Henrywood and Agarwal, Equation (12)

Percentage Accurate: 66.1% → 80.9%

Time: 26.3s

Alternatives: 24

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: 66.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 80.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{-d}\\ t_1 := \frac{\sqrt[3]{h}}{\ell}\\ \mathbf{if}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{t\_0}{\sqrt{-\ell}} \cdot \left(\frac{t\_0}{\sqrt{-h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(D\_m \cdot \frac{\frac{M}{2}}{d}\right)}^{2} \cdot -0.5\right)\right)\right)\\ \mathbf{elif}\;d \leq 3.3 \cdot 10^{-211}:\\ \;\;\;\;-0.125 \cdot \left(\left(\left(\left|t\_1\right| \cdot \sqrt{t\_1}\right) \cdot \frac{{M}^{2}}{d}\right) \cdot {D\_m}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D\_m}{d}\right)}^{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (let* ((t_0 (sqrt (- d))) (t_1 (/ (cbrt h) l)))
   (if (<= d -5e-310)
     (*
      (/ t_0 (sqrt (- l)))
      (*
       (/ t_0 (sqrt (- h)))
       (+ 1.0 (* (/ h l) (* (pow (* D_m (/ (/ M 2.0) d)) 2.0) -0.5)))))
     (if (<= d 3.3e-211)
       (*
        -0.125
        (* (* (* (fabs t_1) (sqrt t_1)) (/ (pow M 2.0) d)) (pow D_m 2.0)))
       (*
        (/ d (* (sqrt h) (sqrt l)))
        (- 1.0 (* 0.5 (* (/ h l) (pow (* (/ M 2.0) (/ D_m d)) 2.0)))))))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double t_0 = sqrt(-d);
	double t_1 = cbrt(h) / l;
	double tmp;
	if (d <= -5e-310) {
		tmp = (t_0 / sqrt(-l)) * ((t_0 / sqrt(-h)) * (1.0 + ((h / l) * (pow((D_m * ((M / 2.0) / d)), 2.0) * -0.5))));
	} else if (d <= 3.3e-211) {
		tmp = -0.125 * (((fabs(t_1) * sqrt(t_1)) * (pow(M, 2.0) / d)) * pow(D_m, 2.0));
	} else {
		tmp = (d / (sqrt(h) * sqrt(l))) * (1.0 - (0.5 * ((h / l) * pow(((M / 2.0) * (D_m / d)), 2.0))));
	}
	return tmp;
}

Java

D_m = Math.abs(D);
assert d < h && h < l && l < M && M < D_m;
public static double code(double d, double h, double l, double M, double D_m) {
	double t_0 = Math.sqrt(-d);
	double t_1 = Math.cbrt(h) / l;
	double tmp;
	if (d <= -5e-310) {
		tmp = (t_0 / Math.sqrt(-l)) * ((t_0 / Math.sqrt(-h)) * (1.0 + ((h / l) * (Math.pow((D_m * ((M / 2.0) / d)), 2.0) * -0.5))));
	} else if (d <= 3.3e-211) {
		tmp = -0.125 * (((Math.abs(t_1) * Math.sqrt(t_1)) * (Math.pow(M, 2.0) / d)) * Math.pow(D_m, 2.0));
	} else {
		tmp = (d / (Math.sqrt(h) * Math.sqrt(l))) * (1.0 - (0.5 * ((h / l) * Math.pow(((M / 2.0) * (D_m / d)), 2.0))));
	}
	return tmp;
}

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	t_0 = sqrt(Float64(-d))
	t_1 = Float64(cbrt(h) / l)
	tmp = 0.0
	if (d <= -5e-310)
		tmp = Float64(Float64(t_0 / sqrt(Float64(-l))) * Float64(Float64(t_0 / sqrt(Float64(-h))) * Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(D_m * Float64(Float64(M / 2.0) / d)) ^ 2.0) * -0.5)))));
	elseif (d <= 3.3e-211)
		tmp = Float64(-0.125 * Float64(Float64(Float64(abs(t_1) * sqrt(t_1)) * Float64((M ^ 2.0) / d)) * (D_m ^ 2.0)));
	else
		tmp = Float64(Float64(d / Float64(sqrt(h) * sqrt(l))) * Float64(1.0 - Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(M / 2.0) * Float64(D_m / d)) ^ 2.0)))));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := Block[{t$95$0 = N[Sqrt[(-d)], $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[h, 1/3], $MachinePrecision] / l), $MachinePrecision]}, If[LessEqual[d, -5e-310], N[(N[(t$95$0 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$0 / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(D$95$m * N[(N[(M / 2.0), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 3.3e-211], N[(-0.125 * N[(N[(N[(N[Abs[t$95$1], $MachinePrecision] * N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision] * N[(N[Power[M, 2.0], $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * N[Power[D$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M / 2.0), $MachinePrecision] * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -4.999999999999985e-310

   1. Initial program 63.6%

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

   3. Simplified 63.7%

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

      1. frac-2neg 63.7%

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

      2. sqrt-div 74.4%

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

   6. Applied egg-rr 74.4%

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

      1. frac-2neg 74.4%

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

      2. sqrt-div 85.1%

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

   8. Applied egg-rr 85.1%

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

   if -4.999999999999985e-310 < d < 3.3000000000000002e-211

   1. Initial program 37.4%

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

   3. Simplified 37.4%

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

   5. Taylor expanded in d around 0 52.6%

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

      1. *-commutative 52.6%

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

      2. associate-/l* 53.5%

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

      3. associate-*l* 53.6%

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

      4. associate-*r* 53.6%

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

      5. *-commutative 53.6%

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

      6. *-commutative 53.6%

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

      7. associate-*l* 53.6%

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

      8. *-commutative 53.6%

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

   7. Simplified 53.6%

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

      1. pow1/2 53.6%

         \[\leadsto -0.125 \cdot \left(\left(\color{blue}{{\left(\frac{h}{{\ell}^{3}}\right)}^{0.5}} \cdot \frac{{M}^{2}}{d}\right) \cdot {D}^{2}\right) \]

      2. add-cube-cbrt 53.5%

         \[\leadsto -0.125 \cdot \left(\left({\color{blue}{\left(\left(\sqrt[3]{\frac{h}{{\ell}^{3}}} \cdot \sqrt[3]{\frac{h}{{\ell}^{3}}}\right) \cdot \sqrt[3]{\frac{h}{{\ell}^{3}}}\right)}}^{0.5} \cdot \frac{{M}^{2}}{d}\right) \cdot {D}^{2}\right) \]

      3. unpow-prod-down 53.5%

         \[\leadsto -0.125 \cdot \left(\left(\color{blue}{\left({\left(\sqrt[3]{\frac{h}{{\ell}^{3}}} \cdot \sqrt[3]{\frac{h}{{\ell}^{3}}}\right)}^{0.5} \cdot {\left(\sqrt[3]{\frac{h}{{\ell}^{3}}}\right)}^{0.5}\right)} \cdot \frac{{M}^{2}}{d}\right) \cdot {D}^{2}\right) \]

      4. pow2 53.5%

         \[\leadsto -0.125 \cdot \left(\left(\left({\color{blue}{\left({\left(\sqrt[3]{\frac{h}{{\ell}^{3}}}\right)}^{2}\right)}}^{0.5} \cdot {\left(\sqrt[3]{\frac{h}{{\ell}^{3}}}\right)}^{0.5}\right) \cdot \frac{{M}^{2}}{d}\right) \cdot {D}^{2}\right) \]

      5. cbrt-div 53.4%

         \[\leadsto -0.125 \cdot \left(\left(\left({\left({\color{blue}{\left(\frac{\sqrt[3]{h}}{\sqrt[3]{{\ell}^{3}}}\right)}}^{2}\right)}^{0.5} \cdot {\left(\sqrt[3]{\frac{h}{{\ell}^{3}}}\right)}^{0.5}\right) \cdot \frac{{M}^{2}}{d}\right) \cdot {D}^{2}\right) \]

      6. unpow3 53.4%

         \[\leadsto -0.125 \cdot \left(\left(\left({\left({\left(\frac{\sqrt[3]{h}}{\sqrt[3]{\color{blue}{\left(\ell \cdot \ell\right) \cdot \ell}}}\right)}^{2}\right)}^{0.5} \cdot {\left(\sqrt[3]{\frac{h}{{\ell}^{3}}}\right)}^{0.5}\right) \cdot \frac{{M}^{2}}{d}\right) \cdot {D}^{2}\right) \]

      7. add-cbrt-cube 53.5%

         \[\leadsto -0.125 \cdot \left(\left(\left({\left({\left(\frac{\sqrt[3]{h}}{\color{blue}{\ell}}\right)}^{2}\right)}^{0.5} \cdot {\left(\sqrt[3]{\frac{h}{{\ell}^{3}}}\right)}^{0.5}\right) \cdot \frac{{M}^{2}}{d}\right) \cdot {D}^{2}\right) \]

      8. cbrt-div 58.7%

         \[\leadsto -0.125 \cdot \left(\left(\left({\left({\left(\frac{\sqrt[3]{h}}{\ell}\right)}^{2}\right)}^{0.5} \cdot {\color{blue}{\left(\frac{\sqrt[3]{h}}{\sqrt[3]{{\ell}^{3}}}\right)}}^{0.5}\right) \cdot \frac{{M}^{2}}{d}\right) \cdot {D}^{2}\right) \]

      9. unpow3 58.7%

         \[\leadsto -0.125 \cdot \left(\left(\left({\left({\left(\frac{\sqrt[3]{h}}{\ell}\right)}^{2}\right)}^{0.5} \cdot {\left(\frac{\sqrt[3]{h}}{\sqrt[3]{\color{blue}{\left(\ell \cdot \ell\right) \cdot \ell}}}\right)}^{0.5}\right) \cdot \frac{{M}^{2}}{d}\right) \cdot {D}^{2}\right) \]

      10. add-cbrt-cube 64.1%

          \[\leadsto -0.125 \cdot \left(\left(\left({\left({\left(\frac{\sqrt[3]{h}}{\ell}\right)}^{2}\right)}^{0.5} \cdot {\left(\frac{\sqrt[3]{h}}{\color{blue}{\ell}}\right)}^{0.5}\right) \cdot \frac{{M}^{2}}{d}\right) \cdot {D}^{2}\right) \]

   9. Applied egg-rr 64.1%

      \[\leadsto -0.125 \cdot \left(\left(\color{blue}{\left({\left({\left(\frac{\sqrt[3]{h}}{\ell}\right)}^{2}\right)}^{0.5} \cdot {\left(\frac{\sqrt[3]{h}}{\ell}\right)}^{0.5}\right)} \cdot \frac{{M}^{2}}{d}\right) \cdot {D}^{2}\right) \]
   10. Step-by-step derivation

       1. unpow1/2 64.1%

          \[\leadsto -0.125 \cdot \left(\left(\left(\color{blue}{\sqrt{{\left(\frac{\sqrt[3]{h}}{\ell}\right)}^{2}}} \cdot {\left(\frac{\sqrt[3]{h}}{\ell}\right)}^{0.5}\right) \cdot \frac{{M}^{2}}{d}\right) \cdot {D}^{2}\right) \]

       2. unpow2 64.1%

          \[\leadsto -0.125 \cdot \left(\left(\left(\sqrt{\color{blue}{\frac{\sqrt[3]{h}}{\ell} \cdot \frac{\sqrt[3]{h}}{\ell}}} \cdot {\left(\frac{\sqrt[3]{h}}{\ell}\right)}^{0.5}\right) \cdot \frac{{M}^{2}}{d}\right) \cdot {D}^{2}\right) \]

       3. rem-sqrt-square 74.1%

          \[\leadsto -0.125 \cdot \left(\left(\left(\color{blue}{\left|\frac{\sqrt[3]{h}}{\ell}\right|} \cdot {\left(\frac{\sqrt[3]{h}}{\ell}\right)}^{0.5}\right) \cdot \frac{{M}^{2}}{d}\right) \cdot {D}^{2}\right) \]

       4. unpow1/2 74.1%

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

   11. Simplified 74.1%

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

   if 3.3000000000000002e-211 < 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 77.8%

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

      1. pow1 77.8%

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

      2. sqrt-unprod 62.1%

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

   6. Applied egg-rr 62.1%

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

      1. unpow1 62.1%

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

      2. *-commutative 62.1%

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

   8. Simplified 62.1%

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

      1. frac-times 48.5%

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

      2. unpow2 48.5%

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

      3. *-commutative 48.5%

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

      4. associate-/l/ 55.9%

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

      5. sqrt-div 66.0%

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

      6. sqrt-div 70.0%

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

      7. sqrt-pow1 91.1%

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

      8. metadata-eval 91.1%

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

      9. pow1 91.1%

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

   10. Applied egg-rr 91.1%

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

       1. associate-/l/ 91.1%

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

   12. Simplified 91.1%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(D \cdot \frac{\frac{M}{2}}{d}\right)}^{2} \cdot -0.5\right)\right)\right)\\ \mathbf{elif}\;d \leq 3.3 \cdot 10^{-211}:\\ \;\;\;\;-0.125 \cdot \left(\left(\left(\left|\frac{\sqrt[3]{h}}{\ell}\right| \cdot \sqrt{\frac{\sqrt[3]{h}}{\ell}}\right) \cdot \frac{{M}^{2}}{d}\right) \cdot {D}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 76.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{D\_m \cdot M}{d \cdot 2}\right)}^{2}\right)\right) \leq 2 \cdot 10^{+215}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\left(1 + \frac{h}{\ell} \cdot \left({\left(D\_m \cdot \frac{\frac{M}{2}}{d}\right)}^{2} \cdot -0.5\right)\right) \cdot \sqrt{\frac{d}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D\_m}{d}\right)}^{2}\right)\right) \cdot \left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (if (<=
      (*
       (* (pow (/ d h) 0.5) (pow (/ d l) 0.5))
       (- 1.0 (* (/ h l) (* 0.5 (pow (/ (* D_m M) (* d 2.0)) 2.0)))))
      2e+215)
   (*
    (sqrt (/ d l))
    (*
     (+ 1.0 (* (/ h l) (* (pow (* D_m (/ (/ M 2.0) d)) 2.0) -0.5)))
     (sqrt (/ d h))))
   (*
    (- 1.0 (* 0.5 (* (/ h l) (pow (* (/ M 2.0) (/ D_m d)) 2.0))))
    (fabs (/ d (sqrt (* l h)))))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double tmp;
	if (((pow((d / h), 0.5) * pow((d / l), 0.5)) * (1.0 - ((h / l) * (0.5 * pow(((D_m * M) / (d * 2.0)), 2.0))))) <= 2e+215) {
		tmp = sqrt((d / l)) * ((1.0 + ((h / l) * (pow((D_m * ((M / 2.0) / d)), 2.0) * -0.5))) * sqrt((d / h)));
	} else {
		tmp = (1.0 - (0.5 * ((h / l) * pow(((M / 2.0) * (D_m / d)), 2.0)))) * fabs((d / sqrt((l * h))));
	}
	return tmp;
}

Fortran

D_m = abs(d)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_m)
    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_m
    real(8) :: tmp
    if (((((d / h) ** 0.5d0) * ((d / l) ** 0.5d0)) * (1.0d0 - ((h / l) * (0.5d0 * (((d_m * m) / (d * 2.0d0)) ** 2.0d0))))) <= 2d+215) then
        tmp = sqrt((d / l)) * ((1.0d0 + ((h / l) * (((d_m * ((m / 2.0d0) / d)) ** 2.0d0) * (-0.5d0)))) * sqrt((d / h)))
    else
        tmp = (1.0d0 - (0.5d0 * ((h / l) * (((m / 2.0d0) * (d_m / d)) ** 2.0d0)))) * abs((d / sqrt((l * h))))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert d < h && h < l && l < M && M < D_m;
public static double code(double d, double h, double l, double M, double D_m) {
	double tmp;
	if (((Math.pow((d / h), 0.5) * Math.pow((d / l), 0.5)) * (1.0 - ((h / l) * (0.5 * Math.pow(((D_m * M) / (d * 2.0)), 2.0))))) <= 2e+215) {
		tmp = Math.sqrt((d / l)) * ((1.0 + ((h / l) * (Math.pow((D_m * ((M / 2.0) / d)), 2.0) * -0.5))) * Math.sqrt((d / h)));
	} else {
		tmp = (1.0 - (0.5 * ((h / l) * Math.pow(((M / 2.0) * (D_m / d)), 2.0)))) * Math.abs((d / Math.sqrt((l * h))));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[d, h, l, M, D_m] = sort([d, h, l, M, D_m])
def code(d, h, l, M, D_m):
	tmp = 0
	if ((math.pow((d / h), 0.5) * math.pow((d / l), 0.5)) * (1.0 - ((h / l) * (0.5 * math.pow(((D_m * M) / (d * 2.0)), 2.0))))) <= 2e+215:
		tmp = math.sqrt((d / l)) * ((1.0 + ((h / l) * (math.pow((D_m * ((M / 2.0) / d)), 2.0) * -0.5))) * math.sqrt((d / h)))
	else:
		tmp = (1.0 - (0.5 * ((h / l) * math.pow(((M / 2.0) * (D_m / d)), 2.0)))) * math.fabs((d / math.sqrt((l * h))))
	return tmp

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	tmp = 0.0
	if (Float64(Float64((Float64(d / h) ^ 0.5) * (Float64(d / l) ^ 0.5)) * Float64(1.0 - Float64(Float64(h / l) * Float64(0.5 * (Float64(Float64(D_m * M) / Float64(d * 2.0)) ^ 2.0))))) <= 2e+215)
		tmp = Float64(sqrt(Float64(d / l)) * Float64(Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(D_m * Float64(Float64(M / 2.0) / d)) ^ 2.0) * -0.5))) * sqrt(Float64(d / h))));
	else
		tmp = Float64(Float64(1.0 - Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(M / 2.0) * Float64(D_m / d)) ^ 2.0)))) * abs(Float64(d / sqrt(Float64(l * h)))));
	end
	return tmp
end

MATLAB

D_m = abs(D);
d, h, l, M, D_m = num2cell(sort([d, h, l, M, D_m])){:}
function tmp_2 = code(d, h, l, M, D_m)
	tmp = 0.0;
	if (((((d / h) ^ 0.5) * ((d / l) ^ 0.5)) * (1.0 - ((h / l) * (0.5 * (((D_m * M) / (d * 2.0)) ^ 2.0))))) <= 2e+215)
		tmp = sqrt((d / l)) * ((1.0 + ((h / l) * (((D_m * ((M / 2.0) / d)) ^ 2.0) * -0.5))) * sqrt((d / h)));
	else
		tmp = (1.0 - (0.5 * ((h / l) * (((M / 2.0) * (D_m / d)) ^ 2.0)))) * abs((d / sqrt((l * h))));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := If[LessEqual[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), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e+215], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(D$95$m * N[(N[(M / 2.0), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M / 2.0), $MachinePrecision] * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Abs[N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 1.99999999999999981e215

