Henrywood and Agarwal, Equation (12)

Percentage Accurate: 65.8% → 78.9%

Time: 30.8s

Alternatives: 19

Speedup: 1.4×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 65.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 78.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h}{\frac{\ell}{-0.125 \cdot {\left(M_m \cdot \frac{D_m}{d}\right)}^{2}}}\right)\\ t_1 := \sqrt{-d}\\ t_2 := \sqrt{-\ell}\\ \mathbf{if}\;\ell \leq -2.4 \cdot 10^{-206}:\\ \;\;\;\;\frac{t_1}{t_2} \cdot \left(\frac{t_1}{\sqrt{-h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{D_m}{2} \cdot \frac{M_m}{d}\right)}^{2} \cdot -0.5\right)\right)\right)\\ \mathbf{elif}\;\ell \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\frac{1}{\frac{t_2}{t_1}} \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0
         (*
          (sqrt (/ d h))
          (+ 1.0 (/ h (/ l (* -0.125 (pow (* M_m (/ D_m d)) 2.0)))))))
        (t_1 (sqrt (- d)))
        (t_2 (sqrt (- l))))
   (if (<= l -2.4e-206)
     (*
      (/ t_1 t_2)
      (*
       (/ t_1 (sqrt (- h)))
       (+ 1.0 (* (/ h l) (* (pow (* (/ D_m 2.0) (/ M_m d)) 2.0) -0.5)))))
     (if (<= l -1e-309)
       (* (/ 1.0 (/ t_2 t_1)) t_0)
       (* t_0 (/ (sqrt d) (sqrt l)))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = sqrt((d / h)) * (1.0 + (h / (l / (-0.125 * pow((M_m * (D_m / d)), 2.0)))));
	double t_1 = sqrt(-d);
	double t_2 = sqrt(-l);
	double tmp;
	if (l <= -2.4e-206) {
		tmp = (t_1 / t_2) * ((t_1 / sqrt(-h)) * (1.0 + ((h / l) * (pow(((D_m / 2.0) * (M_m / d)), 2.0) * -0.5))));
	} else if (l <= -1e-309) {
		tmp = (1.0 / (t_2 / t_1)) * t_0;
	} else {
		tmp = t_0 * (sqrt(d) / sqrt(l));
	}
	return tmp;
}

Fortran

M_m = abs(M)
D_m = abs(D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = sqrt((d / h)) * (1.0d0 + (h / (l / ((-0.125d0) * ((m_m * (d_m / d)) ** 2.0d0)))))
    t_1 = sqrt(-d)
    t_2 = sqrt(-l)
    if (l <= (-2.4d-206)) then
        tmp = (t_1 / t_2) * ((t_1 / sqrt(-h)) * (1.0d0 + ((h / l) * ((((d_m / 2.0d0) * (m_m / d)) ** 2.0d0) * (-0.5d0)))))
    else if (l <= (-1d-309)) then
        tmp = (1.0d0 / (t_2 / t_1)) * t_0
    else
        tmp = t_0 * (sqrt(d) / sqrt(l))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = Math.sqrt((d / h)) * (1.0 + (h / (l / (-0.125 * Math.pow((M_m * (D_m / d)), 2.0)))));
	double t_1 = Math.sqrt(-d);
	double t_2 = Math.sqrt(-l);
	double tmp;
	if (l <= -2.4e-206) {
		tmp = (t_1 / t_2) * ((t_1 / Math.sqrt(-h)) * (1.0 + ((h / l) * (Math.pow(((D_m / 2.0) * (M_m / d)), 2.0) * -0.5))));
	} else if (l <= -1e-309) {
		tmp = (1.0 / (t_2 / t_1)) * t_0;
	} else {
		tmp = t_0 * (Math.sqrt(d) / Math.sqrt(l));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = math.sqrt((d / h)) * (1.0 + (h / (l / (-0.125 * math.pow((M_m * (D_m / d)), 2.0)))))
	t_1 = math.sqrt(-d)
	t_2 = math.sqrt(-l)
	tmp = 0
	if l <= -2.4e-206:
		tmp = (t_1 / t_2) * ((t_1 / math.sqrt(-h)) * (1.0 + ((h / l) * (math.pow(((D_m / 2.0) * (M_m / d)), 2.0) * -0.5))))
	elif l <= -1e-309:
		tmp = (1.0 / (t_2 / t_1)) * t_0
	else:
		tmp = t_0 * (math.sqrt(d) / math.sqrt(l))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(sqrt(Float64(d / h)) * Float64(1.0 + Float64(h / Float64(l / Float64(-0.125 * (Float64(M_m * Float64(D_m / d)) ^ 2.0))))))
	t_1 = sqrt(Float64(-d))
	t_2 = sqrt(Float64(-l))
	tmp = 0.0
	if (l <= -2.4e-206)
		tmp = Float64(Float64(t_1 / t_2) * Float64(Float64(t_1 / sqrt(Float64(-h))) * Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(Float64(D_m / 2.0) * Float64(M_m / d)) ^ 2.0) * -0.5)))));
	elseif (l <= -1e-309)
		tmp = Float64(Float64(1.0 / Float64(t_2 / t_1)) * t_0);
	else
		tmp = Float64(t_0 * Float64(sqrt(d) / sqrt(l)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = sqrt((d / h)) * (1.0 + (h / (l / (-0.125 * ((M_m * (D_m / d)) ^ 2.0)))));
	t_1 = sqrt(-d);
	t_2 = sqrt(-l);
	tmp = 0.0;
	if (l <= -2.4e-206)
		tmp = (t_1 / t_2) * ((t_1 / sqrt(-h)) * (1.0 + ((h / l) * ((((D_m / 2.0) * (M_m / d)) ^ 2.0) * -0.5))));
	elseif (l <= -1e-309)
		tmp = (1.0 / (t_2 / t_1)) * t_0;
	else
		tmp = t_0 * (sqrt(d) / sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(h / N[(l / N[(-0.125 * N[Power[N[(M$95$m * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[(-d)], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[(-l)], $MachinePrecision]}, If[LessEqual[l, -2.4e-206], N[(N[(t$95$1 / t$95$2), $MachinePrecision] * N[(N[(t$95$1 / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(D$95$m / 2.0), $MachinePrecision] * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -1e-309], N[(N[(1.0 / N[(t$95$2 / t$95$1), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], N[(t$95$0 * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -2.4e-206

   1. Initial program 67.4%

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

   3. Simplified 68.9%

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

      1. frac-2neg 68.9%

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

      2. sqrt-div 73.5%

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

   5. Applied egg-rr 73.5%

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

      1. frac-2neg 73.5%

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

      2. sqrt-div 86.0%

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

   7. Applied egg-rr 86.0%

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

   if -2.4e-206 < l < -1.000000000000002e-309

   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 64.0%

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

   5. Applied egg-rr 70.0%

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

      1. clear-num 70.0%

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

      2. sqrt-div 74.1%

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

      3. metadata-eval 74.1%

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

   7. Applied egg-rr 74.1%

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

      1. frac-2neg 74.1%

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

      2. sqrt-div 92.1%

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

   9. Applied egg-rr 92.1%

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

   if -1.000000000000002e-309 < l

   1. Initial program 69.7%

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

   3. Simplified 68.3%

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

   5. Applied egg-rr 71.6%

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

      1. sqrt-div 81.8%

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

      2. div-inv 81.9%

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

   7. Applied egg-rr 81.9%

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

      1. associate-*r/ 81.8%

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

      2. *-rgt-identity 81.8%

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

   9. Simplified 81.8%

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

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.4 \cdot 10^{-206}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot -0.5\right)\right)\right)\\ \mathbf{elif}\;\ell \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{-\ell}}{\sqrt{-d}}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h}{\frac{\ell}{-0.125 \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h}{\frac{\ell}{-0.125 \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}}}\right)\right) \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\\ \end{array} \]

Alternative 2: 77.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{D_m \cdot M_m}{d \cdot 2}\right)}^{2}\right)\right)\\ t_1 := \frac{M_m}{\frac{d}{D_m}}\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;\frac{1}{\sqrt{\frac{\ell}{d}}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h}{-8 \cdot \left(\frac{1}{t_1} \cdot \frac{\ell}{t_1}\right)}\right)\right)\\ \mathbf{elif}\;t_0 \leq 2 \cdot 10^{+190}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq \infty:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}} \cdot \mathsf{fma}\left(h, {\left(M_m \cdot \frac{D_m}{d}\right)}^{2} \cdot \frac{0.125}{\ell}, 1\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\sqrt{h} \cdot \left(-0.125 \cdot {\left(D_m \cdot M_m\right)}^{2}\right)}{{\ell}^{0.75}}}{{\ell}^{0.75}}}{d}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0
         (*
          (* (pow (/ d h) 0.5) (pow (/ d l) 0.5))
          (- 1.0 (* (/ h l) (* 0.5 (pow (/ (* D_m M_m) (* d 2.0)) 2.0))))))
        (t_1 (/ M_m (/ d D_m))))
   (if (<= t_0 (- INFINITY))
     (*
      (/ 1.0 (sqrt (/ l d)))
      (* (sqrt (/ d h)) (+ 1.0 (/ h (* -8.0 (* (/ 1.0 t_1) (/ l t_1)))))))
     (if (<= t_0 2e+190)
       t_0
       (if (<= t_0 INFINITY)
         (fabs
          (*
           (/ d (sqrt (* l h)))
           (fma h (* (pow (* M_m (/ D_m d)) 2.0) (/ 0.125 l)) 1.0)))
         (/
          (/
           (/ (* (sqrt h) (* -0.125 (pow (* D_m M_m) 2.0))) (pow l 0.75))
           (pow l 0.75))
          d))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = (pow((d / h), 0.5) * pow((d / l), 0.5)) * (1.0 - ((h / l) * (0.5 * pow(((D_m * M_m) / (d * 2.0)), 2.0))));
	double t_1 = M_m / (d / D_m);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (1.0 / sqrt((l / d))) * (sqrt((d / h)) * (1.0 + (h / (-8.0 * ((1.0 / t_1) * (l / t_1))))));
	} else if (t_0 <= 2e+190) {
		tmp = t_0;
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = fabs(((d / sqrt((l * h))) * fma(h, (pow((M_m * (D_m / d)), 2.0) * (0.125 / l)), 1.0)));
	} else {
		tmp = (((sqrt(h) * (-0.125 * pow((D_m * M_m), 2.0))) / pow(l, 0.75)) / pow(l, 0.75)) / d;
	}
	return tmp;
}

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	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_m) / Float64(d * 2.0)) ^ 2.0)))))
	t_1 = Float64(M_m / Float64(d / D_m))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(1.0 / sqrt(Float64(l / d))) * Float64(sqrt(Float64(d / h)) * Float64(1.0 + Float64(h / Float64(-8.0 * Float64(Float64(1.0 / t_1) * Float64(l / t_1)))))));
	elseif (t_0 <= 2e+190)
		tmp = t_0;
	elseif (t_0 <= Inf)
		tmp = abs(Float64(Float64(d / sqrt(Float64(l * h))) * fma(h, Float64((Float64(M_m * Float64(D_m / d)) ^ 2.0) * Float64(0.125 / l)), 1.0)));
	else
		tmp = Float64(Float64(Float64(Float64(sqrt(h) * Float64(-0.125 * (Float64(D_m * M_m) ^ 2.0))) / (l ^ 0.75)) / (l ^ 0.75)) / d);
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(N[(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$95$m * M$95$m), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(M$95$m / N[(d / D$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(1.0 / N[Sqrt[N[(l / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(h / N[(-8.0 * N[(N[(1.0 / t$95$1), $MachinePrecision] * N[(l / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e+190], t$95$0, If[LessEqual[t$95$0, Infinity], N[Abs[N[(N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(h * N[(N[Power[N[(M$95$m * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(0.125 / l), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(N[(N[(N[Sqrt[h], $MachinePrecision] * N[(-0.125 * N[Power[N[(D$95$m * M$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[l, 0.75], $MachinePrecision]), $MachinePrecision] / N[Power[l, 0.75], $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 69.1%

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

   3. Simplified 74.0%

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

   5. Applied egg-rr 76.9%

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

      1. clear-num 76.9%

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

      2. sqrt-div 77.6%

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

      3. metadata-eval 77.6%

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

   7. Applied egg-rr 77.6%

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

      1. *-un-lft-identity 77.6%

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

      2. times-frac 77.6%

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

      3. metadata-eval 77.6%

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

      4. associate-*r/ 74.4%

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

      5. associate-*l/ 77.6%

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

      6. *-commutative 77.6%

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

   9. Applied egg-rr 77.6%

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

       1. *-un-lft-identity 77.6%

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

       2. unpow2 77.6%

          \[\leadsto \frac{1}{\sqrt{\frac{\ell}{d}}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h}{-8 \cdot \frac{1 \cdot \ell}{\color{blue}{\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)}}}\right)\right) \]

       3. times-frac 84.5%

          \[\leadsto \frac{1}{\sqrt{\frac{\ell}{d}}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h}{-8 \cdot \color{blue}{\left(\frac{1}{D \cdot \frac{M}{d}} \cdot \frac{\ell}{D \cdot \frac{M}{d}}\right)}}\right)\right) \]

       4. associate-*r/ 79.6%

          \[\leadsto \frac{1}{\sqrt{\frac{\ell}{d}}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h}{-8 \cdot \left(\frac{1}{\color{blue}{\frac{D \cdot M}{d}}} \cdot \frac{\ell}{D \cdot \frac{M}{d}}\right)}\right)\right) \]

       5. *-commutative 79.6%

          \[\leadsto \frac{1}{\sqrt{\frac{\ell}{d}}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h}{-8 \cdot \left(\frac{1}{\frac{\color{blue}{M \cdot D}}{d}} \cdot \frac{\ell}{D \cdot \frac{M}{d}}\right)}\right)\right) \]

       6. associate-/l* 82.8%

          \[\leadsto \frac{1}{\sqrt{\frac{\ell}{d}}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h}{-8 \cdot \left(\frac{1}{\color{blue}{\frac{M}{\frac{d}{D}}}} \cdot \frac{\ell}{D \cdot \frac{M}{d}}\right)}\right)\right) \]

       7. associate-*r/ 79.6%

          \[\leadsto \frac{1}{\sqrt{\frac{\ell}{d}}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h}{-8 \cdot \left(\frac{1}{\frac{M}{\frac{d}{D}}} \cdot \frac{\ell}{\color{blue}{\frac{D \cdot M}{d}}}\right)}\right)\right) \]

