10^{+97}:\\ \;\;\;\;t\_3 \cdot \left(t\_0 \cdot \frac{\sqrt{d}}{\sqrt{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(t\_0 \cdot t\_1\right)\\ \end{array} \end{array} \]

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 (+ 1.0 (* (/ h l) (* (pow (* D (/ (/ M_m 2.0) d)) 2.0) -0.5))))
        (t_1 (sqrt (/ d h)))
        (t_2 (sqrt (- d)))
        (t_3 (sqrt (/ d l))))
   (if (<= l -5e-190)
     (* (* (/ t_2 (sqrt (- h))) t_0) t_3)
     (if (<= l -5e-310)
       (*
        (* (/ t_2 (sqrt (- l))) t_1)
        (- 1.0 (* 0.5 (* h (/ (pow (* D (* 0.5 (/ M_m d))) 2.0) l)))))
       (if (<= l 1.05e+97)
         (* t_3 (* t_0 (/ (sqrt d) (sqrt h))))
         (* (/ (sqrt d) (sqrt l)) (* t_0 t_1)))))))

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 = 1.0 + ((h / l) * (pow((D * ((M_m / 2.0) / d)), 2.0) * -0.5));
	double t_1 = sqrt((d / h));
	double t_2 = sqrt(-d);
	double t_3 = sqrt((d / l));
	double tmp;
	if (l <= -5e-190) {
		tmp = ((t_2 / sqrt(-h)) * t_0) * t_3;
	} else if (l <= -5e-310) {
		tmp = ((t_2 / sqrt(-l)) * t_1) * (1.0 - (0.5 * (h * (pow((D * (0.5 * (M_m / d))), 2.0) / l))));
	} else if (l <= 1.05e+97) {
		tmp = t_3 * (t_0 * (sqrt(d) / sqrt(h)));
	} else {
		tmp = (sqrt(d) / sqrt(l)) * (t_0 * t_1);
	}
	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) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = 1.0d0 + ((h / l) * (((d_1 * ((m_m / 2.0d0) / d)) ** 2.0d0) * (-0.5d0)))
    t_1 = sqrt((d / h))
    t_2 = sqrt(-d)
    t_3 = sqrt((d / l))
    if (l <= (-5d-190)) then
        tmp = ((t_2 / sqrt(-h)) * t_0) * t_3
    else if (l <= (-5d-310)) then
        tmp = ((t_2 / sqrt(-l)) * t_1) * (1.0d0 - (0.5d0 * (h * (((d_1 * (0.5d0 * (m_m / d))) ** 2.0d0) / l))))
    else if (l <= 1.05d+97) then
        tmp = t_3 * (t_0 * (sqrt(d) / sqrt(h)))
    else
        tmp = (sqrt(d) / sqrt(l)) * (t_0 * t_1)
    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 = 1.0 + ((h / l) * (Math.pow((D * ((M_m / 2.0) / d)), 2.0) * -0.5));
	double t_1 = Math.sqrt((d / h));
	double t_2 = Math.sqrt(-d);
	double t_3 = Math.sqrt((d / l));
	double tmp;
	if (l <= -5e-190) {
		tmp = ((t_2 / Math.sqrt(-h)) * t_0) * t_3;
	} else if (l <= -5e-310) {
		tmp = ((t_2 / Math.sqrt(-l)) * t_1) * (1.0 - (0.5 * (h * (Math.pow((D * (0.5 * (M_m / d))), 2.0) / l))));
	} else if (l <= 1.05e+97) {
		tmp = t_3 * (t_0 * (Math.sqrt(d) / Math.sqrt(h)));
	} else {
		tmp = (Math.sqrt(d) / Math.sqrt(l)) * (t_0 * t_1);
	}
	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 = 1.0 + ((h / l) * (math.pow((D * ((M_m / 2.0) / d)), 2.0) * -0.5))
	t_1 = math.sqrt((d / h))
	t_2 = math.sqrt(-d)
	t_3 = math.sqrt((d / l))
	tmp = 0
	if l <= -5e-190:
		tmp = ((t_2 / math.sqrt(-h)) * t_0) * t_3
	elif l <= -5e-310:
		tmp = ((t_2 / math.sqrt(-l)) * t_1) * (1.0 - (0.5 * (h * (math.pow((D * (0.5 * (M_m / d))), 2.0) / l))))
	elif l <= 1.05e+97:
		tmp = t_3 * (t_0 * (math.sqrt(d) / math.sqrt(h)))
	else:
		tmp = (math.sqrt(d) / math.sqrt(l)) * (t_0 * t_1)
	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(1.0 + Float64(Float64(h / l) * Float64((Float64(D * Float64(Float64(M_m / 2.0) / d)) ^ 2.0) * -0.5)))
	t_1 = sqrt(Float64(d / h))
	t_2 = sqrt(Float64(-d))
	t_3 = sqrt(Float64(d / l))
	tmp = 0.0
	if (l <= -5e-190)
		tmp = Float64(Float64(Float64(t_2 / sqrt(Float64(-h))) * t_0) * t_3);
	elseif (l <= -5e-310)
		tmp = Float64(Float64(Float64(t_2 / sqrt(Float64(-l))) * t_1) * Float64(1.0 - Float64(0.5 * Float64(h * Float64((Float64(D * Float64(0.5 * Float64(M_m / d))) ^ 2.0) / l)))));
	elseif (l <= 1.05e+97)
		tmp = Float64(t_3 * Float64(t_0 * Float64(sqrt(d) / sqrt(h))));
	else
		tmp = Float64(Float64(sqrt(d) / sqrt(l)) * Float64(t_0 * t_1));
	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 = 1.0 + ((h / l) * (((D * ((M_m / 2.0) / d)) ^ 2.0) * -0.5));
	t_1 = sqrt((d / h));
	t_2 = sqrt(-d);
	t_3 = sqrt((d / l));
	tmp = 0.0;
	if (l <= -5e-190)
		tmp = ((t_2 / sqrt(-h)) * t_0) * t_3;
	elseif (l <= -5e-310)
		tmp = ((t_2 / sqrt(-l)) * t_1) * (1.0 - (0.5 * (h * (((D * (0.5 * (M_m / d))) ^ 2.0) / l))));
	elseif (l <= 1.05e+97)
		tmp = t_3 * (t_0 * (sqrt(d) / sqrt(h)));
	else
		tmp = (sqrt(d) / sqrt(l)) * (t_0 * t_1);
	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[(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]}, Block[{t$95$1 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[(-d)], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -5e-190], N[(N[(N[(t$95$2 / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$3), $MachinePrecision], If[LessEqual[l, -5e-310], N[(N[(N[(t$95$2 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(h * N[(N[Power[N[(D * N[(0.5 * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.05e+97], N[(t$95$3 * N[(t$95$0 * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * t$95$1), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -5.00000000000000034e-190

   1. Initial program 62.4%

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

   3. Simplified 61.4%

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

      1. frac-2neg 67.5%

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

      2. sqrt-div 83.6%

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

   6. Applied egg-rr 76.3%

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

   if -5.00000000000000034e-190 < l < -4.999999999999985e-310

   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. associate-*r/ 73.4%

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

      2. frac-times 73.4%

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

      3. associate-/l* 73.4%

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

      4. *-commutative 73.4%

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

   6. Applied egg-rr 73.4%

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

      1. *-commutative 73.4%

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

      2. associate-/l* 73.4%

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

      3. associate-*r/ 73.4%

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

      4. *-rgt-identity 73.4%

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

      5. times-frac 73.4%

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

      6. metadata-eval 73.4%

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

      7. *-commutative 73.4%

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

      8. associate-/l* 70.3%

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

      9. associate-*l* 70.3%

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

   8. Simplified 70.3%

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

      1. frac-2neg 66.3%

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

      2. sqrt-div 71.8%

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

   10. Applied egg-rr 87.4%

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

   if -4.999999999999985e-310 < l < 1.05000000000000006e97

   1. Initial program 75.0%

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

   3. Simplified 72.8%

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

      1. sqrt-div 85.2%

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

   6. Applied egg-rr 85.3%

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

   if 1.05000000000000006e97 < l

   1. Initial program 49.1%

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

   3. Simplified 51.1%

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

      1. sqrt-div 72.7%

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

   6. Applied egg-rr 72.7%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5 \cdot 10^{-190}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(D \cdot \frac{\frac{M}{2}}{d}\right)}^{2} \cdot -0.5\right)\right)\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{{\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2}}{\ell}\right)\right)\\ \mathbf{elif}\;\ell \leq 1.05 \cdot 10^{+97}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\left(1 + \frac{h}{\ell} \cdot \left({\left(D \cdot \frac{\frac{M}{2}}{d}\right)}^{2} \cdot -0.5\right)\right) \cdot \frac{\sqrt{d}}{\sqrt{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(\left(1 + \frac{h}{\ell} \cdot \left({\left(D \cdot \frac{\frac{M}{2}}{d}\right)}^{2} \cdot -0.5\right)\right) \cdot \sqrt{\frac{d}{h}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 73.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := D \cdot \frac{M\_m}{d}\\ \mathbf{if}\;d \leq -5.4 \cdot 10^{+274}:\\ \;\;\;\;d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{elif}\;d \leq -2.3 \cdot 10^{-69}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(t\_0 \cdot \left(1 + h \cdot {\left(t\_1 \cdot \sqrt{\frac{-0.125}{\ell}}\right)}^{2}\right)\right)\\ \mathbf{elif}\;d \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(t\_0 \cdot \left(-0.125 \cdot \left(\frac{h}{\ell} \cdot {t\_1}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(\left(1 + \frac{h}{\ell} \cdot \left({\left(D \cdot \frac{\frac{M\_m}{2}}{d}\right)}^{2} \cdot -0.5\right)\right) \cdot \frac{\sqrt{d}}{\sqrt{h}}\right)\\ \end{array} \end{array} \]

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 (sqrt (/ d l))) (t_1 (* D (/ M_m d))))
   (if (<= d -5.4e+274)
     (* d (- (pow (* h l) -0.5)))
     (if (<= d -2.3e-69)
       (*
        (sqrt (/ d h))
        (* t_0 (+ 1.0 (* h (pow (* t_1 (sqrt (/ -0.125 l))) 2.0)))))
       (if (<= d -1e-309)
         (*
          (/ (sqrt (- d)) (sqrt (- h)))
          (* t_0 (* -0.125 (* (/ h l) (pow t_1 2.0)))))
         (*
          t_0
          (*
           (+ 1.0 (* (/ h l) (* (pow (* D (/ (/ M_m 2.0) d)) 2.0) -0.5)))
           (/ (sqrt d) (sqrt h)))))))))

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));
	double t_1 = D * (M_m / d);
	double tmp;
	if (d <= -5.4e+274) {
		tmp = d * -pow((h * l), -0.5);
	} else if (d <= -2.3e-69) {
		tmp = sqrt((d / h)) * (t_0 * (1.0 + (h * pow((t_1 * sqrt((-0.125 / l))), 2.0))));
	} else if (d <= -1e-309) {
		tmp = (sqrt(-d) / sqrt(-h)) * (t_0 * (-0.125 * ((h / l) * pow(t_1, 2.0))));
	} else {
		tmp = t_0 * ((1.0 + ((h / l) * (pow((D * ((M_m / 2.0) / d)), 2.0) * -0.5))) * (sqrt(d) / sqrt(h)));
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = sqrt((d / l))
    t_1 = d_1 * (m_m / d)
    if (d <= (-5.4d+274)) then
        tmp = d * -((h * l) ** (-0.5d0))
    else if (d <= (-2.3d-69)) then
        tmp = sqrt((d / h)) * (t_0 * (1.0d0 + (h * ((t_1 * sqrt(((-0.125d0) / l))) ** 2.0d0))))
    else if (d <= (-1d-309)) then
        tmp = (sqrt(-d) / sqrt(-h)) * (t_0 * ((-0.125d0) * ((h / l) * (t_1 ** 2.0d0))))
    else
        tmp = t_0 * ((1.0d0 + ((h / l) * (((d_1 * ((m_m / 2.0d0) / d)) ** 2.0d0) * (-0.5d0)))) * (sqrt(d) / sqrt(h)))
    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));
	double t_1 = D * (M_m / d);
	double tmp;
	if (d <= -5.4e+274) {
		tmp = d * -Math.pow((h * l), -0.5);
	} else if (d <= -2.3e-69) {
		tmp = Math.sqrt((d / h)) * (t_0 * (1.0 + (h * Math.pow((t_1 * Math.sqrt((-0.125 / l))), 2.0))));
	} else if (d <= -1e-309) {
		tmp = (Math.sqrt(-d) / Math.sqrt(-h)) * (t_0 * (-0.125 * ((h / l) * Math.pow(t_1, 2.0))));
	} else {
		tmp = t_0 * ((1.0 + ((h / l) * (Math.pow((D * ((M_m / 2.0) / d)), 2.0) * -0.5))) * (Math.sqrt(d) / Math.sqrt(h)));
	}
	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))
	t_1 = D * (M_m / d)
	tmp = 0
	if d <= -5.4e+274:
		tmp = d * -math.pow((h * l), -0.5)
	elif d <= -2.3e-69:
		tmp = math.sqrt((d / h)) * (t_0 * (1.0 + (h * math.pow((t_1 * math.sqrt((-0.125 / l))), 2.0))))
	elif d <= -1e-309:
		tmp = (math.sqrt(-d) / math.sqrt(-h)) * (t_0 * (-0.125 * ((h / l) * math.pow(t_1, 2.0))))
	else:
		tmp = t_0 * ((1.0 + ((h / l) * (math.pow((D * ((M_m / 2.0) / d)), 2.0) * -0.5))) * (math.sqrt(d) / math.sqrt(h)))
	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 = sqrt(Float64(d / l))
	t_1 = Float64(D * Float64(M_m / d))
	tmp = 0.0
	if (d <= -5.4e+274)
		tmp = Float64(d * Float64(-(Float64(h * l) ^ -0.5)));
	elseif (d <= -2.3e-69)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(t_0 * Float64(1.0 + Float64(h * (Float64(t_1 * sqrt(Float64(-0.125 / l))) ^ 2.0)))));
	elseif (d <= -1e-309)
		tmp = Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-h))) * Float64(t_0 * Float64(-0.125 * Float64(Float64(h / l) * (t_1 ^ 2.0)))));
	else
		tmp = Float64(t_0 * Float64(Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(D * Float64(Float64(M_m / 2.0) / d)) ^ 2.0) * -0.5))) * Float64(sqrt(d) / sqrt(h))));
	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));
	t_1 = D * (M_m / d);
	tmp = 0.0;
	if (d <= -5.4e+274)
		tmp = d * -((h * l) ^ -0.5);
	elseif (d <= -2.3e-69)
		tmp = sqrt((d / h)) * (t_0 * (1.0 + (h * ((t_1 * sqrt((-0.125 / l))) ^ 2.0))));
	elseif (d <= -1e-309)
		tmp = (sqrt(-d) / sqrt(-h)) * (t_0 * (-0.125 * ((h / l) * (t_1 ^ 2.0))));
	else
		tmp = t_0 * ((1.0 + ((h / l) * (((D * ((M_m / 2.0) / d)) ^ 2.0) * -0.5))) * (sqrt(d) / sqrt(h)));
	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[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(D * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -5.4e+274], N[(d * (-N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision])), $MachinePrecision], If[LessEqual[d, -2.3e-69], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(t$95$0 * N[(1.0 + N[(h * N[Power[N[(t$95$1 * N[Sqrt[N[(-0.125 / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -1e-309], N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * N[(-0.125 * N[(N[(h / l), $MachinePrecision] * N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(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] * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -5.4000000000000003e274

   1. Initial program 40.6%

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

   3. Simplified 41.3%

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

   5. Taylor expanded in d around inf 10.2%

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

   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. associate-*l* 0.0%

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

      2. unpow2 0.0%

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

      3. rem-square-sqrt 89.7%

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

      4. neg-mul-1 89.7%

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

      5. unpow-1 89.7%

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

      6. metadata-eval 89.7%

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

      7. pow-sqr 89.8%

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

      8. rem-sqrt-square 89.8%

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

      9. rem-square-sqrt 89.1%

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

      10. fabs-sqr 89.1%

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

      11. rem-square-sqrt 89.8%

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

   8. Simplified 89.8%

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

   if -5.4000000000000003e274 < d < -2.3000000000000001e-69

   1. Initial program 81.6%

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

   3. Simplified 80.3%

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

      1. fma-undefine 80.3%

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

      2. associate-*r* 80.3%

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

      3. associate-*r/ 80.3%

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

   6. Applied egg-rr 80.3%

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

   7. Taylor expanded in M around 0 64.7%

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

      1. associate-*r/ 64.7%

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

      2. associate-*r* 65.3%

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

      3. associate-*r* 65.3%

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

      4. associate-*l/ 66.7%

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

      5. associate-*r/ 66.7%

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

      6. *-commutative 66.7%

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

      7. associate-*r/ 66.7%

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

      8. *-commutative 66.7%

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

      9. times-frac 71.3%

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

      10. *-commutative 71.3%

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

      11. associate-/l* 71.3%

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

      12. unpow2 71.3%

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

      13. unpow2 71.3%

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

      14. unpow2 71.3%

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

      15. times-frac 75.7%

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

      16. swap-sqr 81.9%

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

      17. unpow2 81.9%

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

   9. Simplified 81.9%

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

       1. add-sqr-sqrt 81.9%

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

       2. pow2 81.9%

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

       3. *-commutative 81.9%

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

       4. associate-*r/ 81.9%

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

       5. sqrt-prod 81.9%

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

       6. sqrt-pow1 84.7%

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

       7. metadata-eval 84.7%

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

       8. pow1 84.7%

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

   11. Applied egg-rr 84.7%

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

   if -2.3000000000000001e-69 < d < -1.000000000000002e-309

   1. Initial program 44.5%

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

   3. Simplified 44.4%

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

   5. Taylor expanded in M around inf 26.9%

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

      1. associate-*r* 29.1%

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

      2. times-frac 27.3%

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

      3. *-commutative 27.3%

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

      4. associate-/l* 25.1%

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

      5. unpow2 25.1%

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

      6. unpow2 25.1%

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

      7. unpow2 25.1%

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

      8. times-frac 31.9%

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

      9. swap-sqr 40.3%

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

      10. unpow2 40.3%

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

      11. associate-*r/ 40.3%

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

      12. *-commutative 40.3%

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

      13. associate-/l* 38.3%

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

   7. Simplified 38.3%

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

      1. frac-2neg 50.6%

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

      2. sqrt-div 77.0%

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

   9. Applied egg-rr 62.4%

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

   if -1.000000000000002e-309 < d

   1. Initial program 66.4%

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

   3. Simplified 65.6%

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

      1. sqrt-div 75.3%

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

   6. Applied egg-rr 75.3%

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

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -5.4 \cdot 10^{+274}:\\ \;\;\;\;d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{elif}\;d \leq -2.3 \cdot 10^{-69}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot {\left(\left(D \cdot \frac{M}{d}\right) \cdot \sqrt{\frac{-0.125}{\ell}}\right)}^{2}\right)\right)\\ \mathbf{elif}\;d \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(-0.125 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\left(1 + \frac{h}{\ell} \cdot \left({\left(D \cdot \frac{\frac{M}{2}}{d}\right)}^{2} \cdot -0.5\right)\right) \cdot \frac{\sqrt{d}}{\sqrt{h}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 69.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ t_1 := \sqrt{\frac{d}{\ell}}\\ t_2 := D \cdot \frac{M\_m}{d}\\ \mathbf{if}\;d \leq -5.7 \cdot 10^{+262}:\\ \;\;\;\;d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{elif}\;d \leq -9.2 \cdot 10^{-70}:\\ \;\;\;\;t\_0 \cdot \left(t\_1 \cdot \left(1 + h \cdot {\left(t\_2 \cdot \sqrt{\frac{-0.125}{\ell}}\right)}^{2}\right)\right)\\ \mathbf{elif}\;d \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(t\_1 \cdot \left(-0.125 \cdot \left(\frac{h}{\ell} \cdot {t\_2}^{2}\right)\right)\right)\\ \mathbf{elif}\;d \leq 2.35 \cdot 10^{-239}:\\ \;\;\;\;-0.125 \cdot \left({\left(\sqrt[3]{D \cdot M\_m}\right)}^{6} \cdot \frac{\sqrt{\frac{h}{{\ell}^{3}}}}{d}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left(h \cdot \frac{{\left(D \cdot \left(0.5 \cdot \frac{M\_m}{d}\right)\right)}^{2}}{\ell}\right)\right) \cdot \left(t\_0 \cdot t\_1\right)\\ \end{array} \end{array} \]