   1. Initial program 87.1%

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

   3. Simplified 86.6%

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

   if 1.99999999999999981e215 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 21.3%

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

   3. Simplified 22.6%

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

      1. pow1 22.6%

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

      2. sqrt-unprod 21.4%

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

   6. Applied egg-rr 21.4%

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

      1. unpow1 21.4%

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

      2. *-commutative 21.4%

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

   8. Simplified 21.4%

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

      1. add-sqr-sqrt 21.4%

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

      2. rem-sqrt-square 21.4%

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

      3. frac-times 21.4%

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

      4. unpow2 21.4%

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

      5. sqrt-div 27.7%

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

      6. sqrt-pow1 60.3%

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

      7. metadata-eval 60.3%

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

      8. pow1 60.3%

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

   10. Applied egg-rr 60.3%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}\right)\right) \leq 2 \cdot 10^{+215}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\left(1 + \frac{h}{\ell} \cdot \left({\left(D \cdot \frac{\frac{M}{2}}{d}\right)}^{2} \cdot -0.5\right)\right) \cdot \sqrt{\frac{d}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}\right)\right) \cdot \left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{-d}\\ \mathbf{if}\;\ell \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\frac{t\_0}{\sqrt{-\ell}} \cdot \left(\frac{t\_0}{\sqrt{-h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(D\_m \cdot \frac{\frac{M}{2}}{d}\right)}^{2} \cdot -0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D\_m}{d}\right)}^{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (let* ((t_0 (sqrt (- d))))
   (if (<= l -4e-310)
     (*
      (/ t_0 (sqrt (- l)))
      (*
       (/ t_0 (sqrt (- h)))
       (+ 1.0 (* (/ h l) (* (pow (* D_m (/ (/ M 2.0) d)) 2.0) -0.5)))))
     (*
      (/ d (* (sqrt h) (sqrt l)))
      (- 1.0 (* 0.5 (* (/ h l) (pow (* (/ M 2.0) (/ D_m d)) 2.0))))))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double t_0 = sqrt(-d);
	double tmp;
	if (l <= -4e-310) {
		tmp = (t_0 / sqrt(-l)) * ((t_0 / sqrt(-h)) * (1.0 + ((h / l) * (pow((D_m * ((M / 2.0) / d)), 2.0) * -0.5))));
	} else {
		tmp = (d / (sqrt(h) * sqrt(l))) * (1.0 - (0.5 * ((h / l) * pow(((M / 2.0) * (D_m / d)), 2.0))));
	}
	return tmp;
}

Fortran

D_m = abs(d)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_m)
    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_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(-d)
    if (l <= (-4d-310)) then
        tmp = (t_0 / sqrt(-l)) * ((t_0 / sqrt(-h)) * (1.0d0 + ((h / l) * (((d_m * ((m / 2.0d0) / d)) ** 2.0d0) * (-0.5d0)))))
    else
        tmp = (d / (sqrt(h) * sqrt(l))) * (1.0d0 - (0.5d0 * ((h / l) * (((m / 2.0d0) * (d_m / d)) ** 2.0d0))))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert d < h && h < l && l < M && M < D_m;
public static double code(double d, double h, double l, double M, double D_m) {
	double t_0 = Math.sqrt(-d);
	double tmp;
	if (l <= -4e-310) {
		tmp = (t_0 / Math.sqrt(-l)) * ((t_0 / Math.sqrt(-h)) * (1.0 + ((h / l) * (Math.pow((D_m * ((M / 2.0) / d)), 2.0) * -0.5))));
	} else {
		tmp = (d / (Math.sqrt(h) * Math.sqrt(l))) * (1.0 - (0.5 * ((h / l) * Math.pow(((M / 2.0) * (D_m / d)), 2.0))));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[d, h, l, M, D_m] = sort([d, h, l, M, D_m])
def code(d, h, l, M, D_m):
	t_0 = math.sqrt(-d)
	tmp = 0
	if l <= -4e-310:
		tmp = (t_0 / math.sqrt(-l)) * ((t_0 / math.sqrt(-h)) * (1.0 + ((h / l) * (math.pow((D_m * ((M / 2.0) / d)), 2.0) * -0.5))))
	else:
		tmp = (d / (math.sqrt(h) * math.sqrt(l))) * (1.0 - (0.5 * ((h / l) * math.pow(((M / 2.0) * (D_m / d)), 2.0))))
	return tmp

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	t_0 = sqrt(Float64(-d))
	tmp = 0.0
	if (l <= -4e-310)
		tmp = Float64(Float64(t_0 / sqrt(Float64(-l))) * Float64(Float64(t_0 / sqrt(Float64(-h))) * Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(D_m * Float64(Float64(M / 2.0) / d)) ^ 2.0) * -0.5)))));
	else
		tmp = Float64(Float64(d / Float64(sqrt(h) * sqrt(l))) * Float64(1.0 - Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(M / 2.0) * Float64(D_m / d)) ^ 2.0)))));
	end
	return tmp
end

MATLAB

D_m = abs(D);
d, h, l, M, D_m = num2cell(sort([d, h, l, M, D_m])){:}
function tmp_2 = code(d, h, l, M, D_m)
	t_0 = sqrt(-d);
	tmp = 0.0;
	if (l <= -4e-310)
		tmp = (t_0 / sqrt(-l)) * ((t_0 / sqrt(-h)) * (1.0 + ((h / l) * (((D_m * ((M / 2.0) / d)) ^ 2.0) * -0.5))));
	else
		tmp = (d / (sqrt(h) * sqrt(l))) * (1.0 - (0.5 * ((h / l) * (((M / 2.0) * (D_m / d)) ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := Block[{t$95$0 = N[Sqrt[(-d)], $MachinePrecision]}, If[LessEqual[l, -4e-310], N[(N[(t$95$0 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$0 / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(D$95$m * N[(N[(M / 2.0), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M / 2.0), $MachinePrecision] * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < -3.999999999999988e-310

   1. Initial program 63.6%

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

   3. Simplified 63.7%

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

      1. frac-2neg 63.7%

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

      2. sqrt-div 74.4%

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

   6. Applied egg-rr 74.4%

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

      1. frac-2neg 74.4%

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

      2. sqrt-div 85.1%

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

   8. Applied egg-rr 85.1%

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

   if -3.999999999999988e-310 < l

   1. Initial program 71.0%

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

   3. Simplified 71.8%

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

      1. pow1 71.8%

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

      2. sqrt-unprod 55.1%

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

   6. Applied egg-rr 55.1%

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

      1. unpow1 55.1%

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

      2. *-commutative 55.1%

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

   8. Simplified 55.1%

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

      1. frac-times 41.4%

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

      2. unpow2 41.4%

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

      3. *-commutative 41.4%

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

      4. associate-/l/ 47.7%

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

      5. sqrt-div 56.3%

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

      6. sqrt-div 59.7%

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

      7. sqrt-pow1 84.7%

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

      8. metadata-eval 84.7%

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

      9. pow1 84.7%

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

   10. Applied egg-rr 84.7%

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

       1. associate-/l/ 84.7%

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

   12. Simplified 84.7%

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

4. Final simplification 84.9%

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

Alternative 4: 78.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{-d}\\ \mathbf{if}\;h \leq -1.22 \cdot 10^{+126}:\\ \;\;\;\;\left(\frac{t\_0}{\sqrt{-h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(D\_m \cdot \frac{\frac{M}{2}}{d}\right)}^{2} \cdot -0.5\right)\right)\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\\ \mathbf{elif}\;h \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\frac{t\_0}{\sqrt{-\ell}} \cdot \left(1 - h \cdot \left(\frac{0.125}{\ell} \cdot {\left(D\_m \cdot \frac{M}{d}\right)}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D\_m}{d}\right)}^{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (let* ((t_0 (sqrt (- d))))
   (if (<= h -1.22e+126)
     (*
      (*
       (/ t_0 (sqrt (- h)))
       (+ 1.0 (* (/ h l) (* (pow (* D_m (/ (/ M 2.0) d)) 2.0) -0.5))))
      (/ 1.0 (sqrt (/ l d))))
     (if (<= h -1e-310)
       (*
        (sqrt (/ d h))
        (*
         (/ t_0 (sqrt (- l)))
         (- 1.0 (* h (* (/ 0.125 l) (pow (* D_m (/ M d)) 2.0))))))
       (*
        (/ d (* (sqrt h) (sqrt l)))
        (- 1.0 (* 0.5 (* (/ h l) (pow (* (/ M 2.0) (/ D_m d)) 2.0)))))))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double t_0 = sqrt(-d);
	double tmp;
	if (h <= -1.22e+126) {
		tmp = ((t_0 / sqrt(-h)) * (1.0 + ((h / l) * (pow((D_m * ((M / 2.0) / d)), 2.0) * -0.5)))) * (1.0 / sqrt((l / d)));
	} else if (h <= -1e-310) {
		tmp = sqrt((d / h)) * ((t_0 / sqrt(-l)) * (1.0 - (h * ((0.125 / l) * pow((D_m * (M / d)), 2.0)))));
	} else {
		tmp = (d / (sqrt(h) * sqrt(l))) * (1.0 - (0.5 * ((h / l) * pow(((M / 2.0) * (D_m / d)), 2.0))));
	}
	return tmp;
}

Fortran

D_m = abs(d)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_m)
    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_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(-d)
    if (h <= (-1.22d+126)) then
        tmp = ((t_0 / sqrt(-h)) * (1.0d0 + ((h / l) * (((d_m * ((m / 2.0d0) / d)) ** 2.0d0) * (-0.5d0))))) * (1.0d0 / sqrt((l / d)))
    else if (h <= (-1d-310)) then
        tmp = sqrt((d / h)) * ((t_0 / sqrt(-l)) * (1.0d0 - (h * ((0.125d0 / l) * ((d_m * (m / d)) ** 2.0d0)))))
    else
        tmp = (d / (sqrt(h) * sqrt(l))) * (1.0d0 - (0.5d0 * ((h / l) * (((m / 2.0d0) * (d_m / d)) ** 2.0d0))))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert d < h && h < l && l < M && M < D_m;
public static double code(double d, double h, double l, double M, double D_m) {
	double t_0 = Math.sqrt(-d);
	double tmp;
	if (h <= -1.22e+126) {
		tmp = ((t_0 / Math.sqrt(-h)) * (1.0 + ((h / l) * (Math.pow((D_m * ((M / 2.0) / d)), 2.0) * -0.5)))) * (1.0 / Math.sqrt((l / d)));
	} else if (h <= -1e-310) {
		tmp = Math.sqrt((d / h)) * ((t_0 / Math.sqrt(-l)) * (1.0 - (h * ((0.125 / l) * Math.pow((D_m * (M / d)), 2.0)))));
	} else {
		tmp = (d / (Math.sqrt(h) * Math.sqrt(l))) * (1.0 - (0.5 * ((h / l) * Math.pow(((M / 2.0) * (D_m / d)), 2.0))));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[d, h, l, M, D_m] = sort([d, h, l, M, D_m])
def code(d, h, l, M, D_m):
	t_0 = math.sqrt(-d)
	tmp = 0
	if h <= -1.22e+126:
		tmp = ((t_0 / math.sqrt(-h)) * (1.0 + ((h / l) * (math.pow((D_m * ((M / 2.0) / d)), 2.0) * -0.5)))) * (1.0 / math.sqrt((l / d)))
	elif h <= -1e-310:
		tmp = math.sqrt((d / h)) * ((t_0 / math.sqrt(-l)) * (1.0 - (h * ((0.125 / l) * math.pow((D_m * (M / d)), 2.0)))))
	else:
		tmp = (d / (math.sqrt(h) * math.sqrt(l))) * (1.0 - (0.5 * ((h / l) * math.pow(((M / 2.0) * (D_m / d)), 2.0))))
	return tmp

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	t_0 = sqrt(Float64(-d))
	tmp = 0.0
	if (h <= -1.22e+126)
		tmp = Float64(Float64(Float64(t_0 / sqrt(Float64(-h))) * Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(D_m * Float64(Float64(M / 2.0) / d)) ^ 2.0) * -0.5)))) * Float64(1.0 / sqrt(Float64(l / d))));
	elseif (h <= -1e-310)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(Float64(t_0 / sqrt(Float64(-l))) * Float64(1.0 - Float64(h * Float64(Float64(0.125 / l) * (Float64(D_m * Float64(M / d)) ^ 2.0))))));
	else
		tmp = Float64(Float64(d / Float64(sqrt(h) * sqrt(l))) * Float64(1.0 - Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(M / 2.0) * Float64(D_m / d)) ^ 2.0)))));
	end
	return tmp
end

MATLAB

D_m = abs(D);
d, h, l, M, D_m = num2cell(sort([d, h, l, M, D_m])){:}
function tmp_2 = code(d, h, l, M, D_m)
	t_0 = sqrt(-d);
	tmp = 0.0;
	if (h <= -1.22e+126)
		tmp = ((t_0 / sqrt(-h)) * (1.0 + ((h / l) * (((D_m * ((M / 2.0) / d)) ^ 2.0) * -0.5)))) * (1.0 / sqrt((l / d)));
	elseif (h <= -1e-310)
		tmp = sqrt((d / h)) * ((t_0 / sqrt(-l)) * (1.0 - (h * ((0.125 / l) * ((D_m * (M / d)) ^ 2.0)))));
	else
		tmp = (d / (sqrt(h) * sqrt(l))) * (1.0 - (0.5 * ((h / l) * (((M / 2.0) * (D_m / d)) ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := Block[{t$95$0 = N[Sqrt[(-d)], $MachinePrecision]}, If[LessEqual[h, -1.22e+126], N[(N[(N[(t$95$0 / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(D$95$m * N[(N[(M / 2.0), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Sqrt[N[(l / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, -1e-310], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[(t$95$0 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(h * N[(N[(0.125 / l), $MachinePrecision] * N[Power[N[(D$95$m * N[(M / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M / 2.0), $MachinePrecision] * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if h < -1.21999999999999995e126

   1. Initial program 53.0%

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

   3. Simplified 55.4%

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

      1. clear-num 55.4%

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

      2. sqrt-div 55.3%

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

      3. metadata-eval 55.3%

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

   6. Applied egg-rr 55.3%

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

      1. frac-2neg 55.5%

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

      2. sqrt-div 82.8%

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

   8. Applied egg-rr 80.3%

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

   if -1.21999999999999995e126 < h < -9.999999999999969e-311

   1. Initial program 68.5%

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

   3. Simplified 68.4%

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

   5. Taylor expanded in h around -inf 43.2%

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

      1. associate-*r* 43.2%

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

      2. neg-mul-1 43.2%

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

      3. sub-neg 43.2%

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

      4. distribute-lft-in 43.2%

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

   7. Simplified 66.5%

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

      1. frac-2neg 67.4%

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

      2. sqrt-div 83.1%

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

   9. Applied egg-rr 85.6%

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

   if -9.999999999999969e-311 < h

   1. Initial program 71.0%

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

   3. Simplified 71.8%

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

      1. pow1 71.8%

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

      2. sqrt-unprod 55.1%

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

   6. Applied egg-rr 55.1%

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

      1. unpow1 55.1%

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

      2. *-commutative 55.1%

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

   8. Simplified 55.1%

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

      1. frac-times 41.4%

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

      2. unpow2 41.4%

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

      3. *-commutative 41.4%

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

      4. associate-/l/ 47.7%

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

      5. sqrt-div 56.3%

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

      6. sqrt-div 59.7%

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

      7. sqrt-pow1 84.7%

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

      8. metadata-eval 84.7%

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

      9. pow1 84.7%

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

   10. Applied egg-rr 84.7%

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

       1. associate-/l/ 84.7%

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

   12. Simplified 84.7%

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

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -1.22 \cdot 10^{+126}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(D \cdot \frac{\frac{M}{2}}{d}\right)}^{2} \cdot -0.5\right)\right)\right) \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\\ \mathbf{elif}\;h \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \left(1 - h \cdot \left(\frac{0.125}{\ell} \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := 1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D\_m}{d}\right)}^{2}\right)\\ t_1 := \sqrt{-d}\\ \mathbf{if}\;h \leq -4.25 \cdot 10^{+127}:\\ \;\;\;\;t\_0 \cdot \left(\frac{t\_1}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\right)\\ \mathbf{elif}\;h \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\frac{t\_1}{\sqrt{-\ell}} \cdot \left(1 - h \cdot \left(\frac{0.125}{\ell} \cdot {\left(D\_m \cdot \frac{M}{d}\right)}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot t\_0\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (let* ((t_0 (- 1.0 (* 0.5 (* (/ h l) (pow (* (/ M 2.0) (/ D_m d)) 2.0)))))
        (t_1 (sqrt (- d))))
   (if (<= h -4.25e+127)
     (* t_0 (* (/ t_1 (sqrt (- h))) (sqrt (/ d l))))
     (if (<= h -1e-310)
       (*
        (sqrt (/ d h))
        (*
         (/ t_1 (sqrt (- l)))
         (- 1.0 (* h (* (/ 0.125 l) (pow (* D_m (/ M d)) 2.0))))))
       (* (/ d (* (sqrt h) (sqrt l))) t_0)))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double t_0 = 1.0 - (0.5 * ((h / l) * pow(((M / 2.0) * (D_m / d)), 2.0)));
	double t_1 = sqrt(-d);
	double tmp;
	if (h <= -4.25e+127) {
		tmp = t_0 * ((t_1 / sqrt(-h)) * sqrt((d / l)));
	} else if (h <= -1e-310) {
		tmp = sqrt((d / h)) * ((t_1 / sqrt(-l)) * (1.0 - (h * ((0.125 / l) * pow((D_m * (M / d)), 2.0)))));
	} else {
		tmp = (d / (sqrt(h) * sqrt(l))) * t_0;
	}
	return tmp;
}

Fortran

D_m = abs(d)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_m)
    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_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 - (0.5d0 * ((h / l) * (((m / 2.0d0) * (d_m / d)) ** 2.0d0)))
    t_1 = sqrt(-d)
    if (h <= (-4.25d+127)) then
        tmp = t_0 * ((t_1 / sqrt(-h)) * sqrt((d / l)))
    else if (h <= (-1d-310)) then
        tmp = sqrt((d / h)) * ((t_1 / sqrt(-l)) * (1.0d0 - (h * ((0.125d0 / l) * ((d_m * (m / d)) ** 2.0d0)))))
    else
        tmp = (d / (sqrt(h) * sqrt(l))) * t_0
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert d < h && h < l && l < M && M < D_m;
public static double code(double d, double h, double l, double M, double D_m) {
	double t_0 = 1.0 - (0.5 * ((h / l) * Math.pow(((M / 2.0) * (D_m / d)), 2.0)));
	double t_1 = Math.sqrt(-d);
	double tmp;
	if (h <= -4.25e+127) {
		tmp = t_0 * ((t_1 / Math.sqrt(-h)) * Math.sqrt((d / l)));
	} else if (h <= -1e-310) {
		tmp = Math.sqrt((d / h)) * ((t_1 / Math.sqrt(-l)) * (1.0 - (h * ((0.125 / l) * Math.pow((D_m * (M / d)), 2.0)))));
	} else {
		tmp = (d / (Math.sqrt(h) * Math.sqrt(l))) * t_0;
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[d, h, l, M, D_m] = sort([d, h, l, M, D_m])
def code(d, h, l, M, D_m):
	t_0 = 1.0 - (0.5 * ((h / l) * math.pow(((M / 2.0) * (D_m / d)), 2.0)))
	t_1 = math.sqrt(-d)
	tmp = 0
	if h <= -4.25e+127:
		tmp = t_0 * ((t_1 / math.sqrt(-h)) * math.sqrt((d / l)))
	elif h <= -1e-310:
		tmp = math.sqrt((d / h)) * ((t_1 / math.sqrt(-l)) * (1.0 - (h * ((0.125 / l) * math.pow((D_m * (M / d)), 2.0)))))
	else:
		tmp = (d / (math.sqrt(h) * math.sqrt(l))) * t_0
	return tmp