       8. *-commutative 79.6%

          \[\leadsto \frac{1}{\sqrt{\frac{\ell}{d}}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h}{-8 \cdot \left(\frac{1}{\frac{M}{\frac{d}{D}}} \cdot \frac{\ell}{\frac{\color{blue}{M \cdot D}}{d}}\right)}\right)\right) \]

       9. associate-/l* 84.4%

          \[\leadsto \frac{1}{\sqrt{\frac{\ell}{d}}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h}{-8 \cdot \left(\frac{1}{\frac{M}{\frac{d}{D}}} \cdot \frac{\ell}{\color{blue}{\frac{M}{\frac{d}{D}}}}\right)}\right)\right) \]

   11. Applied egg-rr 84.4%

       \[\leadsto \frac{1}{\sqrt{\frac{\ell}{d}}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h}{-8 \cdot \color{blue}{\left(\frac{1}{\frac{M}{\frac{d}{D}}} \cdot \frac{\ell}{\frac{M}{\frac{d}{D}}}\right)}}\right)\right) \]

   if -inf.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (-.f64 1 (*.f64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)) (/.f64 h l)))) < 2.0000000000000001e190

   1. Initial program 90.8%

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

   if 2.0000000000000001e190 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (-.f64 1 (*.f64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)) (/.f64 h l)))) < +inf.0

   1. Initial program 64.4%

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

   3. Simplified 64.4%

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

   5. Applied egg-rr 64.4%

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

      1. associate-/r/ 64.4%

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

      2. associate-*r/ 64.4%

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

      3. *-commutative 64.4%

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

      4. associate-*r/ 64.4%

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

   7. Simplified 64.4%

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

   9. Applied egg-rr 49.2%

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

       1. unpow2 49.2%

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

       2. rem-sqrt-square 97.4%

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

       3. +-commutative 97.4%

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

       4. fma-def 97.4%

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

       5. associate-/r/ 97.4%

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

       6. *-commutative 97.4%

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

       7. associate-*r/ 97.4%

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

       8. associate-*l/ 97.4%

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

       9. *-commutative 97.4%

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

   11. Simplified 97.4%

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

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

   1. Initial program 0.0%

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

   3. Simplified 0.0%

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

   4. Taylor expanded in d around 0 12.9%

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

      1. associate-*r* 12.9%

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

      2. *-commutative 12.9%

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

      3. *-commutative 12.9%

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

      4. unpow2 12.9%

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

      5. unpow2 12.9%

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

      6. swap-sqr 15.7%

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

      7. unpow2 15.7%

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

      8. *-commutative 15.7%

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

   6. Simplified 15.7%

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

      1. associate-*r/ 15.7%

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

      2. associate-*r/ 16.2%

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

      3. sqrt-div 13.1%

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

      4. sqrt-pow1 23.4%

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

      5. metadata-eval 23.4%

         \[\leadsto \frac{\frac{\sqrt{h}}{{\ell}^{\color{blue}{1.5}}} \cdot \left(-0.125 \cdot {\left(D \cdot M\right)}^{2}\right)}{d} \]

      6. *-commutative 23.4%

         \[\leadsto \frac{\frac{\sqrt{h}}{{\ell}^{1.5}} \cdot \left(-0.125 \cdot {\color{blue}{\left(M \cdot D\right)}}^{2}\right)}{d} \]

   8. Applied egg-rr 23.4%

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{h}}{{\ell}^{1.5}} \cdot \left(-0.125 \cdot {\left(M \cdot D\right)}^{2}\right)}{d}} \]
   9. Step-by-step derivation

      1. associate-*l/ 28.9%

         \[\leadsto \frac{\color{blue}{\frac{\sqrt{h} \cdot \left(-0.125 \cdot {\left(M \cdot D\right)}^{2}\right)}{{\ell}^{1.5}}}}{d} \]

      2. sqr-pow 28.8%

         \[\leadsto \frac{\frac{\sqrt{h} \cdot \left(-0.125 \cdot {\left(M \cdot D\right)}^{2}\right)}{\color{blue}{{\ell}^{\left(\frac{1.5}{2}\right)} \cdot {\ell}^{\left(\frac{1.5}{2}\right)}}}}{d} \]

      3. associate-/r* 39.5%

         \[\leadsto \frac{\color{blue}{\frac{\frac{\sqrt{h} \cdot \left(-0.125 \cdot {\left(M \cdot D\right)}^{2}\right)}{{\ell}^{\left(\frac{1.5}{2}\right)}}}{{\ell}^{\left(\frac{1.5}{2}\right)}}}}{d} \]

      4. *-commutative 39.5%

         \[\leadsto \frac{\frac{\frac{\sqrt{h} \cdot \left(-0.125 \cdot {\color{blue}{\left(D \cdot M\right)}}^{2}\right)}{{\ell}^{\left(\frac{1.5}{2}\right)}}}{{\ell}^{\left(\frac{1.5}{2}\right)}}}{d} \]

      5. metadata-eval 39.5%

         \[\leadsto \frac{\frac{\frac{\sqrt{h} \cdot \left(-0.125 \cdot {\left(D \cdot M\right)}^{2}\right)}{{\ell}^{\color{blue}{0.75}}}}{{\ell}^{\left(\frac{1.5}{2}\right)}}}{d} \]

      6. metadata-eval 39.5%

         \[\leadsto \frac{\frac{\frac{\sqrt{h} \cdot \left(-0.125 \cdot {\left(D \cdot M\right)}^{2}\right)}{{\ell}^{0.75}}}{{\ell}^{\color{blue}{0.75}}}}{d} \]

   10. Applied egg-rr 39.5%

       \[\leadsto \frac{\color{blue}{\frac{\frac{\sqrt{h} \cdot \left(-0.125 \cdot {\left(D \cdot M\right)}^{2}\right)}{{\ell}^{0.75}}}{{\ell}^{0.75}}}}{d} \]
3. Recombined 4 regimes into one program.

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}\right)\right) \leq -\infty:\\ \;\;\;\;\frac{1}{\sqrt{\frac{\ell}{d}}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h}{-8 \cdot \left(\frac{1}{\frac{M}{\frac{d}{D}}} \cdot \frac{\ell}{\frac{M}{\frac{d}{D}}}\right)}\right)\right)\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}\right)\right) \leq 2 \cdot 10^{+190}:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}\right)\right)\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}\right)\right) \leq \infty:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}} \cdot \mathsf{fma}\left(h, {\left(M \cdot \frac{D}{d}\right)}^{2} \cdot \frac{0.125}{\ell}, 1\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\sqrt{h} \cdot \left(-0.125 \cdot {\left(D \cdot M\right)}^{2}\right)}{{\ell}^{0.75}}}{{\ell}^{0.75}}}{d}\\ \end{array} \]

Alternative 3: 72.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{D_m \cdot M_m}{d \cdot 2}\right)}^{2}\right)\right)\\ t_1 := \frac{M_m}{\frac{d}{D_m}}\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;\frac{1}{\sqrt{\frac{\ell}{d}}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h}{-8 \cdot \left(\frac{1}{t_1} \cdot \frac{\ell}{t_1}\right)}\right)\right)\\ \mathbf{elif}\;t_0 \leq \infty:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\sqrt{h} \cdot \left(-0.125 \cdot {\left(D_m \cdot M_m\right)}^{2}\right)}{{\ell}^{0.75}}}{{\ell}^{0.75}}}{d}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0
         (*
          (* (pow (/ d h) 0.5) (pow (/ d l) 0.5))
          (- 1.0 (* (/ h l) (* 0.5 (pow (/ (* D_m M_m) (* d 2.0)) 2.0))))))
        (t_1 (/ M_m (/ d D_m))))
   (if (<= t_0 (- INFINITY))
     (*
      (/ 1.0 (sqrt (/ l d)))
      (* (sqrt (/ d h)) (+ 1.0 (/ h (* -8.0 (* (/ 1.0 t_1) (/ l t_1)))))))
     (if (<= t_0 INFINITY)
       t_0
       (/
        (/
         (/ (* (sqrt h) (* -0.125 (pow (* D_m M_m) 2.0))) (pow l 0.75))
         (pow l 0.75))
        d)))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = (pow((d / h), 0.5) * pow((d / l), 0.5)) * (1.0 - ((h / l) * (0.5 * pow(((D_m * M_m) / (d * 2.0)), 2.0))));
	double t_1 = M_m / (d / D_m);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (1.0 / sqrt((l / d))) * (sqrt((d / h)) * (1.0 + (h / (-8.0 * ((1.0 / t_1) * (l / t_1))))));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = t_0;
	} else {
		tmp = (((sqrt(h) * (-0.125 * pow((D_m * M_m), 2.0))) / pow(l, 0.75)) / pow(l, 0.75)) / d;
	}
	return tmp;
}

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double 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_m) / (d * 2.0)), 2.0))));
	double t_1 = M_m / (d / D_m);
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = (1.0 / Math.sqrt((l / d))) * (Math.sqrt((d / h)) * (1.0 + (h / (-8.0 * ((1.0 / t_1) * (l / t_1))))));
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = t_0;
	} else {
		tmp = (((Math.sqrt(h) * (-0.125 * Math.pow((D_m * M_m), 2.0))) / Math.pow(l, 0.75)) / Math.pow(l, 0.75)) / d;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	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_m) / (d * 2.0)), 2.0))))
	t_1 = M_m / (d / D_m)
	tmp = 0
	if t_0 <= -math.inf:
		tmp = (1.0 / math.sqrt((l / d))) * (math.sqrt((d / h)) * (1.0 + (h / (-8.0 * ((1.0 / t_1) * (l / t_1))))))
	elif t_0 <= math.inf:
		tmp = t_0
	else:
		tmp = (((math.sqrt(h) * (-0.125 * math.pow((D_m * M_m), 2.0))) / math.pow(l, 0.75)) / math.pow(l, 0.75)) / d
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	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_m) / Float64(d * 2.0)) ^ 2.0)))))
	t_1 = Float64(M_m / Float64(d / D_m))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(1.0 / sqrt(Float64(l / d))) * Float64(sqrt(Float64(d / h)) * Float64(1.0 + Float64(h / Float64(-8.0 * Float64(Float64(1.0 / t_1) * Float64(l / t_1)))))));
	elseif (t_0 <= Inf)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(Float64(sqrt(h) * Float64(-0.125 * (Float64(D_m * M_m) ^ 2.0))) / (l ^ 0.75)) / (l ^ 0.75)) / d);
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = (((d / h) ^ 0.5) * ((d / l) ^ 0.5)) * (1.0 - ((h / l) * (0.5 * (((D_m * M_m) / (d * 2.0)) ^ 2.0))));
	t_1 = M_m / (d / D_m);
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = (1.0 / sqrt((l / d))) * (sqrt((d / h)) * (1.0 + (h / (-8.0 * ((1.0 / t_1) * (l / t_1))))));
	elseif (t_0 <= Inf)
		tmp = t_0;
	else
		tmp = (((sqrt(h) * (-0.125 * ((D_m * M_m) ^ 2.0))) / (l ^ 0.75)) / (l ^ 0.75)) / d;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := 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$95$m * M$95$m), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(M$95$m / N[(d / D$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(1.0 / N[Sqrt[N[(l / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(h / N[(-8.0 * N[(N[(1.0 / t$95$1), $MachinePrecision] * N[(l / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], t$95$0, N[(N[(N[(N[(N[Sqrt[h], $MachinePrecision] * N[(-0.125 * N[Power[N[(D$95$m * M$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[l, 0.75], $MachinePrecision]), $MachinePrecision] / N[Power[l, 0.75], $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 69.1%

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

   3. Simplified 74.0%

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

   5. Applied egg-rr 76.9%

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

      1. clear-num 76.9%

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

      2. sqrt-div 77.6%

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

      3. metadata-eval 77.6%

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

   7. Applied egg-rr 77.6%

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

      1. *-un-lft-identity 77.6%

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

      2. times-frac 77.6%

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

      3. metadata-eval 77.6%

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

      4. associate-*r/ 74.4%

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

      5. associate-*l/ 77.6%

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

      6. *-commutative 77.6%

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

   9. Applied egg-rr 77.6%

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

       1. *-un-lft-identity 77.6%

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

       2. unpow2 77.6%

          \[\leadsto \frac{1}{\sqrt{\frac{\ell}{d}}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h}{-8 \cdot \frac{1 \cdot \ell}{\color{blue}{\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)}}}\right)\right) \]

       3. times-frac 84.5%

          \[\leadsto \frac{1}{\sqrt{\frac{\ell}{d}}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h}{-8 \cdot \color{blue}{\left(\frac{1}{D \cdot \frac{M}{d}} \cdot \frac{\ell}{D \cdot \frac{M}{d}}\right)}}\right)\right) \]

       4. associate-*r/ 79.6%

          \[\leadsto \frac{1}{\sqrt{\frac{\ell}{d}}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h}{-8 \cdot \left(\frac{1}{\color{blue}{\frac{D \cdot M}{d}}} \cdot \frac{\ell}{D \cdot \frac{M}{d}}\right)}\right)\right) \]

       5. *-commutative 79.6%

          \[\leadsto \frac{1}{\sqrt{\frac{\ell}{d}}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h}{-8 \cdot \left(\frac{1}{\frac{\color{blue}{M \cdot D}}{d}} \cdot \frac{\ell}{D \cdot \frac{M}{d}}\right)}\right)\right) \]

       6. associate-/l* 82.8%

          \[\leadsto \frac{1}{\sqrt{\frac{\ell}{d}}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h}{-8 \cdot \left(\frac{1}{\color{blue}{\frac{M}{\frac{d}{D}}}} \cdot \frac{\ell}{D \cdot \frac{M}{d}}\right)}\right)\right) \]

       7. associate-*r/ 79.6%

          \[\leadsto \frac{1}{\sqrt{\frac{\ell}{d}}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h}{-8 \cdot \left(\frac{1}{\frac{M}{\frac{d}{D}}} \cdot \frac{\ell}{\color{blue}{\frac{D \cdot M}{d}}}\right)}\right)\right) \]

       8. *-commutative 79.6%

          \[\leadsto \frac{1}{\sqrt{\frac{\ell}{d}}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h}{-8 \cdot \left(\frac{1}{\frac{M}{\frac{d}{D}}} \cdot \frac{\ell}{\frac{\color{blue}{M \cdot D}}{d}}\right)}\right)\right) \]