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 (sqrt (/ d h))) (t_1 (sqrt (/ d l))) (t_2 (* D (/ M_m d))))
   (if (<= d -5.7e+262)
     (* d (- (pow (* h l) -0.5)))
     (if (<= d -9.2e-70)
       (* t_0 (* t_1 (+ 1.0 (* h (pow (* t_2 (sqrt (/ -0.125 l))) 2.0)))))
       (if (<= d -1e-309)
         (*
          (/ (sqrt (- d)) (sqrt (- h)))
          (* t_1 (* -0.125 (* (/ h l) (pow t_2 2.0)))))
         (if (<= d 2.35e-239)
           (*
            -0.125
            (* (pow (cbrt (* D M_m)) 6.0) (/ (sqrt (/ h (pow l 3.0))) d)))
           (*
            (- 1.0 (* 0.5 (* h (/ (pow (* D (* 0.5 (/ M_m d))) 2.0) l))))
            (* t_0 t_1))))))))

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 / h));
	double t_1 = sqrt((d / l));
	double t_2 = D * (M_m / d);
	double tmp;
	if (d <= -5.7e+262) {
		tmp = d * -pow((h * l), -0.5);
	} else if (d <= -9.2e-70) {
		tmp = t_0 * (t_1 * (1.0 + (h * pow((t_2 * sqrt((-0.125 / l))), 2.0))));
	} else if (d <= -1e-309) {
		tmp = (sqrt(-d) / sqrt(-h)) * (t_1 * (-0.125 * ((h / l) * pow(t_2, 2.0))));
	} else if (d <= 2.35e-239) {
		tmp = -0.125 * (pow(cbrt((D * M_m)), 6.0) * (sqrt((h / pow(l, 3.0))) / d));
	} else {
		tmp = (1.0 - (0.5 * (h * (pow((D * (0.5 * (M_m / d))), 2.0) / l)))) * (t_0 * t_1);
	}
	return tmp;
}

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 / h));
	double t_1 = Math.sqrt((d / l));
	double t_2 = D * (M_m / d);
	double tmp;
	if (d <= -5.7e+262) {
		tmp = d * -Math.pow((h * l), -0.5);
	} else if (d <= -9.2e-70) {
		tmp = t_0 * (t_1 * (1.0 + (h * Math.pow((t_2 * Math.sqrt((-0.125 / l))), 2.0))));
	} else if (d <= -1e-309) {
		tmp = (Math.sqrt(-d) / Math.sqrt(-h)) * (t_1 * (-0.125 * ((h / l) * Math.pow(t_2, 2.0))));
	} else if (d <= 2.35e-239) {
		tmp = -0.125 * (Math.pow(Math.cbrt((D * M_m)), 6.0) * (Math.sqrt((h / Math.pow(l, 3.0))) / d));
	} else {
		tmp = (1.0 - (0.5 * (h * (Math.pow((D * (0.5 * (M_m / d))), 2.0) / l)))) * (t_0 * t_1);
	}
	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 = sqrt(Float64(d / h))
	t_1 = sqrt(Float64(d / l))
	t_2 = Float64(D * Float64(M_m / d))
	tmp = 0.0
	if (d <= -5.7e+262)
		tmp = Float64(d * Float64(-(Float64(h * l) ^ -0.5)));
	elseif (d <= -9.2e-70)
		tmp = Float64(t_0 * Float64(t_1 * Float64(1.0 + Float64(h * (Float64(t_2 * sqrt(Float64(-0.125 / l))) ^ 2.0)))));
	elseif (d <= -1e-309)
		tmp = Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-h))) * Float64(t_1 * Float64(-0.125 * Float64(Float64(h / l) * (t_2 ^ 2.0)))));
	elseif (d <= 2.35e-239)
		tmp = Float64(-0.125 * Float64((cbrt(Float64(D * M_m)) ^ 6.0) * Float64(sqrt(Float64(h / (l ^ 3.0))) / d)));
	else
		tmp = Float64(Float64(1.0 - Float64(0.5 * Float64(h * Float64((Float64(D * Float64(0.5 * Float64(M_m / d))) ^ 2.0) / l)))) * Float64(t_0 * t_1));
	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[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(D * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -5.7e+262], N[(d * (-N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision])), $MachinePrecision], If[LessEqual[d, -9.2e-70], N[(t$95$0 * N[(t$95$1 * N[(1.0 + N[(h * N[Power[N[(t$95$2 * N[Sqrt[N[(-0.125 / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -1e-309], N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * N[(-0.125 * N[(N[(h / l), $MachinePrecision] * N[Power[t$95$2, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 2.35e-239], N[(-0.125 * N[(N[Power[N[Power[N[(D * M$95$m), $MachinePrecision], 1/3], $MachinePrecision], 6.0], $MachinePrecision] * N[(N[Sqrt[N[(h / N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[(0.5 * N[(h * N[(N[Power[N[(D * N[(0.5 * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * t$95$1), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if d < -5.7000000000000002e262

   1. Initial program 40.6%

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

   3. Simplified 41.3%

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

   5. Taylor expanded in d around inf 10.2%

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

   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. associate-*l* 0.0%

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

      2. unpow2 0.0%

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

      3. rem-square-sqrt 89.7%

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

      4. neg-mul-1 89.7%

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

      5. unpow-1 89.7%

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

      6. metadata-eval 89.7%

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

      7. pow-sqr 89.8%

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

      8. rem-sqrt-square 89.8%

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

      9. rem-square-sqrt 89.1%

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

      10. fabs-sqr 89.1%

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

      11. rem-square-sqrt 89.8%

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

   8. Simplified 89.8%

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

   if -5.7000000000000002e262 < d < -9.20000000000000002e-70

   1. Initial program 81.6%

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

   3. Simplified 80.3%

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

      1. fma-undefine 80.3%

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

      2. associate-*r* 80.3%

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

      3. associate-*r/ 80.3%

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

   6. Applied egg-rr 80.3%

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

   7. Taylor expanded in M around 0 64.7%

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

      1. associate-*r/ 64.7%

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

      2. associate-*r* 65.3%

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

      3. associate-*r* 65.3%

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

      4. associate-*l/ 66.7%

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

      5. associate-*r/ 66.7%

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

      6. *-commutative 66.7%

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

      7. associate-*r/ 66.7%

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

      8. *-commutative 66.7%

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

      9. times-frac 71.3%

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

      10. *-commutative 71.3%

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

      11. associate-/l* 71.3%

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

      12. unpow2 71.3%

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

      13. unpow2 71.3%

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

      14. unpow2 71.3%

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

      15. times-frac 75.7%

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

      16. swap-sqr 81.9%

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

      17. unpow2 81.9%

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

   9. Simplified 81.9%

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

       1. add-sqr-sqrt 81.9%

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

       2. pow2 81.9%

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

       3. *-commutative 81.9%

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

       4. associate-*r/ 81.9%

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

       5. sqrt-prod 81.9%

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

       6. sqrt-pow1 84.7%

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

       7. metadata-eval 84.7%

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

       8. pow1 84.7%

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

   11. Applied egg-rr 84.7%

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

   if -9.20000000000000002e-70 < d < -1.000000000000002e-309

   1. Initial program 44.5%

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

   3. Simplified 44.4%

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

   5. Taylor expanded in M around inf 26.9%

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

      1. associate-*r* 29.1%

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

      2. times-frac 27.3%

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

      3. *-commutative 27.3%

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

      4. associate-/l* 25.1%

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

      5. unpow2 25.1%

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

      6. unpow2 25.1%

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

      7. unpow2 25.1%

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

      8. times-frac 31.9%

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

      9. swap-sqr 40.3%

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

      10. unpow2 40.3%

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

      11. associate-*r/ 40.3%

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

      12. *-commutative 40.3%

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

      13. associate-/l* 38.3%

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

   7. Simplified 38.3%

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

      1. frac-2neg 50.6%

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

      2. sqrt-div 77.0%

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

   9. Applied egg-rr 62.4%

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

   if -1.000000000000002e-309 < d < 2.3500000000000001e-239

   1. Initial program 7.0%

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

   3. Simplified 7.0%

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

   5. Taylor expanded in M around inf 5.9%

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

      1. associate-*r* 5.9%

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

      2. times-frac 6.2%

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

      3. *-commutative 6.2%

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

      4. associate-/l* 6.2%

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

      5. unpow2 6.2%

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

      6. unpow2 6.2%

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

      7. unpow2 6.2%

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

      8. times-frac 7.0%

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

      9. swap-sqr 7.0%

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

      10. unpow2 7.0%

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

      11. associate-*r/ 7.0%

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

      12. *-commutative 7.0%

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

      13. associate-/l* 7.0%

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

   7. Simplified 7.0%

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

      1. frac-2neg 7.0%

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

      2. sqrt-div 0.0%

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

   9. Applied egg-rr 0.0%

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

   10. Taylor expanded in d around 0 54.5%

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

       1. associate-*l/ 54.4%

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

       2. associate-/l* 54.4%

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

       3. unpow2 54.4%

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

       4. unpow2 54.4%

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

       5. swap-sqr 60.6%

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

       6. rem-cube-cbrt 60.6%

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

       7. rem-cube-cbrt 60.5%

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

       8. pow-sqr 60.5%

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

       9. metadata-eval 60.5%

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

   12. Simplified 60.5%

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

   if 2.3500000000000001e-239 < d

   1. Initial program 75.1%

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

   3. Simplified 74.2%

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

      1. associate-*r/ 76.9%

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

      2. frac-times 77.3%

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

      3. associate-/l* 76.9%

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

      4. *-commutative 76.9%

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

   6. Applied egg-rr 76.9%

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

      1. *-commutative 76.9%

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

      2. associate-/l* 77.8%

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

      3. associate-*r/ 78.1%

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

      4. *-rgt-identity 78.1%

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

      5. times-frac 78.1%

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

      6. metadata-eval 78.1%

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

      7. *-commutative 78.1%

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

      8. associate-/l* 77.9%

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

      9. associate-*l* 77.9%

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

   8. Simplified 77.9%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(h \cdot \frac{{\left(D \cdot \left(\frac{M}{d} \cdot 0.5\right)\right)}^{2}}{\ell}\right)}\right) \]
3. Recombined 5 regimes into one program.

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -5.7 \cdot 10^{+262}:\\ \;\;\;\;d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{elif}\;d \leq -9.2 \cdot 10^{-70}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot {\left(\left(D \cdot \frac{M}{d}\right) \cdot \sqrt{\frac{-0.125}{\ell}}\right)}^{2}\right)\right)\\ \mathbf{elif}\;d \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(-0.125 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}\right)\right)\right)\\ \mathbf{elif}\;d \leq 2.35 \cdot 10^{-239}:\\ \;\;\;\;-0.125 \cdot \left({\left(\sqrt[3]{D \cdot M}\right)}^{6} \cdot \frac{\sqrt{\frac{h}{{\ell}^{3}}}}{d}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left(h \cdot \frac{{\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2}}{\ell}\right)\right) \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 77.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;h \leq -1 \cdot 10^{-311}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot t\_0\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{{\left(\frac{M\_m}{2} \cdot \frac{D}{d}\right)}^{2}}{\ell}\right)\right)\\ \mathbf{elif}\;h \leq 9.2 \cdot 10^{+51}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(\left(1 + \frac{h}{\ell} \cdot \left({\left(D \cdot \frac{\frac{M\_m}{2}}{d}\right)}^{2} \cdot -0.5\right)\right) \cdot \sqrt{\frac{d}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(t\_0 \cdot \left(1 + {\left(\frac{D}{d} \cdot \left(M\_m \cdot 0.5\right)\right)}^{2} \cdot \frac{h \cdot -0.5}{\ell}\right)\right)\\ \end{array} \end{array} \]

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 (sqrt (/ d l))))
   (if (<= h -1e-311)
     (*
      (* (/ (sqrt (- d)) (sqrt (- h))) t_0)
      (- 1.0 (* 0.5 (* h (/ (pow (* (/ M_m 2.0) (/ D d)) 2.0) l)))))
     (if (<= h 9.2e+51)
       (*
        (/ (sqrt d) (sqrt l))
        (*
         (+ 1.0 (* (/ h l) (* (pow (* D (/ (/ M_m 2.0) d)) 2.0) -0.5)))
         (sqrt (/ d h))))
       (*
        (/ (sqrt d) (sqrt h))
        (*
         t_0
         (+ 1.0 (* (pow (* (/ D d) (* M_m 0.5)) 2.0) (/ (* h -0.5) 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 = sqrt((d / l));
	double tmp;
	if (h <= -1e-311) {
		tmp = ((sqrt(-d) / sqrt(-h)) * t_0) * (1.0 - (0.5 * (h * (pow(((M_m / 2.0) * (D / d)), 2.0) / l))));
	} else if (h <= 9.2e+51) {
		tmp = (sqrt(d) / sqrt(l)) * ((1.0 + ((h / l) * (pow((D * ((M_m / 2.0) / d)), 2.0) * -0.5))) * sqrt((d / h)));
	} else {
		tmp = (sqrt(d) / sqrt(h)) * (t_0 * (1.0 + (pow(((D / d) * (M_m * 0.5)), 2.0) * ((h * -0.5) / 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) :: t_0
    real(8) :: tmp
    t_0 = sqrt((d / l))
    if (h <= (-1d-311)) then
        tmp = ((sqrt(-d) / sqrt(-h)) * t_0) * (1.0d0 - (0.5d0 * (h * ((((m_m / 2.0d0) * (d_1 / d)) ** 2.0d0) / l))))
    else if (h <= 9.2d+51) then
        tmp = (sqrt(d) / sqrt(l)) * ((1.0d0 + ((h / l) * (((d_1 * ((m_m / 2.0d0) / d)) ** 2.0d0) * (-0.5d0)))) * sqrt((d / h)))
    else
        tmp = (sqrt(d) / sqrt(h)) * (t_0 * (1.0d0 + ((((d_1 / d) * (m_m * 0.5d0)) ** 2.0d0) * ((h * (-0.5d0)) / 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 t_0 = Math.sqrt((d / l));
	double tmp;
	if (h <= -1e-311) {
		tmp = ((Math.sqrt(-d) / Math.sqrt(-h)) * t_0) * (1.0 - (0.5 * (h * (Math.pow(((M_m / 2.0) * (D / d)), 2.0) / l))));
	} else if (h <= 9.2e+51) {
		tmp = (Math.sqrt(d) / Math.sqrt(l)) * ((1.0 + ((h / l) * (Math.pow((D * ((M_m / 2.0) / d)), 2.0) * -0.5))) * Math.sqrt((d / h)));
	} else {
		tmp = (Math.sqrt(d) / Math.sqrt(h)) * (t_0 * (1.0 + (Math.pow(((D / d) * (M_m * 0.5)), 2.0) * ((h * -0.5) / 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):
	t_0 = math.sqrt((d / l))
	tmp = 0
	if h <= -1e-311:
		tmp = ((math.sqrt(-d) / math.sqrt(-h)) * t_0) * (1.0 - (0.5 * (h * (math.pow(((M_m / 2.0) * (D / d)), 2.0) / l))))
	elif h <= 9.2e+51:
		tmp = (math.sqrt(d) / math.sqrt(l)) * ((1.0 + ((h / l) * (math.pow((D * ((M_m / 2.0) / d)), 2.0) * -0.5))) * math.sqrt((d / h)))
	else:
		tmp = (math.sqrt(d) / math.sqrt(h)) * (t_0 * (1.0 + (math.pow(((D / d) * (M_m * 0.5)), 2.0) * ((h * -0.5) / 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 = sqrt(Float64(d / l))
	tmp = 0.0
	if (h <= -1e-311)
		tmp = Float64(Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-h))) * t_0) * Float64(1.0 - Float64(0.5 * Float64(h * Float64((Float64(Float64(M_m / 2.0) * Float64(D / d)) ^ 2.0) / l)))));
	elseif (h <= 9.2e+51)
		tmp = Float64(Float64(sqrt(d) / sqrt(l)) * Float64(Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(D * Float64(Float64(M_m / 2.0) / d)) ^ 2.0) * -0.5))) * sqrt(Float64(d / h))));
	else
		tmp = Float64(Float64(sqrt(d) / sqrt(h)) * Float64(t_0 * Float64(1.0 + Float64((Float64(Float64(D / d) * Float64(M_m * 0.5)) ^ 2.0) * Float64(Float64(h * -0.5) / 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)
	t_0 = sqrt((d / l));
	tmp = 0.0;
	if (h <= -1e-311)
		tmp = ((sqrt(-d) / sqrt(-h)) * t_0) * (1.0 - (0.5 * (h * ((((M_m / 2.0) * (D / d)) ^ 2.0) / l))));
	elseif (h <= 9.2e+51)
		tmp = (sqrt(d) / sqrt(l)) * ((1.0 + ((h / l) * (((D * ((M_m / 2.0) / d)) ^ 2.0) * -0.5))) * sqrt((d / h)));
	else
		tmp = (sqrt(d) / sqrt(h)) * (t_0 * (1.0 + ((((D / d) * (M_m * 0.5)) ^ 2.0) * ((h * -0.5) / 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_] := Block[{t$95$0 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[h, -1e-311], N[(N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(h * N[(N[Power[N[(N[(M$95$m / 2.0), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, 9.2e+51], N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * N[(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] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * N[(1.0 + N[(N[Power[N[(N[(D / d), $MachinePrecision] * N[(M$95$m * 0.5), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(h * -0.5), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if h < -9.99999999999948e-312