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	t_0 = Float64(1.0 - Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(M / 2.0) * Float64(D_m / d)) ^ 2.0))))
	t_1 = sqrt(Float64(-d))
	tmp = 0.0
	if (h <= -4.25e+127)
		tmp = Float64(t_0 * Float64(Float64(t_1 / sqrt(Float64(-h))) * sqrt(Float64(d / l))));
	elseif (h <= -1e-310)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(Float64(t_1 / sqrt(Float64(-l))) * Float64(1.0 - Float64(h * Float64(Float64(0.125 / l) * (Float64(D_m * Float64(M / d)) ^ 2.0))))));
	else
		tmp = Float64(Float64(d / Float64(sqrt(h) * sqrt(l))) * t_0);
	end
	return tmp
end

MATLAB

D_m = abs(D);
d, h, l, M, D_m = num2cell(sort([d, h, l, M, D_m])){:}
function tmp_2 = code(d, h, l, M, D_m)
	t_0 = 1.0 - (0.5 * ((h / l) * (((M / 2.0) * (D_m / d)) ^ 2.0)));
	t_1 = sqrt(-d);
	tmp = 0.0;
	if (h <= -4.25e+127)
		tmp = t_0 * ((t_1 / sqrt(-h)) * sqrt((d / l)));
	elseif (h <= -1e-310)
		tmp = sqrt((d / h)) * ((t_1 / sqrt(-l)) * (1.0 - (h * ((0.125 / l) * ((D_m * (M / d)) ^ 2.0)))));
	else
		tmp = (d / (sqrt(h) * sqrt(l))) * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := Block[{t$95$0 = N[(1.0 - N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M / 2.0), $MachinePrecision] * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[(-d)], $MachinePrecision]}, If[LessEqual[h, -4.25e+127], N[(t$95$0 * N[(N[(t$95$1 / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, -1e-310], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[(t$95$1 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(h * N[(N[(0.125 / l), $MachinePrecision] * N[Power[N[(D$95$m * N[(M / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if h < -4.2499999999999998e127

   1. Initial program 53.0%

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

   3. Simplified 55.4%

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

      1. frac-2neg 55.5%

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

      2. sqrt-div 82.8%

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

   6. Applied egg-rr 75.3%

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

   if -4.2499999999999998e127 < h < -9.999999999999969e-311

   1. Initial program 68.5%

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

   3. Simplified 68.4%

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

   5. Taylor expanded in h around -inf 43.2%

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

      1. associate-*r* 43.2%

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

      2. neg-mul-1 43.2%

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

      3. sub-neg 43.2%

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

      4. distribute-lft-in 43.2%

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

   7. Simplified 66.5%

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

      1. frac-2neg 67.4%

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

      2. sqrt-div 83.1%

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

   9. Applied egg-rr 85.6%

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

   if -9.999999999999969e-311 < h

   1. Initial program 71.0%

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

   3. Simplified 71.8%

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

      1. pow1 71.8%

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

      2. sqrt-unprod 55.1%

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

   6. Applied egg-rr 55.1%

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

      1. unpow1 55.1%

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

      2. *-commutative 55.1%

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

   8. Simplified 55.1%

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

      1. frac-times 41.4%

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

      2. unpow2 41.4%

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

      3. *-commutative 41.4%

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

      4. associate-/l/ 47.7%

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

      5. sqrt-div 56.3%

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

      6. sqrt-div 59.7%

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

      7. sqrt-pow1 84.7%

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

      8. metadata-eval 84.7%

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

      9. pow1 84.7%

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

   10. Applied egg-rr 84.7%

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

       1. associate-/l/ 84.7%

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

   12. Simplified 84.7%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -4.25 \cdot 10^{+127}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}\right)\right) \cdot \left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\right)\\ \mathbf{elif}\;h \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \left(1 - h \cdot \left(\frac{0.125}{\ell} \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -1.8 \cdot 10^{-139}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \left(1 - h \cdot \left(\frac{0.125}{\ell} \cdot {\left(D\_m \cdot \frac{M}{d}\right)}^{2}\right)\right)\right)\\ \mathbf{elif}\;d \leq -7.2 \cdot 10^{-300}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(-1 + 0.5 \cdot \frac{h \cdot {\left(\frac{M \cdot \frac{D\_m}{d}}{2}\right)}^{2}}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D\_m}{d}\right)}^{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (if (<= d -1.8e-139)
   (*
    (sqrt (/ d h))
    (*
     (/ (sqrt (- d)) (sqrt (- l)))
     (- 1.0 (* h (* (/ 0.125 l) (pow (* D_m (/ M d)) 2.0))))))
   (if (<= d -7.2e-300)
     (*
      (* d (sqrt (/ 1.0 (* l h))))
      (+ -1.0 (* 0.5 (/ (* h (pow (/ (* M (/ D_m d)) 2.0) 2.0)) l))))
     (*
      (/ d (* (sqrt h) (sqrt l)))
      (- 1.0 (* 0.5 (* (/ h l) (pow (* (/ M 2.0) (/ D_m d)) 2.0))))))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double tmp;
	if (d <= -1.8e-139) {
		tmp = sqrt((d / h)) * ((sqrt(-d) / sqrt(-l)) * (1.0 - (h * ((0.125 / l) * pow((D_m * (M / d)), 2.0)))));
	} else if (d <= -7.2e-300) {
		tmp = (d * sqrt((1.0 / (l * h)))) * (-1.0 + (0.5 * ((h * pow(((M * (D_m / d)) / 2.0), 2.0)) / l)));
	} else {
		tmp = (d / (sqrt(h) * sqrt(l))) * (1.0 - (0.5 * ((h / l) * pow(((M / 2.0) * (D_m / d)), 2.0))));
	}
	return tmp;
}

Fortran

D_m = abs(d)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_m)
    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_m
    real(8) :: tmp
    if (d <= (-1.8d-139)) then
        tmp = sqrt((d / h)) * ((sqrt(-d) / sqrt(-l)) * (1.0d0 - (h * ((0.125d0 / l) * ((d_m * (m / d)) ** 2.0d0)))))
    else if (d <= (-7.2d-300)) then
        tmp = (d * sqrt((1.0d0 / (l * h)))) * ((-1.0d0) + (0.5d0 * ((h * (((m * (d_m / d)) / 2.0d0) ** 2.0d0)) / l)))
    else
        tmp = (d / (sqrt(h) * sqrt(l))) * (1.0d0 - (0.5d0 * ((h / l) * (((m / 2.0d0) * (d_m / d)) ** 2.0d0))))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert d < h && h < l && l < M && M < D_m;
public static double code(double d, double h, double l, double M, double D_m) {
	double tmp;
	if (d <= -1.8e-139) {
		tmp = Math.sqrt((d / h)) * ((Math.sqrt(-d) / Math.sqrt(-l)) * (1.0 - (h * ((0.125 / l) * Math.pow((D_m * (M / d)), 2.0)))));
	} else if (d <= -7.2e-300) {
		tmp = (d * Math.sqrt((1.0 / (l * h)))) * (-1.0 + (0.5 * ((h * Math.pow(((M * (D_m / d)) / 2.0), 2.0)) / l)));
	} else {
		tmp = (d / (Math.sqrt(h) * Math.sqrt(l))) * (1.0 - (0.5 * ((h / l) * Math.pow(((M / 2.0) * (D_m / d)), 2.0))));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[d, h, l, M, D_m] = sort([d, h, l, M, D_m])
def code(d, h, l, M, D_m):
	tmp = 0
	if d <= -1.8e-139:
		tmp = math.sqrt((d / h)) * ((math.sqrt(-d) / math.sqrt(-l)) * (1.0 - (h * ((0.125 / l) * math.pow((D_m * (M / d)), 2.0)))))
	elif d <= -7.2e-300:
		tmp = (d * math.sqrt((1.0 / (l * h)))) * (-1.0 + (0.5 * ((h * math.pow(((M * (D_m / d)) / 2.0), 2.0)) / l)))
	else:
		tmp = (d / (math.sqrt(h) * math.sqrt(l))) * (1.0 - (0.5 * ((h / l) * math.pow(((M / 2.0) * (D_m / d)), 2.0))))
	return tmp

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	tmp = 0.0
	if (d <= -1.8e-139)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-l))) * Float64(1.0 - Float64(h * Float64(Float64(0.125 / l) * (Float64(D_m * Float64(M / d)) ^ 2.0))))));
	elseif (d <= -7.2e-300)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(l * h)))) * Float64(-1.0 + Float64(0.5 * Float64(Float64(h * (Float64(Float64(M * Float64(D_m / d)) / 2.0) ^ 2.0)) / l))));
	else
		tmp = Float64(Float64(d / Float64(sqrt(h) * sqrt(l))) * Float64(1.0 - Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(M / 2.0) * Float64(D_m / d)) ^ 2.0)))));
	end
	return tmp
end

MATLAB

D_m = abs(D);
d, h, l, M, D_m = num2cell(sort([d, h, l, M, D_m])){:}
function tmp_2 = code(d, h, l, M, D_m)
	tmp = 0.0;
	if (d <= -1.8e-139)
		tmp = sqrt((d / h)) * ((sqrt(-d) / sqrt(-l)) * (1.0 - (h * ((0.125 / l) * ((D_m * (M / d)) ^ 2.0)))));
	elseif (d <= -7.2e-300)
		tmp = (d * sqrt((1.0 / (l * h)))) * (-1.0 + (0.5 * ((h * (((M * (D_m / d)) / 2.0) ^ 2.0)) / l)));
	else
		tmp = (d / (sqrt(h) * sqrt(l))) * (1.0 - (0.5 * ((h / l) * (((M / 2.0) * (D_m / d)) ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := If[LessEqual[d, -1.8e-139], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(h * N[(N[(0.125 / l), $MachinePrecision] * N[Power[N[(D$95$m * N[(M / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -7.2e-300], N[(N[(d * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-1.0 + N[(0.5 * N[(N[(h * N[Power[N[(N[(M * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M / 2.0), $MachinePrecision] * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.80000000000000002e-139

   1. Initial program 77.2%

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

   3. Simplified 78.4%

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

   5. Taylor expanded in h around -inf 47.8%

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

      1. associate-*r* 47.8%

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

      2. neg-mul-1 47.8%

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

      3. sub-neg 47.8%

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

      4. distribute-lft-in 47.8%

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

   7. Simplified 77.5%

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

      1. frac-2neg 77.2%

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

      2. sqrt-div 85.3%

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

   9. Applied egg-rr 87.8%

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

   if -1.80000000000000002e-139 < d < -7.20000000000000031e-300

   1. Initial program 35.5%

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

   3. Simplified 35.2%

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

      1. pow1 35.2%

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

      2. sqrt-unprod 32.4%

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

   6. Applied egg-rr 32.4%

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

      1. unpow1 32.4%

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

      2. *-commutative 32.4%

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

      3. associate-*r/ 22.0%

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

      4. associate-*l/ 9.0%

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

      5. unpow2 9.0%

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

   8. Simplified 9.0%

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

      1. associate-*r/ 9.1%

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

      2. associate-*l/ 9.1%

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

   10. Applied egg-rr 9.1%

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

   11. Taylor expanded in d around -inf 69.4%

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

   if -7.20000000000000031e-300 < d

   1. Initial program 69.9%

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

   3. Simplified 70.7%

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

      1. pow1 70.7%

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

      2. sqrt-unprod 54.3%

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

   6. Applied egg-rr 54.3%

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

      1. unpow1 54.3%

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

      2. *-commutative 54.3%

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

   8. Simplified 54.3%

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

      1. frac-times 40.8%

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

      2. unpow2 40.8%

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

      3. *-commutative 40.8%

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

      4. associate-/l/ 47.0%

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

      5. sqrt-div 55.5%

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

      6. sqrt-div 58.8%

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

      7. sqrt-pow1 83.4%

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

      8. metadata-eval 83.4%

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

      9. pow1 83.4%

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

   10. Applied egg-rr 83.4%

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

       1. associate-/l/ 83.4%

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

   12. Simplified 83.4%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.8 \cdot 10^{-139}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \left(1 - h \cdot \left(\frac{0.125}{\ell} \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}\right)\right)\right)\\ \mathbf{elif}\;d \leq -7.2 \cdot 10^{-300}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(-1 + 0.5 \cdot \frac{h \cdot {\left(\frac{M \cdot \frac{D}{d}}{2}\right)}^{2}}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -1.5 \cdot 10^{-102}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\left(1 - h \cdot \left(\frac{0.125}{\ell} \cdot {\left(D\_m \cdot \frac{M}{d}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{\ell}}\right)\\ \mathbf{elif}\;d \leq -7.2 \cdot 10^{-300}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(-1 + 0.5 \cdot \frac{h \cdot {\left(\frac{M \cdot \frac{D\_m}{d}}{2}\right)}^{2}}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D\_m}{d}\right)}^{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (if (<= d -1.5e-102)
   (*
    (sqrt (/ d h))
    (* (- 1.0 (* h (* (/ 0.125 l) (pow (* D_m (/ M d)) 2.0)))) (sqrt (/ d l))))
   (if (<= d -7.2e-300)
     (*
      (* d (sqrt (/ 1.0 (* l h))))
      (+ -1.0 (* 0.5 (/ (* h (pow (/ (* M (/ D_m d)) 2.0) 2.0)) l))))
     (*
      (/ d (* (sqrt h) (sqrt l)))
      (- 1.0 (* 0.5 (* (/ h l) (pow (* (/ M 2.0) (/ D_m d)) 2.0))))))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double tmp;
	if (d <= -1.5e-102) {
		tmp = sqrt((d / h)) * ((1.0 - (h * ((0.125 / l) * pow((D_m * (M / d)), 2.0)))) * sqrt((d / l)));
	} else if (d <= -7.2e-300) {
		tmp = (d * sqrt((1.0 / (l * h)))) * (-1.0 + (0.5 * ((h * pow(((M * (D_m / d)) / 2.0), 2.0)) / l)));
	} else {
		tmp = (d / (sqrt(h) * sqrt(l))) * (1.0 - (0.5 * ((h / l) * pow(((M / 2.0) * (D_m / d)), 2.0))));
	}
	return tmp;
}

Fortran

D_m = abs(d)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_m)
    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_m
    real(8) :: tmp
    if (d <= (-1.5d-102)) then
        tmp = sqrt((d / h)) * ((1.0d0 - (h * ((0.125d0 / l) * ((d_m * (m / d)) ** 2.0d0)))) * sqrt((d / l)))
    else if (d <= (-7.2d-300)) then
        tmp = (d * sqrt((1.0d0 / (l * h)))) * ((-1.0d0) + (0.5d0 * ((h * (((m * (d_m / d)) / 2.0d0) ** 2.0d0)) / l)))
    else
        tmp = (d / (sqrt(h) * sqrt(l))) * (1.0d0 - (0.5d0 * ((h / l) * (((m / 2.0d0) * (d_m / d)) ** 2.0d0))))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert d < h && h < l && l < M && M < D_m;
public static double code(double d, double h, double l, double M, double D_m) {
	double tmp;
	if (d <= -1.5e-102) {
		tmp = Math.sqrt((d / h)) * ((1.0 - (h * ((0.125 / l) * Math.pow((D_m * (M / d)), 2.0)))) * Math.sqrt((d / l)));
	} else if (d <= -7.2e-300) {
		tmp = (d * Math.sqrt((1.0 / (l * h)))) * (-1.0 + (0.5 * ((h * Math.pow(((M * (D_m / d)) / 2.0), 2.0)) / l)));
	} else {
		tmp = (d / (Math.sqrt(h) * Math.sqrt(l))) * (1.0 - (0.5 * ((h / l) * Math.pow(((M / 2.0) * (D_m / d)), 2.0))));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[d, h, l, M, D_m] = sort([d, h, l, M, D_m])
def code(d, h, l, M, D_m):
	tmp = 0
	if d <= -1.5e-102:
		tmp = math.sqrt((d / h)) * ((1.0 - (h * ((0.125 / l) * math.pow((D_m * (M / d)), 2.0)))) * math.sqrt((d / l)))
	elif d <= -7.2e-300:
		tmp = (d * math.sqrt((1.0 / (l * h)))) * (-1.0 + (0.5 * ((h * math.pow(((M * (D_m / d)) / 2.0), 2.0)) / l)))
	else:
		tmp = (d / (math.sqrt(h) * math.sqrt(l))) * (1.0 - (0.5 * ((h / l) * math.pow(((M / 2.0) * (D_m / d)), 2.0))))
	return tmp

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	tmp = 0.0
	if (d <= -1.5e-102)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(Float64(1.0 - Float64(h * Float64(Float64(0.125 / l) * (Float64(D_m * Float64(M / d)) ^ 2.0)))) * sqrt(Float64(d / l))));
	elseif (d <= -7.2e-300)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(l * h)))) * Float64(-1.0 + Float64(0.5 * Float64(Float64(h * (Float64(Float64(M * Float64(D_m / d)) / 2.0) ^ 2.0)) / l))));
	else
		tmp = Float64(Float64(d / Float64(sqrt(h) * sqrt(l))) * Float64(1.0 - Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(M / 2.0) * Float64(D_m / d)) ^ 2.0)))));
	end
	return tmp
end

MATLAB

D_m = abs(D);
d, h, l, M, D_m = num2cell(sort([d, h, l, M, D_m])){:}
function tmp_2 = code(d, h, l, M, D_m)
	tmp = 0.0;
	if (d <= -1.5e-102)
		tmp = sqrt((d / h)) * ((1.0 - (h * ((0.125 / l) * ((D_m * (M / d)) ^ 2.0)))) * sqrt((d / l)));
	elseif (d <= -7.2e-300)
		tmp = (d * sqrt((1.0 / (l * h)))) * (-1.0 + (0.5 * ((h * (((M * (D_m / d)) / 2.0) ^ 2.0)) / l)));
	else
		tmp = (d / (sqrt(h) * sqrt(l))) * (1.0 - (0.5 * ((h / l) * (((M / 2.0) * (D_m / d)) ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := If[LessEqual[d, -1.5e-102], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[(1.0 - N[(h * N[(N[(0.125 / l), $MachinePrecision] * N[Power[N[(D$95$m * N[(M / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -7.2e-300], N[(N[(d * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-1.0 + N[(0.5 * N[(N[(h * N[Power[N[(N[(M * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M / 2.0), $MachinePrecision] * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.5e-102