       9. associate-/l* 84.4%

          \[\leadsto \frac{1}{\sqrt{\frac{\ell}{d}}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h}{-8 \cdot \left(\frac{1}{\frac{M}{\frac{d}{D}}} \cdot \frac{\ell}{\color{blue}{\frac{M}{\frac{d}{D}}}}\right)}\right)\right) \]

   11. Applied egg-rr 84.4%

       \[\leadsto \frac{1}{\sqrt{\frac{\ell}{d}}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h}{-8 \cdot \color{blue}{\left(\frac{1}{\frac{M}{\frac{d}{D}}} \cdot \frac{\ell}{\frac{M}{\frac{d}{D}}}\right)}}\right)\right) \]

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

   1. Initial program 84.7%

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

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

   1. Initial program 0.0%

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

   3. Simplified 0.0%

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

   4. Taylor expanded in d around 0 12.9%

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

      1. associate-*r* 12.9%

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

      2. *-commutative 12.9%

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

      3. *-commutative 12.9%

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

      4. unpow2 12.9%

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

      5. unpow2 12.9%

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

      6. swap-sqr 15.7%

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

      7. unpow2 15.7%

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

      8. *-commutative 15.7%

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

   6. Simplified 15.7%

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

      1. associate-*r/ 15.7%

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

      2. associate-*r/ 16.2%

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

      3. sqrt-div 13.1%

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

      4. sqrt-pow1 23.4%

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

      5. metadata-eval 23.4%

         \[\leadsto \frac{\frac{\sqrt{h}}{{\ell}^{\color{blue}{1.5}}} \cdot \left(-0.125 \cdot {\left(D \cdot M\right)}^{2}\right)}{d} \]

      6. *-commutative 23.4%

         \[\leadsto \frac{\frac{\sqrt{h}}{{\ell}^{1.5}} \cdot \left(-0.125 \cdot {\color{blue}{\left(M \cdot D\right)}}^{2}\right)}{d} \]

   8. Applied egg-rr 23.4%

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{h}}{{\ell}^{1.5}} \cdot \left(-0.125 \cdot {\left(M \cdot D\right)}^{2}\right)}{d}} \]
   9. Step-by-step derivation

      1. associate-*l/ 28.9%

         \[\leadsto \frac{\color{blue}{\frac{\sqrt{h} \cdot \left(-0.125 \cdot {\left(M \cdot D\right)}^{2}\right)}{{\ell}^{1.5}}}}{d} \]

      2. sqr-pow 28.8%

         \[\leadsto \frac{\frac{\sqrt{h} \cdot \left(-0.125 \cdot {\left(M \cdot D\right)}^{2}\right)}{\color{blue}{{\ell}^{\left(\frac{1.5}{2}\right)} \cdot {\ell}^{\left(\frac{1.5}{2}\right)}}}}{d} \]

      3. associate-/r* 39.5%

         \[\leadsto \frac{\color{blue}{\frac{\frac{\sqrt{h} \cdot \left(-0.125 \cdot {\left(M \cdot D\right)}^{2}\right)}{{\ell}^{\left(\frac{1.5}{2}\right)}}}{{\ell}^{\left(\frac{1.5}{2}\right)}}}}{d} \]

      4. *-commutative 39.5%

         \[\leadsto \frac{\frac{\frac{\sqrt{h} \cdot \left(-0.125 \cdot {\color{blue}{\left(D \cdot M\right)}}^{2}\right)}{{\ell}^{\left(\frac{1.5}{2}\right)}}}{{\ell}^{\left(\frac{1.5}{2}\right)}}}{d} \]

      5. metadata-eval 39.5%

         \[\leadsto \frac{\frac{\frac{\sqrt{h} \cdot \left(-0.125 \cdot {\left(D \cdot M\right)}^{2}\right)}{{\ell}^{\color{blue}{0.75}}}}{{\ell}^{\left(\frac{1.5}{2}\right)}}}{d} \]

      6. metadata-eval 39.5%

         \[\leadsto \frac{\frac{\frac{\sqrt{h} \cdot \left(-0.125 \cdot {\left(D \cdot M\right)}^{2}\right)}{{\ell}^{0.75}}}{{\ell}^{\color{blue}{0.75}}}}{d} \]

   10. Applied egg-rr 39.5%

       \[\leadsto \frac{\color{blue}{\frac{\frac{\sqrt{h} \cdot \left(-0.125 \cdot {\left(D \cdot M\right)}^{2}\right)}{{\ell}^{0.75}}}{{\ell}^{0.75}}}}{d} \]
3. Recombined 3 regimes into one program.

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}\right)\right) \leq -\infty:\\ \;\;\;\;\frac{1}{\sqrt{\frac{\ell}{d}}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h}{-8 \cdot \left(\frac{1}{\frac{M}{\frac{d}{D}}} \cdot \frac{\ell}{\frac{M}{\frac{d}{D}}}\right)}\right)\right)\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}\right)\right) \leq \infty:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\sqrt{h} \cdot \left(-0.125 \cdot {\left(D \cdot M\right)}^{2}\right)}{{\ell}^{0.75}}}{{\ell}^{0.75}}}{d}\\ \end{array} \]

Alternative 4: 71.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{D_m \cdot M_m}{d \cdot 2}\right)}^{2}\right)\right)\\ t_1 := \frac{M_m}{\frac{d}{D_m}}\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;\frac{1}{\sqrt{\frac{\ell}{d}}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h}{-8 \cdot \left(\frac{1}{t_1} \cdot \frac{\ell}{t_1}\right)}\right)\right)\\ \mathbf{elif}\;t_0 \leq \infty:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{h} \cdot \left(-0.125 \cdot {\left(D_m \cdot M_m\right)}^{2}\right)}{d \cdot {\ell}^{1.5}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0
         (*
          (* (pow (/ d h) 0.5) (pow (/ d l) 0.5))
          (- 1.0 (* (/ h l) (* 0.5 (pow (/ (* D_m M_m) (* d 2.0)) 2.0))))))
        (t_1 (/ M_m (/ d D_m))))
   (if (<= t_0 (- INFINITY))
     (*
      (/ 1.0 (sqrt (/ l d)))
      (* (sqrt (/ d h)) (+ 1.0 (/ h (* -8.0 (* (/ 1.0 t_1) (/ l t_1)))))))
     (if (<= t_0 INFINITY)
       t_0
       (/ (* (sqrt h) (* -0.125 (pow (* D_m M_m) 2.0))) (* d (pow l 1.5)))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = (pow((d / h), 0.5) * pow((d / l), 0.5)) * (1.0 - ((h / l) * (0.5 * pow(((D_m * M_m) / (d * 2.0)), 2.0))));
	double t_1 = M_m / (d / D_m);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (1.0 / sqrt((l / d))) * (sqrt((d / h)) * (1.0 + (h / (-8.0 * ((1.0 / t_1) * (l / t_1))))));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = t_0;
	} else {
		tmp = (sqrt(h) * (-0.125 * pow((D_m * M_m), 2.0))) / (d * pow(l, 1.5));
	}
	return tmp;
}

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double 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_m) / (d * 2.0)), 2.0))));
	double t_1 = M_m / (d / D_m);
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = (1.0 / Math.sqrt((l / d))) * (Math.sqrt((d / h)) * (1.0 + (h / (-8.0 * ((1.0 / t_1) * (l / t_1))))));
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = t_0;
	} else {
		tmp = (Math.sqrt(h) * (-0.125 * Math.pow((D_m * M_m), 2.0))) / (d * Math.pow(l, 1.5));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = (math.pow((d / h), 0.5) * math.pow((d / l), 0.5)) * (1.0 - ((h / l) * (0.5 * math.pow(((D_m * M_m) / (d * 2.0)), 2.0))))
	t_1 = M_m / (d / D_m)
	tmp = 0
	if t_0 <= -math.inf:
		tmp = (1.0 / math.sqrt((l / d))) * (math.sqrt((d / h)) * (1.0 + (h / (-8.0 * ((1.0 / t_1) * (l / t_1))))))
	elif t_0 <= math.inf:
		tmp = t_0
	else:
		tmp = (math.sqrt(h) * (-0.125 * math.pow((D_m * M_m), 2.0))) / (d * math.pow(l, 1.5))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(Float64((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_m) / Float64(d * 2.0)) ^ 2.0)))))
	t_1 = Float64(M_m / Float64(d / D_m))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(1.0 / sqrt(Float64(l / d))) * Float64(sqrt(Float64(d / h)) * Float64(1.0 + Float64(h / Float64(-8.0 * Float64(Float64(1.0 / t_1) * Float64(l / t_1)))))));
	elseif (t_0 <= Inf)
		tmp = t_0;
	else
		tmp = Float64(Float64(sqrt(h) * Float64(-0.125 * (Float64(D_m * M_m) ^ 2.0))) / Float64(d * (l ^ 1.5)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = (((d / h) ^ 0.5) * ((d / l) ^ 0.5)) * (1.0 - ((h / l) * (0.5 * (((D_m * M_m) / (d * 2.0)) ^ 2.0))));
	t_1 = M_m / (d / D_m);
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = (1.0 / sqrt((l / d))) * (sqrt((d / h)) * (1.0 + (h / (-8.0 * ((1.0 / t_1) * (l / t_1))))));
	elseif (t_0 <= Inf)
		tmp = t_0;
	else
		tmp = (sqrt(h) * (-0.125 * ((D_m * M_m) ^ 2.0))) / (d * (l ^ 1.5));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(N[(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$95$m * M$95$m), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(M$95$m / N[(d / D$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(1.0 / N[Sqrt[N[(l / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(h / N[(-8.0 * N[(N[(1.0 / t$95$1), $MachinePrecision] * N[(l / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], t$95$0, N[(N[(N[Sqrt[h], $MachinePrecision] * N[(-0.125 * N[Power[N[(D$95$m * M$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * N[Power[l, 1.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 69.1%

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

   3. Simplified 74.0%

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

   5. Applied egg-rr 76.9%

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

      1. clear-num 76.9%

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

      2. sqrt-div 77.6%

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

      3. metadata-eval 77.6%

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

   7. Applied egg-rr 77.6%

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

      1. *-un-lft-identity 77.6%

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

      2. times-frac 77.6%

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

      3. metadata-eval 77.6%

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

      4. associate-*r/ 74.4%

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

      5. associate-*l/ 77.6%

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

      6. *-commutative 77.6%

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

   9. Applied egg-rr 77.6%

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

       1. *-un-lft-identity 77.6%

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

       2. unpow2 77.6%

          \[\leadsto \frac{1}{\sqrt{\frac{\ell}{d}}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h}{-8 \cdot \frac{1 \cdot \ell}{\color{blue}{\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)}}}\right)\right) \]

       3. times-frac 84.5%

          \[\leadsto \frac{1}{\sqrt{\frac{\ell}{d}}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h}{-8 \cdot \color{blue}{\left(\frac{1}{D \cdot \frac{M}{d}} \cdot \frac{\ell}{D \cdot \frac{M}{d}}\right)}}\right)\right) \]

       4. associate-*r/ 79.6%

          \[\leadsto \frac{1}{\sqrt{\frac{\ell}{d}}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h}{-8 \cdot \left(\frac{1}{\color{blue}{\frac{D \cdot M}{d}}} \cdot \frac{\ell}{D \cdot \frac{M}{d}}\right)}\right)\right) \]

       5. *-commutative 79.6%

          \[\leadsto \frac{1}{\sqrt{\frac{\ell}{d}}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h}{-8 \cdot \left(\frac{1}{\frac{\color{blue}{M \cdot D}}{d}} \cdot \frac{\ell}{D \cdot \frac{M}{d}}\right)}\right)\right) \]

       6. associate-/l* 82.8%

          \[\leadsto \frac{1}{\sqrt{\frac{\ell}{d}}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h}{-8 \cdot \left(\frac{1}{\color{blue}{\frac{M}{\frac{d}{D}}}} \cdot \frac{\ell}{D \cdot \frac{M}{d}}\right)}\right)\right) \]

       7. associate-*r/ 79.6%

          \[\leadsto \frac{1}{\sqrt{\frac{\ell}{d}}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h}{-8 \cdot \left(\frac{1}{\frac{M}{\frac{d}{D}}} \cdot \frac{\ell}{\color{blue}{\frac{D \cdot M}{d}}}\right)}\right)\right) \]

       8. *-commutative 79.6%

          \[\leadsto \frac{1}{\sqrt{\frac{\ell}{d}}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h}{-8 \cdot \left(\frac{1}{\frac{M}{\frac{d}{D}}} \cdot \frac{\ell}{\frac{\color{blue}{M \cdot D}}{d}}\right)}\right)\right) \]

       9. associate-/l* 84.4%

          \[\leadsto \frac{1}{\sqrt{\frac{\ell}{d}}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h}{-8 \cdot \left(\frac{1}{\frac{M}{\frac{d}{D}}} \cdot \frac{\ell}{\color{blue}{\frac{M}{\frac{d}{D}}}}\right)}\right)\right) \]

   11. Applied egg-rr 84.4%

       \[\leadsto \frac{1}{\sqrt{\frac{\ell}{d}}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h}{-8 \cdot \color{blue}{\left(\frac{1}{\frac{M}{\frac{d}{D}}} \cdot \frac{\ell}{\frac{M}{\frac{d}{D}}}\right)}}\right)\right) \]

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

   1. Initial program 84.7%

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

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

   1. Initial program 0.0%

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

   3. Simplified 0.0%

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

   4. Taylor expanded in d around 0 12.9%

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

      1. associate-*r* 12.9%

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

      2. *-commutative 12.9%

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

      3. *-commutative 12.9%

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

      4. unpow2 12.9%

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

      5. unpow2 12.9%

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

      6. swap-sqr 15.7%

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

      7. unpow2 15.7%

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

      8. *-commutative 15.7%

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

   6. Simplified 15.7%

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

      1. *-commutative 15.7%

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

      2. associate-*r/ 15.7%

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

      3. sqrt-div 12.9%

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

      4. frac-times 13.1%

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

      5. *-commutative 13.1%

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

      6. sqrt-pow1 28.7%

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

      7. metadata-eval 28.7%

         \[\leadsto \frac{\left(-0.125 \cdot {\left(M \cdot D\right)}^{2}\right) \cdot \sqrt{h}}{d \cdot {\ell}^{\color{blue}{1.5}}} \]