   1. Initial program 64.2%

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

   3. Simplified 64.2%

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

      1. associate-*r/ 65.4%

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

      2. frac-times 65.3%

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

      3. associate-/l* 65.4%

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

      4. *-commutative 65.4%

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

   6. Applied egg-rr 65.4%

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

      1. *-commutative 65.4%

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

      2. associate-/l* 65.4%

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

      3. associate-*r/ 65.4%

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

      4. *-rgt-identity 65.4%

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

      5. times-frac 65.4%

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

      6. metadata-eval 65.4%

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

      7. *-commutative 65.4%

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

      8. associate-/l* 64.5%

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

      9. associate-*l* 64.5%

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

   8. Simplified 64.5%

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

      1. *-un-lft-identity 64.5%

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

      2. associate-*l/ 64.5%

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

      3. metadata-eval 64.5%

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

      4. div-inv 64.5%

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

      5. associate-/l/ 64.5%

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

   10. Applied egg-rr 64.5%

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

       1. *-lft-identity 64.5%

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

       2. associate-*r/ 65.4%

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

       3. times-frac 65.4%

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

   12. Simplified 65.4%

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

       1. frac-2neg 68.6%

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

       2. sqrt-div 82.2%

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

   14. Applied egg-rr 77.2%

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

   if -9.99999999999948e-312 < h < 9.2000000000000002e51

   1. Initial program 71.8%

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

   3. Simplified 71.8%

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

      1. sqrt-div 85.0%

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

   6. Applied egg-rr 85.0%

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

   if 9.2000000000000002e51 < h

   1. Initial program 59.8%

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

   3. Simplified 59.6%

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

      1. fma-undefine 59.6%

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

      2. associate-*r* 59.6%

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

      3. associate-*r/ 59.6%

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

   6. Applied egg-rr 59.6%

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

      1. sqrt-div 76.5%

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

   8. Applied egg-rr 76.5%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -1 \cdot 10^{-311}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}}{\ell}\right)\right)\\ \mathbf{elif}\;h \leq 9.2 \cdot 10^{+51}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(\left(1 + \frac{h}{\ell} \cdot \left({\left(D \cdot \frac{\frac{M}{2}}{d}\right)}^{2} \cdot -0.5\right)\right) \cdot \sqrt{\frac{d}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + {\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2} \cdot \frac{h \cdot -0.5}{\ell}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 76.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := 1 + \frac{h}{\ell} \cdot \left({\left(D \cdot \frac{\frac{M\_m}{2}}{d}\right)}^{2} \cdot -0.5\right)\\ t_1 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;h \leq -1 \cdot 10^{-311}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot t\_0\right) \cdot t\_1\\ \mathbf{elif}\;h \leq 5.5 \cdot 10^{+50}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(t\_0 \cdot \sqrt{\frac{d}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(t\_1 \cdot \left(1 + {\left(\frac{D}{d} \cdot \left(M\_m \cdot 0.5\right)\right)}^{2} \cdot \frac{h \cdot -0.5}{\ell}\right)\right)\\ \end{array} \end{array} \]

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 (+ 1.0 (* (/ h l) (* (pow (* D (/ (/ M_m 2.0) d)) 2.0) -0.5))))
        (t_1 (sqrt (/ d l))))
   (if (<= h -1e-311)
     (* (* (/ (sqrt (- d)) (sqrt (- h))) t_0) t_1)
     (if (<= h 5.5e+50)
       (* (/ (sqrt d) (sqrt l)) (* t_0 (sqrt (/ d h))))
       (*
        (/ (sqrt d) (sqrt h))
        (*
         t_1
         (+ 1.0 (* (pow (* (/ D d) (* M_m 0.5)) 2.0) (/ (* h -0.5) 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 = 1.0 + ((h / l) * (pow((D * ((M_m / 2.0) / d)), 2.0) * -0.5));
	double t_1 = sqrt((d / l));
	double tmp;
	if (h <= -1e-311) {
		tmp = ((sqrt(-d) / sqrt(-h)) * t_0) * t_1;
	} else if (h <= 5.5e+50) {
		tmp = (sqrt(d) / sqrt(l)) * (t_0 * sqrt((d / h)));
	} else {
		tmp = (sqrt(d) / sqrt(h)) * (t_1 * (1.0 + (pow(((D / d) * (M_m * 0.5)), 2.0) * ((h * -0.5) / 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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 + ((h / l) * (((d_1 * ((m_m / 2.0d0) / d)) ** 2.0d0) * (-0.5d0)))
    t_1 = sqrt((d / l))
    if (h <= (-1d-311)) then
        tmp = ((sqrt(-d) / sqrt(-h)) * t_0) * t_1
    else if (h <= 5.5d+50) then
        tmp = (sqrt(d) / sqrt(l)) * (t_0 * sqrt((d / h)))
    else
        tmp = (sqrt(d) / sqrt(h)) * (t_1 * (1.0d0 + ((((d_1 / d) * (m_m * 0.5d0)) ** 2.0d0) * ((h * (-0.5d0)) / 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 t_0 = 1.0 + ((h / l) * (Math.pow((D * ((M_m / 2.0) / d)), 2.0) * -0.5));
	double t_1 = Math.sqrt((d / l));
	double tmp;
	if (h <= -1e-311) {
		tmp = ((Math.sqrt(-d) / Math.sqrt(-h)) * t_0) * t_1;
	} else if (h <= 5.5e+50) {
		tmp = (Math.sqrt(d) / Math.sqrt(l)) * (t_0 * Math.sqrt((d / h)));
	} else {
		tmp = (Math.sqrt(d) / Math.sqrt(h)) * (t_1 * (1.0 + (Math.pow(((D / d) * (M_m * 0.5)), 2.0) * ((h * -0.5) / 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):
	t_0 = 1.0 + ((h / l) * (math.pow((D * ((M_m / 2.0) / d)), 2.0) * -0.5))
	t_1 = math.sqrt((d / l))
	tmp = 0
	if h <= -1e-311:
		tmp = ((math.sqrt(-d) / math.sqrt(-h)) * t_0) * t_1
	elif h <= 5.5e+50:
		tmp = (math.sqrt(d) / math.sqrt(l)) * (t_0 * math.sqrt((d / h)))
	else:
		tmp = (math.sqrt(d) / math.sqrt(h)) * (t_1 * (1.0 + (math.pow(((D / d) * (M_m * 0.5)), 2.0) * ((h * -0.5) / 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(1.0 + Float64(Float64(h / l) * Float64((Float64(D * Float64(Float64(M_m / 2.0) / d)) ^ 2.0) * -0.5)))
	t_1 = sqrt(Float64(d / l))
	tmp = 0.0
	if (h <= -1e-311)
		tmp = Float64(Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-h))) * t_0) * t_1);
	elseif (h <= 5.5e+50)
		tmp = Float64(Float64(sqrt(d) / sqrt(l)) * Float64(t_0 * sqrt(Float64(d / h))));
	else
		tmp = Float64(Float64(sqrt(d) / sqrt(h)) * Float64(t_1 * Float64(1.0 + Float64((Float64(Float64(D / d) * Float64(M_m * 0.5)) ^ 2.0) * Float64(Float64(h * -0.5) / 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)
	t_0 = 1.0 + ((h / l) * (((D * ((M_m / 2.0) / d)) ^ 2.0) * -0.5));
	t_1 = sqrt((d / l));
	tmp = 0.0;
	if (h <= -1e-311)
		tmp = ((sqrt(-d) / sqrt(-h)) * t_0) * t_1;
	elseif (h <= 5.5e+50)
		tmp = (sqrt(d) / sqrt(l)) * (t_0 * sqrt((d / h)));
	else
		tmp = (sqrt(d) / sqrt(h)) * (t_1 * (1.0 + ((((D / d) * (M_m * 0.5)) ^ 2.0) * ((h * -0.5) / 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_] := Block[{t$95$0 = 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]}, Block[{t$95$1 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[h, -1e-311], N[(N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[h, 5.5e+50], N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * N[(1.0 + N[(N[Power[N[(N[(D / d), $MachinePrecision] * N[(M$95$m * 0.5), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(h * -0.5), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if h < -9.99999999999948e-312

   1. Initial program 64.2%

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

   3. Simplified 62.7%

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

      1. frac-2neg 68.6%

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

      2. sqrt-div 82.2%

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

   6. Applied egg-rr 75.4%

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

   if -9.99999999999948e-312 < h < 5.4999999999999998e50

   1. Initial program 71.8%

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

   3. Simplified 71.8%

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

      1. sqrt-div 85.0%

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

   6. Applied egg-rr 85.0%

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

   if 5.4999999999999998e50 < h

   1. Initial program 59.8%

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

   3. Simplified 59.6%

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

      1. fma-undefine 59.6%

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

      2. associate-*r* 59.6%

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

      3. associate-*r/ 59.6%

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

   6. Applied egg-rr 59.6%

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

      1. sqrt-div 76.5%

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

   8. Applied egg-rr 76.5%

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

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -1 \cdot 10^{-311}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(D \cdot \frac{\frac{M}{2}}{d}\right)}^{2} \cdot -0.5\right)\right)\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;h \leq 5.5 \cdot 10^{+50}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(\left(1 + \frac{h}{\ell} \cdot \left({\left(D \cdot \frac{\frac{M}{2}}{d}\right)}^{2} \cdot -0.5\right)\right) \cdot \sqrt{\frac{d}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + {\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2} \cdot \frac{h \cdot -0.5}{\ell}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 77.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;h \leq -1 \cdot 10^{-311}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(t\_0 \cdot \left(1 + h \cdot \left(\frac{-0.125}{\ell} \cdot {\left(\frac{D \cdot M\_m}{d}\right)}^{2}\right)\right)\right)\\ \mathbf{elif}\;h \leq 9 \cdot 10^{+51}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(\left(1 + \frac{h}{\ell} \cdot \left({\left(D \cdot \frac{\frac{M\_m}{2}}{d}\right)}^{2} \cdot -0.5\right)\right) \cdot \sqrt{\frac{d}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(t\_0 \cdot \left(1 + {\left(\frac{D}{d} \cdot \left(M\_m \cdot 0.5\right)\right)}^{2} \cdot \frac{h \cdot -0.5}{\ell}\right)\right)\\ \end{array} \end{array} \]

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 (sqrt (/ d l))))
   (if (<= h -1e-311)
     (*
      (/ (sqrt (- d)) (sqrt (- h)))
      (* t_0 (+ 1.0 (* h (* (/ -0.125 l) (pow (/ (* D M_m) d) 2.0))))))
     (if (<= h 9e+51)
       (*
        (/ (sqrt d) (sqrt l))
        (*
         (+ 1.0 (* (/ h l) (* (pow (* D (/ (/ M_m 2.0) d)) 2.0) -0.5)))
         (sqrt (/ d h))))
       (*
        (/ (sqrt d) (sqrt h))
        (*
         t_0
         (+ 1.0 (* (pow (* (/ D d) (* M_m 0.5)) 2.0) (/ (* h -0.5) 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 = sqrt((d / l));
	double tmp;
	if (h <= -1e-311) {
		tmp = (sqrt(-d) / sqrt(-h)) * (t_0 * (1.0 + (h * ((-0.125 / l) * pow(((D * M_m) / d), 2.0)))));
	} else if (h <= 9e+51) {
		tmp = (sqrt(d) / sqrt(l)) * ((1.0 + ((h / l) * (pow((D * ((M_m / 2.0) / d)), 2.0) * -0.5))) * sqrt((d / h)));
	} else {
		tmp = (sqrt(d) / sqrt(h)) * (t_0 * (1.0 + (pow(((D / d) * (M_m * 0.5)), 2.0) * ((h * -0.5) / 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) :: t_0
    real(8) :: tmp
    t_0 = sqrt((d / l))
    if (h <= (-1d-311)) then
        tmp = (sqrt(-d) / sqrt(-h)) * (t_0 * (1.0d0 + (h * (((-0.125d0) / l) * (((d_1 * m_m) / d) ** 2.0d0)))))
    else if (h <= 9d+51) then
        tmp = (sqrt(d) / sqrt(l)) * ((1.0d0 + ((h / l) * (((d_1 * ((m_m / 2.0d0) / d)) ** 2.0d0) * (-0.5d0)))) * sqrt((d / h)))
    else
        tmp = (sqrt(d) / sqrt(h)) * (t_0 * (1.0d0 + ((((d_1 / d) * (m_m * 0.5d0)) ** 2.0d0) * ((h * (-0.5d0)) / 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 t_0 = Math.sqrt((d / l));
	double tmp;
	if (h <= -1e-311) {
		tmp = (Math.sqrt(-d) / Math.sqrt(-h)) * (t_0 * (1.0 + (h * ((-0.125 / l) * Math.pow(((D * M_m) / d), 2.0)))));
	} else if (h <= 9e+51) {
		tmp = (Math.sqrt(d) / Math.sqrt(l)) * ((1.0 + ((h / l) * (Math.pow((D * ((M_m / 2.0) / d)), 2.0) * -0.5))) * Math.sqrt((d / h)));
	} else {
		tmp = (Math.sqrt(d) / Math.sqrt(h)) * (t_0 * (1.0 + (Math.pow(((D / d) * (M_m * 0.5)), 2.0) * ((h * -0.5) / 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):
	t_0 = math.sqrt((d / l))
	tmp = 0
	if h <= -1e-311:
		tmp = (math.sqrt(-d) / math.sqrt(-h)) * (t_0 * (1.0 + (h * ((-0.125 / l) * math.pow(((D * M_m) / d), 2.0)))))
	elif h <= 9e+51:
		tmp = (math.sqrt(d) / math.sqrt(l)) * ((1.0 + ((h / l) * (math.pow((D * ((M_m / 2.0) / d)), 2.0) * -0.5))) * math.sqrt((d / h)))
	else:
		tmp = (math.sqrt(d) / math.sqrt(h)) * (t_0 * (1.0 + (math.pow(((D / d) * (M_m * 0.5)), 2.0) * ((h * -0.5) / 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 = sqrt(Float64(d / l))
	tmp = 0.0
	if (h <= -1e-311)
		tmp = Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-h))) * Float64(t_0 * Float64(1.0 + Float64(h * Float64(Float64(-0.125 / l) * (Float64(Float64(D * M_m) / d) ^ 2.0))))));
	elseif (h <= 9e+51)
		tmp = Float64(Float64(sqrt(d) / sqrt(l)) * Float64(Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(D * Float64(Float64(M_m / 2.0) / d)) ^ 2.0) * -0.5))) * sqrt(Float64(d / h))));
	else
		tmp = Float64(Float64(sqrt(d) / sqrt(h)) * Float64(t_0 * Float64(1.0 + Float64((Float64(Float64(D / d) * Float64(M_m * 0.5)) ^ 2.0) * Float64(Float64(h * -0.5) / 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)
	t_0 = sqrt((d / l));
	tmp = 0.0;
	if (h <= -1e-311)
		tmp = (sqrt(-d) / sqrt(-h)) * (t_0 * (1.0 + (h * ((-0.125 / l) * (((D * M_m) / d) ^ 2.0)))));
	elseif (h <= 9e+51)
		tmp = (sqrt(d) / sqrt(l)) * ((1.0 + ((h / l) * (((D * ((M_m / 2.0) / d)) ^ 2.0) * -0.5))) * sqrt((d / h)));
	else
		tmp = (sqrt(d) / sqrt(h)) * (t_0 * (1.0 + ((((D / d) * (M_m * 0.5)) ^ 2.0) * ((h * -0.5) / 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_] := Block[{t$95$0 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[h, -1e-311], N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * N[(1.0 + N[(h * N[(N[(-0.125 / l), $MachinePrecision] * N[Power[N[(N[(D * M$95$m), $MachinePrecision] / d), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, 9e+51], N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * N[(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] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * N[(1.0 + N[(N[Power[N[(N[(D / d), $MachinePrecision] * N[(M$95$m * 0.5), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(h * -0.5), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if h < -9.99999999999948e-312

   1. Initial program 64.2%

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

   3. Simplified 63.6%

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

      1. fma-undefine 63.6%

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

      2. associate-*r* 63.6%

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

      3. associate-*r/ 63.6%

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

   6. Applied egg-rr 63.6%

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

   7. Taylor expanded in M around 0 48.5%

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

      1. associate-*r/ 48.5%

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

      2. associate-*r* 50.5%

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

      3. associate-*r* 50.5%

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

      4. associate-*l/ 49.8%

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

      5. associate-*r/ 49.8%

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

      6. *-commutative 49.8%

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

      7. associate-*r/ 49.8%

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

      8. *-commutative 49.8%

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

      9. times-frac 53.0%

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

      10. *-commutative 53.0%

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

      11. associate-/l* 52.2%

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

      12. unpow2 52.2%

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

      13. unpow2 52.2%

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

      14. unpow2 52.2%

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

      15. times-frac 58.8%

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

      16. swap-sqr 64.7%

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

      17. unpow2 64.7%

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

   9. Simplified 64.7%

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

       1. frac-2neg 68.6%

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

       2. sqrt-div 82.2%

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

   11. Applied egg-rr 75.8%

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

   if -9.99999999999948e-312 < h < 8.9999999999999999e51

   1. Initial program 71.8%

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

   3. Simplified 71.8%

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

      1. sqrt-div 85.0%

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

   6. Applied egg-rr 85.0%

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

   if 8.9999999999999999e51 < h

   1. Initial program 59.8%

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

   3. Simplified 59.6%

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

      1. fma-undefine 59.6%

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

      2. associate-*r* 59.6%

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

      3. associate-*r/ 59.6%

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

   6. Applied egg-rr 59.6%

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

      1. sqrt-div 76.5%

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

   8. Applied egg-rr 76.5%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -1 \cdot 10^{-311}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \left(\frac{-0.125}{\ell} \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)\right)\right)\\ \mathbf{elif}\;h \leq 9 \cdot 10^{+51}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(\left(1 + \frac{h}{\ell} \cdot \left({\left(D \cdot \frac{\frac{M}{2}}{d}\right)}^{2} \cdot -0.5\right)\right) \cdot \sqrt{\frac{d}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + {\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2} \cdot \frac{h \cdot -0.5}{\ell}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 70.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.55 \cdot 10^{+199}:\\ \;\;\;\;d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{elif}\;\ell \leq 1.9 \cdot 10^{+135}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\left(1 + h \cdot \left(\frac{-0.125}{\ell} \cdot {\left(\frac{D \cdot M\_m}{d}\right)}^{2}\right)\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \end{array} \]