   1. Initial program 81.5%

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

   3. Simplified 82.8%

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

   5. Taylor expanded in h around -inf 49.2%

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

      1. associate-*r* 49.2%

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

      2. neg-mul-1 49.2%

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

      3. sub-neg 49.2%

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

      4. distribute-lft-in 49.2%

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

   7. Simplified 83.0%

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

   if -1.5e-102 < d < -7.20000000000000031e-300

   1. Initial program 38.1%

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

   3. Simplified 37.9%

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

      1. pow1 37.9%

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

      2. sqrt-unprod 35.7%

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

   6. Applied egg-rr 35.7%

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

      1. unpow1 35.7%

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

      2. *-commutative 35.7%

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

      3. associate-*r/ 25.7%

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

      4. associate-*l/ 15.6%

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

      5. unpow2 15.6%

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

   8. Simplified 15.6%

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

      1. associate-*r/ 15.6%

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

      2. associate-*l/ 15.6%

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

   10. Applied egg-rr 15.6%

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

   11. Taylor expanded in d around -inf 68.4%

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

   if -7.20000000000000031e-300 < d

   1. Initial program 69.9%

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

   3. Simplified 70.7%

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

      1. pow1 70.7%

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

      2. sqrt-unprod 54.3%

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

   6. Applied egg-rr 54.3%

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

      1. unpow1 54.3%

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

      2. *-commutative 54.3%

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

   8. Simplified 54.3%

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

      1. frac-times 40.8%

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

      2. unpow2 40.8%

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

      3. *-commutative 40.8%

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

      4. associate-/l/ 47.0%

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

      5. sqrt-div 55.5%

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

      6. sqrt-div 58.8%

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

      7. sqrt-pow1 83.4%

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

      8. metadata-eval 83.4%

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

      9. pow1 83.4%

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

   10. Applied egg-rr 83.4%

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

       1. associate-/l/ 83.4%

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

   12. Simplified 83.4%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.5 \cdot 10^{-102}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\left(1 - h \cdot \left(\frac{0.125}{\ell} \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{\ell}}\right)\\ \mathbf{elif}\;d \leq -7.2 \cdot 10^{-300}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(-1 + 0.5 \cdot \frac{h \cdot {\left(\frac{M \cdot \frac{D}{d}}{2}\right)}^{2}}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 72.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;h \leq -4.9 \cdot 10^{-232}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(-1 + 0.5 \cdot \frac{h \cdot {\left(\frac{M \cdot \frac{D\_m}{d}}{2}\right)}^{2}}{\ell}\right)\\ \mathbf{elif}\;h \leq 2.1 \cdot 10^{-254}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D\_m}{d \cdot \frac{2}{M}}\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left(h, -0.5 \cdot \frac{{\left(M \cdot \frac{\frac{D\_m}{2}}{d}\right)}^{2}}{\ell}, 1\right)}{\sqrt{\ell \cdot h}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (if (<= h -4.9e-232)
   (*
    (* d (sqrt (/ 1.0 (* l h))))
    (+ -1.0 (* 0.5 (/ (* h (pow (/ (* M (/ D_m d)) 2.0) 2.0)) l))))
   (if (<= h 2.1e-254)
     (*
      (sqrt (* (/ d h) (/ d l)))
      (- 1.0 (* 0.5 (* (/ h l) (pow (/ D_m (* d (/ 2.0 M))) 2.0)))))
     (*
      d
      (/
       (fma h (* -0.5 (/ (pow (* M (/ (/ D_m 2.0) d)) 2.0) l)) 1.0)
       (sqrt (* l h)))))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double tmp;
	if (h <= -4.9e-232) {
		tmp = (d * sqrt((1.0 / (l * h)))) * (-1.0 + (0.5 * ((h * pow(((M * (D_m / d)) / 2.0), 2.0)) / l)));
	} else if (h <= 2.1e-254) {
		tmp = sqrt(((d / h) * (d / l))) * (1.0 - (0.5 * ((h / l) * pow((D_m / (d * (2.0 / M))), 2.0))));
	} else {
		tmp = d * (fma(h, (-0.5 * (pow((M * ((D_m / 2.0) / d)), 2.0) / l)), 1.0) / sqrt((l * h)));
	}
	return tmp;
}

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	tmp = 0.0
	if (h <= -4.9e-232)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(l * h)))) * Float64(-1.0 + Float64(0.5 * Float64(Float64(h * (Float64(Float64(M * Float64(D_m / d)) / 2.0) ^ 2.0)) / l))));
	elseif (h <= 2.1e-254)
		tmp = Float64(sqrt(Float64(Float64(d / h) * Float64(d / l))) * Float64(1.0 - Float64(0.5 * Float64(Float64(h / l) * (Float64(D_m / Float64(d * Float64(2.0 / M))) ^ 2.0)))));
	else
		tmp = Float64(d * Float64(fma(h, Float64(-0.5 * Float64((Float64(M * Float64(Float64(D_m / 2.0) / d)) ^ 2.0) / l)), 1.0) / sqrt(Float64(l * h))));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := If[LessEqual[h, -4.9e-232], N[(N[(d * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-1.0 + N[(0.5 * N[(N[(h * N[Power[N[(N[(M * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, 2.1e-254], N[(N[Sqrt[N[(N[(d / h), $MachinePrecision] * N[(d / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(D$95$m / N[(d * N[(2.0 / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[(h * N[(-0.5 * N[(N[Power[N[(M * N[(N[(D$95$m / 2.0), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if h < -4.9000000000000003e-232

   1. Initial program 61.5%

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

   3. Simplified 62.2%

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

      1. pow1 62.2%

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

      2. sqrt-unprod 55.4%

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

   6. Applied egg-rr 55.4%

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

      1. unpow1 55.4%

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

      2. *-commutative 55.4%

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

      3. associate-*r/ 50.1%

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

      4. associate-*l/ 43.1%

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

      5. unpow2 43.1%

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

   8. Simplified 43.1%

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

      1. associate-*r/ 42.3%

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

      2. associate-*l/ 42.3%

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

   10. Applied egg-rr 42.3%

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

   11. Taylor expanded in d around -inf 70.2%

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

   if -4.9000000000000003e-232 < h < 2.09999999999999997e-254

   1. Initial program 76.1%

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

   3. Simplified 76.1%

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

      1. pow1 76.1%

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

      2. sqrt-unprod 72.3%

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

   6. Applied egg-rr 72.3%

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

      1. unpow1 72.3%

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

      2. *-commutative 72.3%

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

   8. Simplified 72.3%

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

      1. clear-num 72.2%

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

      2. frac-times 64.1%

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

      3. *-un-lft-identity 64.1%

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

   10. Applied egg-rr 64.1%

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

   if 2.09999999999999997e-254 < h

   1. Initial program 70.9%

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

   3. Simplified 71.8%

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

      1. pow1 71.8%

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

      2. sqrt-unprod 54.1%

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

   6. Applied egg-rr 54.1%

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

      1. unpow1 54.1%

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

      2. *-commutative 54.1%

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

   8. Simplified 54.1%

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

   10. Applied egg-rr 72.7%

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

       1. unpow1 72.7%

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

       2. associate-*l/ 74.3%

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

       3. associate-/l* 74.3%

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

   12. Simplified 75.9%

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

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -4.9 \cdot 10^{-232}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(-1 + 0.5 \cdot \frac{h \cdot {\left(\frac{M \cdot \frac{D}{d}}{2}\right)}^{2}}{\ell}\right)\\ \mathbf{elif}\;h \leq 2.1 \cdot 10^{-254}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d \cdot \frac{2}{M}}\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left(h, -0.5 \cdot \frac{{\left(M \cdot \frac{\frac{D}{2}}{d}\right)}^{2}}{\ell}, 1\right)}{\sqrt{\ell \cdot h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 76.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;h \leq 1.5 \cdot 10^{-305}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(-1 + 0.5 \cdot \frac{h \cdot {\left(\frac{M \cdot \frac{D\_m}{d}}{2}\right)}^{2}}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D\_m}{d}\right)}^{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (if (<= h 1.5e-305)
   (*
    (* d (sqrt (/ 1.0 (* l h))))
    (+ -1.0 (* 0.5 (/ (* h (pow (/ (* M (/ D_m d)) 2.0) 2.0)) l))))
   (*
    (/ d (* (sqrt h) (sqrt l)))
    (- 1.0 (* 0.5 (* (/ h l) (pow (* (/ M 2.0) (/ D_m d)) 2.0)))))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double tmp;
	if (h <= 1.5e-305) {
		tmp = (d * sqrt((1.0 / (l * h)))) * (-1.0 + (0.5 * ((h * pow(((M * (D_m / d)) / 2.0), 2.0)) / l)));
	} else {
		tmp = (d / (sqrt(h) * sqrt(l))) * (1.0 - (0.5 * ((h / l) * pow(((M / 2.0) * (D_m / d)), 2.0))));
	}
	return tmp;
}

Fortran

D_m = abs(d)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_m)
    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_m
    real(8) :: tmp
    if (h <= 1.5d-305) then
        tmp = (d * sqrt((1.0d0 / (l * h)))) * ((-1.0d0) + (0.5d0 * ((h * (((m * (d_m / d)) / 2.0d0) ** 2.0d0)) / l)))
    else
        tmp = (d / (sqrt(h) * sqrt(l))) * (1.0d0 - (0.5d0 * ((h / l) * (((m / 2.0d0) * (d_m / d)) ** 2.0d0))))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert d < h && h < l && l < M && M < D_m;
public static double code(double d, double h, double l, double M, double D_m) {
	double tmp;
	if (h <= 1.5e-305) {
		tmp = (d * Math.sqrt((1.0 / (l * h)))) * (-1.0 + (0.5 * ((h * Math.pow(((M * (D_m / d)) / 2.0), 2.0)) / l)));
	} else {
		tmp = (d / (Math.sqrt(h) * Math.sqrt(l))) * (1.0 - (0.5 * ((h / l) * Math.pow(((M / 2.0) * (D_m / d)), 2.0))));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[d, h, l, M, D_m] = sort([d, h, l, M, D_m])
def code(d, h, l, M, D_m):
	tmp = 0
	if h <= 1.5e-305:
		tmp = (d * math.sqrt((1.0 / (l * h)))) * (-1.0 + (0.5 * ((h * math.pow(((M * (D_m / d)) / 2.0), 2.0)) / l)))
	else:
		tmp = (d / (math.sqrt(h) * math.sqrt(l))) * (1.0 - (0.5 * ((h / l) * math.pow(((M / 2.0) * (D_m / d)), 2.0))))
	return tmp

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	tmp = 0.0
	if (h <= 1.5e-305)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(l * h)))) * Float64(-1.0 + Float64(0.5 * Float64(Float64(h * (Float64(Float64(M * Float64(D_m / d)) / 2.0) ^ 2.0)) / l))));
	else
		tmp = Float64(Float64(d / Float64(sqrt(h) * sqrt(l))) * Float64(1.0 - Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(M / 2.0) * Float64(D_m / d)) ^ 2.0)))));
	end
	return tmp
end

MATLAB

D_m = abs(D);
d, h, l, M, D_m = num2cell(sort([d, h, l, M, D_m])){:}
function tmp_2 = code(d, h, l, M, D_m)
	tmp = 0.0;
	if (h <= 1.5e-305)
		tmp = (d * sqrt((1.0 / (l * h)))) * (-1.0 + (0.5 * ((h * (((M * (D_m / d)) / 2.0) ^ 2.0)) / l)));
	else
		tmp = (d / (sqrt(h) * sqrt(l))) * (1.0 - (0.5 * ((h / l) * (((M / 2.0) * (D_m / d)) ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := If[LessEqual[h, 1.5e-305], N[(N[(d * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-1.0 + N[(0.5 * N[(N[(h * N[Power[N[(N[(M * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M / 2.0), $MachinePrecision] * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if h < 1.5000000000000001e-305

   1. Initial program 63.1%

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

   3. Simplified 63.8%

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

      1. pow1 63.8%

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

      2. sqrt-unprod 57.2%

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

   6. Applied egg-rr 57.2%

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

      1. unpow1 57.2%

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

      2. *-commutative 57.2%

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

      3. associate-*r/ 50.4%

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

      4. associate-*l/ 43.6%

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

      5. unpow2 43.6%

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

   8. Simplified 43.6%

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

      1. associate-*r/ 42.9%

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

      2. associate-*l/ 42.9%

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

   10. Applied egg-rr 42.9%

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

   11. Taylor expanded in d around -inf 69.2%

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

   if 1.5000000000000001e-305 < h

   1. Initial program 71.6%

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

   3. Simplified 72.3%

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

      1. pow1 72.3%

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

      2. sqrt-unprod 55.6%

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

   6. Applied egg-rr 55.6%

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

      1. unpow1 55.6%

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

      2. *-commutative 55.6%

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

   8. Simplified 55.6%

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

      1. frac-times 41.7%

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

      2. unpow2 41.7%

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

      3. *-commutative 41.7%

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

      4. associate-/l/ 48.1%

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

      5. sqrt-div 56.8%

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

      6. sqrt-div 60.2%

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

      7. sqrt-pow1 85.4%

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

      8. metadata-eval 85.4%

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

      9. pow1 85.4%

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

   10. Applied egg-rr 85.4%

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

       1. associate-/l/ 85.4%

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

   12. Simplified 85.4%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq 1.5 \cdot 10^{-305}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(-1 + 0.5 \cdot \frac{h \cdot {\left(\frac{M \cdot \frac{D}{d}}{2}\right)}^{2}}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 69.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ t_1 := 1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D\_m}{d}\right)}^{2}\right)\\ \mathbf{if}\;h \leq -4.4 \cdot 10^{-232}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(-1 + 0.5 \cdot \frac{h \cdot {\left(\frac{M \cdot \frac{D\_m}{d}}{2}\right)}^{2}}{\ell}\right)\\ \mathbf{elif}\;h \leq 2.6 \cdot 10^{-254}:\\ \;\;\;\;t\_0 \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D\_m}{d \cdot \frac{2}{M}}\right)}^{2}\right)\right)\\ \mathbf{elif}\;h \leq 6 \cdot 10^{+212}:\\ \;\;\;\;t\_1 \cdot \frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot t\_0\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (let* ((t_0 (sqrt (* (/ d h) (/ d l))))
        (t_1 (- 1.0 (* 0.5 (* (/ h l) (pow (* (/ M 2.0) (/ D_m d)) 2.0))))))
   (if (<= h -4.4e-232)
     (*
      (* d (sqrt (/ 1.0 (* l h))))
      (+ -1.0 (* 0.5 (/ (* h (pow (/ (* M (/ D_m d)) 2.0) 2.0)) l))))
     (if (<= h 2.6e-254)
       (* t_0 (- 1.0 (* 0.5 (* (/ h l) (pow (/ D_m (* d (/ 2.0 M))) 2.0)))))
       (if (<= h 6e+212) (* t_1 (/ d (sqrt (* l h)))) (* t_1 t_0))))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double t_0 = sqrt(((d / h) * (d / l)));
	double t_1 = 1.0 - (0.5 * ((h / l) * pow(((M / 2.0) * (D_m / d)), 2.0)));
	double tmp;
	if (h <= -4.4e-232) {
		tmp = (d * sqrt((1.0 / (l * h)))) * (-1.0 + (0.5 * ((h * pow(((M * (D_m / d)) / 2.0), 2.0)) / l)));
	} else if (h <= 2.6e-254) {
		tmp = t_0 * (1.0 - (0.5 * ((h / l) * pow((D_m / (d * (2.0 / M))), 2.0))));
	} else if (h <= 6e+212) {
		tmp = t_1 * (d / sqrt((l * h)));
	} else {
		tmp = t_1 * t_0;
	}
	return tmp;
}

Fortran

D_m = abs(d)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_m)
    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_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sqrt(((d / h) * (d / l)))
    t_1 = 1.0d0 - (0.5d0 * ((h / l) * (((m / 2.0d0) * (d_m / d)) ** 2.0d0)))
    if (h <= (-4.4d-232)) then
        tmp = (d * sqrt((1.0d0 / (l * h)))) * ((-1.0d0) + (0.5d0 * ((h * (((m * (d_m / d)) / 2.0d0) ** 2.0d0)) / l)))
    else if (h <= 2.6d-254) then
        tmp = t_0 * (1.0d0 - (0.5d0 * ((h / l) * ((d_m / (d * (2.0d0 / m))) ** 2.0d0))))
    else if (h <= 6d+212) then
        tmp = t_1 * (d / sqrt((l * h)))
    else
        tmp = t_1 * t_0
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert d < h && h < l && l < M && M < D_m;
public static double code(double d, double h, double l, double M, double D_m) {
	double t_0 = Math.sqrt(((d / h) * (d / l)));
	double t_1 = 1.0 - (0.5 * ((h / l) * Math.pow(((M / 2.0) * (D_m / d)), 2.0)));
	double tmp;
	if (h <= -4.4e-232) {
		tmp = (d * Math.sqrt((1.0 / (l * h)))) * (-1.0 + (0.5 * ((h * Math.pow(((M * (D_m / d)) / 2.0), 2.0)) / l)));
	} else if (h <= 2.6e-254) {
		tmp = t_0 * (1.0 - (0.5 * ((h / l) * Math.pow((D_m / (d * (2.0 / M))), 2.0))));
	} else if (h <= 6e+212) {
		tmp = t_1 * (d / Math.sqrt((l * h)));
	} else {
		tmp = t_1 * t_0;
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[d, h, l, M, D_m] = sort([d, h, l, M, D_m])
def code(d, h, l, M, D_m):
	t_0 = math.sqrt(((d / h) * (d / l)))
	t_1 = 1.0 - (0.5 * ((h / l) * math.pow(((M / 2.0) * (D_m / d)), 2.0)))
	tmp = 0
	if h <= -4.4e-232:
		tmp = (d * math.sqrt((1.0 / (l * h)))) * (-1.0 + (0.5 * ((h * math.pow(((M * (D_m / d)) / 2.0), 2.0)) / l)))
	elif h <= 2.6e-254:
		tmp = t_0 * (1.0 - (0.5 * ((h / l) * math.pow((D_m / (d * (2.0 / M))), 2.0))))
	elif h <= 6e+212:
		tmp = t_1 * (d / math.sqrt((l * h)))
	else:
		tmp = t_1 * t_0
	return tmp