   8. Applied egg-rr 28.7%

      \[\leadsto \color{blue}{\frac{\left(-0.125 \cdot {\left(M \cdot D\right)}^{2}\right) \cdot \sqrt{h}}{d \cdot {\ell}^{1.5}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}\right)\right) \leq -\infty:\\ \;\;\;\;\frac{1}{\sqrt{\frac{\ell}{d}}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h}{-8 \cdot \left(\frac{1}{\frac{M}{\frac{d}{D}}} \cdot \frac{\ell}{\frac{M}{\frac{d}{D}}}\right)}\right)\right)\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}\right)\right) \leq \infty:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{h} \cdot \left(-0.125 \cdot {\left(D \cdot M\right)}^{2}\right)}{d \cdot {\ell}^{1.5}}\\ \end{array} \]

Alternative 5: 75.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := 1 + \frac{h}{\frac{\ell}{-0.125 \cdot {\left(M_m \cdot \frac{D_m}{d}\right)}^{2}}}\\ t_1 := \sqrt{\frac{d}{h}} \cdot t_0\\ t_2 := \sqrt{-d}\\ \mathbf{if}\;h \leq -8 \cdot 10^{-114}:\\ \;\;\;\;\frac{t_2}{\sqrt{-\ell}} \cdot t_1\\ \mathbf{elif}\;h \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\frac{t_2}{\sqrt{-h}} \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ h (/ l (* -0.125 (pow (* M_m (/ D_m d)) 2.0))))))
        (t_1 (* (sqrt (/ d h)) t_0))
        (t_2 (sqrt (- d))))
   (if (<= h -8e-114)
     (* (/ t_2 (sqrt (- l))) t_1)
     (if (<= h -1e-309)
       (* (sqrt (/ d l)) (* (/ t_2 (sqrt (- h))) t_0))
       (* t_1 (/ (sqrt d) (sqrt l)))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = 1.0 + (h / (l / (-0.125 * pow((M_m * (D_m / d)), 2.0))));
	double t_1 = sqrt((d / h)) * t_0;
	double t_2 = sqrt(-d);
	double tmp;
	if (h <= -8e-114) {
		tmp = (t_2 / sqrt(-l)) * t_1;
	} else if (h <= -1e-309) {
		tmp = sqrt((d / l)) * ((t_2 / sqrt(-h)) * t_0);
	} else {
		tmp = t_1 * (sqrt(d) / sqrt(l));
	}
	return tmp;
}

Fortran

M_m = abs(M)
D_m = abs(D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = 1.0d0 + (h / (l / ((-0.125d0) * ((m_m * (d_m / d)) ** 2.0d0))))
    t_1 = sqrt((d / h)) * t_0
    t_2 = sqrt(-d)
    if (h <= (-8d-114)) then
        tmp = (t_2 / sqrt(-l)) * t_1
    else if (h <= (-1d-309)) then
        tmp = sqrt((d / l)) * ((t_2 / sqrt(-h)) * t_0)
    else
        tmp = t_1 * (sqrt(d) / sqrt(l))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = 1.0 + (h / (l / (-0.125 * Math.pow((M_m * (D_m / d)), 2.0))));
	double t_1 = Math.sqrt((d / h)) * t_0;
	double t_2 = Math.sqrt(-d);
	double tmp;
	if (h <= -8e-114) {
		tmp = (t_2 / Math.sqrt(-l)) * t_1;
	} else if (h <= -1e-309) {
		tmp = Math.sqrt((d / l)) * ((t_2 / Math.sqrt(-h)) * t_0);
	} else {
		tmp = t_1 * (Math.sqrt(d) / Math.sqrt(l));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = 1.0 + (h / (l / (-0.125 * math.pow((M_m * (D_m / d)), 2.0))))
	t_1 = math.sqrt((d / h)) * t_0
	t_2 = math.sqrt(-d)
	tmp = 0
	if h <= -8e-114:
		tmp = (t_2 / math.sqrt(-l)) * t_1
	elif h <= -1e-309:
		tmp = math.sqrt((d / l)) * ((t_2 / math.sqrt(-h)) * t_0)
	else:
		tmp = t_1 * (math.sqrt(d) / math.sqrt(l))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(1.0 + Float64(h / Float64(l / Float64(-0.125 * (Float64(M_m * Float64(D_m / d)) ^ 2.0)))))
	t_1 = Float64(sqrt(Float64(d / h)) * t_0)
	t_2 = sqrt(Float64(-d))
	tmp = 0.0
	if (h <= -8e-114)
		tmp = Float64(Float64(t_2 / sqrt(Float64(-l))) * t_1);
	elseif (h <= -1e-309)
		tmp = Float64(sqrt(Float64(d / l)) * Float64(Float64(t_2 / sqrt(Float64(-h))) * t_0));
	else
		tmp = Float64(t_1 * Float64(sqrt(d) / sqrt(l)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = 1.0 + (h / (l / (-0.125 * ((M_m * (D_m / d)) ^ 2.0))));
	t_1 = sqrt((d / h)) * t_0;
	t_2 = sqrt(-d);
	tmp = 0.0;
	if (h <= -8e-114)
		tmp = (t_2 / sqrt(-l)) * t_1;
	elseif (h <= -1e-309)
		tmp = sqrt((d / l)) * ((t_2 / sqrt(-h)) * t_0);
	else
		tmp = t_1 * (sqrt(d) / sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(1.0 + N[(h / N[(l / N[(-0.125 * N[Power[N[(M$95$m * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[(-d)], $MachinePrecision]}, If[LessEqual[h, -8e-114], N[(N[(t$95$2 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[h, -1e-309], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(N[(t$95$2 / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if h < -8.0000000000000004e-114

   1. Initial program 65.9%

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

   3. Simplified 67.7%

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

   5. Applied egg-rr 69.8%

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

      1. frac-2neg 67.7%

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

      2. sqrt-div 74.1%

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

   7. Applied egg-rr 79.6%

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

   if -8.0000000000000004e-114 < h < -1.000000000000002e-309

   1. Initial program 69.6%

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

   3. Simplified 69.6%

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

   5. Applied egg-rr 67.2%

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

      1. frac-2neg 69.7%

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

      2. sqrt-div 88.9%

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

   7. Applied egg-rr 86.7%

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

   if -1.000000000000002e-309 < h

   1. Initial program 69.7%

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

   3. Simplified 68.3%

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

   5. Applied egg-rr 71.6%

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

      1. sqrt-div 81.8%

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

      2. div-inv 81.9%

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

   7. Applied egg-rr 81.9%

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

      1. associate-*r/ 81.8%

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

      2. *-rgt-identity 81.8%

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

   9. Simplified 81.8%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -8 \cdot 10^{-114}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h}{\frac{\ell}{-0.125 \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}}}\right)\right)\\ \mathbf{elif}\;h \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(1 + \frac{h}{\frac{\ell}{-0.125 \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h}{\frac{\ell}{-0.125 \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}}}\right)\right) \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\\ \end{array} \]

Alternative 6: 77.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{D_m}{2} \cdot \frac{M_m}{d}\right)}^{2} \cdot -0.5\right)\right)\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;d \leq 1.36 \cdot 10^{-209}:\\ \;\;\;\;\frac{\frac{\frac{\sqrt{h} \cdot \left(-0.125 \cdot {\left(D_m \cdot M_m\right)}^{2}\right)}{{\ell}^{0.75}}}{{\ell}^{0.75}}}{d}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h}{\frac{\ell}{-0.125 \cdot {\left(M_m \cdot \frac{D_m}{d}\right)}^{2}}}\right)\right) \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= d -2e-310)
   (*
    (*
     (/ (sqrt (- d)) (sqrt (- h)))
     (+ 1.0 (* (/ h l) (* (pow (* (/ D_m 2.0) (/ M_m d)) 2.0) -0.5))))
    (sqrt (/ d l)))
   (if (<= d 1.36e-209)
     (/
      (/
       (/ (* (sqrt h) (* -0.125 (pow (* D_m M_m) 2.0))) (pow l 0.75))
       (pow l 0.75))
      d)
     (*
      (*
       (sqrt (/ d h))
       (+ 1.0 (/ h (/ l (* -0.125 (pow (* M_m (/ D_m d)) 2.0))))))
      (/ (sqrt d) (sqrt l))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= -2e-310) {
		tmp = ((sqrt(-d) / sqrt(-h)) * (1.0 + ((h / l) * (pow(((D_m / 2.0) * (M_m / d)), 2.0) * -0.5)))) * sqrt((d / l));
	} else if (d <= 1.36e-209) {
		tmp = (((sqrt(h) * (-0.125 * pow((D_m * M_m), 2.0))) / pow(l, 0.75)) / pow(l, 0.75)) / d;
	} else {
		tmp = (sqrt((d / h)) * (1.0 + (h / (l / (-0.125 * pow((M_m * (D_m / d)), 2.0)))))) * (sqrt(d) / sqrt(l));
	}
	return tmp;
}

Fortran

M_m = abs(M)
D_m = abs(D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (d <= (-2d-310)) then
        tmp = ((sqrt(-d) / sqrt(-h)) * (1.0d0 + ((h / l) * ((((d_m / 2.0d0) * (m_m / d)) ** 2.0d0) * (-0.5d0))))) * sqrt((d / l))
    else if (d <= 1.36d-209) then
        tmp = (((sqrt(h) * ((-0.125d0) * ((d_m * m_m) ** 2.0d0))) / (l ** 0.75d0)) / (l ** 0.75d0)) / d
    else
        tmp = (sqrt((d / h)) * (1.0d0 + (h / (l / ((-0.125d0) * ((m_m * (d_m / d)) ** 2.0d0)))))) * (sqrt(d) / sqrt(l))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= -2e-310) {
		tmp = ((Math.sqrt(-d) / Math.sqrt(-h)) * (1.0 + ((h / l) * (Math.pow(((D_m / 2.0) * (M_m / d)), 2.0) * -0.5)))) * Math.sqrt((d / l));
	} else if (d <= 1.36e-209) {
		tmp = (((Math.sqrt(h) * (-0.125 * Math.pow((D_m * M_m), 2.0))) / Math.pow(l, 0.75)) / Math.pow(l, 0.75)) / d;
	} else {
		tmp = (Math.sqrt((d / h)) * (1.0 + (h / (l / (-0.125 * Math.pow((M_m * (D_m / d)), 2.0)))))) * (Math.sqrt(d) / Math.sqrt(l));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if d <= -2e-310:
		tmp = ((math.sqrt(-d) / math.sqrt(-h)) * (1.0 + ((h / l) * (math.pow(((D_m / 2.0) * (M_m / d)), 2.0) * -0.5)))) * math.sqrt((d / l))
	elif d <= 1.36e-209:
		tmp = (((math.sqrt(h) * (-0.125 * math.pow((D_m * M_m), 2.0))) / math.pow(l, 0.75)) / math.pow(l, 0.75)) / d
	else:
		tmp = (math.sqrt((d / h)) * (1.0 + (h / (l / (-0.125 * math.pow((M_m * (D_m / d)), 2.0)))))) * (math.sqrt(d) / math.sqrt(l))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (d <= -2e-310)
		tmp = Float64(Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-h))) * Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(Float64(D_m / 2.0) * Float64(M_m / d)) ^ 2.0) * -0.5)))) * sqrt(Float64(d / l)));
	elseif (d <= 1.36e-209)
		tmp = Float64(Float64(Float64(Float64(sqrt(h) * Float64(-0.125 * (Float64(D_m * M_m) ^ 2.0))) / (l ^ 0.75)) / (l ^ 0.75)) / d);
	else
		tmp = Float64(Float64(sqrt(Float64(d / h)) * Float64(1.0 + Float64(h / Float64(l / Float64(-0.125 * (Float64(M_m * Float64(D_m / d)) ^ 2.0)))))) * Float64(sqrt(d) / sqrt(l)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (d <= -2e-310)
		tmp = ((sqrt(-d) / sqrt(-h)) * (1.0 + ((h / l) * ((((D_m / 2.0) * (M_m / d)) ^ 2.0) * -0.5)))) * sqrt((d / l));
	elseif (d <= 1.36e-209)
		tmp = (((sqrt(h) * (-0.125 * ((D_m * M_m) ^ 2.0))) / (l ^ 0.75)) / (l ^ 0.75)) / d;
	else
		tmp = (sqrt((d / h)) * (1.0 + (h / (l / (-0.125 * ((M_m * (D_m / d)) ^ 2.0)))))) * (sqrt(d) / sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[d, -2e-310], N[(N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(D$95$m / 2.0), $MachinePrecision] * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.36e-209], N[(N[(N[(N[(N[Sqrt[h], $MachinePrecision] * N[(-0.125 * N[Power[N[(D$95$m * M$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[l, 0.75], $MachinePrecision]), $MachinePrecision] / N[Power[l, 0.75], $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], N[(N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(h / N[(l / N[(-0.125 * N[Power[N[(M$95$m * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.999999999999994e-310

   1. Initial program 66.9%

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

   3. Simplified 68.2%

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

      1. frac-2neg 72.9%

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

      2. sqrt-div 83.5%

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

   5. Applied egg-rr 77.1%

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

   if -1.999999999999994e-310 < d < 1.3599999999999999e-209

   1. Initial program 40.4%

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

   3. Simplified 40.4%

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

   4. Taylor expanded in d around 0 44.9%

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

      1. associate-*r* 44.9%

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

      2. *-commutative 44.9%

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

      3. *-commutative 44.9%

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

      4. unpow2 44.9%

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

      5. unpow2 44.9%

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

      6. swap-sqr 58.7%

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

      7. unpow2 58.7%

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

      8. *-commutative 58.7%

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

   6. Simplified 58.7%

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

      1. associate-*r/ 58.7%

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

      2. associate-*r/ 58.8%

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

      3. sqrt-div 58.8%

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

      4. sqrt-pow1 72.2%

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

      5. metadata-eval 72.2%

         \[\leadsto \frac{\frac{\sqrt{h}}{{\ell}^{\color{blue}{1.5}}} \cdot \left(-0.125 \cdot {\left(D \cdot M\right)}^{2}\right)}{d} \]

      6. *-commutative 72.2%

         \[\leadsto \frac{\frac{\sqrt{h}}{{\ell}^{1.5}} \cdot \left(-0.125 \cdot {\color{blue}{\left(M \cdot D\right)}}^{2}\right)}{d} \]

   8. Applied egg-rr 72.2%

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{h}}{{\ell}^{1.5}} \cdot \left(-0.125 \cdot {\left(M \cdot D\right)}^{2}\right)}{d}} \]
   9. Step-by-step derivation