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 -1.55e+199)
   (* d (- (pow (* h l) -0.5)))
   (if (<= l 1.9e+135)
     (*
      (sqrt (/ d h))
      (*
       (+ 1.0 (* h (* (/ -0.125 l) (pow (/ (* D M_m) d) 2.0))))
       (/ 1.0 (sqrt (/ l 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 (l <= -1.55e+199) {
		tmp = d * -pow((h * l), -0.5);
	} else if (l <= 1.9e+135) {
		tmp = sqrt((d / h)) * ((1.0 + (h * ((-0.125 / l) * pow(((D * M_m) / d), 2.0)))) * (1.0 / sqrt((l / d))));
	} 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 <= (-1.55d+199)) then
        tmp = d * -((h * l) ** (-0.5d0))
    else if (l <= 1.9d+135) then
        tmp = sqrt((d / h)) * ((1.0d0 + (h * (((-0.125d0) / l) * (((d_1 * m_m) / d) ** 2.0d0)))) * (1.0d0 / sqrt((l / d))))
    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 <= -1.55e+199) {
		tmp = d * -Math.pow((h * l), -0.5);
	} else if (l <= 1.9e+135) {
		tmp = Math.sqrt((d / h)) * ((1.0 + (h * ((-0.125 / l) * Math.pow(((D * M_m) / d), 2.0)))) * (1.0 / Math.sqrt((l / d))));
	} 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 <= -1.55e+199:
		tmp = d * -math.pow((h * l), -0.5)
	elif l <= 1.9e+135:
		tmp = math.sqrt((d / h)) * ((1.0 + (h * ((-0.125 / l) * math.pow(((D * M_m) / d), 2.0)))) * (1.0 / math.sqrt((l / d))))
	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 <= -1.55e+199)
		tmp = Float64(d * Float64(-(Float64(h * l) ^ -0.5)));
	elseif (l <= 1.9e+135)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(Float64(1.0 + Float64(h * Float64(Float64(-0.125 / l) * (Float64(Float64(D * M_m) / d) ^ 2.0)))) * Float64(1.0 / sqrt(Float64(l / d)))));
	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 <= -1.55e+199)
		tmp = d * -((h * l) ^ -0.5);
	elseif (l <= 1.9e+135)
		tmp = sqrt((d / h)) * ((1.0 + (h * ((-0.125 / l) * (((D * M_m) / d) ^ 2.0)))) * (1.0 / sqrt((l / d))));
	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, -1.55e+199], N[(d * (-N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision])), $MachinePrecision], If[LessEqual[l, 1.9e+135], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[(1.0 + N[(h * N[(N[(-0.125 / l), $MachinePrecision] * N[Power[N[(N[(D * M$95$m), $MachinePrecision] / d), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Sqrt[N[(l / d), $MachinePrecision]], $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 3 regimes

2. if l < -1.54999999999999993e199

   1. Initial program 53.3%

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

   3. Simplified 53.7%

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

   5. Taylor expanded in d around inf 17.8%

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

   6. Taylor expanded in h around -inf 0.0%

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

      1. associate-*l* 0.0%

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

      2. unpow2 0.0%

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

      3. rem-square-sqrt 79.3%

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

      4. neg-mul-1 79.3%

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

      5. unpow-1 79.3%

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

      6. metadata-eval 79.3%

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

      7. pow-sqr 79.4%

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

      8. rem-sqrt-square 79.4%

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

      9. rem-square-sqrt 78.9%

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

      10. fabs-sqr 78.9%

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

      11. rem-square-sqrt 79.4%

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

   8. Simplified 79.4%

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

   if -1.54999999999999993e199 < l < 1.9000000000000001e135

   1. Initial program 70.0%

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

   3. Simplified 68.5%

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

      1. fma-undefine 68.5%

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

      2. associate-*r* 68.5%

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

      3. associate-*r/ 68.5%

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

   6. Applied egg-rr 68.5%

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

   7. Taylor expanded in M around 0 51.1%

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

      1. associate-*r/ 51.1%

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

      2. associate-*r* 52.9%

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

      3. associate-*r* 52.9%

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

      4. associate-*l/ 52.5%

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

      5. associate-*r/ 52.5%

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

      6. *-commutative 52.5%

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

      7. associate-*r/ 52.5%

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

      8. *-commutative 52.5%

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

      9. times-frac 54.5%

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

      10. *-commutative 54.5%

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

      11. associate-/l* 54.5%

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

      12. unpow2 54.5%

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

      13. unpow2 54.5%

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

      14. unpow2 54.5%

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

      15. times-frac 64.7%

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

      16. swap-sqr 70.8%

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

      17. unpow2 70.8%

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

   9. Simplified 71.4%

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

       1. clear-num 71.4%

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

       2. sqrt-div 72.0%

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

       3. metadata-eval 72.0%

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

   11. Applied egg-rr 72.0%

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

   if 1.9000000000000001e135 < l

   1. Initial program 47.6%

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

   3. Simplified 49.9%

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

      1. pow1 49.9%

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

      2. associate-*r* 49.9%

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

      3. sqrt-unprod 40.7%

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

      4. associate-*r* 40.7%

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

      5. associate-*r/ 40.7%

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

   6. Applied egg-rr 40.7%

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

   8. Simplified 35.7%

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

   9. Taylor expanded in d around inf 45.3%

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

       1. unpow-1 45.3%

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

       2. metadata-eval 45.3%

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

       3. pow-sqr 45.3%

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

       4. rem-sqrt-square 45.3%

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

       5. rem-square-sqrt 45.1%

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

       6. fabs-sqr 45.1%

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

       7. rem-square-sqrt 45.3%

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

   11. Simplified 45.3%

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

       1. *-commutative 45.3%

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

       2. unpow-prod-down 67.1%

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

   13. Applied egg-rr 67.1%

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

4. Final simplification 71.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.55 \cdot 10^{+199}:\\ \;\;\;\;d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{elif}\;\ell \leq 1.9 \cdot 10^{+135}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\left(1 + h \cdot \left(\frac{-0.125}{\ell} \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 70.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.4 \cdot 10^{+196}:\\ \;\;\;\;d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{elif}\;\ell \leq 3.3 \cdot 10^{+137}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \left(\frac{-0.125}{\ell} \cdot {\left(\frac{D \cdot M\_m}{d}\right)}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \end{array} \]

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 -1.4e+196)
   (* d (- (pow (* h l) -0.5)))
   (if (<= l 3.3e+137)
     (*
      (sqrt (/ d h))
      (*
       (sqrt (/ d l))
       (+ 1.0 (* h (* (/ -0.125 l) (pow (/ (* D M_m) d) 2.0))))))
     (* 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 tmp;
	if (l <= -1.4e+196) {
		tmp = d * -pow((h * l), -0.5);
	} else if (l <= 3.3e+137) {
		tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0 + (h * ((-0.125 / l) * pow(((D * M_m) / d), 2.0)))));
	} 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 <= (-1.4d+196)) then
        tmp = d * -((h * l) ** (-0.5d0))
    else if (l <= 3.3d+137) then
        tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0d0 + (h * (((-0.125d0) / l) * (((d_1 * m_m) / d) ** 2.0d0)))))
    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 <= -1.4e+196) {
		tmp = d * -Math.pow((h * l), -0.5);
	} else if (l <= 3.3e+137) {
		tmp = Math.sqrt((d / h)) * (Math.sqrt((d / l)) * (1.0 + (h * ((-0.125 / l) * Math.pow(((D * M_m) / d), 2.0)))));
	} 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 <= -1.4e+196:
		tmp = d * -math.pow((h * l), -0.5)
	elif l <= 3.3e+137:
		tmp = math.sqrt((d / h)) * (math.sqrt((d / l)) * (1.0 + (h * ((-0.125 / l) * math.pow(((D * M_m) / d), 2.0)))))
	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 <= -1.4e+196)
		tmp = Float64(d * Float64(-(Float64(h * l) ^ -0.5)));
	elseif (l <= 3.3e+137)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(sqrt(Float64(d / l)) * Float64(1.0 + Float64(h * Float64(Float64(-0.125 / l) * (Float64(Float64(D * M_m) / d) ^ 2.0))))));
	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 <= -1.4e+196)
		tmp = d * -((h * l) ^ -0.5);
	elseif (l <= 3.3e+137)
		tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0 + (h * ((-0.125 / l) * (((D * M_m) / d) ^ 2.0)))));
	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, -1.4e+196], N[(d * (-N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision])), $MachinePrecision], If[LessEqual[l, 3.3e+137], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(h * N[(N[(-0.125 / l), $MachinePrecision] * N[Power[N[(N[(D * M$95$m), $MachinePrecision] / d), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $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 3 regimes

2. if l < -1.4000000000000001e196

   1. Initial program 53.3%

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

   3. Simplified 53.7%

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

   5. Taylor expanded in d around inf 17.8%

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

   6. Taylor expanded in h around -inf 0.0%

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

      1. associate-*l* 0.0%

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

      2. unpow2 0.0%

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

      3. rem-square-sqrt 79.3%

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

      4. neg-mul-1 79.3%

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

      5. unpow-1 79.3%

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

      6. metadata-eval 79.3%

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

      7. pow-sqr 79.4%

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

      8. rem-sqrt-square 79.4%

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

      9. rem-square-sqrt 78.9%

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

      10. fabs-sqr 78.9%

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

      11. rem-square-sqrt 79.4%

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

   8. Simplified 79.4%

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

   if -1.4000000000000001e196 < l < 3.30000000000000003e137

   1. Initial program 70.0%

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

   3. Simplified 68.5%

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

      1. fma-undefine 68.5%

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

      2. associate-*r* 68.5%

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

      3. associate-*r/ 68.5%

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

   6. Applied egg-rr 68.5%

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

   7. Taylor expanded in M around 0 51.1%

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

      1. associate-*r/ 51.1%

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

      2. associate-*r* 52.9%

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

      3. associate-*r* 52.9%

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

      4. associate-*l/ 52.5%

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

      5. associate-*r/ 52.5%

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

      6. *-commutative 52.5%

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

      7. associate-*r/ 52.5%

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

      8. *-commutative 52.5%

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

      9. times-frac 54.5%

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

      10. *-commutative 54.5%

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

      11. associate-/l* 54.5%

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

      12. unpow2 54.5%

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

      13. unpow2 54.5%

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

      14. unpow2 54.5%

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

      15. times-frac 64.7%

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

      16. swap-sqr 70.8%

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

      17. unpow2 70.8%

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

   9. Simplified 71.4%

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

   if 3.30000000000000003e137 < l

   1. Initial program 47.6%

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

   3. Simplified 49.9%

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

      1. pow1 49.9%

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

      2. associate-*r* 49.9%

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

      3. sqrt-unprod 40.7%

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

      4. associate-*r* 40.7%

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

      5. associate-*r/ 40.7%

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

   6. Applied egg-rr 40.7%

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

   8. Simplified 35.7%

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

   9. Taylor expanded in d around inf 45.3%

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

       1. unpow-1 45.3%

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

       2. metadata-eval 45.3%

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

       3. pow-sqr 45.3%

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

       4. rem-sqrt-square 45.3%

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

       5. rem-square-sqrt 45.1%

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

       6. fabs-sqr 45.1%

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

       7. rem-square-sqrt 45.3%

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

   11. Simplified 45.3%

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

       1. *-commutative 45.3%

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

       2. unpow-prod-down 67.1%

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

   13. Applied egg-rr 67.1%

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

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.4 \cdot 10^{+196}:\\ \;\;\;\;d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{elif}\;\ell \leq 3.3 \cdot 10^{+137}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \left(\frac{-0.125}{\ell} \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 61.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -5.2 \cdot 10^{+184}:\\ \;\;\;\;d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{elif}\;\ell \leq 5.7 \cdot 10^{+25}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \frac{-0.5 \cdot \left(h \cdot {\left(\frac{D \cdot \left(M\_m \cdot 0.5\right)}{d}\right)}^{2}\right)}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \end{array} \]

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 -5.2e+184)
   (* d (- (pow (* h l) -0.5)))
   (if (<= l 5.7e+25)
     (*
      (sqrt (* (/ d h) (/ d l)))
      (+ 1.0 (/ (* -0.5 (* h (pow (/ (* D (* M_m 0.5)) d) 2.0))) 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 <= -5.2e+184) {
		tmp = d * -pow((h * l), -0.5);
	} else if (l <= 5.7e+25) {
		tmp = sqrt(((d / h) * (d / l))) * (1.0 + ((-0.5 * (h * pow(((D * (M_m * 0.5)) / d), 2.0))) / 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 <= (-5.2d+184)) then
        tmp = d * -((h * l) ** (-0.5d0))
    else if (l <= 5.7d+25) then
        tmp = sqrt(((d / h) * (d / l))) * (1.0d0 + (((-0.5d0) * (h * (((d_1 * (m_m * 0.5d0)) / d) ** 2.0d0))) / 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 <= -5.2e+184) {
		tmp = d * -Math.pow((h * l), -0.5);
	} else if (l <= 5.7e+25) {
		tmp = Math.sqrt(((d / h) * (d / l))) * (1.0 + ((-0.5 * (h * Math.pow(((D * (M_m * 0.5)) / d), 2.0))) / 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 <= -5.2e+184:
		tmp = d * -math.pow((h * l), -0.5)
	elif l <= 5.7e+25:
		tmp = math.sqrt(((d / h) * (d / l))) * (1.0 + ((-0.5 * (h * math.pow(((D * (M_m * 0.5)) / d), 2.0))) / 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 <= -5.2e+184)
		tmp = Float64(d * Float64(-(Float64(h * l) ^ -0.5)));
	elseif (l <= 5.7e+25)
		tmp = Float64(sqrt(Float64(Float64(d / h) * Float64(d / l))) * Float64(1.0 + Float64(Float64(-0.5 * Float64(h * (Float64(Float64(D * Float64(M_m * 0.5)) / d) ^ 2.0))) / 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 <= -5.2e+184)
		tmp = d * -((h * l) ^ -0.5);
	elseif (l <= 5.7e+25)
		tmp = sqrt(((d / h) * (d / l))) * (1.0 + ((-0.5 * (h * (((D * (M_m * 0.5)) / d) ^ 2.0))) / 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, -5.2e+184], N[(d * (-N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision])), $MachinePrecision], If[LessEqual[l, 5.7e+25], N[(N[Sqrt[N[(N[(d / h), $MachinePrecision] * N[(d / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(N[(-0.5 * N[(h * N[Power[N[(N[(D * N[(M$95$m * 0.5), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $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 3 regimes

2. if l < -5.19999999999999986e184

   1. Initial program 50.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 51.3%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 17.1%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

   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. associate-*l* 0.0%

         \[\leadsto \color{blue}{d \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. unpow2 0.0%

         \[\leadsto d \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      3. rem-square-sqrt 75.4%

         \[\leadsto d \cdot \left(\color{blue}{-1} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      4. neg-mul-1 75.4%

         \[\leadsto d \cdot \color{blue}{\left(-\sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      5. unpow-1 75.4%

         \[\leadsto d \cdot \left(-\sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}}\right) \]

      6. metadata-eval 75.4%

         \[\leadsto d \cdot \left(-\sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}}\right) \]

      7. pow-sqr 75.5%

         \[\leadsto d \cdot \left(-\sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}}\right) \]

      8. rem-sqrt-square 75.5%

         \[\leadsto d \cdot \left(-\color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|}\right) \]

      9. rem-square-sqrt 75.0%

         \[\leadsto d \cdot \left(-\left|\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right|\right) \]

      10. fabs-sqr 75.0%

          \[\leadsto d \cdot \left(-\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right) \]

      11. rem-square-sqrt 75.5%

          \[\leadsto d \cdot \left(-\color{blue}{{\left(h \cdot \ell\right)}^{-0.5}}\right) \]

   8. Simplified 75.5%

      \[\leadsto \color{blue}{d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)} \]

   if -5.19999999999999986e184 < l < 5.6999999999999996e25

   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 68.9%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r/ 72.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot h}{\ell}}\right) \]

      2. frac-times 72.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot h}{\ell}\right) \]

      3. associate-/l* 72.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)}}^{2} \cdot h}{\ell}\right) \]

      4. *-commutative 72.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{{\left(M \cdot \frac{D}{\color{blue}{d \cdot 2}}\right)}^{2} \cdot h}{\ell}\right) \]

   6. Applied egg-rr 72.0%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot h}{\ell}}\right) \]
   7. Step-by-step derivation

      1. *-commutative 72.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{\color{blue}{h \cdot {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}}}{\ell}\right) \]

      2. associate-/l* 72.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(h \cdot \frac{{\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}}{\ell}\right)}\right) \]

      3. associate-*r/ 72.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{{\color{blue}{\left(\frac{M \cdot D}{d \cdot 2}\right)}}^{2}}{\ell}\right)\right) \]

      4. *-rgt-identity 72.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{{\left(\frac{\color{blue}{\left(M \cdot D\right) \cdot 1}}{d \cdot 2}\right)}^{2}}{\ell}\right)\right) \]

      5. times-frac 72.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{{\color{blue}{\left(\frac{M \cdot D}{d} \cdot \frac{1}{2}\right)}}^{2}}{\ell}\right)\right) \]

      6. metadata-eval 72.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{{\left(\frac{M \cdot D}{d} \cdot \color{blue}{0.5}\right)}^{2}}{\ell}\right)\right) \]

      7. *-commutative 72.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{{\left(\frac{\color{blue}{D \cdot M}}{d} \cdot 0.5\right)}^{2}}{\ell}\right)\right) \]

      8. associate-/l* 70.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{{\left(\color{blue}{\left(D \cdot \frac{M}{d}\right)} \cdot 0.5\right)}^{2}}{\ell}\right)\right) \]

      9. associate-*l* 70.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{{\color{blue}{\left(D \cdot \left(\frac{M}{d} \cdot 0.5\right)\right)}}^{2}}{\ell}\right)\right) \]

   8. Simplified 70.9%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(h \cdot \frac{{\left(D \cdot \left(\frac{M}{d} \cdot 0.5\right)\right)}^{2}}{\ell}\right)}\right) \]
   9. Step-by-step derivation

      1. *-un-lft-identity 70.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \color{blue}{\left(1 \cdot \frac{{\left(D \cdot \left(\frac{M}{d} \cdot 0.5\right)\right)}^{2}}{\ell}\right)}\right)\right) \]

      2. associate-*l/ 70.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \left(1 \cdot \frac{{\left(D \cdot \color{blue}{\frac{M \cdot 0.5}{d}}\right)}^{2}}{\ell}\right)\right)\right) \]