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	t_0 = sqrt(Float64(Float64(d / h) * Float64(d / l)))
	t_1 = Float64(1.0 - Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(M / 2.0) * Float64(D_m / d)) ^ 2.0))))
	tmp = 0.0
	if (h <= -4.4e-232)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(l * h)))) * Float64(-1.0 + Float64(0.5 * Float64(Float64(h * (Float64(Float64(M * Float64(D_m / d)) / 2.0) ^ 2.0)) / l))));
	elseif (h <= 2.6e-254)
		tmp = Float64(t_0 * Float64(1.0 - Float64(0.5 * Float64(Float64(h / l) * (Float64(D_m / Float64(d * Float64(2.0 / M))) ^ 2.0)))));
	elseif (h <= 6e+212)
		tmp = Float64(t_1 * Float64(d / sqrt(Float64(l * h))));
	else
		tmp = Float64(t_1 * t_0);
	end
	return tmp
end

MATLAB

D_m = abs(D);
d, h, l, M, D_m = num2cell(sort([d, h, l, M, D_m])){:}
function tmp_2 = code(d, h, l, M, D_m)
	t_0 = sqrt(((d / h) * (d / l)));
	t_1 = 1.0 - (0.5 * ((h / l) * (((M / 2.0) * (D_m / d)) ^ 2.0)));
	tmp = 0.0;
	if (h <= -4.4e-232)
		tmp = (d * sqrt((1.0 / (l * h)))) * (-1.0 + (0.5 * ((h * (((M * (D_m / d)) / 2.0) ^ 2.0)) / l)));
	elseif (h <= 2.6e-254)
		tmp = t_0 * (1.0 - (0.5 * ((h / l) * ((D_m / (d * (2.0 / M))) ^ 2.0))));
	elseif (h <= 6e+212)
		tmp = t_1 * (d / sqrt((l * h)));
	else
		tmp = t_1 * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := Block[{t$95$0 = N[Sqrt[N[(N[(d / h), $MachinePrecision] * N[(d / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M / 2.0), $MachinePrecision] * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[h, -4.4e-232], N[(N[(d * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-1.0 + N[(0.5 * N[(N[(h * N[Power[N[(N[(M * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, 2.6e-254], N[(t$95$0 * N[(1.0 - N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(D$95$m / N[(d * N[(2.0 / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, 6e+212], N[(t$95$1 * N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * t$95$0), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if h < -4.40000000000000004e-232

   1. Initial program 61.5%

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

   3. Simplified 62.2%

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

      1. pow1 62.2%

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

      2. sqrt-unprod 55.4%

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

   6. Applied egg-rr 55.4%

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

      1. unpow1 55.4%

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

      2. *-commutative 55.4%

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

      3. associate-*r/ 50.1%

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

      4. associate-*l/ 43.1%

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

      5. unpow2 43.1%

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

   8. Simplified 43.1%

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

      1. associate-*r/ 42.3%

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

      2. associate-*l/ 42.3%

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

   10. Applied egg-rr 42.3%

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

   11. Taylor expanded in d around -inf 70.2%

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

   if -4.40000000000000004e-232 < h < 2.6e-254

   1. Initial program 76.1%

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

   3. Simplified 76.1%

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

      1. pow1 76.1%

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

      2. sqrt-unprod 72.3%

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

   6. Applied egg-rr 72.3%

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

      1. unpow1 72.3%

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

      2. *-commutative 72.3%

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

   8. Simplified 72.3%

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

      1. clear-num 72.2%

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

      2. frac-times 64.1%

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

      3. *-un-lft-identity 64.1%

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

   10. Applied egg-rr 64.1%

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

   if 2.6e-254 < h < 6e212

   1. Initial program 72.5%

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

   3. Simplified 72.5%

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

      1. pow1 72.5%

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

      2. sqrt-unprod 52.6%

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

   6. Applied egg-rr 52.6%

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

      1. unpow1 52.6%

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

      2. *-commutative 52.6%

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

   8. Simplified 52.6%

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

      1. frac-times 44.9%

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

      2. unpow2 44.9%

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

      3. *-commutative 44.9%

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

      4. associate-/l/ 46.9%

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

      5. *-un-lft-identity 46.9%

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

      6. associate-/l/ 44.9%

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

      7. *-commutative 44.9%

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

      8. sqrt-div 54.7%

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

      9. sqrt-pow1 79.4%

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

      10. metadata-eval 79.4%

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

      11. pow1 79.4%

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

   10. Applied egg-rr 79.4%

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

       1. *-lft-identity 79.4%

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

   12. Simplified 79.4%

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

   if 6e212 < h

   1. Initial program 63.9%

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

   3. Simplified 68.5%

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

      1. pow1 68.5%

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

      2. sqrt-unprod 61.0%

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

   6. Applied egg-rr 61.0%

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

      1. unpow1 61.0%

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

      2. *-commutative 61.0%

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

   8. Simplified 61.0%

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

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -4.4 \cdot 10^{-232}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(-1 + 0.5 \cdot \frac{h \cdot {\left(\frac{M \cdot \frac{D}{d}}{2}\right)}^{2}}{\ell}\right)\\ \mathbf{elif}\;h \leq 2.6 \cdot 10^{-254}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d \cdot \frac{2}{M}}\right)}^{2}\right)\right)\\ \mathbf{elif}\;h \leq 6 \cdot 10^{+212}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}\right)\right) \cdot \frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 66.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D\_m}{d}\right)}^{2}\right)\\ t_1 := 1 - t\_0\\ t_2 := \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{if}\;h \leq -5 \cdot 10^{-158}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(t\_0 + -1\right)\\ \mathbf{elif}\;h \leq 6.5 \cdot 10^{-252}:\\ \;\;\;\;t\_2 \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D\_m}{d \cdot \frac{2}{M}}\right)}^{2}\right)\right)\\ \mathbf{elif}\;h \leq 1.2 \cdot 10^{+206}:\\ \;\;\;\;t\_1 \cdot \frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot t\_2\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (let* ((t_0 (* 0.5 (* (/ h l) (pow (* (/ M 2.0) (/ D_m d)) 2.0))))
        (t_1 (- 1.0 t_0))
        (t_2 (sqrt (* (/ d h) (/ d l)))))
   (if (<= h -5e-158)
     (* (* d (sqrt (/ 1.0 (* l h)))) (+ t_0 -1.0))
     (if (<= h 6.5e-252)
       (* t_2 (- 1.0 (* 0.5 (* (/ h l) (pow (/ D_m (* d (/ 2.0 M))) 2.0)))))
       (if (<= h 1.2e+206) (* t_1 (/ d (sqrt (* l h)))) (* t_1 t_2))))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double t_0 = 0.5 * ((h / l) * pow(((M / 2.0) * (D_m / d)), 2.0));
	double t_1 = 1.0 - t_0;
	double t_2 = sqrt(((d / h) * (d / l)));
	double tmp;
	if (h <= -5e-158) {
		tmp = (d * sqrt((1.0 / (l * h)))) * (t_0 + -1.0);
	} else if (h <= 6.5e-252) {
		tmp = t_2 * (1.0 - (0.5 * ((h / l) * pow((D_m / (d * (2.0 / M))), 2.0))));
	} else if (h <= 1.2e+206) {
		tmp = t_1 * (d / sqrt((l * h)));
	} else {
		tmp = t_1 * t_2;
	}
	return tmp;
}

Fortran

D_m = abs(d)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_m)
    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_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = 0.5d0 * ((h / l) * (((m / 2.0d0) * (d_m / d)) ** 2.0d0))
    t_1 = 1.0d0 - t_0
    t_2 = sqrt(((d / h) * (d / l)))
    if (h <= (-5d-158)) then
        tmp = (d * sqrt((1.0d0 / (l * h)))) * (t_0 + (-1.0d0))
    else if (h <= 6.5d-252) then
        tmp = t_2 * (1.0d0 - (0.5d0 * ((h / l) * ((d_m / (d * (2.0d0 / m))) ** 2.0d0))))
    else if (h <= 1.2d+206) then
        tmp = t_1 * (d / sqrt((l * h)))
    else
        tmp = t_1 * t_2
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert d < h && h < l && l < M && M < D_m;
public static double code(double d, double h, double l, double M, double D_m) {
	double t_0 = 0.5 * ((h / l) * Math.pow(((M / 2.0) * (D_m / d)), 2.0));
	double t_1 = 1.0 - t_0;
	double t_2 = Math.sqrt(((d / h) * (d / l)));
	double tmp;
	if (h <= -5e-158) {
		tmp = (d * Math.sqrt((1.0 / (l * h)))) * (t_0 + -1.0);
	} else if (h <= 6.5e-252) {
		tmp = t_2 * (1.0 - (0.5 * ((h / l) * Math.pow((D_m / (d * (2.0 / M))), 2.0))));
	} else if (h <= 1.2e+206) {
		tmp = t_1 * (d / Math.sqrt((l * h)));
	} else {
		tmp = t_1 * t_2;
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[d, h, l, M, D_m] = sort([d, h, l, M, D_m])
def code(d, h, l, M, D_m):
	t_0 = 0.5 * ((h / l) * math.pow(((M / 2.0) * (D_m / d)), 2.0))
	t_1 = 1.0 - t_0
	t_2 = math.sqrt(((d / h) * (d / l)))
	tmp = 0
	if h <= -5e-158:
		tmp = (d * math.sqrt((1.0 / (l * h)))) * (t_0 + -1.0)
	elif h <= 6.5e-252:
		tmp = t_2 * (1.0 - (0.5 * ((h / l) * math.pow((D_m / (d * (2.0 / M))), 2.0))))
	elif h <= 1.2e+206:
		tmp = t_1 * (d / math.sqrt((l * h)))
	else:
		tmp = t_1 * t_2
	return tmp

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	t_0 = Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(M / 2.0) * Float64(D_m / d)) ^ 2.0)))
	t_1 = Float64(1.0 - t_0)
	t_2 = sqrt(Float64(Float64(d / h) * Float64(d / l)))
	tmp = 0.0
	if (h <= -5e-158)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(l * h)))) * Float64(t_0 + -1.0));
	elseif (h <= 6.5e-252)
		tmp = Float64(t_2 * Float64(1.0 - Float64(0.5 * Float64(Float64(h / l) * (Float64(D_m / Float64(d * Float64(2.0 / M))) ^ 2.0)))));
	elseif (h <= 1.2e+206)
		tmp = Float64(t_1 * Float64(d / sqrt(Float64(l * h))));
	else
		tmp = Float64(t_1 * t_2);
	end
	return tmp
end

MATLAB

D_m = abs(D);
d, h, l, M, D_m = num2cell(sort([d, h, l, M, D_m])){:}
function tmp_2 = code(d, h, l, M, D_m)
	t_0 = 0.5 * ((h / l) * (((M / 2.0) * (D_m / d)) ^ 2.0));
	t_1 = 1.0 - t_0;
	t_2 = sqrt(((d / h) * (d / l)));
	tmp = 0.0;
	if (h <= -5e-158)
		tmp = (d * sqrt((1.0 / (l * h)))) * (t_0 + -1.0);
	elseif (h <= 6.5e-252)
		tmp = t_2 * (1.0 - (0.5 * ((h / l) * ((D_m / (d * (2.0 / M))) ^ 2.0))));
	elseif (h <= 1.2e+206)
		tmp = t_1 * (d / sqrt((l * h)));
	else
		tmp = t_1 * t_2;
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := Block[{t$95$0 = N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M / 2.0), $MachinePrecision] * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(N[(d / h), $MachinePrecision] * N[(d / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[h, -5e-158], N[(N[(d * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, 6.5e-252], N[(t$95$2 * N[(1.0 - N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(D$95$m / N[(d * N[(2.0 / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, 1.2e+206], N[(t$95$1 * N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * t$95$2), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if h < -4.99999999999999972e-158

   1. Initial program 62.2%

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

   3. Simplified 63.1%

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

      1. pow1 63.1%

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

      2. sqrt-unprod 55.1%

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

   6. Applied egg-rr 55.1%

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

      1. unpow1 55.1%

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

      2. *-commutative 55.1%

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

      3. associate-*r/ 49.0%

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

      4. associate-*l/ 41.8%

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

      5. unpow2 41.8%

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

   8. Simplified 41.8%

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

   9. Taylor expanded in d around -inf 68.1%

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

   if -4.99999999999999972e-158 < h < 6.4999999999999998e-252

   1. Initial program 68.6%

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

   3. Simplified 68.5%

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

      1. pow1 68.5%

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

      2. sqrt-unprod 66.3%

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

   6. Applied egg-rr 66.3%

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

      1. unpow1 66.3%

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

      2. *-commutative 66.3%

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

   8. Simplified 66.3%

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

      1. clear-num 66.2%

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

      2. frac-times 61.4%

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

      3. *-un-lft-identity 61.4%

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

   10. Applied egg-rr 61.4%

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

   if 6.4999999999999998e-252 < h < 1.2e206

   1. Initial program 72.5%

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

   3. Simplified 72.5%

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

      1. pow1 72.5%

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

      2. sqrt-unprod 52.6%

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

   6. Applied egg-rr 52.6%

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

      1. unpow1 52.6%

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

      2. *-commutative 52.6%

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

   8. Simplified 52.6%

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

      1. frac-times 44.9%

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

      2. unpow2 44.9%

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

      3. *-commutative 44.9%

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

      4. associate-/l/ 46.9%

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

      5. *-un-lft-identity 46.9%

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

      6. associate-/l/ 44.9%

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

      7. *-commutative 44.9%

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

      8. sqrt-div 54.7%

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

      9. sqrt-pow1 79.4%

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

      10. metadata-eval 79.4%

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

      11. pow1 79.4%

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

   10. Applied egg-rr 79.4%

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

       1. *-lft-identity 79.4%

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

   12. Simplified 79.4%

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

   if 1.2e206 < h

   1. Initial program 63.9%

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

   3. Simplified 68.5%

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

      1. pow1 68.5%

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

      2. sqrt-unprod 61.0%

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

   6. Applied egg-rr 61.0%

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

      1. unpow1 61.0%

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

      2. *-commutative 61.0%

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

   8. Simplified 61.0%

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

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -5 \cdot 10^{-158}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}\right) + -1\right)\\ \mathbf{elif}\;h \leq 6.5 \cdot 10^{-252}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d \cdot \frac{2}{M}}\right)}^{2}\right)\right)\\ \mathbf{elif}\;h \leq 1.2 \cdot 10^{+206}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}\right)\right) \cdot \frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 61.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := 1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D\_m}{d}\right)}^{2}\right)\\ t_1 := \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{if}\;h \leq 8.8 \cdot 10^{-254}:\\ \;\;\;\;t\_1 \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D\_m}{d \cdot \frac{2}{M}}\right)}^{2}\right)\right)\\ \mathbf{elif}\;h \leq 3 \cdot 10^{+210}:\\ \;\;\;\;t\_0 \cdot \frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot t\_1\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (let* ((t_0 (- 1.0 (* 0.5 (* (/ h l) (pow (* (/ M 2.0) (/ D_m d)) 2.0)))))
        (t_1 (sqrt (* (/ d h) (/ d l)))))
   (if (<= h 8.8e-254)
     (* t_1 (- 1.0 (* 0.5 (* (/ h l) (pow (/ D_m (* d (/ 2.0 M))) 2.0)))))
     (if (<= h 3e+210) (* t_0 (/ d (sqrt (* l h)))) (* t_0 t_1)))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double t_0 = 1.0 - (0.5 * ((h / l) * pow(((M / 2.0) * (D_m / d)), 2.0)));
	double t_1 = sqrt(((d / h) * (d / l)));
	double tmp;
	if (h <= 8.8e-254) {
		tmp = t_1 * (1.0 - (0.5 * ((h / l) * pow((D_m / (d * (2.0 / M))), 2.0))));
	} else if (h <= 3e+210) {
		tmp = t_0 * (d / sqrt((l * h)));
	} else {
		tmp = t_0 * t_1;
	}
	return tmp;
}

Fortran

D_m = abs(d)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_m)
    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_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 - (0.5d0 * ((h / l) * (((m / 2.0d0) * (d_m / d)) ** 2.0d0)))
    t_1 = sqrt(((d / h) * (d / l)))
    if (h <= 8.8d-254) then
        tmp = t_1 * (1.0d0 - (0.5d0 * ((h / l) * ((d_m / (d * (2.0d0 / m))) ** 2.0d0))))
    else if (h <= 3d+210) then
        tmp = t_0 * (d / sqrt((l * h)))
    else
        tmp = t_0 * t_1
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert d < h && h < l && l < M && M < D_m;
public static double code(double d, double h, double l, double M, double D_m) {
	double t_0 = 1.0 - (0.5 * ((h / l) * Math.pow(((M / 2.0) * (D_m / d)), 2.0)));
	double t_1 = Math.sqrt(((d / h) * (d / l)));
	double tmp;
	if (h <= 8.8e-254) {
		tmp = t_1 * (1.0 - (0.5 * ((h / l) * Math.pow((D_m / (d * (2.0 / M))), 2.0))));
	} else if (h <= 3e+210) {
		tmp = t_0 * (d / Math.sqrt((l * h)));
	} else {
		tmp = t_0 * t_1;
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[d, h, l, M, D_m] = sort([d, h, l, M, D_m])
def code(d, h, l, M, D_m):
	t_0 = 1.0 - (0.5 * ((h / l) * math.pow(((M / 2.0) * (D_m / d)), 2.0)))
	t_1 = math.sqrt(((d / h) * (d / l)))
	tmp = 0
	if h <= 8.8e-254:
		tmp = t_1 * (1.0 - (0.5 * ((h / l) * math.pow((D_m / (d * (2.0 / M))), 2.0))))
	elif h <= 3e+210:
		tmp = t_0 * (d / math.sqrt((l * h)))
	else:
		tmp = t_0 * t_1
	return tmp

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	t_0 = Float64(1.0 - Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(M / 2.0) * Float64(D_m / d)) ^ 2.0))))
	t_1 = sqrt(Float64(Float64(d / h) * Float64(d / l)))
	tmp = 0.0
	if (h <= 8.8e-254)
		tmp = Float64(t_1 * Float64(1.0 - Float64(0.5 * Float64(Float64(h / l) * (Float64(D_m / Float64(d * Float64(2.0 / M))) ^ 2.0)))));
	elseif (h <= 3e+210)
		tmp = Float64(t_0 * Float64(d / sqrt(Float64(l * h))));
	else
		tmp = Float64(t_0 * t_1);
	end
	return tmp
end

MATLAB

D_m = abs(D);
d, h, l, M, D_m = num2cell(sort([d, h, l, M, D_m])){:}
function tmp_2 = code(d, h, l, M, D_m)
	t_0 = 1.0 - (0.5 * ((h / l) * (((M / 2.0) * (D_m / d)) ^ 2.0)));
	t_1 = sqrt(((d / h) * (d / l)));
	tmp = 0.0;
	if (h <= 8.8e-254)
		tmp = t_1 * (1.0 - (0.5 * ((h / l) * ((D_m / (d * (2.0 / M))) ^ 2.0))));
	elseif (h <= 3e+210)
		tmp = t_0 * (d / sqrt((l * h)));
	else
		tmp = t_0 * t_1;
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := Block[{t$95$0 = N[(1.0 - N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M / 2.0), $MachinePrecision] * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(N[(d / h), $MachinePrecision] * N[(d / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[h, 8.8e-254], N[(t$95$1 * N[(1.0 - N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(D$95$m / N[(d * N[(2.0 / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, 3e+210], N[(t$95$0 * N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if h < 8.8000000000000004e-254