      1. associate-*l/ 76.8%

         \[\leadsto \frac{\color{blue}{\frac{\sqrt{h} \cdot \left(-0.125 \cdot {\left(M \cdot D\right)}^{2}\right)}{{\ell}^{1.5}}}}{d} \]

      2. sqr-pow 76.7%

         \[\leadsto \frac{\frac{\sqrt{h} \cdot \left(-0.125 \cdot {\left(M \cdot D\right)}^{2}\right)}{\color{blue}{{\ell}^{\left(\frac{1.5}{2}\right)} \cdot {\ell}^{\left(\frac{1.5}{2}\right)}}}}{d} \]

      3. associate-/r* 86.2%

         \[\leadsto \frac{\color{blue}{\frac{\frac{\sqrt{h} \cdot \left(-0.125 \cdot {\left(M \cdot D\right)}^{2}\right)}{{\ell}^{\left(\frac{1.5}{2}\right)}}}{{\ell}^{\left(\frac{1.5}{2}\right)}}}}{d} \]

      4. *-commutative 86.2%

         \[\leadsto \frac{\frac{\frac{\sqrt{h} \cdot \left(-0.125 \cdot {\color{blue}{\left(D \cdot M\right)}}^{2}\right)}{{\ell}^{\left(\frac{1.5}{2}\right)}}}{{\ell}^{\left(\frac{1.5}{2}\right)}}}{d} \]

      5. metadata-eval 86.2%

         \[\leadsto \frac{\frac{\frac{\sqrt{h} \cdot \left(-0.125 \cdot {\left(D \cdot M\right)}^{2}\right)}{{\ell}^{\color{blue}{0.75}}}}{{\ell}^{\left(\frac{1.5}{2}\right)}}}{d} \]

      6. metadata-eval 86.2%

         \[\leadsto \frac{\frac{\frac{\sqrt{h} \cdot \left(-0.125 \cdot {\left(D \cdot M\right)}^{2}\right)}{{\ell}^{0.75}}}{{\ell}^{\color{blue}{0.75}}}}{d} \]

   10. Applied egg-rr 86.2%

       \[\leadsto \frac{\color{blue}{\frac{\frac{\sqrt{h} \cdot \left(-0.125 \cdot {\left(D \cdot M\right)}^{2}\right)}{{\ell}^{0.75}}}{{\ell}^{0.75}}}}{d} \]

   if 1.3599999999999999e-209 < d

   1. Initial program 76.2%

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

   3. Simplified 75.4%

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

   5. Applied egg-rr 78.4%

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

      1. sqrt-div 85.8%

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

      2. div-inv 85.8%

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

   7. Applied egg-rr 85.8%

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

      1. associate-*r/ 85.8%

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

      2. *-rgt-identity 85.8%

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

   9. Simplified 85.8%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot -0.5\right)\right)\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;d \leq 1.36 \cdot 10^{-209}:\\ \;\;\;\;\frac{\frac{\frac{\sqrt{h} \cdot \left(-0.125 \cdot {\left(D \cdot M\right)}^{2}\right)}{{\ell}^{0.75}}}{{\ell}^{0.75}}}{d}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h}{\frac{\ell}{-0.125 \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}}}\right)\right) \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\\ \end{array} \]

Alternative 7: 70.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \frac{M_m}{\frac{d}{D_m}}\\ \mathbf{if}\;d \leq -3.3 \cdot 10^{-307} \lor \neg \left(d \leq 8.2 \cdot 10^{-220}\right):\\ \;\;\;\;\frac{1}{\sqrt{\frac{\ell}{d}}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h}{-8 \cdot \left(\frac{1}{t_0} \cdot \frac{\ell}{t_0}\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{h} \cdot \left(-0.125 \cdot {\left(D_m \cdot M_m\right)}^{2}\right)}{d \cdot {\ell}^{1.5}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (/ M_m (/ d D_m))))
   (if (or (<= d -3.3e-307) (not (<= d 8.2e-220)))
     (*
      (/ 1.0 (sqrt (/ l d)))
      (* (sqrt (/ d h)) (+ 1.0 (/ h (* -8.0 (* (/ 1.0 t_0) (/ l t_0)))))))
     (/ (* (sqrt h) (* -0.125 (pow (* D_m M_m) 2.0))) (* d (pow l 1.5))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = M_m / (d / D_m);
	double tmp;
	if ((d <= -3.3e-307) || !(d <= 8.2e-220)) {
		tmp = (1.0 / sqrt((l / d))) * (sqrt((d / h)) * (1.0 + (h / (-8.0 * ((1.0 / t_0) * (l / t_0))))));
	} else {
		tmp = (sqrt(h) * (-0.125 * pow((D_m * M_m), 2.0))) / (d * pow(l, 1.5));
	}
	return tmp;
}

Fortran

M_m = abs(M)
D_m = abs(D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = m_m / (d / d_m)
    if ((d <= (-3.3d-307)) .or. (.not. (d <= 8.2d-220))) then
        tmp = (1.0d0 / sqrt((l / d))) * (sqrt((d / h)) * (1.0d0 + (h / ((-8.0d0) * ((1.0d0 / t_0) * (l / t_0))))))
    else
        tmp = (sqrt(h) * ((-0.125d0) * ((d_m * m_m) ** 2.0d0))) / (d * (l ** 1.5d0))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = M_m / (d / D_m);
	double tmp;
	if ((d <= -3.3e-307) || !(d <= 8.2e-220)) {
		tmp = (1.0 / Math.sqrt((l / d))) * (Math.sqrt((d / h)) * (1.0 + (h / (-8.0 * ((1.0 / t_0) * (l / t_0))))));
	} else {
		tmp = (Math.sqrt(h) * (-0.125 * Math.pow((D_m * M_m), 2.0))) / (d * Math.pow(l, 1.5));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = M_m / (d / D_m)
	tmp = 0
	if (d <= -3.3e-307) or not (d <= 8.2e-220):
		tmp = (1.0 / math.sqrt((l / d))) * (math.sqrt((d / h)) * (1.0 + (h / (-8.0 * ((1.0 / t_0) * (l / t_0))))))
	else:
		tmp = (math.sqrt(h) * (-0.125 * math.pow((D_m * M_m), 2.0))) / (d * math.pow(l, 1.5))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(M_m / Float64(d / D_m))
	tmp = 0.0
	if ((d <= -3.3e-307) || !(d <= 8.2e-220))
		tmp = Float64(Float64(1.0 / sqrt(Float64(l / d))) * Float64(sqrt(Float64(d / h)) * Float64(1.0 + Float64(h / Float64(-8.0 * Float64(Float64(1.0 / t_0) * Float64(l / t_0)))))));
	else
		tmp = Float64(Float64(sqrt(h) * Float64(-0.125 * (Float64(D_m * M_m) ^ 2.0))) / Float64(d * (l ^ 1.5)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = M_m / (d / D_m);
	tmp = 0.0;
	if ((d <= -3.3e-307) || ~((d <= 8.2e-220)))
		tmp = (1.0 / sqrt((l / d))) * (sqrt((d / h)) * (1.0 + (h / (-8.0 * ((1.0 / t_0) * (l / t_0))))));
	else
		tmp = (sqrt(h) * (-0.125 * ((D_m * M_m) ^ 2.0))) / (d * (l ^ 1.5));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(M$95$m / N[(d / D$95$m), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[d, -3.3e-307], N[Not[LessEqual[d, 8.2e-220]], $MachinePrecision]], N[(N[(1.0 / N[Sqrt[N[(l / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(h / N[(-8.0 * N[(N[(1.0 / t$95$0), $MachinePrecision] * N[(l / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[h], $MachinePrecision] * N[(-0.125 * N[Power[N[(D$95$m * M$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * N[Power[l, 1.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if d < -3.3e-307 or 8.19999999999999982e-220 < d

   1. Initial program 70.5%

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

   3. Simplified 71.0%

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

   5. Applied egg-rr 72.7%

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

      1. clear-num 72.3%

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

      2. sqrt-div 73.0%

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

      3. metadata-eval 73.0%

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

   7. Applied egg-rr 73.0%

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

      1. *-un-lft-identity 73.0%

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

      2. times-frac 73.0%

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

      3. metadata-eval 73.0%

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

      4. associate-*r/ 72.2%

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

      5. associate-*l/ 72.3%

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

      6. *-commutative 72.3%

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

   9. Applied egg-rr 72.3%

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

       1. *-un-lft-identity 72.3%

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

       2. unpow2 72.3%

          \[\leadsto \frac{1}{\sqrt{\frac{\ell}{d}}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h}{-8 \cdot \frac{1 \cdot \ell}{\color{blue}{\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)}}}\right)\right) \]

       3. times-frac 73.9%

          \[\leadsto \frac{1}{\sqrt{\frac{\ell}{d}}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h}{-8 \cdot \color{blue}{\left(\frac{1}{D \cdot \frac{M}{d}} \cdot \frac{\ell}{D \cdot \frac{M}{d}}\right)}}\right)\right) \]

       4. associate-*r/ 72.8%

          \[\leadsto \frac{1}{\sqrt{\frac{\ell}{d}}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h}{-8 \cdot \left(\frac{1}{\color{blue}{\frac{D \cdot M}{d}}} \cdot \frac{\ell}{D \cdot \frac{M}{d}}\right)}\right)\right) \]

       5. *-commutative 72.8%

          \[\leadsto \frac{1}{\sqrt{\frac{\ell}{d}}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h}{-8 \cdot \left(\frac{1}{\frac{\color{blue}{M \cdot D}}{d}} \cdot \frac{\ell}{D \cdot \frac{M}{d}}\right)}\right)\right) \]

       6. associate-/l* 73.6%

          \[\leadsto \frac{1}{\sqrt{\frac{\ell}{d}}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h}{-8 \cdot \left(\frac{1}{\color{blue}{\frac{M}{\frac{d}{D}}}} \cdot \frac{\ell}{D \cdot \frac{M}{d}}\right)}\right)\right) \]

       7. associate-*r/ 73.5%

          \[\leadsto \frac{1}{\sqrt{\frac{\ell}{d}}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h}{-8 \cdot \left(\frac{1}{\frac{M}{\frac{d}{D}}} \cdot \frac{\ell}{\color{blue}{\frac{D \cdot M}{d}}}\right)}\right)\right) \]

       8. *-commutative 73.5%

          \[\leadsto \frac{1}{\sqrt{\frac{\ell}{d}}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h}{-8 \cdot \left(\frac{1}{\frac{M}{\frac{d}{D}}} \cdot \frac{\ell}{\frac{\color{blue}{M \cdot D}}{d}}\right)}\right)\right) \]

       9. associate-/l* 74.7%

          \[\leadsto \frac{1}{\sqrt{\frac{\ell}{d}}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h}{-8 \cdot \left(\frac{1}{\frac{M}{\frac{d}{D}}} \cdot \frac{\ell}{\color{blue}{\frac{M}{\frac{d}{D}}}}\right)}\right)\right) \]

   11. Applied egg-rr 74.7%

       \[\leadsto \frac{1}{\sqrt{\frac{\ell}{d}}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h}{-8 \cdot \color{blue}{\left(\frac{1}{\frac{M}{\frac{d}{D}}} \cdot \frac{\ell}{\frac{M}{\frac{d}{D}}}\right)}}\right)\right) \]

   if -3.3e-307 < d < 8.19999999999999982e-220

   1. Initial program 42.5%

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

   3. Simplified 42.5%

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

   4. Taylor expanded in d around 0 47.3%

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

      1. associate-*r* 47.3%

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

      2. *-commutative 47.3%

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

      3. *-commutative 47.3%

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

      4. unpow2 47.3%

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

      5. unpow2 47.3%

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

      6. swap-sqr 61.7%

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

      7. unpow2 61.7%

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

      8. *-commutative 61.7%

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

   6. Simplified 61.7%

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

      1. *-commutative 61.7%

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

      2. associate-*r/ 61.7%

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

      3. sqrt-div 61.5%

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

      4. frac-times 61.6%

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

      5. *-commutative 61.6%

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

      6. sqrt-pow1 80.5%

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

      7. metadata-eval 80.5%

         \[\leadsto \frac{\left(-0.125 \cdot {\left(M \cdot D\right)}^{2}\right) \cdot \sqrt{h}}{d \cdot {\ell}^{\color{blue}{1.5}}} \]

   8. Applied egg-rr 80.5%

      \[\leadsto \color{blue}{\frac{\left(-0.125 \cdot {\left(M \cdot D\right)}^{2}\right) \cdot \sqrt{h}}{d \cdot {\ell}^{1.5}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -3.3 \cdot 10^{-307} \lor \neg \left(d \leq 8.2 \cdot 10^{-220}\right):\\ \;\;\;\;\frac{1}{\sqrt{\frac{\ell}{d}}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h}{-8 \cdot \left(\frac{1}{\frac{M}{\frac{d}{D}}} \cdot \frac{\ell}{\frac{M}{\frac{d}{D}}}\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{h} \cdot \left(-0.125 \cdot {\left(D \cdot M\right)}^{2}\right)}{d \cdot {\ell}^{1.5}}\\ \end{array} \]

Alternative 8: 69.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \frac{M_m}{\frac{d}{D_m}}\\ \mathbf{if}\;\ell \leq 4 \cdot 10^{+234}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{\ell}{d}}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h}{-8 \cdot \left(\frac{1}{t_0} \cdot \frac{\ell}{t_0}\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (/ M_m (/ d D_m))))
   (if (<= l 4e+234)
     (*
      (/ 1.0 (sqrt (/ l d)))
      (* (sqrt (/ d h)) (+ 1.0 (/ h (* -8.0 (* (/ 1.0 t_0) (/ l t_0)))))))
     (* d (/ (sqrt (/ 1.0 h)) (sqrt l))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = M_m / (d / D_m);
	double tmp;
	if (l <= 4e+234) {
		tmp = (1.0 / sqrt((l / d))) * (sqrt((d / h)) * (1.0 + (h / (-8.0 * ((1.0 / t_0) * (l / t_0))))));
	} else {
		tmp = d * (sqrt((1.0 / h)) / sqrt(l));
	}
	return tmp;
}