      3. metadata-eval 70.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \left(1 \cdot \frac{{\left(D \cdot \frac{M \cdot \color{blue}{\frac{1}{2}}}{d}\right)}^{2}}{\ell}\right)\right)\right) \]

      4. div-inv 70.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \left(1 \cdot \frac{{\left(D \cdot \frac{\color{blue}{\frac{M}{2}}}{d}\right)}^{2}}{\ell}\right)\right)\right) \]

      5. associate-/l/ 70.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \left(1 \cdot \frac{{\left(D \cdot \color{blue}{\frac{M}{d \cdot 2}}\right)}^{2}}{\ell}\right)\right)\right) \]

   10. Applied egg-rr 70.9%

       \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \color{blue}{\left(1 \cdot \frac{{\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}}{\ell}\right)}\right)\right) \]
   11. Step-by-step derivation

       1. *-lft-identity 70.9%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \color{blue}{\frac{{\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}}{\ell}}\right)\right) \]

       2. associate-*r/ 72.2%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{{\color{blue}{\left(\frac{D \cdot M}{d \cdot 2}\right)}}^{2}}{\ell}\right)\right) \]

       3. times-frac 72.0%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{{\color{blue}{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}}^{2}}{\ell}\right)\right) \]

   12. Simplified 72.0%

       \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \color{blue}{\frac{{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}}{\ell}}\right)\right) \]
   13. Step-by-step derivation

       1. pow1 72.0%

          \[\leadsto \color{blue}{{\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}}{\ell}\right)\right)\right)}^{1}} \]

       2. sqrt-unprod 63.5%

          \[\leadsto {\left(\color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}}{\ell}\right)\right)\right)}^{1} \]

       3. cancel-sign-sub-inv 63.5%

          \[\leadsto {\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \color{blue}{\left(1 + \left(-0.5\right) \cdot \left(h \cdot \frac{{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}}{\ell}\right)\right)}\right)}^{1} \]

       4. metadata-eval 63.5%

          \[\leadsto {\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \color{blue}{-0.5} \cdot \left(h \cdot \frac{{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}}{\ell}\right)\right)\right)}^{1} \]

       5. associate-*r/ 64.0%

          \[\leadsto {\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \color{blue}{\frac{h \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}}{\ell}}\right)\right)}^{1} \]

       6. div-inv 64.0%

          \[\leadsto {\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \frac{h \cdot {\left(\frac{D}{d} \cdot \color{blue}{\left(M \cdot \frac{1}{2}\right)}\right)}^{2}}{\ell}\right)\right)}^{1} \]

       7. metadata-eval 64.0%

          \[\leadsto {\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \frac{h \cdot {\left(\frac{D}{d} \cdot \left(M \cdot \color{blue}{0.5}\right)\right)}^{2}}{\ell}\right)\right)}^{1} \]

   14. Applied egg-rr 64.0%

       \[\leadsto \color{blue}{{\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \frac{h \cdot {\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2}}{\ell}\right)\right)}^{1}} \]
   15. Step-by-step derivation

       1. unpow1 64.0%

          \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \frac{h \cdot {\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2}}{\ell}\right)} \]

       2. associate-*r/ 64.0%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \color{blue}{\frac{-0.5 \cdot \left(h \cdot {\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2}\right)}{\ell}}\right) \]

       3. associate-*l/ 64.1%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \frac{-0.5 \cdot \left(h \cdot {\color{blue}{\left(\frac{D \cdot \left(M \cdot 0.5\right)}{d}\right)}}^{2}\right)}{\ell}\right) \]

   16. Simplified 64.1%

       \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \frac{-0.5 \cdot \left(h \cdot {\left(\frac{D \cdot \left(M \cdot 0.5\right)}{d}\right)}^{2}\right)}{\ell}\right)} \]

   if 5.6999999999999996e25 < l

   1. Initial program 58.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 57.8%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. pow1 57.8%

         \[\leadsto \color{blue}{{\left(\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)\right)}^{1}} \]

      2. associate-*r* 57.8%

         \[\leadsto {\color{blue}{\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)}}^{1} \]

      3. sqrt-unprod 42.9%

         \[\leadsto {\left(\color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)}^{1} \]

      4. associate-*r* 42.9%

         \[\leadsto {\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \mathsf{fma}\left({\color{blue}{\left(\left(0.5 \cdot M\right) \cdot \frac{D}{d}\right)}}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)}^{1} \]

      5. associate-*r/ 42.9%

         \[\leadsto {\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(\left(0.5 \cdot M\right) \cdot \frac{D}{d}\right)}^{2}, \color{blue}{\frac{-0.5 \cdot h}{\ell}}, 1\right)\right)}^{1} \]

   6. Applied egg-rr 42.9%

      \[\leadsto \color{blue}{{\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(\left(0.5 \cdot M\right) \cdot \frac{D}{d}\right)}^{2}, \frac{-0.5 \cdot h}{\ell}, 1\right)\right)}^{1}} \]

   8. Simplified 36.8%

      \[\leadsto \color{blue}{\sqrt{\frac{\frac{{d}^{2}}{h}}{\ell}} \cdot \mathsf{fma}\left(h \cdot \frac{-0.5}{\ell}, {\left(D \cdot \left(\frac{M}{d} \cdot 0.5\right)\right)}^{2}, 1\right)} \]

   9. Taylor expanded in d around inf 44.1%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   10. Step-by-step derivation

       1. unpow-1 44.1%

          \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \]

       2. metadata-eval 44.1%

          \[\leadsto d \cdot \sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \]

       3. pow-sqr 44.2%

          \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}} \]

       4. rem-sqrt-square 44.2%

          \[\leadsto d \cdot \color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|} \]

       5. rem-square-sqrt 44.0%

          \[\leadsto d \cdot \left|\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right| \]

       6. fabs-sqr 44.0%

          \[\leadsto d \cdot \color{blue}{\left(\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}\right)} \]

       7. rem-square-sqrt 44.2%

          \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

   11. Simplified 44.2%

       \[\leadsto \color{blue}{d \cdot {\left(h \cdot \ell\right)}^{-0.5}} \]
   12. Step-by-step derivation

       1. *-commutative 44.2%

          \[\leadsto d \cdot {\color{blue}{\left(\ell \cdot h\right)}}^{-0.5} \]

       2. unpow-prod-down 65.7%

          \[\leadsto d \cdot \color{blue}{\left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)} \]

   13. Applied egg-rr 65.7%

       \[\leadsto d \cdot \color{blue}{\left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 16: 53.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -1.9 \cdot 10^{+20}:\\ \;\;\;\;d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{elif}\;d \leq -3.5 \cdot 10^{-301}:\\ \;\;\;\;d \cdot {0}^{-0.5}\\ \mathbf{elif}\;d \leq 0.7:\\ \;\;\;\;d \cdot \frac{-0.125 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M\_m}{d}\right)}^{2}\right)}{\sqrt{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \end{array} \]

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 (<= d -1.9e+20)
   (* d (- (pow (* h l) -0.5)))
   (if (<= d -3.5e-301)
     (* d (pow 0.0 -0.5))
     (if (<= d 0.7)
       (*
        d
        (/ (* -0.125 (* (/ h l) (pow (* D (/ M_m d)) 2.0))) (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 (d <= -1.9e+20) {
		tmp = d * -pow((h * l), -0.5);
	} else if (d <= -3.5e-301) {
		tmp = d * pow(0.0, -0.5);
	} else if (d <= 0.7) {
		tmp = d * ((-0.125 * ((h / l) * pow((D * (M_m / d)), 2.0))) / 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 (d <= (-1.9d+20)) then
        tmp = d * -((h * l) ** (-0.5d0))
    else if (d <= (-3.5d-301)) then
        tmp = d * (0.0d0 ** (-0.5d0))
    else if (d <= 0.7d0) then
        tmp = d * (((-0.125d0) * ((h / l) * ((d_1 * (m_m / d)) ** 2.0d0))) / 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 (d <= -1.9e+20) {
		tmp = d * -Math.pow((h * l), -0.5);
	} else if (d <= -3.5e-301) {
		tmp = d * Math.pow(0.0, -0.5);
	} else if (d <= 0.7) {
		tmp = d * ((-0.125 * ((h / l) * Math.pow((D * (M_m / d)), 2.0))) / 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 d <= -1.9e+20:
		tmp = d * -math.pow((h * l), -0.5)
	elif d <= -3.5e-301:
		tmp = d * math.pow(0.0, -0.5)
	elif d <= 0.7:
		tmp = d * ((-0.125 * ((h / l) * math.pow((D * (M_m / d)), 2.0))) / 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 (d <= -1.9e+20)
		tmp = Float64(d * Float64(-(Float64(h * l) ^ -0.5)));
	elseif (d <= -3.5e-301)
		tmp = Float64(d * (0.0 ^ -0.5));
	elseif (d <= 0.7)
		tmp = Float64(d * Float64(Float64(-0.125 * Float64(Float64(h / l) * (Float64(D * Float64(M_m / d)) ^ 2.0))) / 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 (d <= -1.9e+20)
		tmp = d * -((h * l) ^ -0.5);
	elseif (d <= -3.5e-301)
		tmp = d * (0.0 ^ -0.5);
	elseif (d <= 0.7)
		tmp = d * ((-0.125 * ((h / l) * ((D * (M_m / d)) ^ 2.0))) / 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[d, -1.9e+20], N[(d * (-N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision])), $MachinePrecision], If[LessEqual[d, -3.5e-301], N[(d * N[Power[0.0, -0.5], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 0.7], N[(d * N[(N[(-0.125 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(D * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 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 4 regimes

2. if d < -1.9e20

   1. Initial program 74.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 72.7%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 8.7%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

   6. Taylor expanded in h around -inf 0.0%

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   7. Step-by-step derivation

      1. associate-*l* 0.0%

         \[\leadsto \color{blue}{d \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. unpow2 0.0%

         \[\leadsto d \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      3. rem-square-sqrt 63.6%

         \[\leadsto d \cdot \left(\color{blue}{-1} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      4. neg-mul-1 63.6%

         \[\leadsto d \cdot \color{blue}{\left(-\sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      5. unpow-1 63.6%

         \[\leadsto d \cdot \left(-\sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}}\right) \]

      6. metadata-eval 63.6%

         \[\leadsto d \cdot \left(-\sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}}\right) \]

      7. pow-sqr 63.6%

         \[\leadsto d \cdot \left(-\sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}}\right) \]

      8. rem-sqrt-square 63.6%

         \[\leadsto d \cdot \left(-\color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|}\right) \]

      9. rem-square-sqrt 63.2%

         \[\leadsto d \cdot \left(-\left|\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right|\right) \]

      10. fabs-sqr 63.2%

          \[\leadsto d \cdot \left(-\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right) \]

      11. rem-square-sqrt 63.6%

          \[\leadsto d \cdot \left(-\color{blue}{{\left(h \cdot \ell\right)}^{-0.5}}\right) \]

   8. Simplified 63.6%

      \[\leadsto \color{blue}{d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)} \]

   if -1.9e20 < d < -3.49999999999999992e-301

   1. Initial program 55.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 55.8%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. pow1 55.8%

         \[\leadsto \color{blue}{{\left(\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)\right)}^{1}} \]

      2. associate-*r* 55.8%

         \[\leadsto {\color{blue}{\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)}}^{1} \]

      3. sqrt-unprod 43.9%

         \[\leadsto {\left(\color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)}^{1} \]

      4. associate-*r* 43.9%

         \[\leadsto {\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \mathsf{fma}\left({\color{blue}{\left(\left(0.5 \cdot M\right) \cdot \frac{D}{d}\right)}}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)}^{1} \]

      5. associate-*r/ 43.9%

         \[\leadsto {\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(\left(0.5 \cdot M\right) \cdot \frac{D}{d}\right)}^{2}, \color{blue}{\frac{-0.5 \cdot h}{\ell}}, 1\right)\right)}^{1} \]

   6. Applied egg-rr 43.9%

      \[\leadsto \color{blue}{{\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(\left(0.5 \cdot M\right) \cdot \frac{D}{d}\right)}^{2}, \frac{-0.5 \cdot h}{\ell}, 1\right)\right)}^{1}} \]

   8. Simplified 32.5%

      \[\leadsto \color{blue}{\sqrt{\frac{\frac{{d}^{2}}{h}}{\ell}} \cdot \mathsf{fma}\left(h \cdot \frac{-0.5}{\ell}, {\left(D \cdot \left(\frac{M}{d} \cdot 0.5\right)\right)}^{2}, 1\right)} \]

   9. Taylor expanded in d around inf 10.9%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   10. Step-by-step derivation

       1. unpow-1 10.9%

          \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \]

       2. metadata-eval 10.9%

          \[\leadsto d \cdot \sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \]

       3. pow-sqr 10.9%

          \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}} \]

       4. rem-sqrt-square 10.9%

          \[\leadsto d \cdot \color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|} \]

       5. rem-square-sqrt 10.9%

          \[\leadsto d \cdot \left|\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right| \]

       6. fabs-sqr 10.9%

          \[\leadsto d \cdot \color{blue}{\left(\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}\right)} \]

       7. rem-square-sqrt 10.9%

          \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

   11. Simplified 10.9%

       \[\leadsto \color{blue}{d \cdot {\left(h \cdot \ell\right)}^{-0.5}} \]
   12. Step-by-step derivation

       1. expm1-log1p-u 10.9%

          \[\leadsto d \cdot {\color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(h \cdot \ell\right)\right)\right)}}^{-0.5} \]

       2. expm1-undefine 30.7%

          \[\leadsto d \cdot {\color{blue}{\left(e^{\mathsf{log1p}\left(h \cdot \ell\right)} - 1\right)}}^{-0.5} \]

   13. Applied egg-rr 30.7%

       \[\leadsto d \cdot {\color{blue}{\left(e^{\mathsf{log1p}\left(h \cdot \ell\right)} - 1\right)}}^{-0.5} \]
   14. Step-by-step derivation

       1. sub-neg 30.7%

          \[\leadsto d \cdot {\color{blue}{\left(e^{\mathsf{log1p}\left(h \cdot \ell\right)} + \left(-1\right)\right)}}^{-0.5} \]

       2. metadata-eval 30.7%

          \[\leadsto d \cdot {\left(e^{\mathsf{log1p}\left(h \cdot \ell\right)} + \color{blue}{-1}\right)}^{-0.5} \]

       3. +-commutative 30.7%

          \[\leadsto d \cdot {\color{blue}{\left(-1 + e^{\mathsf{log1p}\left(h \cdot \ell\right)}\right)}}^{-0.5} \]

       4. log1p-undefine 30.7%

          \[\leadsto d \cdot {\left(-1 + e^{\color{blue}{\log \left(1 + h \cdot \ell\right)}}\right)}^{-0.5} \]

       5. rem-exp-log 30.7%

          \[\leadsto d \cdot {\left(-1 + \color{blue}{\left(1 + h \cdot \ell\right)}\right)}^{-0.5} \]

       6. +-commutative 30.7%

          \[\leadsto d \cdot {\left(-1 + \color{blue}{\left(h \cdot \ell + 1\right)}\right)}^{-0.5} \]

       7. fma-define 30.7%

          \[\leadsto d \cdot {\left(-1 + \color{blue}{\mathsf{fma}\left(h, \ell, 1\right)}\right)}^{-0.5} \]

   15. Simplified 30.7%

       \[\leadsto d \cdot {\color{blue}{\left(-1 + \mathsf{fma}\left(h, \ell, 1\right)\right)}}^{-0.5} \]

   16. Taylor expanded in h around 0 45.9%

       \[\leadsto d \cdot {\left(-1 + \color{blue}{1}\right)}^{-0.5} \]

   if -3.49999999999999992e-301 < d < 0.69999999999999996

   1. Initial program 55.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 52.7%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in M around inf 30.0%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(-0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)}\right) \]
   6. Step-by-step derivation

      1. associate-*r* 30.1%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(-0.125 \cdot \frac{\color{blue}{\left({D}^{2} \cdot {M}^{2}\right) \cdot h}}{{d}^{2} \cdot \ell}\right)\right) \]

      2. times-frac 30.1%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(-0.125 \cdot \color{blue}{\left(\frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}} \cdot \frac{h}{\ell}\right)}\right)\right) \]

      3. *-commutative 30.1%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(-0.125 \cdot \left(\frac{\color{blue}{{M}^{2} \cdot {D}^{2}}}{{d}^{2}} \cdot \frac{h}{\ell}\right)\right)\right) \]

      4. associate-/l* 30.0%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(-0.125 \cdot \left(\color{blue}{\left({M}^{2} \cdot \frac{{D}^{2}}{{d}^{2}}\right)} \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. unpow2 30.0%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(-0.125 \cdot \left(\left(\color{blue}{\left(M \cdot M\right)} \cdot \frac{{D}^{2}}{{d}^{2}}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      6. unpow2 30.0%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(-0.125 \cdot \left(\left(\left(M \cdot M\right) \cdot \frac{\color{blue}{D \cdot D}}{{d}^{2}}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      7. unpow2 30.0%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(-0.125 \cdot \left(\left(\left(M \cdot M\right) \cdot \frac{D \cdot D}{\color{blue}{d \cdot d}}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      8. times-frac 33.5%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(-0.125 \cdot \left(\left(\left(M \cdot M\right) \cdot \color{blue}{\left(\frac{D}{d} \cdot \frac{D}{d}\right)}\right) \cdot \frac{h}{\ell}\right)\right)\right) \]

      9. swap-sqr 37.6%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(-0.125 \cdot \left(\color{blue}{\left(\left(M \cdot \frac{D}{d}\right) \cdot \left(M \cdot \frac{D}{d}\right)\right)} \cdot \frac{h}{\ell}\right)\right)\right) \]

      10. unpow2 37.6%

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(-0.125 \cdot \left(\color{blue}{{\left(M \cdot \frac{D}{d}\right)}^{2}} \cdot \frac{h}{\ell}\right)\right)\right) \]

      11. associate-*r/ 39.0%

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(-0.125 \cdot \left({\color{blue}{\left(\frac{M \cdot D}{d}\right)}}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      12. *-commutative 39.0%

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(-0.125 \cdot \left({\left(\frac{\color{blue}{D \cdot M}}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      13. associate-/l* 37.7%

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(-0.125 \cdot \left({\color{blue}{\left(D \cdot \frac{M}{d}\right)}}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

   7. Simplified 37.7%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(-0.125 \cdot \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)}\right) \]
   8. Step-by-step derivation

      1. frac-2neg 54.1%

         \[\leadsto \sqrt{\color{blue}{\frac{-d}{-\ell}}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(D \cdot \frac{\frac{M}{2}}{d}\right)}^{2} \cdot -0.5\right)\right)\right) \]

      2. sqrt-div 0.1%

         \[\leadsto \color{blue}{\frac{\sqrt{-d}}{\sqrt{-\ell}}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(D \cdot \frac{\frac{M}{2}}{d}\right)}^{2} \cdot -0.5\right)\right)\right) \]