   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 64.7%

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

      1. pow1 64.7%

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

      2. sqrt-unprod 58.4%

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

   6. Applied egg-rr 58.4%

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

      1. unpow1 58.4%

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

      2. *-commutative 58.4%

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

   8. Simplified 58.4%

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

      1. clear-num 58.4%

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

      2. frac-times 57.7%

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

      3. *-un-lft-identity 57.7%

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

   10. Applied egg-rr 57.7%

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

   if 8.8000000000000004e-254 < h < 3.00000000000000022e210

   1. Initial program 72.5%

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

   3. Simplified 72.5%

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

      1. pow1 72.5%

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

      2. sqrt-unprod 52.6%

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

   6. Applied egg-rr 52.6%

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

      1. unpow1 52.6%

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

      2. *-commutative 52.6%

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

   8. Simplified 52.6%

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

      1. frac-times 44.9%

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

      2. unpow2 44.9%

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

      3. *-commutative 44.9%

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

      4. associate-/l/ 46.9%

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

      5. *-un-lft-identity 46.9%

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

      6. associate-/l/ 44.9%

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

      7. *-commutative 44.9%

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

      8. sqrt-div 54.7%

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

      9. sqrt-pow1 79.4%

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

      10. metadata-eval 79.4%

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

      11. pow1 79.4%

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

   10. Applied egg-rr 79.4%

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

       1. *-lft-identity 79.4%

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

   12. Simplified 79.4%

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

   if 3.00000000000000022e210 < h

   1. Initial program 63.9%

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

   3. Simplified 68.5%

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

      1. pow1 68.5%

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

      2. sqrt-unprod 61.0%

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

   6. Applied egg-rr 61.0%

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

      1. unpow1 61.0%

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

      2. *-commutative 61.0%

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

   8. Simplified 61.0%

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

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq 8.8 \cdot 10^{-254}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d \cdot \frac{2}{M}}\right)}^{2}\right)\right)\\ \mathbf{elif}\;h \leq 3 \cdot 10^{+210}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}\right)\right) \cdot \frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 63.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := 1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D\_m}{d}\right)}^{2}\right)\\ \mathbf{if}\;\ell \leq 7.8 \cdot 10^{-221}:\\ \;\;\;\;t\_0 \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{elif}\;\ell \leq 1.7 \cdot 10^{+175}:\\ \;\;\;\;t\_0 \cdot \frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (let* ((t_0 (- 1.0 (* 0.5 (* (/ h l) (pow (* (/ M 2.0) (/ D_m d)) 2.0))))))
   (if (<= l 7.8e-221)
     (* t_0 (sqrt (* (/ d h) (/ d l))))
     (if (<= l 1.7e+175)
       (* t_0 (/ d (sqrt (* l h))))
       (* d (* (pow l -0.5) (pow h -0.5)))))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double t_0 = 1.0 - (0.5 * ((h / l) * pow(((M / 2.0) * (D_m / d)), 2.0)));
	double tmp;
	if (l <= 7.8e-221) {
		tmp = t_0 * sqrt(((d / h) * (d / l)));
	} else if (l <= 1.7e+175) {
		tmp = t_0 * (d / sqrt((l * h)));
	} else {
		tmp = d * (pow(l, -0.5) * pow(h, -0.5));
	}
	return tmp;
}

Fortran

D_m = abs(d)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_m)
    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_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 - (0.5d0 * ((h / l) * (((m / 2.0d0) * (d_m / d)) ** 2.0d0)))
    if (l <= 7.8d-221) then
        tmp = t_0 * sqrt(((d / h) * (d / l)))
    else if (l <= 1.7d+175) then
        tmp = t_0 * (d / sqrt((l * h)))
    else
        tmp = d * ((l ** (-0.5d0)) * (h ** (-0.5d0)))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert d < h && h < l && l < M && M < D_m;
public static double code(double d, double h, double l, double M, double D_m) {
	double t_0 = 1.0 - (0.5 * ((h / l) * Math.pow(((M / 2.0) * (D_m / d)), 2.0)));
	double tmp;
	if (l <= 7.8e-221) {
		tmp = t_0 * Math.sqrt(((d / h) * (d / l)));
	} else if (l <= 1.7e+175) {
		tmp = t_0 * (d / Math.sqrt((l * h)));
	} else {
		tmp = d * (Math.pow(l, -0.5) * Math.pow(h, -0.5));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[d, h, l, M, D_m] = sort([d, h, l, M, D_m])
def code(d, h, l, M, D_m):
	t_0 = 1.0 - (0.5 * ((h / l) * math.pow(((M / 2.0) * (D_m / d)), 2.0)))
	tmp = 0
	if l <= 7.8e-221:
		tmp = t_0 * math.sqrt(((d / h) * (d / l)))
	elif l <= 1.7e+175:
		tmp = t_0 * (d / math.sqrt((l * h)))
	else:
		tmp = d * (math.pow(l, -0.5) * math.pow(h, -0.5))
	return tmp

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	t_0 = Float64(1.0 - Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(M / 2.0) * Float64(D_m / d)) ^ 2.0))))
	tmp = 0.0
	if (l <= 7.8e-221)
		tmp = Float64(t_0 * sqrt(Float64(Float64(d / h) * Float64(d / l))));
	elseif (l <= 1.7e+175)
		tmp = Float64(t_0 * Float64(d / sqrt(Float64(l * h))));
	else
		tmp = Float64(d * Float64((l ^ -0.5) * (h ^ -0.5)));
	end
	return tmp
end

MATLAB

D_m = abs(D);
d, h, l, M, D_m = num2cell(sort([d, h, l, M, D_m])){:}
function tmp_2 = code(d, h, l, M, D_m)
	t_0 = 1.0 - (0.5 * ((h / l) * (((M / 2.0) * (D_m / d)) ^ 2.0)));
	tmp = 0.0;
	if (l <= 7.8e-221)
		tmp = t_0 * sqrt(((d / h) * (d / l)));
	elseif (l <= 1.7e+175)
		tmp = t_0 * (d / sqrt((l * h)));
	else
		tmp = d * ((l ^ -0.5) * (h ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := Block[{t$95$0 = N[(1.0 - N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M / 2.0), $MachinePrecision] * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, 7.8e-221], N[(t$95$0 * N[Sqrt[N[(N[(d / h), $MachinePrecision] * N[(d / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.7e+175], N[(t$95$0 * N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[l, -0.5], $MachinePrecision] * N[Power[h, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < 7.7999999999999997e-221

   1. Initial program 66.3%

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

   3. Simplified 66.9%

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

      1. pow1 66.9%

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

      2. sqrt-unprod 60.4%

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

   6. Applied egg-rr 60.4%

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

      1. unpow1 60.4%

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

      2. *-commutative 60.4%

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

   8. Simplified 60.4%

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

   if 7.7999999999999997e-221 < l < 1.70000000000000014e175

   1. Initial program 71.1%

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

   3. Simplified 72.3%

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

      1. pow1 72.3%

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

      2. sqrt-unprod 52.1%

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

   6. Applied egg-rr 52.1%

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

      1. unpow1 52.1%

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

      2. *-commutative 52.1%

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

   8. Simplified 52.1%

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

      1. frac-times 42.5%

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

      2. unpow2 42.5%

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

      3. *-commutative 42.5%

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

      4. associate-/l/ 48.1%

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

      5. *-un-lft-identity 48.1%

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

      6. associate-/l/ 42.5%

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

      7. *-commutative 42.5%

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

      8. sqrt-div 51.3%

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

      9. sqrt-pow1 75.6%

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

      10. metadata-eval 75.6%

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

      11. pow1 75.6%

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

   10. Applied egg-rr 75.6%

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

       1. *-lft-identity 75.6%

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

   12. Simplified 75.6%

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

   if 1.70000000000000014e175 < l

   1. Initial program 59.4%

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

   3. Simplified 59.4%

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

   5. Taylor expanded in d around inf 43.3%

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

      1. *-commutative 43.3%

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

   7. Simplified 43.3%

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

   8. Taylor expanded in l around 0 43.3%

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

      1. *-commutative 43.3%

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

      2. unpow-1 43.3%

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

      3. metadata-eval 43.3%

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

      4. pow-sqr 43.3%

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

      5. rem-sqrt-square 43.3%

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

      6. rem-square-sqrt 43.2%

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

      7. fabs-sqr 43.2%

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

      8. rem-square-sqrt 43.3%

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

   10. Simplified 43.3%

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

       1. unpow-prod-down 71.1%

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

   12. Applied egg-rr 71.1%

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

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 7.8 \cdot 10^{-221}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{elif}\;\ell \leq 1.7 \cdot 10^{+175}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}\right)\right) \cdot \frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 54.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -3.05 \cdot 10^{+59}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{\ell}{d}}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;d \leq -4.7 \cdot 10^{-297}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(-0.25 \cdot \left({\left(D\_m \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D\_m}{d}\right)}^{2}\right)\right) \cdot \frac{d}{\sqrt{\ell \cdot h}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (if (<= d -3.05e+59)
   (* (/ 1.0 (sqrt (/ l d))) (sqrt (/ d h)))
   (if (<= d -4.7e-297)
     (*
      (sqrt (* (/ d h) (/ d l)))
      (* -0.25 (* (pow (* D_m (/ M d)) 2.0) (* (/ h l) 0.5))))
     (*
      (- 1.0 (* 0.5 (* (/ h l) (pow (* (/ M 2.0) (/ D_m d)) 2.0))))
      (/ d (sqrt (* l h)))))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double tmp;
	if (d <= -3.05e+59) {
		tmp = (1.0 / sqrt((l / d))) * sqrt((d / h));
	} else if (d <= -4.7e-297) {
		tmp = sqrt(((d / h) * (d / l))) * (-0.25 * (pow((D_m * (M / d)), 2.0) * ((h / l) * 0.5)));
	} else {
		tmp = (1.0 - (0.5 * ((h / l) * pow(((M / 2.0) * (D_m / d)), 2.0)))) * (d / sqrt((l * h)));
	}
	return tmp;
}

Fortran

D_m = abs(d)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_m)
    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_m
    real(8) :: tmp
    if (d <= (-3.05d+59)) then
        tmp = (1.0d0 / sqrt((l / d))) * sqrt((d / h))
    else if (d <= (-4.7d-297)) then
        tmp = sqrt(((d / h) * (d / l))) * ((-0.25d0) * (((d_m * (m / d)) ** 2.0d0) * ((h / l) * 0.5d0)))
    else
        tmp = (1.0d0 - (0.5d0 * ((h / l) * (((m / 2.0d0) * (d_m / d)) ** 2.0d0)))) * (d / sqrt((l * h)))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert d < h && h < l && l < M && M < D_m;
public static double code(double d, double h, double l, double M, double D_m) {
	double tmp;
	if (d <= -3.05e+59) {
		tmp = (1.0 / Math.sqrt((l / d))) * Math.sqrt((d / h));
	} else if (d <= -4.7e-297) {
		tmp = Math.sqrt(((d / h) * (d / l))) * (-0.25 * (Math.pow((D_m * (M / d)), 2.0) * ((h / l) * 0.5)));
	} else {
		tmp = (1.0 - (0.5 * ((h / l) * Math.pow(((M / 2.0) * (D_m / d)), 2.0)))) * (d / Math.sqrt((l * h)));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[d, h, l, M, D_m] = sort([d, h, l, M, D_m])
def code(d, h, l, M, D_m):
	tmp = 0
	if d <= -3.05e+59:
		tmp = (1.0 / math.sqrt((l / d))) * math.sqrt((d / h))
	elif d <= -4.7e-297:
		tmp = math.sqrt(((d / h) * (d / l))) * (-0.25 * (math.pow((D_m * (M / d)), 2.0) * ((h / l) * 0.5)))
	else:
		tmp = (1.0 - (0.5 * ((h / l) * math.pow(((M / 2.0) * (D_m / d)), 2.0)))) * (d / math.sqrt((l * h)))
	return tmp

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	tmp = 0.0
	if (d <= -3.05e+59)
		tmp = Float64(Float64(1.0 / sqrt(Float64(l / d))) * sqrt(Float64(d / h)));
	elseif (d <= -4.7e-297)
		tmp = Float64(sqrt(Float64(Float64(d / h) * Float64(d / l))) * Float64(-0.25 * Float64((Float64(D_m * Float64(M / d)) ^ 2.0) * Float64(Float64(h / l) * 0.5))));
	else
		tmp = Float64(Float64(1.0 - Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(M / 2.0) * Float64(D_m / d)) ^ 2.0)))) * Float64(d / sqrt(Float64(l * h))));
	end
	return tmp
end

MATLAB

D_m = abs(D);
d, h, l, M, D_m = num2cell(sort([d, h, l, M, D_m])){:}
function tmp_2 = code(d, h, l, M, D_m)
	tmp = 0.0;
	if (d <= -3.05e+59)
		tmp = (1.0 / sqrt((l / d))) * sqrt((d / h));
	elseif (d <= -4.7e-297)
		tmp = sqrt(((d / h) * (d / l))) * (-0.25 * (((D_m * (M / d)) ^ 2.0) * ((h / l) * 0.5)));
	else
		tmp = (1.0 - (0.5 * ((h / l) * (((M / 2.0) * (D_m / d)) ^ 2.0)))) * (d / sqrt((l * h)));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := If[LessEqual[d, -3.05e+59], N[(N[(1.0 / N[Sqrt[N[(l / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -4.7e-297], N[(N[Sqrt[N[(N[(d / h), $MachinePrecision] * N[(d / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-0.25 * N[(N[Power[N[(D$95$m * N[(M / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(h / l), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M / 2.0), $MachinePrecision] * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -3.04999999999999986e59

   1. Initial program 79.3%

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

   3. Simplified 83.6%

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

   5. Taylor expanded in M around 0 60.2%

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

      1. *-rgt-identity 60.2%

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

      2. clear-num 60.2%

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

      3. sqrt-div 61.5%

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

      4. metadata-eval 61.5%

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

   7. Applied egg-rr 61.5%

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

   if -3.04999999999999986e59 < d < -4.69999999999999986e-297

   1. Initial program 56.9%

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

   3. Simplified 55.5%

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

      1. pow1 55.5%

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

      2. sqrt-unprod 50.6%

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

   6. Applied egg-rr 50.6%

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

      1. unpow1 50.6%

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

      2. *-commutative 50.6%

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

   8. Simplified 50.6%

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

      1. add-sqr-sqrt 50.6%

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

      2. pow2 50.6%

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

      3. *-commutative 50.6%

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

      4. sqrt-prod 50.6%

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

      5. sqrt-prod 50.6%

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

      6. sqrt-pow1 52.9%

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

      7. metadata-eval 52.9%

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

      8. pow1 52.9%

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

      9. div-inv 52.9%

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

      10. metadata-eval 52.9%

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

   10. Applied egg-rr 52.9%

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

   11. Taylor expanded in M around inf 26.1%

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

       1. associate-/l* 26.1%

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

       2. times-frac 27.4%

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

       3. associate-*r* 27.4%

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

       4. unpow2 27.4%

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

       5. unpow2 27.4%

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

       6. unpow2 27.4%

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

       7. times-frac 34.0%

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

       8. swap-sqr 40.6%

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

       9. associate-/l* 40.7%

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

       10. associate-/l* 41.8%

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

       11. unpow2 41.8%

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

       12. associate-/l* 40.6%

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

       13. *-commutative 40.6%

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

       14. unpow2 40.6%

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

       15. rem-square-sqrt 40.7%

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

       16. associate-*r/ 40.7%

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

       17. *-commutative 40.7%

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

   13. Simplified 40.7%

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

   if -4.69999999999999986e-297 < d

   1. Initial program 69.4%

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

   3. Simplified 70.1%

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

      1. pow1 70.1%

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

      2. sqrt-unprod 53.9%

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

   6. Applied egg-rr 53.9%

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

      1. unpow1 53.9%

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

      2. *-commutative 53.9%

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

   8. Simplified 53.9%

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

      1. frac-times 40.4%

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

      2. unpow2 40.4%

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

      3. *-commutative 40.4%

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

      4. associate-/l/ 46.6%

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

      5. *-un-lft-identity 46.6%

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

      6. associate-/l/ 40.4%

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

      7. *-commutative 40.4%

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

      8. sqrt-div 48.6%

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

      9. sqrt-pow1 68.8%

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

      10. metadata-eval 68.8%

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

      11. pow1 68.8%

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

   10. Applied egg-rr 68.8%

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

       1. *-lft-identity 68.8%

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

   12. Simplified 68.8%

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

4. Final simplification 58.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -3.05 \cdot 10^{+59}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{\ell}{d}}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;d \leq -4.7 \cdot 10^{-297}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(-0.25 \cdot \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}\right)\right) \cdot \frac{d}{\sqrt{\ell \cdot h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 50.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;D\_m \leq 2.3 \cdot 10^{+170}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(-0.25 \cdot \left({\left(D\_m \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot 0.5\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (if (<= D_m 2.3e+170)
   (fabs (/ d (sqrt (* l h))))
   (*
    (sqrt (* (/ d h) (/ d l)))
    (* -0.25 (* (pow (* D_m (/ M d)) 2.0) (* (/ h l) 0.5))))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double tmp;
	if (D_m <= 2.3e+170) {
		tmp = fabs((d / sqrt((l * h))));
	} else {
		tmp = sqrt(((d / h) * (d / l))) * (-0.25 * (pow((D_m * (M / d)), 2.0) * ((h / l) * 0.5)));
	}
	return tmp;
}

Fortran

D_m = abs(d)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_m)
    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_m
    real(8) :: tmp
    if (d_m <= 2.3d+170) then
        tmp = abs((d / sqrt((l * h))))
    else
        tmp = sqrt(((d / h) * (d / l))) * ((-0.25d0) * (((d_m * (m / d)) ** 2.0d0) * ((h / l) * 0.5d0)))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert d < h && h < l && l < M && M < D_m;
public static double code(double d, double h, double l, double M, double D_m) {
	double tmp;
	if (D_m <= 2.3e+170) {
		tmp = Math.abs((d / Math.sqrt((l * h))));
	} else {
		tmp = Math.sqrt(((d / h) * (d / l))) * (-0.25 * (Math.pow((D_m * (M / d)), 2.0) * ((h / l) * 0.5)));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[d, h, l, M, D_m] = sort([d, h, l, M, D_m])
def code(d, h, l, M, D_m):
	tmp = 0
	if D_m <= 2.3e+170:
		tmp = math.fabs((d / math.sqrt((l * h))))
	else:
		tmp = math.sqrt(((d / h) * (d / l))) * (-0.25 * (math.pow((D_m * (M / d)), 2.0) * ((h / l) * 0.5)))
	return tmp