Fortran

M_m = abs(M)
D_m = abs(D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = m_m / (d / d_m)
    if (l <= 4d+234) then
        tmp = (1.0d0 / sqrt((l / d))) * (sqrt((d / h)) * (1.0d0 + (h / ((-8.0d0) * ((1.0d0 / t_0) * (l / t_0))))))
    else
        tmp = d * (sqrt((1.0d0 / h)) / sqrt(l))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = M_m / (d / D_m);
	double tmp;
	if (l <= 4e+234) {
		tmp = (1.0 / Math.sqrt((l / d))) * (Math.sqrt((d / h)) * (1.0 + (h / (-8.0 * ((1.0 / t_0) * (l / t_0))))));
	} else {
		tmp = d * (Math.sqrt((1.0 / h)) / Math.sqrt(l));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = M_m / (d / D_m)
	tmp = 0
	if l <= 4e+234:
		tmp = (1.0 / math.sqrt((l / d))) * (math.sqrt((d / h)) * (1.0 + (h / (-8.0 * ((1.0 / t_0) * (l / t_0))))))
	else:
		tmp = d * (math.sqrt((1.0 / h)) / math.sqrt(l))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(M_m / Float64(d / D_m))
	tmp = 0.0
	if (l <= 4e+234)
		tmp = Float64(Float64(1.0 / sqrt(Float64(l / d))) * Float64(sqrt(Float64(d / h)) * Float64(1.0 + Float64(h / Float64(-8.0 * Float64(Float64(1.0 / t_0) * Float64(l / t_0)))))));
	else
		tmp = Float64(d * Float64(sqrt(Float64(1.0 / h)) / sqrt(l)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = M_m / (d / D_m);
	tmp = 0.0;
	if (l <= 4e+234)
		tmp = (1.0 / sqrt((l / d))) * (sqrt((d / h)) * (1.0 + (h / (-8.0 * ((1.0 / t_0) * (l / t_0))))));
	else
		tmp = d * (sqrt((1.0 / h)) / sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(M$95$m / N[(d / D$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, 4e+234], N[(N[(1.0 / N[Sqrt[N[(l / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(h / N[(-8.0 * N[(N[(1.0 / t$95$0), $MachinePrecision] * N[(l / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Sqrt[N[(1.0 / h), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < 4.00000000000000007e234

   1. Initial program 69.6%

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

   3. Simplified 70.1%

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

   5. Applied egg-rr 71.8%

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

      1. clear-num 71.4%

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

      2. sqrt-div 72.1%

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

      3. metadata-eval 72.1%

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

   7. Applied egg-rr 72.1%

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

      1. *-un-lft-identity 72.1%

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

      2. times-frac 72.1%

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

      3. metadata-eval 72.1%

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

      4. associate-*r/ 71.3%

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

      5. associate-*l/ 71.3%

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

      6. *-commutative 71.3%

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

   9. Applied egg-rr 71.3%

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

       1. *-un-lft-identity 71.3%

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

       2. unpow2 71.3%

          \[\leadsto \frac{1}{\sqrt{\frac{\ell}{d}}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h}{-8 \cdot \frac{1 \cdot \ell}{\color{blue}{\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)}}}\right)\right) \]

       3. times-frac 73.0%

          \[\leadsto \frac{1}{\sqrt{\frac{\ell}{d}}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h}{-8 \cdot \color{blue}{\left(\frac{1}{D \cdot \frac{M}{d}} \cdot \frac{\ell}{D \cdot \frac{M}{d}}\right)}}\right)\right) \]

       4. associate-*r/ 71.8%

          \[\leadsto \frac{1}{\sqrt{\frac{\ell}{d}}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h}{-8 \cdot \left(\frac{1}{\color{blue}{\frac{D \cdot M}{d}}} \cdot \frac{\ell}{D \cdot \frac{M}{d}}\right)}\right)\right) \]

       5. *-commutative 71.8%

          \[\leadsto \frac{1}{\sqrt{\frac{\ell}{d}}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h}{-8 \cdot \left(\frac{1}{\frac{\color{blue}{M \cdot D}}{d}} \cdot \frac{\ell}{D \cdot \frac{M}{d}}\right)}\right)\right) \]

       6. associate-/l* 72.6%

          \[\leadsto \frac{1}{\sqrt{\frac{\ell}{d}}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h}{-8 \cdot \left(\frac{1}{\color{blue}{\frac{M}{\frac{d}{D}}}} \cdot \frac{\ell}{D \cdot \frac{M}{d}}\right)}\right)\right) \]

       7. associate-*r/ 72.6%

          \[\leadsto \frac{1}{\sqrt{\frac{\ell}{d}}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h}{-8 \cdot \left(\frac{1}{\frac{M}{\frac{d}{D}}} \cdot \frac{\ell}{\color{blue}{\frac{D \cdot M}{d}}}\right)}\right)\right) \]

       8. *-commutative 72.6%

          \[\leadsto \frac{1}{\sqrt{\frac{\ell}{d}}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h}{-8 \cdot \left(\frac{1}{\frac{M}{\frac{d}{D}}} \cdot \frac{\ell}{\frac{\color{blue}{M \cdot D}}{d}}\right)}\right)\right) \]

       9. associate-/l* 73.7%

          \[\leadsto \frac{1}{\sqrt{\frac{\ell}{d}}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h}{-8 \cdot \left(\frac{1}{\frac{M}{\frac{d}{D}}} \cdot \frac{\ell}{\color{blue}{\frac{M}{\frac{d}{D}}}}\right)}\right)\right) \]

   11. Applied egg-rr 73.7%

       \[\leadsto \frac{1}{\sqrt{\frac{\ell}{d}}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h}{-8 \cdot \color{blue}{\left(\frac{1}{\frac{M}{\frac{d}{D}}} \cdot \frac{\ell}{\frac{M}{\frac{d}{D}}}\right)}}\right)\right) \]

   if 4.00000000000000007e234 < l

   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 53.2%

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

   4. Taylor expanded in d around inf 46.2%

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

      1. associate-/r* 46.2%

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

      2. sqrt-div 68.8%

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

   6. Applied egg-rr 68.8%

      \[\leadsto d \cdot \color{blue}{\frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 4 \cdot 10^{+234}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{\ell}{d}}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h}{-8 \cdot \left(\frac{1}{\frac{M}{\frac{d}{D}}} \cdot \frac{\ell}{\frac{M}{\frac{d}{D}}}\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \]

Alternative 9: 46.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \frac{\frac{\sqrt{h}}{{\ell}^{1.5}} \cdot \left(-0.125 \cdot \left(D_m \cdot \left(M_m \cdot \left(D_m \cdot M_m\right)\right)\right)\right)}{d}\\ t_1 := \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;M_m \leq 1.15 \cdot 10^{-75}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;M_m \leq 1.9 \cdot 10^{-57}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;M_m \leq 1.6 \cdot 10^{-16}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;M_m \leq 1.55 \cdot 10^{+132}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}} \cdot \left(1 + h \cdot \left(\frac{0.125}{\ell} \cdot {\left(D_m \cdot \frac{M_m}{d}\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0
         (/
          (* (/ (sqrt h) (pow l 1.5)) (* -0.125 (* D_m (* M_m (* D_m M_m)))))
          d))
        (t_1 (* (sqrt (/ d h)) (sqrt (/ d l)))))
   (if (<= M_m 1.15e-75)
     t_1
     (if (<= M_m 1.9e-57)
       t_0
       (if (<= M_m 1.6e-16)
         t_1
         (if (<= M_m 1.55e+132)
           (*
            (/ d (sqrt (* l h)))
            (+ 1.0 (* h (* (/ 0.125 l) (pow (* D_m (/ M_m d)) 2.0)))))
           t_0))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = ((sqrt(h) / pow(l, 1.5)) * (-0.125 * (D_m * (M_m * (D_m * M_m))))) / d;
	double t_1 = sqrt((d / h)) * sqrt((d / l));
	double tmp;
	if (M_m <= 1.15e-75) {
		tmp = t_1;
	} else if (M_m <= 1.9e-57) {
		tmp = t_0;
	} else if (M_m <= 1.6e-16) {
		tmp = t_1;
	} else if (M_m <= 1.55e+132) {
		tmp = (d / sqrt((l * h))) * (1.0 + (h * ((0.125 / l) * pow((D_m * (M_m / d)), 2.0))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

M_m = abs(M)
D_m = abs(D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((sqrt(h) / (l ** 1.5d0)) * ((-0.125d0) * (d_m * (m_m * (d_m * m_m))))) / d
    t_1 = sqrt((d / h)) * sqrt((d / l))
    if (m_m <= 1.15d-75) then
        tmp = t_1
    else if (m_m <= 1.9d-57) then
        tmp = t_0
    else if (m_m <= 1.6d-16) then
        tmp = t_1
    else if (m_m <= 1.55d+132) then
        tmp = (d / sqrt((l * h))) * (1.0d0 + (h * ((0.125d0 / l) * ((d_m * (m_m / d)) ** 2.0d0))))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = ((Math.sqrt(h) / Math.pow(l, 1.5)) * (-0.125 * (D_m * (M_m * (D_m * M_m))))) / d;
	double t_1 = Math.sqrt((d / h)) * Math.sqrt((d / l));
	double tmp;
	if (M_m <= 1.15e-75) {
		tmp = t_1;
	} else if (M_m <= 1.9e-57) {
		tmp = t_0;
	} else if (M_m <= 1.6e-16) {
		tmp = t_1;
	} else if (M_m <= 1.55e+132) {
		tmp = (d / Math.sqrt((l * h))) * (1.0 + (h * ((0.125 / l) * Math.pow((D_m * (M_m / d)), 2.0))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = ((math.sqrt(h) / math.pow(l, 1.5)) * (-0.125 * (D_m * (M_m * (D_m * M_m))))) / d
	t_1 = math.sqrt((d / h)) * math.sqrt((d / l))
	tmp = 0
	if M_m <= 1.15e-75:
		tmp = t_1
	elif M_m <= 1.9e-57:
		tmp = t_0
	elif M_m <= 1.6e-16:
		tmp = t_1
	elif M_m <= 1.55e+132:
		tmp = (d / math.sqrt((l * h))) * (1.0 + (h * ((0.125 / l) * math.pow((D_m * (M_m / d)), 2.0))))
	else:
		tmp = t_0
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(Float64(Float64(sqrt(h) / (l ^ 1.5)) * Float64(-0.125 * Float64(D_m * Float64(M_m * Float64(D_m * M_m))))) / d)
	t_1 = Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l)))
	tmp = 0.0
	if (M_m <= 1.15e-75)
		tmp = t_1;
	elseif (M_m <= 1.9e-57)
		tmp = t_0;
	elseif (M_m <= 1.6e-16)
		tmp = t_1;
	elseif (M_m <= 1.55e+132)
		tmp = Float64(Float64(d / sqrt(Float64(l * h))) * Float64(1.0 + Float64(h * Float64(Float64(0.125 / l) * (Float64(D_m * Float64(M_m / d)) ^ 2.0)))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = ((sqrt(h) / (l ^ 1.5)) * (-0.125 * (D_m * (M_m * (D_m * M_m))))) / d;
	t_1 = sqrt((d / h)) * sqrt((d / l));
	tmp = 0.0;
	if (M_m <= 1.15e-75)
		tmp = t_1;
	elseif (M_m <= 1.9e-57)
		tmp = t_0;
	elseif (M_m <= 1.6e-16)
		tmp = t_1;
	elseif (M_m <= 1.55e+132)
		tmp = (d / sqrt((l * h))) * (1.0 + (h * ((0.125 / l) * ((D_m * (M_m / d)) ^ 2.0))));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(N[(N[(N[Sqrt[h], $MachinePrecision] / N[Power[l, 1.5], $MachinePrecision]), $MachinePrecision] * N[(-0.125 * N[(D$95$m * N[(M$95$m * N[(D$95$m * M$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[M$95$m, 1.15e-75], t$95$1, If[LessEqual[M$95$m, 1.9e-57], t$95$0, If[LessEqual[M$95$m, 1.6e-16], t$95$1, If[LessEqual[M$95$m, 1.55e+132], N[(N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(h * N[(N[(0.125 / l), $MachinePrecision] * N[Power[N[(D$95$m * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if M < 1.15e-75 or 1.8999999999999999e-57 < M < 1.60000000000000011e-16

   1. Initial program 74.2%

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

   3. Simplified 75.2%

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

   4. Taylor expanded in h around 0 54.5%

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

   if 1.15e-75 < M < 1.8999999999999999e-57 or 1.5499999999999999e132 < M

   1. Initial program 48.1%

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

   3. Simplified 45.5%

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

   4. Taylor expanded in d around 0 13.3%

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

      1. associate-*r* 13.3%

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

      2. *-commutative 13.3%

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

      3. *-commutative 13.3%

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

      4. unpow2 13.3%

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

      5. unpow2 13.3%

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

      6. swap-sqr 18.7%

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

      7. unpow2 18.7%

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

      8. *-commutative 18.7%

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

   6. Simplified 18.7%

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

      1. associate-*r/ 18.3%

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

      2. associate-*r/ 18.5%

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

      3. sqrt-div 17.8%

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

      4. sqrt-pow1 21.4%

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

      5. metadata-eval 21.4%

         \[\leadsto \frac{\frac{\sqrt{h}}{{\ell}^{\color{blue}{1.5}}} \cdot \left(-0.125 \cdot {\left(D \cdot M\right)}^{2}\right)}{d} \]

      6. *-commutative 21.4%

         \[\leadsto \frac{\frac{\sqrt{h}}{{\ell}^{1.5}} \cdot \left(-0.125 \cdot {\color{blue}{\left(M \cdot D\right)}}^{2}\right)}{d} \]

   8. Applied egg-rr 21.4%

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{h}}{{\ell}^{1.5}} \cdot \left(-0.125 \cdot {\left(M \cdot D\right)}^{2}\right)}{d}} \]
   9. Step-by-step derivation

      1. unpow2 21.4%

         \[\leadsto \frac{\frac{\sqrt{h}}{{\ell}^{1.5}} \cdot \left(-0.125 \cdot \color{blue}{\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right)}\right)}{d} \]

      2. associate-*r* 20.5%

         \[\leadsto \frac{\frac{\sqrt{h}}{{\ell}^{1.5}} \cdot \left(-0.125 \cdot \color{blue}{\left(\left(\left(M \cdot D\right) \cdot M\right) \cdot D\right)}\right)}{d} \]

      3. *-commutative 20.5%

         \[\leadsto \frac{\frac{\sqrt{h}}{{\ell}^{1.5}} \cdot \left(-0.125 \cdot \left(\left(\color{blue}{\left(D \cdot M\right)} \cdot M\right) \cdot D\right)\right)}{d} \]