   9. Applied egg-rr 0.1%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{\frac{\sqrt{-d}}{\sqrt{-\ell}}} \cdot \left(-0.125 \cdot \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]
   10. Step-by-step derivation

       1. sqrt-div 0.0%

          \[\leadsto \color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot \left(\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \left(-0.125 \cdot \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

       2. associate-*l/ 0.0%

          \[\leadsto \frac{\sqrt{d}}{\sqrt{h}} \cdot \color{blue}{\frac{\sqrt{-d} \cdot \left(-0.125 \cdot \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)}{\sqrt{-\ell}}} \]

       3. frac-times 0.0%

          \[\leadsto \color{blue}{\frac{\sqrt{d} \cdot \left(\sqrt{-d} \cdot \left(-0.125 \cdot \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)}{\sqrt{h} \cdot \sqrt{-\ell}}} \]

       4. add-sqr-sqrt 0.0%

          \[\leadsto \frac{\sqrt{d} \cdot \left(\sqrt{\color{blue}{\sqrt{-d} \cdot \sqrt{-d}}} \cdot \left(-0.125 \cdot \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)}{\sqrt{h} \cdot \sqrt{-\ell}} \]

       5. sqrt-unprod 0.0%

          \[\leadsto \frac{\sqrt{d} \cdot \left(\sqrt{\color{blue}{\sqrt{\left(-d\right) \cdot \left(-d\right)}}} \cdot \left(-0.125 \cdot \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)}{\sqrt{h} \cdot \sqrt{-\ell}} \]

       6. sqr-neg 0.0%

          \[\leadsto \frac{\sqrt{d} \cdot \left(\sqrt{\sqrt{\color{blue}{d \cdot d}}} \cdot \left(-0.125 \cdot \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)}{\sqrt{h} \cdot \sqrt{-\ell}} \]

       7. sqrt-unprod 0.0%

          \[\leadsto \frac{\sqrt{d} \cdot \left(\sqrt{\color{blue}{\sqrt{d} \cdot \sqrt{d}}} \cdot \left(-0.125 \cdot \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)}{\sqrt{h} \cdot \sqrt{-\ell}} \]

       8. add-sqr-sqrt 0.0%

          \[\leadsto \frac{\sqrt{d} \cdot \left(\sqrt{\color{blue}{d}} \cdot \left(-0.125 \cdot \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)}{\sqrt{h} \cdot \sqrt{-\ell}} \]

       9. add-sqr-sqrt 0.0%

          \[\leadsto \frac{\sqrt{d} \cdot \left(\sqrt{d} \cdot \left(-0.125 \cdot \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)}{\sqrt{h} \cdot \sqrt{\color{blue}{\sqrt{-\ell} \cdot \sqrt{-\ell}}}} \]

       10. sqrt-unprod 50.1%

           \[\leadsto \frac{\sqrt{d} \cdot \left(\sqrt{d} \cdot \left(-0.125 \cdot \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)}{\sqrt{h} \cdot \sqrt{\color{blue}{\sqrt{\left(-\ell\right) \cdot \left(-\ell\right)}}}} \]

       11. sqr-neg 50.1%

           \[\leadsto \frac{\sqrt{d} \cdot \left(\sqrt{d} \cdot \left(-0.125 \cdot \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)}{\sqrt{h} \cdot \sqrt{\sqrt{\color{blue}{\ell \cdot \ell}}}} \]

       12. sqrt-unprod 59.7%

           \[\leadsto \frac{\sqrt{d} \cdot \left(\sqrt{d} \cdot \left(-0.125 \cdot \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)}{\sqrt{h} \cdot \sqrt{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}}} \]

   11. Applied egg-rr 51.2%

       \[\leadsto \color{blue}{\frac{\sqrt{d} \cdot \left(\sqrt{d} \cdot \left(-0.125 \cdot \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)}{\sqrt{h \cdot \ell}}} \]
   12. Step-by-step derivation

       1. *-commutative 51.2%

          \[\leadsto \frac{\color{blue}{\left(\sqrt{d} \cdot \left(-0.125 \cdot \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \cdot \sqrt{d}}}{\sqrt{h \cdot \ell}} \]

       2. *-commutative 51.2%

          \[\leadsto \frac{\color{blue}{\sqrt{d} \cdot \left(\sqrt{d} \cdot \left(-0.125 \cdot \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)}}{\sqrt{h \cdot \ell}} \]

       3. associate-*r* 51.2%

          \[\leadsto \frac{\color{blue}{\left(\sqrt{d} \cdot \sqrt{d}\right) \cdot \left(-0.125 \cdot \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)}}{\sqrt{h \cdot \ell}} \]

       4. rem-square-sqrt 51.3%

          \[\leadsto \frac{\color{blue}{d} \cdot \left(-0.125 \cdot \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)}{\sqrt{h \cdot \ell}} \]

       5. associate-/l* 50.0%

          \[\leadsto \color{blue}{d \cdot \frac{-0.125 \cdot \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)}{\sqrt{h \cdot \ell}}} \]

   13. Simplified 50.0%

       \[\leadsto \color{blue}{d \cdot \frac{-0.125 \cdot \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)}{\sqrt{h \cdot \ell}}} \]

   if 0.69999999999999996 < d

   1. Initial program 78.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 80.1%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. pow1 80.1%

         \[\leadsto \color{blue}{{\left(\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)\right)}^{1}} \]

      2. associate-*r* 80.1%

         \[\leadsto {\color{blue}{\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)}}^{1} \]

      3. sqrt-unprod 67.9%

         \[\leadsto {\left(\color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)}^{1} \]

      4. associate-*r* 67.9%

         \[\leadsto {\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \mathsf{fma}\left({\color{blue}{\left(\left(0.5 \cdot M\right) \cdot \frac{D}{d}\right)}}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)}^{1} \]

      5. associate-*r/ 67.9%

         \[\leadsto {\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(\left(0.5 \cdot M\right) \cdot \frac{D}{d}\right)}^{2}, \color{blue}{\frac{-0.5 \cdot h}{\ell}}, 1\right)\right)}^{1} \]

   6. Applied egg-rr 67.9%

      \[\leadsto \color{blue}{{\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(\left(0.5 \cdot M\right) \cdot \frac{D}{d}\right)}^{2}, \frac{-0.5 \cdot h}{\ell}, 1\right)\right)}^{1}} \]

   8. Simplified 56.8%

      \[\leadsto \color{blue}{\sqrt{\frac{\frac{{d}^{2}}{h}}{\ell}} \cdot \mathsf{fma}\left(h \cdot \frac{-0.5}{\ell}, {\left(D \cdot \left(\frac{M}{d} \cdot 0.5\right)\right)}^{2}, 1\right)} \]

   9. Taylor expanded in d around inf 60.7%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   10. Step-by-step derivation

       1. unpow-1 60.7%

          \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \]

       2. metadata-eval 60.7%

          \[\leadsto d \cdot \sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \]

       3. pow-sqr 60.7%

          \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}} \]

       4. rem-sqrt-square 60.7%

          \[\leadsto d \cdot \color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|} \]

       5. rem-square-sqrt 60.5%

          \[\leadsto d \cdot \left|\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right| \]

       6. fabs-sqr 60.5%

          \[\leadsto d \cdot \color{blue}{\left(\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}\right)} \]

       7. rem-square-sqrt 60.7%

          \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

   11. Simplified 60.7%

       \[\leadsto \color{blue}{d \cdot {\left(h \cdot \ell\right)}^{-0.5}} \]
   12. Step-by-step derivation

       1. *-commutative 60.7%

          \[\leadsto d \cdot {\color{blue}{\left(\ell \cdot h\right)}}^{-0.5} \]

       2. unpow-prod-down 79.5%

          \[\leadsto d \cdot \color{blue}{\left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)} \]

   13. Applied egg-rr 79.5%

       \[\leadsto d \cdot \color{blue}{\left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 59.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.9 \cdot 10^{+20}:\\ \;\;\;\;d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{elif}\;d \leq -3.5 \cdot 10^{-301}:\\ \;\;\;\;d \cdot {0}^{-0.5}\\ \mathbf{elif}\;d \leq 0.7:\\ \;\;\;\;d \cdot \frac{-0.125 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}\right)}{\sqrt{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 49.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -6.2 \cdot 10^{+19}:\\ \;\;\;\;d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{elif}\;d \leq -1 \cdot 10^{-309}:\\ \;\;\;\;d \cdot {0}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \end{array} \]

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 (<= d -6.2e+19)
   (* d (- (pow (* h l) -0.5)))
   (if (<= d -1e-309)
     (* d (pow 0.0 -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 (d <= -6.2e+19) {
		tmp = d * -pow((h * l), -0.5);
	} else if (d <= -1e-309) {
		tmp = d * pow(0.0, -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 (d <= (-6.2d+19)) then
        tmp = d * -((h * l) ** (-0.5d0))
    else if (d <= (-1d-309)) then
        tmp = d * (0.0d0 ** (-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 (d <= -6.2e+19) {
		tmp = d * -Math.pow((h * l), -0.5);
	} else if (d <= -1e-309) {
		tmp = d * Math.pow(0.0, -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 d <= -6.2e+19:
		tmp = d * -math.pow((h * l), -0.5)
	elif d <= -1e-309:
		tmp = d * math.pow(0.0, -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 (d <= -6.2e+19)
		tmp = Float64(d * Float64(-(Float64(h * l) ^ -0.5)));
	elseif (d <= -1e-309)
		tmp = Float64(d * (0.0 ^ -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 (d <= -6.2e+19)
		tmp = d * -((h * l) ^ -0.5);
	elseif (d <= -1e-309)
		tmp = d * (0.0 ^ -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[d, -6.2e+19], N[(d * (-N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision])), $MachinePrecision], If[LessEqual[d, -1e-309], N[(d * N[Power[0.0, -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 3 regimes

2. if d < -6.2e19

   1. Initial program 74.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 72.7%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 8.7%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

   6. Taylor expanded in h around -inf 0.0%

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   7. Step-by-step derivation

      1. associate-*l* 0.0%

         \[\leadsto \color{blue}{d \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. unpow2 0.0%

         \[\leadsto d \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      3. rem-square-sqrt 63.6%

         \[\leadsto d \cdot \left(\color{blue}{-1} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      4. neg-mul-1 63.6%

         \[\leadsto d \cdot \color{blue}{\left(-\sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      5. unpow-1 63.6%

         \[\leadsto d \cdot \left(-\sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}}\right) \]

      6. metadata-eval 63.6%

         \[\leadsto d \cdot \left(-\sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}}\right) \]

      7. pow-sqr 63.6%

         \[\leadsto d \cdot \left(-\sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}}\right) \]

      8. rem-sqrt-square 63.6%

         \[\leadsto d \cdot \left(-\color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|}\right) \]

      9. rem-square-sqrt 63.2%

         \[\leadsto d \cdot \left(-\left|\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right|\right) \]

      10. fabs-sqr 63.2%

          \[\leadsto d \cdot \left(-\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right) \]

      11. rem-square-sqrt 63.6%

          \[\leadsto d \cdot \left(-\color{blue}{{\left(h \cdot \ell\right)}^{-0.5}}\right) \]

   8. Simplified 63.6%

      \[\leadsto \color{blue}{d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)} \]

   if -6.2e19 < d < -1.000000000000002e-309

   1. Initial program 55.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 55.0%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. pow1 55.0%

         \[\leadsto \color{blue}{{\left(\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)\right)}^{1}} \]

      2. associate-*r* 55.0%

         \[\leadsto {\color{blue}{\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)}}^{1} \]

      3. sqrt-unprod 43.3%

         \[\leadsto {\left(\color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)}^{1} \]

      4. associate-*r* 43.3%

         \[\leadsto {\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \mathsf{fma}\left({\color{blue}{\left(\left(0.5 \cdot M\right) \cdot \frac{D}{d}\right)}}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)}^{1} \]

      5. associate-*r/ 43.3%

         \[\leadsto {\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(\left(0.5 \cdot M\right) \cdot \frac{D}{d}\right)}^{2}, \color{blue}{\frac{-0.5 \cdot h}{\ell}}, 1\right)\right)}^{1} \]

   6. Applied egg-rr 43.3%

      \[\leadsto \color{blue}{{\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(\left(0.5 \cdot M\right) \cdot \frac{D}{d}\right)}^{2}, \frac{-0.5 \cdot h}{\ell}, 1\right)\right)}^{1}} \]

   8. Simplified 32.1%

      \[\leadsto \color{blue}{\sqrt{\frac{\frac{{d}^{2}}{h}}{\ell}} \cdot \mathsf{fma}\left(h \cdot \frac{-0.5}{\ell}, {\left(D \cdot \left(\frac{M}{d} \cdot 0.5\right)\right)}^{2}, 1\right)} \]

   9. Taylor expanded in d around inf 10.9%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   10. Step-by-step derivation

       1. unpow-1 10.9%

          \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \]

       2. metadata-eval 10.9%

          \[\leadsto d \cdot \sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \]

       3. pow-sqr 10.9%

          \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}} \]

       4. rem-sqrt-square 10.9%

          \[\leadsto d \cdot \color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|} \]

       5. rem-square-sqrt 10.9%

          \[\leadsto d \cdot \left|\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right| \]

       6. fabs-sqr 10.9%

          \[\leadsto d \cdot \color{blue}{\left(\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}\right)} \]

       7. rem-square-sqrt 10.9%

          \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

   11. Simplified 10.9%

       \[\leadsto \color{blue}{d \cdot {\left(h \cdot \ell\right)}^{-0.5}} \]
   12. Step-by-step derivation

       1. expm1-log1p-u 10.9%

          \[\leadsto d \cdot {\color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(h \cdot \ell\right)\right)\right)}}^{-0.5} \]

       2. expm1-undefine 30.3%

          \[\leadsto d \cdot {\color{blue}{\left(e^{\mathsf{log1p}\left(h \cdot \ell\right)} - 1\right)}}^{-0.5} \]

   13. Applied egg-rr 30.3%

       \[\leadsto d \cdot {\color{blue}{\left(e^{\mathsf{log1p}\left(h \cdot \ell\right)} - 1\right)}}^{-0.5} \]
   14. Step-by-step derivation

       1. sub-neg 30.3%

          \[\leadsto d \cdot {\color{blue}{\left(e^{\mathsf{log1p}\left(h \cdot \ell\right)} + \left(-1\right)\right)}}^{-0.5} \]

       2. metadata-eval 30.3%

          \[\leadsto d \cdot {\left(e^{\mathsf{log1p}\left(h \cdot \ell\right)} + \color{blue}{-1}\right)}^{-0.5} \]

       3. +-commutative 30.3%

          \[\leadsto d \cdot {\color{blue}{\left(-1 + e^{\mathsf{log1p}\left(h \cdot \ell\right)}\right)}}^{-0.5} \]

       4. log1p-undefine 30.3%

          \[\leadsto d \cdot {\left(-1 + e^{\color{blue}{\log \left(1 + h \cdot \ell\right)}}\right)}^{-0.5} \]

       5. rem-exp-log 30.3%

          \[\leadsto d \cdot {\left(-1 + \color{blue}{\left(1 + h \cdot \ell\right)}\right)}^{-0.5} \]

       6. +-commutative 30.3%

          \[\leadsto d \cdot {\left(-1 + \color{blue}{\left(h \cdot \ell + 1\right)}\right)}^{-0.5} \]

       7. fma-define 30.3%

          \[\leadsto d \cdot {\left(-1 + \color{blue}{\mathsf{fma}\left(h, \ell, 1\right)}\right)}^{-0.5} \]

   15. Simplified 30.3%

       \[\leadsto d \cdot {\color{blue}{\left(-1 + \mathsf{fma}\left(h, \ell, 1\right)\right)}}^{-0.5} \]

   16. Taylor expanded in h around 0 45.2%

       \[\leadsto d \cdot {\left(-1 + \color{blue}{1}\right)}^{-0.5} \]

   if -1.000000000000002e-309 < d

   1. Initial program 66.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 65.5%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. pow1 65.5%

         \[\leadsto \color{blue}{{\left(\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)\right)}^{1}} \]

      2. associate-*r* 65.5%

         \[\leadsto {\color{blue}{\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)}}^{1} \]

      3. sqrt-unprod 55.0%

         \[\leadsto {\left(\color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)}^{1} \]

      4. associate-*r* 55.0%

         \[\leadsto {\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \mathsf{fma}\left({\color{blue}{\left(\left(0.5 \cdot M\right) \cdot \frac{D}{d}\right)}}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)}^{1} \]

      5. associate-*r/ 55.0%

         \[\leadsto {\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(\left(0.5 \cdot M\right) \cdot \frac{D}{d}\right)}^{2}, \color{blue}{\frac{-0.5 \cdot h}{\ell}}, 1\right)\right)}^{1} \]

   6. Applied egg-rr 55.0%

      \[\leadsto \color{blue}{{\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(\left(0.5 \cdot M\right) \cdot \frac{D}{d}\right)}^{2}, \frac{-0.5 \cdot h}{\ell}, 1\right)\right)}^{1}} \]

   8. Simplified 47.0%

      \[\leadsto \color{blue}{\sqrt{\frac{\frac{{d}^{2}}{h}}{\ell}} \cdot \mathsf{fma}\left(h \cdot \frac{-0.5}{\ell}, {\left(D \cdot \left(\frac{M}{d} \cdot 0.5\right)\right)}^{2}, 1\right)} \]

   9. Taylor expanded in d around inf 39.7%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   10. Step-by-step derivation

       1. unpow-1 39.7%

          \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \]

       2. metadata-eval 39.7%

          \[\leadsto d \cdot \sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \]

       3. pow-sqr 39.8%

          \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}} \]

       4. rem-sqrt-square 39.8%

          \[\leadsto d \cdot \color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|} \]

       5. rem-square-sqrt 39.6%

          \[\leadsto d \cdot \left|\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right| \]

       6. fabs-sqr 39.6%

          \[\leadsto d \cdot \color{blue}{\left(\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}\right)} \]

       7. rem-square-sqrt 39.8%

          \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

   11. Simplified 39.8%

       \[\leadsto \color{blue}{d \cdot {\left(h \cdot \ell\right)}^{-0.5}} \]
   12. Step-by-step derivation

       1. *-commutative 39.8%

          \[\leadsto d \cdot {\color{blue}{\left(\ell \cdot h\right)}}^{-0.5} \]

       2. unpow-prod-down 53.3%

          \[\leadsto d \cdot \color{blue}{\left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)} \]