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	tmp = 0.0
	if (D_m <= 2.3e+170)
		tmp = abs(Float64(d / sqrt(Float64(l * h))));
	else
		tmp = Float64(sqrt(Float64(Float64(d / h) * Float64(d / l))) * Float64(-0.25 * Float64((Float64(D_m * Float64(M / d)) ^ 2.0) * Float64(Float64(h / l) * 0.5))));
	end
	return tmp
end

MATLAB

D_m = abs(D);
d, h, l, M, D_m = num2cell(sort([d, h, l, M, D_m])){:}
function tmp_2 = code(d, h, l, M, D_m)
	tmp = 0.0;
	if (D_m <= 2.3e+170)
		tmp = abs((d / sqrt((l * h))));
	else
		tmp = sqrt(((d / h) * (d / l))) * (-0.25 * (((D_m * (M / d)) ^ 2.0) * ((h / l) * 0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := If[LessEqual[D$95$m, 2.3e+170], N[Abs[N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(N[(d / h), $MachinePrecision] * N[(d / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-0.25 * N[(N[Power[N[(D$95$m * N[(M / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(h / l), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if D < 2.3000000000000001e170

   1. Initial program 66.6%

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

   3. Simplified 67.5%

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

   5. Taylor expanded in M around 0 36.4%

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

      1. *-rgt-identity 36.4%

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

      2. *-commutative 36.4%

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

      3. sqrt-prod 30.1%

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

      4. add-sqr-sqrt 30.1%

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

      5. rem-sqrt-square 30.1%

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

      6. frac-times 20.0%

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

      7. sqrt-div 24.2%

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

      8. sqrt-unprod 19.0%

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

      9. add-sqr-sqrt 36.5%

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

      10. *-commutative 36.5%

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

   7. Applied egg-rr 36.5%

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

   if 2.3000000000000001e170 < D

   1. Initial program 71.5%

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

   3. Simplified 71.5%

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

      1. pow1 71.5%

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

      2. sqrt-unprod 65.8%

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

   6. Applied egg-rr 65.8%

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

      1. unpow1 65.8%

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

      2. *-commutative 65.8%

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

   8. Simplified 65.8%

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

      1. add-sqr-sqrt 65.8%

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

      2. pow2 65.8%

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

      3. *-commutative 65.8%

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

      4. sqrt-prod 65.8%

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

      5. sqrt-prod 65.8%

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

      6. sqrt-pow1 65.8%

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

      7. metadata-eval 65.8%

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

      8. pow1 65.8%

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

      9. div-inv 65.8%

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

      10. metadata-eval 65.8%

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

   10. Applied egg-rr 65.8%

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

   11. Taylor expanded in M around inf 40.1%

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

       1. associate-/l* 40.1%

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

       2. times-frac 40.1%

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

       3. associate-*r* 40.1%

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

       4. unpow2 40.1%

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

       5. unpow2 40.1%

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

       6. unpow2 40.1%

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

       7. times-frac 48.7%

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

       8. swap-sqr 57.4%

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

       9. associate-/l* 57.3%

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

       10. associate-/l* 57.3%

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

       11. unpow2 57.3%

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

       12. associate-/l* 57.4%

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

       13. *-commutative 57.4%

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

       14. unpow2 57.4%

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

       15. rem-square-sqrt 57.4%

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

       16. associate-*r/ 57.4%

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

       17. *-commutative 57.4%

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

   13. Simplified 57.4%

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

4. Final simplification 39.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq 2.3 \cdot 10^{+170}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(-0.25 \cdot \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot 0.5\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 46.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -5.6 \cdot 10^{-19}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{\ell}{d}}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;d \cdot {\left(-1 + \mathsf{fma}\left(\ell, h, 1\right)\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (if (<= d -5.6e-19)
   (* (/ 1.0 (sqrt (/ l d))) (sqrt (/ d h)))
   (if (<= d -5e-310)
     (* d (pow (+ -1.0 (fma l h 1.0)) -0.5))
     (* d (* (pow l -0.5) (pow h -0.5))))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double tmp;
	if (d <= -5.6e-19) {
		tmp = (1.0 / sqrt((l / d))) * sqrt((d / h));
	} else if (d <= -5e-310) {
		tmp = d * pow((-1.0 + fma(l, h, 1.0)), -0.5);
	} else {
		tmp = d * (pow(l, -0.5) * pow(h, -0.5));
	}
	return tmp;
}

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	tmp = 0.0
	if (d <= -5.6e-19)
		tmp = Float64(Float64(1.0 / sqrt(Float64(l / d))) * sqrt(Float64(d / h)));
	elseif (d <= -5e-310)
		tmp = Float64(d * (Float64(-1.0 + fma(l, h, 1.0)) ^ -0.5));
	else
		tmp = Float64(d * Float64((l ^ -0.5) * (h ^ -0.5)));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := If[LessEqual[d, -5.6e-19], N[(N[(1.0 / N[Sqrt[N[(l / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -5e-310], N[(d * N[Power[N[(-1.0 + N[(l * h + 1.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[l, -0.5], $MachinePrecision] * N[Power[h, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -5.60000000000000005e-19

   1. Initial program 79.9%

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

   3. Simplified 83.2%

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

   5. Taylor expanded in M around 0 52.3%

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

      1. *-rgt-identity 52.3%

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

      2. clear-num 52.3%

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

      3. sqrt-div 53.3%

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

      4. metadata-eval 53.3%

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

   7. Applied egg-rr 53.3%

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

   if -5.60000000000000005e-19 < d < -4.999999999999985e-310

   1. Initial program 48.8%

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

   3. Simplified 47.1%

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

   5. Taylor expanded in d around inf 16.6%

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

      1. *-commutative 16.6%

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

   7. Simplified 16.6%

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

   8. Taylor expanded in l around 0 16.6%

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

      1. *-commutative 16.6%

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

      2. unpow-1 16.6%

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

      3. metadata-eval 16.6%

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

      4. pow-sqr 16.6%

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

      5. rem-sqrt-square 16.6%

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

      6. rem-square-sqrt 16.6%

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

      7. fabs-sqr 16.6%

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

      8. rem-square-sqrt 16.6%

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

   10. Simplified 16.6%

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

       1. expm1-log1p-u 16.6%

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

       2. expm1-undefine 35.3%

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

   12. Applied egg-rr 35.3%

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

       1. sub-neg 35.3%

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

       2. metadata-eval 35.3%

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

       3. +-commutative 35.3%

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

       4. log1p-undefine 35.3%

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

       5. rem-exp-log 35.3%

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

       6. +-commutative 35.3%

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

       7. fma-define 35.3%

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

   14. Simplified 35.3%

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

   if -4.999999999999985e-310 < d

   1. Initial program 71.0%

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

   3. Simplified 71.8%

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

   5. Taylor expanded in d around inf 34.6%

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

      1. *-commutative 34.6%

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

   7. Simplified 34.6%

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

   8. Taylor expanded in l around 0 34.6%

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

      1. *-commutative 34.6%

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

      2. unpow-1 34.6%

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

      3. metadata-eval 34.6%

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

      4. pow-sqr 34.6%

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

      5. rem-sqrt-square 34.6%

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

      6. rem-square-sqrt 34.5%

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

      7. fabs-sqr 34.5%

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

      8. rem-square-sqrt 34.6%

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

   10. Simplified 34.6%

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

       1. unpow-prod-down 45.7%

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

   12. Applied egg-rr 45.7%

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

4. Final simplification 44.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -5.6 \cdot 10^{-19}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{\ell}{d}}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;d \cdot {\left(-1 + \mathsf{fma}\left(\ell, h, 1\right)\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 46.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -1.58 \cdot 10^{-18}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;d \cdot {\left(-1 + \mathsf{fma}\left(\ell, h, 1\right)\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (if (<= d -1.58e-18)
   (* (sqrt (/ d h)) (sqrt (/ d l)))
   (if (<= d -5e-310)
     (* d (pow (+ -1.0 (fma l h 1.0)) -0.5))
     (* d (* (pow l -0.5) (pow h -0.5))))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double tmp;
	if (d <= -1.58e-18) {
		tmp = sqrt((d / h)) * sqrt((d / l));
	} else if (d <= -5e-310) {
		tmp = d * pow((-1.0 + fma(l, h, 1.0)), -0.5);
	} else {
		tmp = d * (pow(l, -0.5) * pow(h, -0.5));
	}
	return tmp;
}

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	tmp = 0.0
	if (d <= -1.58e-18)
		tmp = Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l)));
	elseif (d <= -5e-310)
		tmp = Float64(d * (Float64(-1.0 + fma(l, h, 1.0)) ^ -0.5));
	else
		tmp = Float64(d * Float64((l ^ -0.5) * (h ^ -0.5)));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := If[LessEqual[d, -1.58e-18], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -5e-310], N[(d * N[Power[N[(-1.0 + N[(l * h + 1.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[l, -0.5], $MachinePrecision] * N[Power[h, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.5800000000000001e-18

   1. Initial program 79.9%

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

   3. Simplified 83.2%

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

   5. Taylor expanded in M around 0 52.3%

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

   6. Taylor expanded in d around 0 52.3%

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

   if -1.5800000000000001e-18 < d < -4.999999999999985e-310

   1. Initial program 48.8%

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

   3. Simplified 47.1%

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

   5. Taylor expanded in d around inf 16.6%

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

      1. *-commutative 16.6%

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

   7. Simplified 16.6%

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

   8. Taylor expanded in l around 0 16.6%

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

      1. *-commutative 16.6%

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

      2. unpow-1 16.6%

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

      3. metadata-eval 16.6%

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

      4. pow-sqr 16.6%

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

      5. rem-sqrt-square 16.6%

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

      6. rem-square-sqrt 16.6%

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

      7. fabs-sqr 16.6%

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

      8. rem-square-sqrt 16.6%

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

   10. Simplified 16.6%

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

       1. expm1-log1p-u 16.6%

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

       2. expm1-undefine 35.3%

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

   12. Applied egg-rr 35.3%

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

       1. sub-neg 35.3%

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

       2. metadata-eval 35.3%

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

       3. +-commutative 35.3%

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

       4. log1p-undefine 35.3%

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

       5. rem-exp-log 35.3%

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

       6. +-commutative 35.3%

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

       7. fma-define 35.3%

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

   14. Simplified 35.3%

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

   if -4.999999999999985e-310 < d

   1. Initial program 71.0%

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

   3. Simplified 71.8%

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

   5. Taylor expanded in d around inf 34.6%

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

      1. *-commutative 34.6%

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

   7. Simplified 34.6%

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

   8. Taylor expanded in l around 0 34.6%

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

      1. *-commutative 34.6%

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

      2. unpow-1 34.6%

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

      3. metadata-eval 34.6%

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

      4. pow-sqr 34.6%

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

      5. rem-sqrt-square 34.6%

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

      6. rem-square-sqrt 34.5%

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

      7. fabs-sqr 34.5%

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

      8. rem-square-sqrt 34.6%

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

   10. Simplified 34.6%

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

       1. unpow-prod-down 45.7%

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

   12. Applied egg-rr 45.7%

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

Alternative 18: 47.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -6.2 \cdot 10^{-215}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{elif}\;\ell \leq -4 \cdot 10^{-310}:\\ \;\;\;\;d \cdot {\left({\left(\ell \cdot h\right)}^{2}\right)}^{-0.25}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (if (<= l -6.2e-215)
   (* (- d) (sqrt (/ (/ 1.0 l) h)))
   (if (<= l -4e-310)
     (* d (pow (pow (* l h) 2.0) -0.25))
     (* d (* (pow l -0.5) (pow h -0.5))))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double tmp;
	if (l <= -6.2e-215) {
		tmp = -d * sqrt(((1.0 / l) / h));
	} else if (l <= -4e-310) {
		tmp = d * pow(pow((l * h), 2.0), -0.25);
	} else {
		tmp = d * (pow(l, -0.5) * pow(h, -0.5));
	}
	return tmp;
}

Fortran

D_m = abs(d)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_m)
    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_m
    real(8) :: tmp
    if (l <= (-6.2d-215)) then
        tmp = -d * sqrt(((1.0d0 / l) / h))
    else if (l <= (-4d-310)) then
        tmp = d * (((l * h) ** 2.0d0) ** (-0.25d0))
    else
        tmp = d * ((l ** (-0.5d0)) * (h ** (-0.5d0)))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert d < h && h < l && l < M && M < D_m;
public static double code(double d, double h, double l, double M, double D_m) {
	double tmp;
	if (l <= -6.2e-215) {
		tmp = -d * Math.sqrt(((1.0 / l) / h));
	} else if (l <= -4e-310) {
		tmp = d * Math.pow(Math.pow((l * h), 2.0), -0.25);
	} else {
		tmp = d * (Math.pow(l, -0.5) * Math.pow(h, -0.5));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[d, h, l, M, D_m] = sort([d, h, l, M, D_m])
def code(d, h, l, M, D_m):
	tmp = 0
	if l <= -6.2e-215:
		tmp = -d * math.sqrt(((1.0 / l) / h))
	elif l <= -4e-310:
		tmp = d * math.pow(math.pow((l * h), 2.0), -0.25)
	else:
		tmp = d * (math.pow(l, -0.5) * math.pow(h, -0.5))
	return tmp

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	tmp = 0.0
	if (l <= -6.2e-215)
		tmp = Float64(Float64(-d) * sqrt(Float64(Float64(1.0 / l) / h)));
	elseif (l <= -4e-310)
		tmp = Float64(d * ((Float64(l * h) ^ 2.0) ^ -0.25));
	else
		tmp = Float64(d * Float64((l ^ -0.5) * (h ^ -0.5)));
	end
	return tmp
end

MATLAB

D_m = abs(D);
d, h, l, M, D_m = num2cell(sort([d, h, l, M, D_m])){:}
function tmp_2 = code(d, h, l, M, D_m)
	tmp = 0.0;
	if (l <= -6.2e-215)
		tmp = -d * sqrt(((1.0 / l) / h));
	elseif (l <= -4e-310)
		tmp = d * (((l * h) ^ 2.0) ^ -0.25);
	else
		tmp = d * ((l ^ -0.5) * (h ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := If[LessEqual[l, -6.2e-215], N[((-d) * N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -4e-310], N[(d * N[Power[N[Power[N[(l * h), $MachinePrecision], 2.0], $MachinePrecision], -0.25], $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[l, -0.5], $MachinePrecision] * N[Power[h, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -6.19999999999999987e-215

   1. Initial program 62.1%

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

   3. Simplified 62.9%

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

   5. Taylor expanded in d around inf 6.4%

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

      1. *-commutative 6.4%

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

   7. Simplified 6.4%

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

   8. Taylor expanded in l around 0 6.4%

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

      1. *-commutative 6.4%

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

      2. unpow-1 6.4%

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

      3. metadata-eval 6.4%

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

      4. pow-sqr 6.4%

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

      5. rem-sqrt-square 6.4%

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

      6. rem-square-sqrt 6.4%

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

      7. fabs-sqr 6.4%

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

      8. rem-square-sqrt 6.4%

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

   10. Simplified 6.4%

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

   11. Taylor expanded in l around -inf 0.0%

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

       1. *-commutative 0.0%

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

       2. *-commutative 0.0%

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

       3. associate-/r* 0.0%

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

       4. *-commutative 0.0%

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

       5. unpow2 0.0%

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

       6. rem-square-sqrt 36.5%

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

       7. neg-mul-1 36.5%

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

   13. Simplified 36.5%

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

   if -6.19999999999999987e-215 < l < -3.999999999999988e-310

   1. Initial program 72.7%

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

   3. Simplified 72.7%

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

   5. Taylor expanded in d around inf 40.4%

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

      1. *-commutative 40.4%

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

   7. Simplified 40.4%

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

   8. Taylor expanded in l around -inf 0.0%

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

      1. associate-*l* 0.0%

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

      2. unpow2 0.0%

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

      3. rem-square-sqrt 6.8%

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

      4. associate-*r* 6.8%

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

      5. *-commutative 6.8%

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

      6. neg-mul-1 6.8%

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

      7. distribute-lft-neg-in 6.8%

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

      8. distribute-rgt-neg-in 6.8%

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

      9. *-commutative 6.8%

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

      10. unpow-1 6.8%

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

      11. metadata-eval 6.8%

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

      12. pow-sqr 6.8%

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

      13. rem-sqrt-square 6.8%

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

      14. rem-square-sqrt 6.8%

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

      15. fabs-sqr 6.8%

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

      16. rem-square-sqrt 6.8%

          \[\leadsto d \cdot \left(-\color{blue}{{\left(\ell \cdot h\right)}^{-0.5}}\right) \]

   10. Simplified 6.8%

       \[\leadsto \color{blue}{d \cdot \left(-{\left(\ell \cdot h\right)}^{-0.5}\right)} \]
   11. Step-by-step derivation

       1. add-sqr-sqrt 0.0%

          \[\leadsto d \cdot \color{blue}{\left(\sqrt{-{\left(\ell \cdot h\right)}^{-0.5}} \cdot \sqrt{-{\left(\ell \cdot h\right)}^{-0.5}}\right)} \]

       2. sqrt-unprod 40.4%

          \[\leadsto d \cdot \color{blue}{\sqrt{\left(-{\left(\ell \cdot h\right)}^{-0.5}\right) \cdot \left(-{\left(\ell \cdot h\right)}^{-0.5}\right)}} \]

       3. sqr-neg 40.4%

          \[\leadsto d \cdot \sqrt{\color{blue}{{\left(\ell \cdot h\right)}^{-0.5} \cdot {\left(\ell \cdot h\right)}^{-0.5}}} \]

       4. sqrt-unprod 40.4%

          \[\leadsto d \cdot \color{blue}{\left(\sqrt{{\left(\ell \cdot h\right)}^{-0.5}} \cdot \sqrt{{\left(\ell \cdot h\right)}^{-0.5}}\right)} \]

       5. add-sqr-sqrt 40.4%

          \[\leadsto d \cdot \color{blue}{{\left(\ell \cdot h\right)}^{-0.5}} \]

       6. sqr-pow 40.4%

          \[\leadsto d \cdot \color{blue}{\left({\left(\ell \cdot h\right)}^{\left(\frac{-0.5}{2}\right)} \cdot {\left(\ell \cdot h\right)}^{\left(\frac{-0.5}{2}\right)}\right)} \]

       7. pow-prod-down 51.1%

          \[\leadsto d \cdot \color{blue}{{\left(\left(\ell \cdot h\right) \cdot \left(\ell \cdot h\right)\right)}^{\left(\frac{-0.5}{2}\right)}} \]

       8. pow2 51.1%

          \[\leadsto d \cdot {\color{blue}{\left({\left(\ell \cdot h\right)}^{2}\right)}}^{\left(\frac{-0.5}{2}\right)} \]