   10. Applied egg-rr 20.5%

       \[\leadsto \frac{\frac{\sqrt{h}}{{\ell}^{1.5}} \cdot \left(-0.125 \cdot \color{blue}{\left(\left(\left(D \cdot M\right) \cdot M\right) \cdot D\right)}\right)}{d} \]

   if 1.60000000000000011e-16 < M < 1.5499999999999999e132

   1. Initial program 61.3%

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

   3. Simplified 63.8%

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

   5. Applied egg-rr 63.8%

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

      1. associate-/r/ 64.0%

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

      2. associate-*r/ 61.5%

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

      3. *-commutative 61.5%

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

      4. associate-*r/ 61.0%

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

   7. Simplified 61.0%

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

   9. Applied egg-rr 37.9%

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

       1. *-rgt-identity 37.9%

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

       2. distribute-lft-in 37.9%

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

       3. associate-/r/ 38.0%

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

   11. Simplified 38.0%

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

4. Final simplification 46.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 1.15 \cdot 10^{-75}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;M \leq 1.9 \cdot 10^{-57}:\\ \;\;\;\;\frac{\frac{\sqrt{h}}{{\ell}^{1.5}} \cdot \left(-0.125 \cdot \left(D \cdot \left(M \cdot \left(D \cdot M\right)\right)\right)\right)}{d}\\ \mathbf{elif}\;M \leq 1.6 \cdot 10^{-16}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;M \leq 1.55 \cdot 10^{+132}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}} \cdot \left(1 + h \cdot \left(\frac{0.125}{\ell} \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt{h}}{{\ell}^{1.5}} \cdot \left(-0.125 \cdot \left(D \cdot \left(M \cdot \left(D \cdot M\right)\right)\right)\right)}{d}\\ \end{array} \]

Alternative 10: 45.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 1.5 \cdot 10^{-280}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;d \leq 4.4 \cdot 10^{-226}:\\ \;\;\;\;\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \left(\left(M_m \cdot \frac{D_m}{d}\right) \cdot \left(D_m \cdot M_m\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= d 1.5e-280)
   (* (sqrt (/ d h)) (sqrt (/ d l)))
   (if (<= d 4.4e-226)
     (* (sqrt (/ h (pow l 3.0))) (* -0.125 (* (* M_m (/ D_m d)) (* D_m M_m))))
     (/ (/ d (sqrt l)) (sqrt h)))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= 1.5e-280) {
		tmp = sqrt((d / h)) * sqrt((d / l));
	} else if (d <= 4.4e-226) {
		tmp = sqrt((h / pow(l, 3.0))) * (-0.125 * ((M_m * (D_m / d)) * (D_m * M_m)));
	} else {
		tmp = (d / sqrt(l)) / sqrt(h);
	}
	return tmp;
}

Fortran

M_m = abs(M)
D_m = abs(D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (d <= 1.5d-280) then
        tmp = sqrt((d / h)) * sqrt((d / l))
    else if (d <= 4.4d-226) then
        tmp = sqrt((h / (l ** 3.0d0))) * ((-0.125d0) * ((m_m * (d_m / d)) * (d_m * m_m)))
    else
        tmp = (d / sqrt(l)) / sqrt(h)
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= 1.5e-280) {
		tmp = Math.sqrt((d / h)) * Math.sqrt((d / l));
	} else if (d <= 4.4e-226) {
		tmp = Math.sqrt((h / Math.pow(l, 3.0))) * (-0.125 * ((M_m * (D_m / d)) * (D_m * M_m)));
	} else {
		tmp = (d / Math.sqrt(l)) / Math.sqrt(h);
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if d <= 1.5e-280:
		tmp = math.sqrt((d / h)) * math.sqrt((d / l))
	elif d <= 4.4e-226:
		tmp = math.sqrt((h / math.pow(l, 3.0))) * (-0.125 * ((M_m * (D_m / d)) * (D_m * M_m)))
	else:
		tmp = (d / math.sqrt(l)) / math.sqrt(h)
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (d <= 1.5e-280)
		tmp = Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l)));
	elseif (d <= 4.4e-226)
		tmp = Float64(sqrt(Float64(h / (l ^ 3.0))) * Float64(-0.125 * Float64(Float64(M_m * Float64(D_m / d)) * Float64(D_m * M_m))));
	else
		tmp = Float64(Float64(d / sqrt(l)) / sqrt(h));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (d <= 1.5e-280)
		tmp = sqrt((d / h)) * sqrt((d / l));
	elseif (d <= 4.4e-226)
		tmp = sqrt((h / (l ^ 3.0))) * (-0.125 * ((M_m * (D_m / d)) * (D_m * M_m)));
	else
		tmp = (d / sqrt(l)) / sqrt(h);
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[d, 1.5e-280], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 4.4e-226], N[(N[Sqrt[N[(h / N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-0.125 * N[(N[(M$95$m * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision] * N[(D$95$m * M$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(d / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < 1.49999999999999994e-280

   1. Initial program 64.3%

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

   3. Simplified 66.3%

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

   4. Taylor expanded in h around 0 45.0%

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

   if 1.49999999999999994e-280 < d < 4.4e-226

   1. Initial program 56.8%

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

   3. Simplified 56.8%

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

   4. Taylor expanded in d around 0 48.5%

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

      1. associate-*r* 48.5%

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

      2. *-commutative 48.5%

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

      3. *-commutative 48.5%

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

      4. unpow2 48.5%

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

      5. unpow2 48.5%

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

      6. swap-sqr 73.7%

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

      7. unpow2 73.7%

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

      8. *-commutative 73.7%

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

   6. Simplified 73.7%

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

      1. div-inv 73.7%

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

      2. unpow2 73.7%

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

      3. associate-*l* 73.7%

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

      4. *-commutative 73.7%

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

      5. *-commutative 73.7%

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

      6. associate-*r* 73.7%

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

      7. div-inv 73.7%

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

   8. Applied egg-rr 73.7%

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

   if 4.4e-226 < d

   1. Initial program 74.9%

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

   3. Simplified 73.9%

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

   4. Taylor expanded in d around inf 48.3%

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

      1. sqrt-div 48.2%

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

      2. metadata-eval 48.2%

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

      3. un-div-inv 48.2%

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

   6. Applied egg-rr 48.2%

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

      1. *-un-lft-identity 48.2%

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

      2. sqrt-prod 62.2%

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

      3. times-frac 62.2%

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

   8. Applied egg-rr 62.2%

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

      1. associate-*l/ 62.3%

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

      2. *-lft-identity 62.3%

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

   10. Simplified 62.3%

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

4. Final simplification 53.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 1.5 \cdot 10^{-280}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;d \leq 4.4 \cdot 10^{-226}:\\ \;\;\;\;\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(-0.125 \cdot \left(\left(M \cdot \frac{D}{d}\right) \cdot \left(D \cdot M\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \]

Alternative 11: 48.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;d \leq 1.6 \cdot 10^{-99}:\\ \;\;\;\;\frac{\frac{\sqrt{h}}{{\ell}^{1.5}} \cdot \left(-0.125 \cdot \left(D_m \cdot \left(M_m \cdot \left(D_m \cdot M_m\right)\right)\right)\right)}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= d -2e-310)
   (* (sqrt (/ d h)) (sqrt (/ d l)))
   (if (<= d 1.6e-99)
     (/ (* (/ (sqrt h) (pow l 1.5)) (* -0.125 (* D_m (* M_m (* D_m M_m))))) d)
     (/ (/ d (sqrt l)) (sqrt h)))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= -2e-310) {
		tmp = sqrt((d / h)) * sqrt((d / l));
	} else if (d <= 1.6e-99) {
		tmp = ((sqrt(h) / pow(l, 1.5)) * (-0.125 * (D_m * (M_m * (D_m * M_m))))) / d;
	} else {
		tmp = (d / sqrt(l)) / sqrt(h);
	}
	return tmp;
}

Fortran

M_m = abs(M)
D_m = abs(D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (d <= (-2d-310)) then
        tmp = sqrt((d / h)) * sqrt((d / l))
    else if (d <= 1.6d-99) then
        tmp = ((sqrt(h) / (l ** 1.5d0)) * ((-0.125d0) * (d_m * (m_m * (d_m * m_m))))) / d
    else
        tmp = (d / sqrt(l)) / sqrt(h)
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= -2e-310) {
		tmp = Math.sqrt((d / h)) * Math.sqrt((d / l));
	} else if (d <= 1.6e-99) {
		tmp = ((Math.sqrt(h) / Math.pow(l, 1.5)) * (-0.125 * (D_m * (M_m * (D_m * M_m))))) / d;
	} else {
		tmp = (d / Math.sqrt(l)) / Math.sqrt(h);
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if d <= -2e-310:
		tmp = math.sqrt((d / h)) * math.sqrt((d / l))
	elif d <= 1.6e-99:
		tmp = ((math.sqrt(h) / math.pow(l, 1.5)) * (-0.125 * (D_m * (M_m * (D_m * M_m))))) / d
	else:
		tmp = (d / math.sqrt(l)) / math.sqrt(h)
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (d <= -2e-310)
		tmp = Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l)));
	elseif (d <= 1.6e-99)
		tmp = Float64(Float64(Float64(sqrt(h) / (l ^ 1.5)) * Float64(-0.125 * Float64(D_m * Float64(M_m * Float64(D_m * M_m))))) / d);
	else
		tmp = Float64(Float64(d / sqrt(l)) / sqrt(h));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (d <= -2e-310)
		tmp = sqrt((d / h)) * sqrt((d / l));
	elseif (d <= 1.6e-99)
		tmp = ((sqrt(h) / (l ^ 1.5)) * (-0.125 * (D_m * (M_m * (D_m * M_m))))) / d;
	else
		tmp = (d / sqrt(l)) / sqrt(h);
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[d, -2e-310], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.6e-99], N[(N[(N[(N[Sqrt[h], $MachinePrecision] / N[Power[l, 1.5], $MachinePrecision]), $MachinePrecision] * N[(-0.125 * N[(D$95$m * N[(M$95$m * N[(D$95$m * M$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], N[(N[(d / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.999999999999994e-310

   1. Initial program 66.9%

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

   3. Simplified 68.9%

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

   4. Taylor expanded in h around 0 45.8%

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

   if -1.999999999999994e-310 < d < 1.6e-99

   1. Initial program 48.9%

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

   3. Simplified 48.9%

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

   4. Taylor expanded in d around 0 36.4%

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

      1. associate-*r* 36.4%

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

      2. *-commutative 36.4%

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

      3. *-commutative 36.4%

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

      4. unpow2 36.4%

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

      5. unpow2 36.4%

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

      6. swap-sqr 43.2%

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

      7. unpow2 43.2%

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

      8. *-commutative 43.2%

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

   6. Simplified 43.2%

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

      1. associate-*r/ 42.8%

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

      2. associate-*r/ 44.9%

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

      3. sqrt-div 44.9%

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

      4. sqrt-pow1 56.8%

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

      5. metadata-eval 56.8%

         \[\leadsto \frac{\frac{\sqrt{h}}{{\ell}^{\color{blue}{1.5}}} \cdot \left(-0.125 \cdot {\left(D \cdot M\right)}^{2}\right)}{d} \]

      6. *-commutative 56.8%

         \[\leadsto \frac{\frac{\sqrt{h}}{{\ell}^{1.5}} \cdot \left(-0.125 \cdot {\color{blue}{\left(M \cdot D\right)}}^{2}\right)}{d} \]

   8. Applied egg-rr 56.8%

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{h}}{{\ell}^{1.5}} \cdot \left(-0.125 \cdot {\left(M \cdot D\right)}^{2}\right)}{d}} \]
   9. Step-by-step derivation

      1. unpow2 56.8%

         \[\leadsto \frac{\frac{\sqrt{h}}{{\ell}^{1.5}} \cdot \left(-0.125 \cdot \color{blue}{\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right)}\right)}{d} \]

      2. associate-*r* 56.1%

         \[\leadsto \frac{\frac{\sqrt{h}}{{\ell}^{1.5}} \cdot \left(-0.125 \cdot \color{blue}{\left(\left(\left(M \cdot D\right) \cdot M\right) \cdot D\right)}\right)}{d} \]

      3. *-commutative 56.1%

         \[\leadsto \frac{\frac{\sqrt{h}}{{\ell}^{1.5}} \cdot \left(-0.125 \cdot \left(\left(\color{blue}{\left(D \cdot M\right)} \cdot M\right) \cdot D\right)\right)}{d} \]

   10. Applied egg-rr 56.1%

       \[\leadsto \frac{\frac{\sqrt{h}}{{\ell}^{1.5}} \cdot \left(-0.125 \cdot \color{blue}{\left(\left(\left(D \cdot M\right) \cdot M\right) \cdot D\right)}\right)}{d} \]

   if 1.6e-99 < d

   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 80.5%

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

   4. Taylor expanded in d around inf 55.1%

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

      1. sqrt-div 54.9%

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

      2. metadata-eval 54.9%

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

      3. un-div-inv 55.0%

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

   6. Applied egg-rr 55.0%

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

      1. *-un-lft-identity 55.0%

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

      2. sqrt-prod 70.0%

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

      3. times-frac 69.9%

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

   8. Applied egg-rr 69.9%

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

      1. associate-*l/ 70.1%

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

      2. *-lft-identity 70.1%

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

   10. Simplified 70.1%

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

4. Final simplification 54.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;d \leq 1.6 \cdot 10^{-99}:\\ \;\;\;\;\frac{\frac{\sqrt{h}}{{\ell}^{1.5}} \cdot \left(-0.125 \cdot \left(D \cdot \left(M \cdot \left(D \cdot M\right)\right)\right)\right)}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \]

Alternative 12: 38.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 2.3 \cdot 10^{-299}:\\ \;\;\;\;\sqrt{\frac{{d}^{2}}{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= l 2.3e-299)
   (sqrt (/ (pow d 2.0) (* l h)))
   (* d (/ (sqrt (/ 1.0 h)) (sqrt l)))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= 2.3e-299) {
		tmp = sqrt((pow(d, 2.0) / (l * h)));
	} else {
		tmp = d * (sqrt((1.0 / h)) / sqrt(l));
	}
	return tmp;
}