   13. Applied egg-rr 53.3%

       \[\leadsto d \cdot \color{blue}{\left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 53.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -6.2 \cdot 10^{+19}:\\ \;\;\;\;d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{elif}\;d \leq -1 \cdot 10^{-309}:\\ \;\;\;\;d \cdot {0}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 45.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -2 \cdot 10^{+19}:\\ \;\;\;\;d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{elif}\;d \leq -4.8 \cdot 10^{-297}:\\ \;\;\;\;d \cdot {0}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \end{array} \end{array} \]

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 (<= d -2e+19)
   (* d (- (pow (* h l) -0.5)))
   (if (<= d -4.8e-297) (* d (pow 0.0 -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 (d <= -2e+19) {
		tmp = d * -pow((h * l), -0.5);
	} else if (d <= -4.8e-297) {
		tmp = d * pow(0.0, -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 (d <= (-2d+19)) then
        tmp = d * -((h * l) ** (-0.5d0))
    else if (d <= (-4.8d-297)) then
        tmp = d * (0.0d0 ** (-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 (d <= -2e+19) {
		tmp = d * -Math.pow((h * l), -0.5);
	} else if (d <= -4.8e-297) {
		tmp = d * Math.pow(0.0, -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 d <= -2e+19:
		tmp = d * -math.pow((h * l), -0.5)
	elif d <= -4.8e-297:
		tmp = d * math.pow(0.0, -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 (d <= -2e+19)
		tmp = Float64(d * Float64(-(Float64(h * l) ^ -0.5)));
	elseif (d <= -4.8e-297)
		tmp = Float64(d * (0.0 ^ -0.5));
	else
		tmp = Float64(d * sqrt(Float64(Float64(1.0 / 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 (d <= -2e+19)
		tmp = d * -((h * l) ^ -0.5);
	elseif (d <= -4.8e-297)
		tmp = d * (0.0 ^ -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[d, -2e+19], N[(d * (-N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision])), $MachinePrecision], If[LessEqual[d, -4.8e-297], N[(d * N[Power[0.0, -0.5], $MachinePrecision]), $MachinePrecision], N[(d * N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -2e19

   1. Initial program 74.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 72.7%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 8.7%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

   6. Taylor expanded in h around -inf 0.0%

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   7. Step-by-step derivation

      1. associate-*l* 0.0%

         \[\leadsto \color{blue}{d \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. unpow2 0.0%

         \[\leadsto d \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      3. rem-square-sqrt 63.6%

         \[\leadsto d \cdot \left(\color{blue}{-1} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      4. neg-mul-1 63.6%

         \[\leadsto d \cdot \color{blue}{\left(-\sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      5. unpow-1 63.6%

         \[\leadsto d \cdot \left(-\sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}}\right) \]

      6. metadata-eval 63.6%

         \[\leadsto d \cdot \left(-\sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}}\right) \]

      7. pow-sqr 63.6%

         \[\leadsto d \cdot \left(-\sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}}\right) \]

      8. rem-sqrt-square 63.6%

         \[\leadsto d \cdot \left(-\color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|}\right) \]

      9. rem-square-sqrt 63.2%

         \[\leadsto d \cdot \left(-\left|\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right|\right) \]

      10. fabs-sqr 63.2%

          \[\leadsto d \cdot \left(-\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right) \]

      11. rem-square-sqrt 63.6%

          \[\leadsto d \cdot \left(-\color{blue}{{\left(h \cdot \ell\right)}^{-0.5}}\right) \]

   8. Simplified 63.6%

      \[\leadsto \color{blue}{d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)} \]

   if -2e19 < d < -4.7999999999999999e-297

   1. Initial program 54.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 54.4%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. pow1 54.4%

         \[\leadsto \color{blue}{{\left(\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)\right)}^{1}} \]

      2. associate-*r* 54.3%

         \[\leadsto {\color{blue}{\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)}}^{1} \]

      3. sqrt-unprod 43.7%

         \[\leadsto {\left(\color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)}^{1} \]

      4. associate-*r* 43.7%

         \[\leadsto {\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \mathsf{fma}\left({\color{blue}{\left(\left(0.5 \cdot M\right) \cdot \frac{D}{d}\right)}}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)}^{1} \]

      5. associate-*r/ 43.7%

         \[\leadsto {\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(\left(0.5 \cdot M\right) \cdot \frac{D}{d}\right)}^{2}, \color{blue}{\frac{-0.5 \cdot h}{\ell}}, 1\right)\right)}^{1} \]

   6. Applied egg-rr 43.7%

      \[\leadsto \color{blue}{{\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(\left(0.5 \cdot M\right) \cdot \frac{D}{d}\right)}^{2}, \frac{-0.5 \cdot h}{\ell}, 1\right)\right)}^{1}} \]

   8. Simplified 31.9%

      \[\leadsto \color{blue}{\sqrt{\frac{\frac{{d}^{2}}{h}}{\ell}} \cdot \mathsf{fma}\left(h \cdot \frac{-0.5}{\ell}, {\left(D \cdot \left(\frac{M}{d} \cdot 0.5\right)\right)}^{2}, 1\right)} \]

   9. Taylor expanded in d around inf 9.6%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   10. Step-by-step derivation

       1. unpow-1 9.6%

          \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \]

       2. metadata-eval 9.6%

          \[\leadsto d \cdot \sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \]

       3. pow-sqr 9.6%

          \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}} \]

       4. rem-sqrt-square 9.6%

          \[\leadsto d \cdot \color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|} \]

       5. rem-square-sqrt 9.6%

          \[\leadsto d \cdot \left|\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right| \]

       6. fabs-sqr 9.6%

          \[\leadsto d \cdot \color{blue}{\left(\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}\right)} \]

       7. rem-square-sqrt 9.6%

          \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

   11. Simplified 9.6%

       \[\leadsto \color{blue}{d \cdot {\left(h \cdot \ell\right)}^{-0.5}} \]
   12. Step-by-step derivation

       1. expm1-log1p-u 9.6%

          \[\leadsto d \cdot {\color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(h \cdot \ell\right)\right)\right)}}^{-0.5} \]

       2. expm1-undefine 28.5%

          \[\leadsto d \cdot {\color{blue}{\left(e^{\mathsf{log1p}\left(h \cdot \ell\right)} - 1\right)}}^{-0.5} \]

   13. Applied egg-rr 28.5%

       \[\leadsto d \cdot {\color{blue}{\left(e^{\mathsf{log1p}\left(h \cdot \ell\right)} - 1\right)}}^{-0.5} \]
   14. Step-by-step derivation

       1. sub-neg 28.5%

          \[\leadsto d \cdot {\color{blue}{\left(e^{\mathsf{log1p}\left(h \cdot \ell\right)} + \left(-1\right)\right)}}^{-0.5} \]

       2. metadata-eval 28.5%

          \[\leadsto d \cdot {\left(e^{\mathsf{log1p}\left(h \cdot \ell\right)} + \color{blue}{-1}\right)}^{-0.5} \]

       3. +-commutative 28.5%

          \[\leadsto d \cdot {\color{blue}{\left(-1 + e^{\mathsf{log1p}\left(h \cdot \ell\right)}\right)}}^{-0.5} \]

       4. log1p-undefine 28.5%

          \[\leadsto d \cdot {\left(-1 + e^{\color{blue}{\log \left(1 + h \cdot \ell\right)}}\right)}^{-0.5} \]

       5. rem-exp-log 28.5%

          \[\leadsto d \cdot {\left(-1 + \color{blue}{\left(1 + h \cdot \ell\right)}\right)}^{-0.5} \]

       6. +-commutative 28.5%

          \[\leadsto d \cdot {\left(-1 + \color{blue}{\left(h \cdot \ell + 1\right)}\right)}^{-0.5} \]

       7. fma-define 28.5%

          \[\leadsto d \cdot {\left(-1 + \color{blue}{\mathsf{fma}\left(h, \ell, 1\right)}\right)}^{-0.5} \]

   15. Simplified 28.5%

       \[\leadsto d \cdot {\color{blue}{\left(-1 + \mathsf{fma}\left(h, \ell, 1\right)\right)}}^{-0.5} \]

   16. Taylor expanded in h around 0 45.7%

       \[\leadsto d \cdot {\left(-1 + \color{blue}{1}\right)}^{-0.5} \]

   if -4.7999999999999999e-297 < d

   1. Initial program 66.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 65.6%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. pow1 65.6%

         \[\leadsto \color{blue}{{\left(\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)\right)}^{1}} \]

      2. associate-*r* 65.6%

         \[\leadsto {\color{blue}{\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)}}^{1} \]

      3. sqrt-unprod 54.6%

         \[\leadsto {\left(\color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)}^{1} \]

      4. associate-*r* 54.6%

         \[\leadsto {\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \mathsf{fma}\left({\color{blue}{\left(\left(0.5 \cdot M\right) \cdot \frac{D}{d}\right)}}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)}^{1} \]

      5. associate-*r/ 54.6%

         \[\leadsto {\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(\left(0.5 \cdot M\right) \cdot \frac{D}{d}\right)}^{2}, \color{blue}{\frac{-0.5 \cdot h}{\ell}}, 1\right)\right)}^{1} \]

   6. Applied egg-rr 54.6%

      \[\leadsto \color{blue}{{\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(\left(0.5 \cdot M\right) \cdot \frac{D}{d}\right)}^{2}, \frac{-0.5 \cdot h}{\ell}, 1\right)\right)}^{1}} \]

   8. Simplified 46.7%

      \[\leadsto \color{blue}{\sqrt{\frac{\frac{{d}^{2}}{h}}{\ell}} \cdot \mathsf{fma}\left(h \cdot \frac{-0.5}{\ell}, {\left(D \cdot \left(\frac{M}{d} \cdot 0.5\right)\right)}^{2}, 1\right)} \]

   9. Taylor expanded in d around inf 39.7%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   10. Step-by-step derivation

       1. unpow-1 39.7%

          \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \]

       2. metadata-eval 39.7%

          \[\leadsto d \cdot \sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \]

       3. pow-sqr 39.7%

          \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}} \]

       4. rem-sqrt-square 39.7%

          \[\leadsto d \cdot \color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|} \]

       5. rem-square-sqrt 39.6%

          \[\leadsto d \cdot \left|\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right| \]

       6. fabs-sqr 39.6%

          \[\leadsto d \cdot \color{blue}{\left(\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}\right)} \]

       7. rem-square-sqrt 39.7%

          \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

   11. Simplified 39.7%

       \[\leadsto \color{blue}{d \cdot {\left(h \cdot \ell\right)}^{-0.5}} \]
   12. Step-by-step derivation

       1. expm1-log1p-u 38.6%

          \[\leadsto d \cdot {\color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(h \cdot \ell\right)\right)\right)}}^{-0.5} \]

       2. expm1-undefine 27.1%

          \[\leadsto d \cdot {\color{blue}{\left(e^{\mathsf{log1p}\left(h \cdot \ell\right)} - 1\right)}}^{-0.5} \]

   13. Applied egg-rr 27.1%

       \[\leadsto d \cdot {\color{blue}{\left(e^{\mathsf{log1p}\left(h \cdot \ell\right)} - 1\right)}}^{-0.5} \]
   14. Step-by-step derivation

       1. sub-neg 27.1%

          \[\leadsto d \cdot {\color{blue}{\left(e^{\mathsf{log1p}\left(h \cdot \ell\right)} + \left(-1\right)\right)}}^{-0.5} \]

       2. metadata-eval 27.1%

          \[\leadsto d \cdot {\left(e^{\mathsf{log1p}\left(h \cdot \ell\right)} + \color{blue}{-1}\right)}^{-0.5} \]

       3. +-commutative 27.1%

          \[\leadsto d \cdot {\color{blue}{\left(-1 + e^{\mathsf{log1p}\left(h \cdot \ell\right)}\right)}}^{-0.5} \]

       4. log1p-undefine 27.1%

          \[\leadsto d \cdot {\left(-1 + e^{\color{blue}{\log \left(1 + h \cdot \ell\right)}}\right)}^{-0.5} \]

       5. rem-exp-log 28.2%

          \[\leadsto d \cdot {\left(-1 + \color{blue}{\left(1 + h \cdot \ell\right)}\right)}^{-0.5} \]

       6. +-commutative 28.2%

          \[\leadsto d \cdot {\left(-1 + \color{blue}{\left(h \cdot \ell + 1\right)}\right)}^{-0.5} \]

       7. fma-define 28.2%

          \[\leadsto d \cdot {\left(-1 + \color{blue}{\mathsf{fma}\left(h, \ell, 1\right)}\right)}^{-0.5} \]

   15. Simplified 28.2%

       \[\leadsto d \cdot {\color{blue}{\left(-1 + \mathsf{fma}\left(h, \ell, 1\right)\right)}}^{-0.5} \]

   16. Taylor expanded in d around 0 39.7%

       \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   17. Step-by-step derivation

       1. associate-/r* 41.8%

          \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \]

   18. Simplified 41.8%

       \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 47.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2 \cdot 10^{+19}:\\ \;\;\;\;d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{elif}\;d \leq -4.8 \cdot 10^{-297}:\\ \;\;\;\;d \cdot {0}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 45.7% accurate, 2.9× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := {\left(h \cdot \ell\right)}^{-0.5}\\ \mathbf{if}\;d \leq -3.2 \cdot 10^{+20}:\\ \;\;\;\;d \cdot \left(-t\_0\right)\\ \mathbf{elif}\;d \leq -4.4 \cdot 10^{-297}:\\ \;\;\;\;d \cdot {0}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;d \cdot t\_0\\ \end{array} \end{array} \]

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 (pow (* h l) -0.5)))
   (if (<= d -3.2e+20)
     (* d (- t_0))
     (if (<= d -4.4e-297) (* d (pow 0.0 -0.5)) (* d 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((h * l), -0.5);
	double tmp;
	if (d <= -3.2e+20) {
		tmp = d * -t_0;
	} else if (d <= -4.4e-297) {
		tmp = d * pow(0.0, -0.5);
	} else {
		tmp = d * 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 = (h * l) ** (-0.5d0)
    if (d <= (-3.2d+20)) then
        tmp = d * -t_0
    else if (d <= (-4.4d-297)) then
        tmp = d * (0.0d0 ** (-0.5d0))
    else
        tmp = d * 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((h * l), -0.5);
	double tmp;
	if (d <= -3.2e+20) {
		tmp = d * -t_0;
	} else if (d <= -4.4e-297) {
		tmp = d * Math.pow(0.0, -0.5);
	} else {
		tmp = d * 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((h * l), -0.5)
	tmp = 0
	if d <= -3.2e+20:
		tmp = d * -t_0
	elif d <= -4.4e-297:
		tmp = d * math.pow(0.0, -0.5)
	else:
		tmp = d * 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(h * l) ^ -0.5
	tmp = 0.0
	if (d <= -3.2e+20)
		tmp = Float64(d * Float64(-t_0));
	elseif (d <= -4.4e-297)
		tmp = Float64(d * (0.0 ^ -0.5));
	else
		tmp = Float64(d * 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 = (h * l) ^ -0.5;
	tmp = 0.0;
	if (d <= -3.2e+20)
		tmp = d * -t_0;
	elseif (d <= -4.4e-297)
		tmp = d * (0.0 ^ -0.5);
	else
		tmp = d * 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[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision]}, If[LessEqual[d, -3.2e+20], N[(d * (-t$95$0)), $MachinePrecision], If[LessEqual[d, -4.4e-297], N[(d * N[Power[0.0, -0.5], $MachinePrecision]), $MachinePrecision], N[(d * t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if d < -3.2e20

   1. Initial program 74.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 72.7%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 8.7%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

   6. Taylor expanded in h around -inf 0.0%

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   7. Step-by-step derivation

      1. associate-*l* 0.0%

         \[\leadsto \color{blue}{d \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. unpow2 0.0%

         \[\leadsto d \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      3. rem-square-sqrt 63.6%

         \[\leadsto d \cdot \left(\color{blue}{-1} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      4. neg-mul-1 63.6%

         \[\leadsto d \cdot \color{blue}{\left(-\sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      5. unpow-1 63.6%

         \[\leadsto d \cdot \left(-\sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}}\right) \]

      6. metadata-eval 63.6%

         \[\leadsto d \cdot \left(-\sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}}\right) \]

      7. pow-sqr 63.6%

         \[\leadsto d \cdot \left(-\sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}}\right) \]

      8. rem-sqrt-square 63.6%

         \[\leadsto d \cdot \left(-\color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|}\right) \]

      9. rem-square-sqrt 63.2%

         \[\leadsto d \cdot \left(-\left|\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right|\right) \]

      10. fabs-sqr 63.2%

          \[\leadsto d \cdot \left(-\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right) \]

      11. rem-square-sqrt 63.6%

          \[\leadsto d \cdot \left(-\color{blue}{{\left(h \cdot \ell\right)}^{-0.5}}\right) \]

   8. Simplified 63.6%

      \[\leadsto \color{blue}{d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)} \]

   if -3.2e20 < d < -4.3999999999999997e-297

   1. Initial program 54.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 54.4%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. pow1 54.4%

         \[\leadsto \color{blue}{{\left(\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)\right)}^{1}} \]

      2. associate-*r* 54.3%

         \[\leadsto {\color{blue}{\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)}}^{1} \]

      3. sqrt-unprod 43.7%

         \[\leadsto {\left(\color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)}^{1} \]

      4. associate-*r* 43.7%

         \[\leadsto {\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \mathsf{fma}\left({\color{blue}{\left(\left(0.5 \cdot M\right) \cdot \frac{D}{d}\right)}}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)}^{1} \]

      5. associate-*r/ 43.7%

         \[\leadsto {\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(\left(0.5 \cdot M\right) \cdot \frac{D}{d}\right)}^{2}, \color{blue}{\frac{-0.5 \cdot h}{\ell}}, 1\right)\right)}^{1} \]

   6. Applied egg-rr 43.7%

      \[\leadsto \color{blue}{{\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(\left(0.5 \cdot M\right) \cdot \frac{D}{d}\right)}^{2}, \frac{-0.5 \cdot h}{\ell}, 1\right)\right)}^{1}} \]

   8. Simplified 31.9%

      \[\leadsto \color{blue}{\sqrt{\frac{\frac{{d}^{2}}{h}}{\ell}} \cdot \mathsf{fma}\left(h \cdot \frac{-0.5}{\ell}, {\left(D \cdot \left(\frac{M}{d} \cdot 0.5\right)\right)}^{2}, 1\right)} \]

   9. Taylor expanded in d around inf 9.6%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   10. Step-by-step derivation

       1. unpow-1 9.6%

          \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \]

       2. metadata-eval 9.6%

          \[\leadsto d \cdot \sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \]

       3. pow-sqr 9.6%

          \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}} \]

       4. rem-sqrt-square 9.6%

          \[\leadsto d \cdot \color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|} \]

       5. rem-square-sqrt 9.6%

          \[\leadsto d \cdot \left|\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right| \]

       6. fabs-sqr 9.6%

          \[\leadsto d \cdot \color{blue}{\left(\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}\right)} \]