Fortran

M_m = abs(M)
D_m = abs(D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (l <= 2.3d-299) then
        tmp = sqrt(((d ** 2.0d0) / (l * h)))
    else
        tmp = d * (sqrt((1.0d0 / h)) / sqrt(l))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= 2.3e-299) {
		tmp = Math.sqrt((Math.pow(d, 2.0) / (l * h)));
	} else {
		tmp = d * (Math.sqrt((1.0 / h)) / Math.sqrt(l));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if l <= 2.3e-299:
		tmp = math.sqrt((math.pow(d, 2.0) / (l * h)))
	else:
		tmp = d * (math.sqrt((1.0 / h)) / math.sqrt(l))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (l <= 2.3e-299)
		tmp = sqrt(Float64((d ^ 2.0) / Float64(l * h)));
	else
		tmp = Float64(d * Float64(sqrt(Float64(1.0 / h)) / sqrt(l)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (l <= 2.3e-299)
		tmp = sqrt(((d ^ 2.0) / (l * h)));
	else
		tmp = d * (sqrt((1.0 / h)) / sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[l, 2.3e-299], N[Sqrt[N[(N[Power[d, 2.0], $MachinePrecision] / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(d * N[(N[Sqrt[N[(1.0 / h), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 2.3000000000000001e-299

   1. Initial program 66.7%

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

   3. Simplified 68.7%

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

   4. Taylor expanded in d around inf 6.7%

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

      1. sqrt-div 6.7%

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

      2. metadata-eval 6.7%

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

      3. un-div-inv 6.7%

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

      4. add-sqr-sqrt 1.5%

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

      5. sqrt-prod 1.5%

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

      6. frac-times 1.5%

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

      7. sqrt-div 1.5%

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

      8. sqrt-div 45.9%

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

      9. pow1 45.9%

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

      10. add-sqr-sqrt 45.7%

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

      11. sqrt-unprod 33.3%

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

      12. sqrt-pow2 33.3%

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

   6. Applied egg-rr 27.7%

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

      1. unpow1/2 27.7%

         \[\leadsto \color{blue}{\sqrt{\frac{{d}^{2}}{h \cdot \ell}}} \]

   8. Simplified 27.7%

      \[\leadsto \color{blue}{\sqrt{\frac{{d}^{2}}{h \cdot \ell}}} \]

   if 2.3000000000000001e-299 < l

   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 69.2%

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

   4. Taylor expanded in d around inf 44.9%

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

      1. associate-/r* 44.9%

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

      2. sqrt-div 56.9%

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

   6. Applied egg-rr 56.9%

      \[\leadsto d \cdot \color{blue}{\frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 41.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2.3 \cdot 10^{-299}:\\ \;\;\;\;\sqrt{\frac{{d}^{2}}{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \]

Alternative 13: 44.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 1.55 \cdot 10^{-282}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= d 1.55e-282)
   (* (sqrt (/ d h)) (sqrt (/ d l)))
   (* d (/ (sqrt (/ 1.0 h)) (sqrt l)))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= 1.55e-282) {
		tmp = sqrt((d / h)) * sqrt((d / l));
	} else {
		tmp = d * (sqrt((1.0 / h)) / sqrt(l));
	}
	return tmp;
}

Fortran

M_m = abs(M)
D_m = abs(D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (d <= 1.55d-282) then
        tmp = sqrt((d / h)) * sqrt((d / l))
    else
        tmp = d * (sqrt((1.0d0 / h)) / sqrt(l))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= 1.55e-282) {
		tmp = Math.sqrt((d / h)) * Math.sqrt((d / l));
	} else {
		tmp = d * (Math.sqrt((1.0 / h)) / Math.sqrt(l));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if d <= 1.55e-282:
		tmp = math.sqrt((d / h)) * math.sqrt((d / l))
	else:
		tmp = d * (math.sqrt((1.0 / h)) / math.sqrt(l))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (d <= 1.55e-282)
		tmp = Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l)));
	else
		tmp = Float64(d * Float64(sqrt(Float64(1.0 / h)) / sqrt(l)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (d <= 1.55e-282)
		tmp = sqrt((d / h)) * sqrt((d / l));
	else
		tmp = d * (sqrt((1.0 / h)) / sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[d, 1.55e-282], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Sqrt[N[(1.0 / h), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 1.55000000000000007e-282

   1. Initial program 64.8%

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

   3. Simplified 66.7%

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

   4. Taylor expanded in h around 0 45.3%

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

   if 1.55000000000000007e-282 < d

   1. Initial program 72.4%

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

   3. Simplified 71.5%

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

   4. Taylor expanded in d around inf 46.4%

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

      1. associate-/r* 46.4%

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

      2. sqrt-div 58.8%

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

   6. Applied egg-rr 58.8%

      \[\leadsto d \cdot \color{blue}{\frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 51.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 1.55 \cdot 10^{-282}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \]

Alternative 14: 37.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;h \leq 1.05 \cdot 10^{-303}:\\ \;\;\;\;\sqrt{\frac{{d}^{2}}{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= h 1.05e-303)
   (sqrt (/ (pow d 2.0) (* l h)))
   (/ (/ d (sqrt l)) (sqrt h))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (h <= 1.05e-303) {
		tmp = sqrt((pow(d, 2.0) / (l * h)));
	} else {
		tmp = (d / sqrt(l)) / sqrt(h);
	}
	return tmp;
}

Fortran

M_m = abs(M)
D_m = abs(D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (h <= 1.05d-303) then
        tmp = sqrt(((d ** 2.0d0) / (l * h)))
    else
        tmp = (d / sqrt(l)) / sqrt(h)
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (h <= 1.05e-303) {
		tmp = Math.sqrt((Math.pow(d, 2.0) / (l * h)));
	} else {
		tmp = (d / Math.sqrt(l)) / Math.sqrt(h);
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if h <= 1.05e-303:
		tmp = math.sqrt((math.pow(d, 2.0) / (l * h)))
	else:
		tmp = (d / math.sqrt(l)) / math.sqrt(h)
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (h <= 1.05e-303)
		tmp = sqrt(Float64((d ^ 2.0) / Float64(l * h)));
	else
		tmp = Float64(Float64(d / sqrt(l)) / sqrt(h));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (h <= 1.05e-303)
		tmp = sqrt(((d ^ 2.0) / (l * h)));
	else
		tmp = (d / sqrt(l)) / sqrt(h);
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[h, 1.05e-303], N[Sqrt[N[(N[Power[d, 2.0], $MachinePrecision] / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(d / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if h < 1.05e-303

   1. Initial program 67.1%

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

   3. Simplified 69.2%

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

   4. Taylor expanded in d around inf 6.0%

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

      1. sqrt-div 6.0%

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

      2. metadata-eval 6.0%

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

      3. un-div-inv 6.0%

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

      4. add-sqr-sqrt 0.7%

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

      5. sqrt-prod 0.7%

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

      6. frac-times 0.7%

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

      7. sqrt-div 0.7%

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

      8. sqrt-div 46.2%

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

      9. pow1 46.2%

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

      10. add-sqr-sqrt 46.0%

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

      11. sqrt-unprod 33.5%

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

      12. sqrt-pow2 33.5%

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

   6. Applied egg-rr 27.1%

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

      1. unpow1/2 27.1%

         \[\leadsto \color{blue}{\sqrt{\frac{{d}^{2}}{h \cdot \ell}}} \]

   8. Simplified 27.1%

      \[\leadsto \color{blue}{\sqrt{\frac{{d}^{2}}{h \cdot \ell}}} \]

   if 1.05e-303 < h

   1. Initial program 69.5%

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

   3. Simplified 68.6%

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

   4. Taylor expanded in d around inf 45.4%

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

      1. sqrt-div 45.3%

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

      2. metadata-eval 45.3%

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

      3. un-div-inv 45.3%

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

   6. Applied egg-rr 45.3%

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

      1. *-un-lft-identity 45.3%

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

      2. sqrt-prod 57.3%

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

      3. times-frac 57.1%

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

   8. Applied egg-rr 57.1%

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

      1. associate-*l/ 57.2%

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

      2. *-lft-identity 57.2%

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

   10. Simplified 57.2%

       \[\leadsto \color{blue}{\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 41.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq 1.05 \cdot 10^{-303}:\\ \;\;\;\;\sqrt{\frac{{d}^{2}}{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \]

Alternative 15: 29.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 6.2 \cdot 10^{-209}:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{h} \cdot \frac{1}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= d 6.2e-209)
   (* d (sqrt (* (/ 1.0 h) (/ 1.0 l))))
   (/ (/ d (sqrt h)) (sqrt l))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= 6.2e-209) {
		tmp = d * sqrt(((1.0 / h) * (1.0 / l)));
	} else {
		tmp = (d / sqrt(h)) / sqrt(l);
	}
	return tmp;
}

Fortran

M_m = abs(M)
D_m = abs(D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (d <= 6.2d-209) then
        tmp = d * sqrt(((1.0d0 / h) * (1.0d0 / l)))
    else
        tmp = (d / sqrt(h)) / sqrt(l)
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= 6.2e-209) {
		tmp = d * Math.sqrt(((1.0 / h) * (1.0 / l)));
	} else {
		tmp = (d / Math.sqrt(h)) / Math.sqrt(l);
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if d <= 6.2e-209:
		tmp = d * math.sqrt(((1.0 / h) * (1.0 / l)))
	else:
		tmp = (d / math.sqrt(h)) / math.sqrt(l)
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (d <= 6.2e-209)
		tmp = Float64(d * sqrt(Float64(Float64(1.0 / h) * Float64(1.0 / l))));
	else
		tmp = Float64(Float64(d / sqrt(h)) / sqrt(l));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (d <= 6.2e-209)
		tmp = d * sqrt(((1.0 / h) * (1.0 / l)));
	else
		tmp = (d / sqrt(h)) / sqrt(l);
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[d, 6.2e-209], N[(d * N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] * N[(1.0 / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(d / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 6.2e-209

   1. Initial program 62.8%

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

   3. Simplified 64.5%

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

   4. Taylor expanded in d around inf 9.8%

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

      1. associate-/r* 9.8%

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

      2. div-inv 9.8%

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

   6. Applied egg-rr 9.8%

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

   if 6.2e-209 < d

   1. Initial program 76.9%

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

   3. Simplified 75.9%

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

   4. Taylor expanded in d around inf 48.2%

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

      1. add-log-exp 14.7%

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

      2. sqrt-div 14.6%

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

      3. metadata-eval 14.6%

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

      4. un-div-inv 14.6%

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

   6. Applied egg-rr 14.6%

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

      1. add-log-exp 48.1%

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

      2. sqrt-prod 62.7%

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

      3. associate-/r* 62.6%

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

   8. Applied egg-rr 62.6%

      \[\leadsto \color{blue}{\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 30.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 6.2 \cdot 10^{-209}:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{h} \cdot \frac{1}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}}\\ \end{array} \]

Alternative 16: 29.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 1.4 \cdot 10^{-208}:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{h} \cdot \frac{1}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= d 1.4e-208)
   (* d (sqrt (* (/ 1.0 h) (/ 1.0 l))))
   (/ (/ d (sqrt l)) (sqrt h))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= 1.4e-208) {
		tmp = d * sqrt(((1.0 / h) * (1.0 / l)));
	} else {
		tmp = (d / sqrt(l)) / sqrt(h);
	}
	return tmp;
}

Fortran

M_m = abs(M)
D_m = abs(D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (d <= 1.4d-208) then
        tmp = d * sqrt(((1.0d0 / h) * (1.0d0 / l)))
    else
        tmp = (d / sqrt(l)) / sqrt(h)
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= 1.4e-208) {
		tmp = d * Math.sqrt(((1.0 / h) * (1.0 / l)));
	} else {
		tmp = (d / Math.sqrt(l)) / Math.sqrt(h);
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if d <= 1.4e-208:
		tmp = d * math.sqrt(((1.0 / h) * (1.0 / l)))
	else:
		tmp = (d / math.sqrt(l)) / math.sqrt(h)
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (d <= 1.4e-208)
		tmp = Float64(d * sqrt(Float64(Float64(1.0 / h) * Float64(1.0 / l))));
	else
		tmp = Float64(Float64(d / sqrt(l)) / sqrt(h));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (d <= 1.4e-208)
		tmp = d * sqrt(((1.0 / h) * (1.0 / l)));
	else
		tmp = (d / sqrt(l)) / sqrt(h);
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[d, 1.4e-208], N[(d * N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] * N[(1.0 / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(d / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 1.40000000000000001e-208

   1. Initial program 62.8%

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

   3. Simplified 64.5%

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

   4. Taylor expanded in d around inf 9.8%

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

      1. associate-/r* 9.8%

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

      2. div-inv 9.8%

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

   6. Applied egg-rr 9.8%

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

   if 1.40000000000000001e-208 < d

   1. Initial program 76.9%

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

   3. Simplified 75.9%

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

   4. Taylor expanded in d around inf 48.2%

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

      1. sqrt-div 48.1%

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

      2. metadata-eval 48.1%

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

      3. un-div-inv 48.1%

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

   6. Applied egg-rr 48.1%

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

      1. *-un-lft-identity 48.1%

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

      2. sqrt-prod 62.7%

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

      3. times-frac 62.7%

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

   8. Applied egg-rr 62.7%

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

      1. associate-*l/ 62.8%

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

      2. *-lft-identity 62.8%

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

   10. Simplified 62.8%

       \[\leadsto \color{blue}{\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 30.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 1.4 \cdot 10^{-208}:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{h} \cdot \frac{1}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \]

Alternative 17: 25.7% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.2%

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

3. Simplified 68.9%

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

4. Taylor expanded in d around inf 24.6%

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

5. Final simplification 24.6%

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

Alternative 18: 25.6% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.2%

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

3. Simplified 68.9%

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

4. Taylor expanded in d around inf 24.6%

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

   1. sqrt-div 24.6%

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

   2. metadata-eval 24.6%

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

   3. un-div-inv 24.6%

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

6. Applied egg-rr 24.6%

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

7. Final simplification 24.6%

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

Alternative 19: 4.3% accurate, 332.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ 0 \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m) :precision binary64 0.0)

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	return 0.0;
}

Fortran

M_m = abs(M)
D_m = abs(D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    code = 0.0d0
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	return 0.0;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	return 0.0

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	return 0.0
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp = code(d, h, l, M_m, D_m)
	tmp = 0.0;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := 0.0

Derivation

1. Initial program 68.2%

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

3. Simplified 68.9%

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

4. Taylor expanded in d around inf 24.6%

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

   1. add-log-exp 14.5%

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

   2. sqrt-div 14.5%

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

   3. metadata-eval 14.5%

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

   4. un-div-inv 14.5%

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

6. Applied egg-rr 14.5%

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

7. Taylor expanded in d around 0 6.1%

   \[\leadsto \log \color{blue}{1} \]

8. Final simplification 6.1%

   \[\leadsto 0 \]

Reproduce

herbie shell --seed 2023326 
(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)))))