       7. rem-square-sqrt 9.6%

          \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

   11. Simplified 9.6%

       \[\leadsto \color{blue}{d \cdot {\left(h \cdot \ell\right)}^{-0.5}} \]
   12. Step-by-step derivation

       1. expm1-log1p-u 9.6%

          \[\leadsto d \cdot {\color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(h \cdot \ell\right)\right)\right)}}^{-0.5} \]

       2. expm1-undefine 28.5%

          \[\leadsto d \cdot {\color{blue}{\left(e^{\mathsf{log1p}\left(h \cdot \ell\right)} - 1\right)}}^{-0.5} \]

   13. Applied egg-rr 28.5%

       \[\leadsto d \cdot {\color{blue}{\left(e^{\mathsf{log1p}\left(h \cdot \ell\right)} - 1\right)}}^{-0.5} \]
   14. Step-by-step derivation

       1. sub-neg 28.5%

          \[\leadsto d \cdot {\color{blue}{\left(e^{\mathsf{log1p}\left(h \cdot \ell\right)} + \left(-1\right)\right)}}^{-0.5} \]

       2. metadata-eval 28.5%

          \[\leadsto d \cdot {\left(e^{\mathsf{log1p}\left(h \cdot \ell\right)} + \color{blue}{-1}\right)}^{-0.5} \]

       3. +-commutative 28.5%

          \[\leadsto d \cdot {\color{blue}{\left(-1 + e^{\mathsf{log1p}\left(h \cdot \ell\right)}\right)}}^{-0.5} \]

       4. log1p-undefine 28.5%

          \[\leadsto d \cdot {\left(-1 + e^{\color{blue}{\log \left(1 + h \cdot \ell\right)}}\right)}^{-0.5} \]

       5. rem-exp-log 28.5%

          \[\leadsto d \cdot {\left(-1 + \color{blue}{\left(1 + h \cdot \ell\right)}\right)}^{-0.5} \]

       6. +-commutative 28.5%

          \[\leadsto d \cdot {\left(-1 + \color{blue}{\left(h \cdot \ell + 1\right)}\right)}^{-0.5} \]

       7. fma-define 28.5%

          \[\leadsto d \cdot {\left(-1 + \color{blue}{\mathsf{fma}\left(h, \ell, 1\right)}\right)}^{-0.5} \]

   15. Simplified 28.5%

       \[\leadsto d \cdot {\color{blue}{\left(-1 + \mathsf{fma}\left(h, \ell, 1\right)\right)}}^{-0.5} \]

   16. Taylor expanded in h around 0 45.7%

       \[\leadsto d \cdot {\left(-1 + \color{blue}{1}\right)}^{-0.5} \]

   if -4.3999999999999997e-297 < d

   1. Initial program 66.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 65.6%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. pow1 65.6%

         \[\leadsto \color{blue}{{\left(\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)\right)}^{1}} \]

      2. associate-*r* 65.6%

         \[\leadsto {\color{blue}{\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)}}^{1} \]

      3. sqrt-unprod 54.6%

         \[\leadsto {\left(\color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)}^{1} \]

      4. associate-*r* 54.6%

         \[\leadsto {\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \mathsf{fma}\left({\color{blue}{\left(\left(0.5 \cdot M\right) \cdot \frac{D}{d}\right)}}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)}^{1} \]

      5. associate-*r/ 54.6%

         \[\leadsto {\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(\left(0.5 \cdot M\right) \cdot \frac{D}{d}\right)}^{2}, \color{blue}{\frac{-0.5 \cdot h}{\ell}}, 1\right)\right)}^{1} \]

   6. Applied egg-rr 54.6%

      \[\leadsto \color{blue}{{\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(\left(0.5 \cdot M\right) \cdot \frac{D}{d}\right)}^{2}, \frac{-0.5 \cdot h}{\ell}, 1\right)\right)}^{1}} \]

   8. Simplified 46.7%

      \[\leadsto \color{blue}{\sqrt{\frac{\frac{{d}^{2}}{h}}{\ell}} \cdot \mathsf{fma}\left(h \cdot \frac{-0.5}{\ell}, {\left(D \cdot \left(\frac{M}{d} \cdot 0.5\right)\right)}^{2}, 1\right)} \]

   9. Taylor expanded in d around inf 39.7%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   10. Step-by-step derivation

       1. unpow-1 39.7%

          \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \]

       2. metadata-eval 39.7%

          \[\leadsto d \cdot \sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \]

       3. pow-sqr 39.7%

          \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}} \]

       4. rem-sqrt-square 39.7%

          \[\leadsto d \cdot \color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|} \]

       5. rem-square-sqrt 39.6%

          \[\leadsto d \cdot \left|\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right| \]

       6. fabs-sqr 39.6%

          \[\leadsto d \cdot \color{blue}{\left(\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}\right)} \]

       7. rem-square-sqrt 39.7%

          \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

   11. Simplified 39.7%

       \[\leadsto \color{blue}{d \cdot {\left(h \cdot \ell\right)}^{-0.5}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 46.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -3.2 \cdot 10^{+20}:\\ \;\;\;\;d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{elif}\;d \leq -4.4 \cdot 10^{-297}:\\ \;\;\;\;d \cdot {0}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;d \cdot {\left(h \cdot \ell\right)}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 37.1% accurate, 3.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -4.4 \cdot 10^{-297}:\\ \;\;\;\;d \cdot {0}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;d \cdot {\left(h \cdot \ell\right)}^{-0.5}\\ \end{array} \end{array} \]

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 -4.4e-297) (* d (pow 0.0 -0.5)) (* 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) {
	double tmp;
	if (d <= -4.4e-297) {
		tmp = d * pow(0.0, -0.5);
	} else {
		tmp = d * pow((h * l), -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 (d <= (-4.4d-297)) then
        tmp = d * (0.0d0 ** (-0.5d0))
    else
        tmp = d * ((h * l) ** (-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 (d <= -4.4e-297) {
		tmp = d * Math.pow(0.0, -0.5);
	} else {
		tmp = d * Math.pow((h * l), -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 d <= -4.4e-297:
		tmp = d * math.pow(0.0, -0.5)
	else:
		tmp = d * math.pow((h * l), -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 <= -4.4e-297)
		tmp = Float64(d * (0.0 ^ -0.5));
	else
		tmp = Float64(d * (Float64(h * l) ^ -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 (d <= -4.4e-297)
		tmp = d * (0.0 ^ -0.5);
	else
		tmp = d * ((h * l) ^ -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[d, -4.4e-297], N[(d * N[Power[0.0, -0.5], $MachinePrecision]), $MachinePrecision], N[(d * N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -4.3999999999999997e-297

   1. Initial program 64.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 63.5%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. pow1 63.5%

         \[\leadsto \color{blue}{{\left(\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)\right)}^{1}} \]

      2. associate-*r* 64.1%

         \[\leadsto {\color{blue}{\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)}}^{1} \]

      3. sqrt-unprod 56.5%

         \[\leadsto {\left(\color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)}^{1} \]

      4. associate-*r* 56.5%

         \[\leadsto {\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \mathsf{fma}\left({\color{blue}{\left(\left(0.5 \cdot M\right) \cdot \frac{D}{d}\right)}}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)}^{1} \]

      5. associate-*r/ 56.5%

         \[\leadsto {\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(\left(0.5 \cdot M\right) \cdot \frac{D}{d}\right)}^{2}, \color{blue}{\frac{-0.5 \cdot h}{\ell}}, 1\right)\right)}^{1} \]

   6. Applied egg-rr 56.5%

      \[\leadsto \color{blue}{{\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(\left(0.5 \cdot M\right) \cdot \frac{D}{d}\right)}^{2}, \frac{-0.5 \cdot h}{\ell}, 1\right)\right)}^{1}} \]

   8. Simplified 45.8%

      \[\leadsto \color{blue}{\sqrt{\frac{\frac{{d}^{2}}{h}}{\ell}} \cdot \mathsf{fma}\left(h \cdot \frac{-0.5}{\ell}, {\left(D \cdot \left(\frac{M}{d} \cdot 0.5\right)\right)}^{2}, 1\right)} \]

   9. Taylor expanded in d around inf 9.1%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   10. Step-by-step derivation

       1. unpow-1 9.1%

          \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \]

       2. metadata-eval 9.1%

          \[\leadsto d \cdot \sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \]

       3. pow-sqr 9.1%

          \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}} \]

       4. rem-sqrt-square 9.1%

          \[\leadsto d \cdot \color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|} \]

       5. rem-square-sqrt 9.1%

          \[\leadsto d \cdot \left|\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right| \]

       6. fabs-sqr 9.1%

          \[\leadsto d \cdot \color{blue}{\left(\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}\right)} \]

       7. rem-square-sqrt 9.1%

          \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

   11. Simplified 9.1%

       \[\leadsto \color{blue}{d \cdot {\left(h \cdot \ell\right)}^{-0.5}} \]
   12. Step-by-step derivation

       1. expm1-log1p-u 9.1%

          \[\leadsto d \cdot {\color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(h \cdot \ell\right)\right)\right)}}^{-0.5} \]

       2. expm1-undefine 21.8%

          \[\leadsto d \cdot {\color{blue}{\left(e^{\mathsf{log1p}\left(h \cdot \ell\right)} - 1\right)}}^{-0.5} \]

   13. Applied egg-rr 21.8%

       \[\leadsto d \cdot {\color{blue}{\left(e^{\mathsf{log1p}\left(h \cdot \ell\right)} - 1\right)}}^{-0.5} \]
   14. Step-by-step derivation

       1. sub-neg 21.8%

          \[\leadsto d \cdot {\color{blue}{\left(e^{\mathsf{log1p}\left(h \cdot \ell\right)} + \left(-1\right)\right)}}^{-0.5} \]

       2. metadata-eval 21.8%

          \[\leadsto d \cdot {\left(e^{\mathsf{log1p}\left(h \cdot \ell\right)} + \color{blue}{-1}\right)}^{-0.5} \]

       3. +-commutative 21.8%

          \[\leadsto d \cdot {\color{blue}{\left(-1 + e^{\mathsf{log1p}\left(h \cdot \ell\right)}\right)}}^{-0.5} \]

       4. log1p-undefine 21.8%

          \[\leadsto d \cdot {\left(-1 + e^{\color{blue}{\log \left(1 + h \cdot \ell\right)}}\right)}^{-0.5} \]

       5. rem-exp-log 21.8%

          \[\leadsto d \cdot {\left(-1 + \color{blue}{\left(1 + h \cdot \ell\right)}\right)}^{-0.5} \]

       6. +-commutative 21.8%

          \[\leadsto d \cdot {\left(-1 + \color{blue}{\left(h \cdot \ell + 1\right)}\right)}^{-0.5} \]

       7. fma-define 21.8%

          \[\leadsto d \cdot {\left(-1 + \color{blue}{\mathsf{fma}\left(h, \ell, 1\right)}\right)}^{-0.5} \]

   15. Simplified 21.8%

       \[\leadsto d \cdot {\color{blue}{\left(-1 + \mathsf{fma}\left(h, \ell, 1\right)\right)}}^{-0.5} \]

   16. Taylor expanded in h around 0 33.4%

       \[\leadsto d \cdot {\left(-1 + \color{blue}{1}\right)}^{-0.5} \]

   if -4.3999999999999997e-297 < d

   1. Initial program 66.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 65.6%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. pow1 65.6%

         \[\leadsto \color{blue}{{\left(\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)\right)}^{1}} \]

      2. associate-*r* 65.6%

         \[\leadsto {\color{blue}{\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)}}^{1} \]

      3. sqrt-unprod 54.6%

         \[\leadsto {\left(\color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)}^{1} \]

      4. associate-*r* 54.6%

         \[\leadsto {\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \mathsf{fma}\left({\color{blue}{\left(\left(0.5 \cdot M\right) \cdot \frac{D}{d}\right)}}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)}^{1} \]

      5. associate-*r/ 54.6%

         \[\leadsto {\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(\left(0.5 \cdot M\right) \cdot \frac{D}{d}\right)}^{2}, \color{blue}{\frac{-0.5 \cdot h}{\ell}}, 1\right)\right)}^{1} \]

   6. Applied egg-rr 54.6%

      \[\leadsto \color{blue}{{\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(\left(0.5 \cdot M\right) \cdot \frac{D}{d}\right)}^{2}, \frac{-0.5 \cdot h}{\ell}, 1\right)\right)}^{1}} \]

   8. Simplified 46.7%

      \[\leadsto \color{blue}{\sqrt{\frac{\frac{{d}^{2}}{h}}{\ell}} \cdot \mathsf{fma}\left(h \cdot \frac{-0.5}{\ell}, {\left(D \cdot \left(\frac{M}{d} \cdot 0.5\right)\right)}^{2}, 1\right)} \]

   9. Taylor expanded in d around inf 39.7%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   10. Step-by-step derivation

       1. unpow-1 39.7%

          \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \]

       2. metadata-eval 39.7%

          \[\leadsto d \cdot \sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \]

       3. pow-sqr 39.7%

          \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}} \]

       4. rem-sqrt-square 39.7%

          \[\leadsto d \cdot \color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|} \]

       5. rem-square-sqrt 39.6%

          \[\leadsto d \cdot \left|\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right| \]

       6. fabs-sqr 39.6%

          \[\leadsto d \cdot \color{blue}{\left(\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}\right)} \]

       7. rem-square-sqrt 39.7%

          \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

   11. Simplified 39.7%

       \[\leadsto \color{blue}{d \cdot {\left(h \cdot \ell\right)}^{-0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 36.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -4.4 \cdot 10^{-297}:\\ \;\;\;\;d \cdot {0}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;d \cdot {\left(h \cdot \ell\right)}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 26.2% 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 65.3%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

3. Simplified 64.6%

   \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. pow1 64.6%

      \[\leadsto \color{blue}{{\left(\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)\right)}^{1}} \]

   2. associate-*r* 64.9%

      \[\leadsto {\color{blue}{\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)}}^{1} \]

   3. sqrt-unprod 55.5%

      \[\leadsto {\left(\color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)}^{1} \]

   4. associate-*r* 55.5%

      \[\leadsto {\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \mathsf{fma}\left({\color{blue}{\left(\left(0.5 \cdot M\right) \cdot \frac{D}{d}\right)}}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)}^{1} \]

   5. associate-*r/ 55.5%

      \[\leadsto {\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(\left(0.5 \cdot M\right) \cdot \frac{D}{d}\right)}^{2}, \color{blue}{\frac{-0.5 \cdot h}{\ell}}, 1\right)\right)}^{1} \]

6. Applied egg-rr 55.5%

   \[\leadsto \color{blue}{{\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(\left(0.5 \cdot M\right) \cdot \frac{D}{d}\right)}^{2}, \frac{-0.5 \cdot h}{\ell}, 1\right)\right)}^{1}} \]

8. Simplified 46.3%

   \[\leadsto \color{blue}{\sqrt{\frac{\frac{{d}^{2}}{h}}{\ell}} \cdot \mathsf{fma}\left(h \cdot \frac{-0.5}{\ell}, {\left(D \cdot \left(\frac{M}{d} \cdot 0.5\right)\right)}^{2}, 1\right)} \]

9. Taylor expanded in d around inf 25.2%

   \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
10. Step-by-step derivation

    1. unpow-1 25.2%

       \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \]

    2. metadata-eval 25.2%

       \[\leadsto d \cdot \sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \]

    3. pow-sqr 25.3%

       \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}} \]

    4. rem-sqrt-square 25.3%

       \[\leadsto d \cdot \color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|} \]

    5. rem-square-sqrt 25.2%

       \[\leadsto d \cdot \left|\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right| \]

    6. fabs-sqr 25.2%

       \[\leadsto d \cdot \color{blue}{\left(\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}\right)} \]

    7. rem-square-sqrt 25.3%

       \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

11. Simplified 25.3%

    \[\leadsto \color{blue}{d \cdot {\left(h \cdot \ell\right)}^{-0.5}} \]
12. Add Preprocessing

Alternative 22: 4.3% accurate, 332.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ 0 \end{array} \]

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 0.0)

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) {
	return 0.0;
}

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 = 0.0d0
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 0.0;
}

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 0.0

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 0.0
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 = 0.0;
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_] := 0.0

Derivation

1. Initial program 65.3%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

3. Simplified 64.6%

   \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in d around inf 25.2%

   \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
6. Step-by-step derivation

   1. expm1-log1p-u 21.7%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)\right)} \]

   2. expm1-undefine 16.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} - 1} \]

   3. pow1/2 16.2%

      \[\leadsto e^{\mathsf{log1p}\left(d \cdot \color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{0.5}}\right)} - 1 \]

   4. inv-pow 16.2%

      \[\leadsto e^{\mathsf{log1p}\left(d \cdot {\color{blue}{\left({\left(h \cdot \ell\right)}^{-1}\right)}}^{0.5}\right)} - 1 \]

   5. pow-pow 16.2%

      \[\leadsto e^{\mathsf{log1p}\left(d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{\left(-1 \cdot 0.5\right)}}\right)} - 1 \]

   6. metadata-eval 16.2%

      \[\leadsto e^{\mathsf{log1p}\left(d \cdot {\left(h \cdot \ell\right)}^{\color{blue}{-0.5}}\right)} - 1 \]

7. Applied egg-rr 16.2%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right)} - 1} \]
8. Step-by-step derivation

   1. sub-neg 16.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right)} + \left(-1\right)} \]

   2. metadata-eval 16.2%

      \[\leadsto e^{\mathsf{log1p}\left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right)} + \color{blue}{-1} \]

   3. +-commutative 16.2%

      \[\leadsto \color{blue}{-1 + e^{\mathsf{log1p}\left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right)}} \]

   4. log1p-undefine 16.2%

      \[\leadsto -1 + e^{\color{blue}{\log \left(1 + d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right)}} \]

   5. rem-exp-log 19.7%

      \[\leadsto -1 + \color{blue}{\left(1 + d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right)} \]

   6. +-commutative 19.7%

      \[\leadsto -1 + \color{blue}{\left(d \cdot {\left(h \cdot \ell\right)}^{-0.5} + 1\right)} \]

   7. fma-define 19.7%

      \[\leadsto -1 + \color{blue}{\mathsf{fma}\left(d, {\left(h \cdot \ell\right)}^{-0.5}, 1\right)} \]

9. Simplified 19.7%

   \[\leadsto \color{blue}{-1 + \mathsf{fma}\left(d, {\left(h \cdot \ell\right)}^{-0.5}, 1\right)} \]

10. Taylor expanded in d around 0 4.5%

    \[\leadsto -1 + \color{blue}{1} \]
11. Step-by-step derivation

    1. metadata-eval 4.5%

       \[\leadsto \color{blue}{0} \]

12. Applied egg-rr 4.5%

    \[\leadsto \color{blue}{0} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024155 
(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)))))