       9. *-commutative 51.1%

          \[\leadsto d \cdot {\left({\color{blue}{\left(h \cdot \ell\right)}}^{2}\right)}^{\left(\frac{-0.5}{2}\right)} \]

       10. metadata-eval 51.1%

           \[\leadsto d \cdot {\left({\left(h \cdot \ell\right)}^{2}\right)}^{\color{blue}{-0.25}} \]

   12. Applied egg-rr 51.1%

       \[\leadsto d \cdot \color{blue}{{\left({\left(h \cdot \ell\right)}^{2}\right)}^{-0.25}} \]

   if -3.999999999999988e-310 < l

   1. Initial program 71.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 71.8%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 34.6%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. *-commutative 34.6%

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

   7. Simplified 34.6%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

   8. Taylor expanded in l around 0 34.6%

      \[\leadsto d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]
   9. Step-by-step derivation

      1. *-commutative 34.6%

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      2. unpow-1 34.6%

         \[\leadsto d \cdot \sqrt{\color{blue}{{\left(\ell \cdot h\right)}^{-1}}} \]

      3. metadata-eval 34.6%

         \[\leadsto d \cdot \sqrt{{\left(\ell \cdot h\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \]

      4. pow-sqr 34.6%

         \[\leadsto d \cdot \sqrt{\color{blue}{{\left(\ell \cdot h\right)}^{-0.5} \cdot {\left(\ell \cdot h\right)}^{-0.5}}} \]

      5. rem-sqrt-square 34.6%

         \[\leadsto d \cdot \color{blue}{\left|{\left(\ell \cdot h\right)}^{-0.5}\right|} \]

      6. rem-square-sqrt 34.5%

         \[\leadsto d \cdot \left|\color{blue}{\sqrt{{\left(\ell \cdot h\right)}^{-0.5}} \cdot \sqrt{{\left(\ell \cdot h\right)}^{-0.5}}}\right| \]

      7. fabs-sqr 34.5%

         \[\leadsto d \cdot \color{blue}{\left(\sqrt{{\left(\ell \cdot h\right)}^{-0.5}} \cdot \sqrt{{\left(\ell \cdot h\right)}^{-0.5}}\right)} \]

      8. rem-square-sqrt 34.6%

         \[\leadsto d \cdot \color{blue}{{\left(\ell \cdot h\right)}^{-0.5}} \]

   10. Simplified 34.6%

       \[\leadsto d \cdot \color{blue}{{\left(\ell \cdot h\right)}^{-0.5}} \]
   11. Step-by-step derivation

       1. unpow-prod-down 45.7%

          \[\leadsto d \cdot \color{blue}{\left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)} \]

   12. Applied egg-rr 45.7%

       \[\leadsto d \cdot \color{blue}{\left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 42.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -6.2 \cdot 10^{-215}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{elif}\;\ell \leq -4 \cdot 10^{-310}:\\ \;\;\;\;d \cdot {\left({\left(\ell \cdot h\right)}^{2}\right)}^{-0.25}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 46.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -2.7 \cdot 10^{-215}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{elif}\;\ell \leq 7.5 \cdot 10^{-288}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (if (<= l -2.7e-215)
   (* (- d) (sqrt (/ (/ 1.0 l) h)))
   (if (<= l 7.5e-288)
     (* d (sqrt (/ (/ 1.0 h) l)))
     (* d (* (pow l -0.5) (pow h -0.5))))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double tmp;
	if (l <= -2.7e-215) {
		tmp = -d * sqrt(((1.0 / l) / h));
	} else if (l <= 7.5e-288) {
		tmp = d * sqrt(((1.0 / h) / l));
	} else {
		tmp = d * (pow(l, -0.5) * pow(h, -0.5));
	}
	return tmp;
}

Fortran

D_m = abs(d)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_m)
    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_m
    real(8) :: tmp
    if (l <= (-2.7d-215)) then
        tmp = -d * sqrt(((1.0d0 / l) / h))
    else if (l <= 7.5d-288) then
        tmp = d * sqrt(((1.0d0 / h) / l))
    else
        tmp = d * ((l ** (-0.5d0)) * (h ** (-0.5d0)))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert d < h && h < l && l < M && M < D_m;
public static double code(double d, double h, double l, double M, double D_m) {
	double tmp;
	if (l <= -2.7e-215) {
		tmp = -d * Math.sqrt(((1.0 / l) / h));
	} else if (l <= 7.5e-288) {
		tmp = d * Math.sqrt(((1.0 / h) / l));
	} else {
		tmp = d * (Math.pow(l, -0.5) * Math.pow(h, -0.5));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[d, h, l, M, D_m] = sort([d, h, l, M, D_m])
def code(d, h, l, M, D_m):
	tmp = 0
	if l <= -2.7e-215:
		tmp = -d * math.sqrt(((1.0 / l) / h))
	elif l <= 7.5e-288:
		tmp = d * math.sqrt(((1.0 / h) / l))
	else:
		tmp = d * (math.pow(l, -0.5) * math.pow(h, -0.5))
	return tmp

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	tmp = 0.0
	if (l <= -2.7e-215)
		tmp = Float64(Float64(-d) * sqrt(Float64(Float64(1.0 / l) / h)));
	elseif (l <= 7.5e-288)
		tmp = Float64(d * sqrt(Float64(Float64(1.0 / h) / l)));
	else
		tmp = Float64(d * Float64((l ^ -0.5) * (h ^ -0.5)));
	end
	return tmp
end

MATLAB

D_m = abs(D);
d, h, l, M, D_m = num2cell(sort([d, h, l, M, D_m])){:}
function tmp_2 = code(d, h, l, M, D_m)
	tmp = 0.0;
	if (l <= -2.7e-215)
		tmp = -d * sqrt(((1.0 / l) / h));
	elseif (l <= 7.5e-288)
		tmp = d * sqrt(((1.0 / h) / l));
	else
		tmp = d * ((l ^ -0.5) * (h ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := If[LessEqual[l, -2.7e-215], N[((-d) * N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 7.5e-288], N[(d * N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[l, -0.5], $MachinePrecision] * N[Power[h, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -2.70000000000000018e-215

   1. Initial program 62.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 62.9%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 6.4%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. *-commutative 6.4%

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

   7. Simplified 6.4%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

   8. Taylor expanded in l around 0 6.4%

      \[\leadsto d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]
   9. Step-by-step derivation

      1. *-commutative 6.4%

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      2. unpow-1 6.4%

         \[\leadsto d \cdot \sqrt{\color{blue}{{\left(\ell \cdot h\right)}^{-1}}} \]

      3. metadata-eval 6.4%

         \[\leadsto d \cdot \sqrt{{\left(\ell \cdot h\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \]

      4. pow-sqr 6.4%

         \[\leadsto d \cdot \sqrt{\color{blue}{{\left(\ell \cdot h\right)}^{-0.5} \cdot {\left(\ell \cdot h\right)}^{-0.5}}} \]

      5. rem-sqrt-square 6.4%

         \[\leadsto d \cdot \color{blue}{\left|{\left(\ell \cdot h\right)}^{-0.5}\right|} \]

      6. rem-square-sqrt 6.4%

         \[\leadsto d \cdot \left|\color{blue}{\sqrt{{\left(\ell \cdot h\right)}^{-0.5}} \cdot \sqrt{{\left(\ell \cdot h\right)}^{-0.5}}}\right| \]

      7. fabs-sqr 6.4%

         \[\leadsto d \cdot \color{blue}{\left(\sqrt{{\left(\ell \cdot h\right)}^{-0.5}} \cdot \sqrt{{\left(\ell \cdot h\right)}^{-0.5}}\right)} \]

      8. rem-square-sqrt 6.4%

         \[\leadsto d \cdot \color{blue}{{\left(\ell \cdot h\right)}^{-0.5}} \]

   10. Simplified 6.4%

       \[\leadsto d \cdot \color{blue}{{\left(\ell \cdot h\right)}^{-0.5}} \]

   11. Taylor expanded in l around -inf 0.0%

       \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   12. Step-by-step derivation

       1. *-commutative 0.0%

          \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]

       2. *-commutative 0.0%

          \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

       3. associate-/r* 0.0%

          \[\leadsto \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

       4. *-commutative 0.0%

          \[\leadsto \sqrt{\frac{\frac{1}{\ell}}{h}} \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot d\right)} \]

       5. unpow2 0.0%

          \[\leadsto \sqrt{\frac{\frac{1}{\ell}}{h}} \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right) \]

       6. rem-square-sqrt 36.5%

          \[\leadsto \sqrt{\frac{\frac{1}{\ell}}{h}} \cdot \left(\color{blue}{-1} \cdot d\right) \]

       7. neg-mul-1 36.5%

          \[\leadsto \sqrt{\frac{\frac{1}{\ell}}{h}} \cdot \color{blue}{\left(-d\right)} \]

   13. Simplified 36.5%

       \[\leadsto \color{blue}{\sqrt{\frac{\frac{1}{\ell}}{h}} \cdot \left(-d\right)} \]

   if -2.70000000000000018e-215 < l < 7.4999999999999998e-288

   1. Initial program 71.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 71.8%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 44.1%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. associate-/r* 44.1%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \]

   7. Simplified 44.1%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \]

   if 7.4999999999999998e-288 < l

   1. Initial program 71.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 71.9%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 33.9%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. *-commutative 33.9%

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

   7. Simplified 33.9%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

   8. Taylor expanded in l around 0 33.9%

      \[\leadsto d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]
   9. Step-by-step derivation

      1. *-commutative 33.9%

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      2. unpow-1 33.9%

         \[\leadsto d \cdot \sqrt{\color{blue}{{\left(\ell \cdot h\right)}^{-1}}} \]

      3. metadata-eval 33.9%

         \[\leadsto d \cdot \sqrt{{\left(\ell \cdot h\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \]

      4. pow-sqr 33.9%

         \[\leadsto d \cdot \sqrt{\color{blue}{{\left(\ell \cdot h\right)}^{-0.5} \cdot {\left(\ell \cdot h\right)}^{-0.5}}} \]

      5. rem-sqrt-square 33.9%

         \[\leadsto d \cdot \color{blue}{\left|{\left(\ell \cdot h\right)}^{-0.5}\right|} \]

      6. rem-square-sqrt 33.8%

         \[\leadsto d \cdot \left|\color{blue}{\sqrt{{\left(\ell \cdot h\right)}^{-0.5}} \cdot \sqrt{{\left(\ell \cdot h\right)}^{-0.5}}}\right| \]

      7. fabs-sqr 33.8%

         \[\leadsto d \cdot \color{blue}{\left(\sqrt{{\left(\ell \cdot h\right)}^{-0.5}} \cdot \sqrt{{\left(\ell \cdot h\right)}^{-0.5}}\right)} \]

      8. rem-square-sqrt 33.9%

         \[\leadsto d \cdot \color{blue}{{\left(\ell \cdot h\right)}^{-0.5}} \]

   10. Simplified 33.9%

       \[\leadsto d \cdot \color{blue}{{\left(\ell \cdot h\right)}^{-0.5}} \]
   11. Step-by-step derivation

       1. unpow-prod-down 45.2%

          \[\leadsto d \cdot \color{blue}{\left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)} \]

   12. Applied egg-rr 45.2%

       \[\leadsto d \cdot \color{blue}{\left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 41.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.7 \cdot 10^{-215}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{elif}\;\ell \leq 7.5 \cdot 10^{-288}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 42.7% accurate, 2.9× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -5.6 \cdot 10^{-215}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (if (<= l -5.6e-215)
   (* (- d) (sqrt (/ (/ 1.0 l) h)))
   (* d (sqrt (/ (/ 1.0 h) l)))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double tmp;
	if (l <= -5.6e-215) {
		tmp = -d * sqrt(((1.0 / l) / h));
	} else {
		tmp = d * sqrt(((1.0 / h) / l));
	}
	return tmp;
}

Fortran

D_m = abs(d)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_m)
    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_m
    real(8) :: tmp
    if (l <= (-5.6d-215)) then
        tmp = -d * sqrt(((1.0d0 / l) / h))
    else
        tmp = d * sqrt(((1.0d0 / h) / l))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert d < h && h < l && l < M && M < D_m;
public static double code(double d, double h, double l, double M, double D_m) {
	double tmp;
	if (l <= -5.6e-215) {
		tmp = -d * Math.sqrt(((1.0 / l) / h));
	} else {
		tmp = d * Math.sqrt(((1.0 / h) / l));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[d, h, l, M, D_m] = sort([d, h, l, M, D_m])
def code(d, h, l, M, D_m):
	tmp = 0
	if l <= -5.6e-215:
		tmp = -d * math.sqrt(((1.0 / l) / h))
	else:
		tmp = d * math.sqrt(((1.0 / h) / l))
	return tmp

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	tmp = 0.0
	if (l <= -5.6e-215)
		tmp = Float64(Float64(-d) * sqrt(Float64(Float64(1.0 / l) / h)));
	else
		tmp = Float64(d * sqrt(Float64(Float64(1.0 / h) / l)));
	end
	return tmp
end

MATLAB

D_m = abs(D);
d, h, l, M, D_m = num2cell(sort([d, h, l, M, D_m])){:}
function tmp_2 = code(d, h, l, M, D_m)
	tmp = 0.0;
	if (l <= -5.6e-215)
		tmp = -d * sqrt(((1.0 / l) / h));
	else
		tmp = d * sqrt(((1.0 / h) / l));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := If[LessEqual[l, -5.6e-215], N[((-d) * N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d * N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -5.59999999999999972e-215

   1. Initial program 62.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 62.9%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 6.4%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. *-commutative 6.4%

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

   7. Simplified 6.4%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

   8. Taylor expanded in l around 0 6.4%

      \[\leadsto d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]
   9. Step-by-step derivation

      1. *-commutative 6.4%

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      2. unpow-1 6.4%

         \[\leadsto d \cdot \sqrt{\color{blue}{{\left(\ell \cdot h\right)}^{-1}}} \]

      3. metadata-eval 6.4%

         \[\leadsto d \cdot \sqrt{{\left(\ell \cdot h\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \]

      4. pow-sqr 6.4%

         \[\leadsto d \cdot \sqrt{\color{blue}{{\left(\ell \cdot h\right)}^{-0.5} \cdot {\left(\ell \cdot h\right)}^{-0.5}}} \]

      5. rem-sqrt-square 6.4%

         \[\leadsto d \cdot \color{blue}{\left|{\left(\ell \cdot h\right)}^{-0.5}\right|} \]

      6. rem-square-sqrt 6.4%

         \[\leadsto d \cdot \left|\color{blue}{\sqrt{{\left(\ell \cdot h\right)}^{-0.5}} \cdot \sqrt{{\left(\ell \cdot h\right)}^{-0.5}}}\right| \]

      7. fabs-sqr 6.4%

         \[\leadsto d \cdot \color{blue}{\left(\sqrt{{\left(\ell \cdot h\right)}^{-0.5}} \cdot \sqrt{{\left(\ell \cdot h\right)}^{-0.5}}\right)} \]

      8. rem-square-sqrt 6.4%

         \[\leadsto d \cdot \color{blue}{{\left(\ell \cdot h\right)}^{-0.5}} \]

   10. Simplified 6.4%

       \[\leadsto d \cdot \color{blue}{{\left(\ell \cdot h\right)}^{-0.5}} \]

   11. Taylor expanded in l around -inf 0.0%

       \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   12. Step-by-step derivation

       1. *-commutative 0.0%

          \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]

       2. *-commutative 0.0%

          \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

       3. associate-/r* 0.0%

          \[\leadsto \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

       4. *-commutative 0.0%

          \[\leadsto \sqrt{\frac{\frac{1}{\ell}}{h}} \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot d\right)} \]

       5. unpow2 0.0%

          \[\leadsto \sqrt{\frac{\frac{1}{\ell}}{h}} \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right) \]

       6. rem-square-sqrt 36.5%

          \[\leadsto \sqrt{\frac{\frac{1}{\ell}}{h}} \cdot \left(\color{blue}{-1} \cdot d\right) \]

       7. neg-mul-1 36.5%

          \[\leadsto \sqrt{\frac{\frac{1}{\ell}}{h}} \cdot \color{blue}{\left(-d\right)} \]

   13. Simplified 36.5%

       \[\leadsto \color{blue}{\sqrt{\frac{\frac{1}{\ell}}{h}} \cdot \left(-d\right)} \]

   if -5.59999999999999972e-215 < l

   1. Initial program 71.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 71.9%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 35.3%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. associate-/r* 37.1%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \]

   7. Simplified 37.1%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 36.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5.6 \cdot 10^{-215}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 42.7% accurate, 2.9× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{if}\;\ell \leq -5.2 \cdot 10^{-215}:\\ \;\;\;\;d \cdot \left(-t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot t\_0\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (let* ((t_0 (sqrt (/ (/ 1.0 h) l))))
   (if (<= l -5.2e-215) (* d (- t_0)) (* d t_0))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double t_0 = sqrt(((1.0 / h) / l));
	double tmp;
	if (l <= -5.2e-215) {
		tmp = d * -t_0;
	} else {
		tmp = d * t_0;
	}
	return tmp;
}

Fortran

D_m = abs(d)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_m)
    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_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(((1.0d0 / h) / l))
    if (l <= (-5.2d-215)) then
        tmp = d * -t_0
    else
        tmp = d * t_0
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert d < h && h < l && l < M && M < D_m;
public static double code(double d, double h, double l, double M, double D_m) {
	double t_0 = Math.sqrt(((1.0 / h) / l));
	double tmp;
	if (l <= -5.2e-215) {
		tmp = d * -t_0;
	} else {
		tmp = d * t_0;
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[d, h, l, M, D_m] = sort([d, h, l, M, D_m])
def code(d, h, l, M, D_m):
	t_0 = math.sqrt(((1.0 / h) / l))
	tmp = 0
	if l <= -5.2e-215:
		tmp = d * -t_0
	else:
		tmp = d * t_0
	return tmp

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	t_0 = sqrt(Float64(Float64(1.0 / h) / l))
	tmp = 0.0
	if (l <= -5.2e-215)
		tmp = Float64(d * Float64(-t_0));
	else
		tmp = Float64(d * t_0);
	end
	return tmp
end

MATLAB

D_m = abs(D);
d, h, l, M, D_m = num2cell(sort([d, h, l, M, D_m])){:}
function tmp_2 = code(d, h, l, M, D_m)
	t_0 = sqrt(((1.0 / h) / l));
	tmp = 0.0;
	if (l <= -5.2e-215)
		tmp = d * -t_0;
	else
		tmp = d * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := Block[{t$95$0 = N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -5.2e-215], N[(d * (-t$95$0)), $MachinePrecision], N[(d * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < -5.2e-215

   1. Initial program 62.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 62.9%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in l around -inf 0.0%

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. *-commutative 0.0%

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]

      2. associate-/r* 0.0%

         \[\leadsto \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

      3. *-commutative 0.0%

         \[\leadsto \sqrt{\frac{\frac{1}{h}}{\ell}} \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot d\right)} \]

      4. unpow2 0.0%

         \[\leadsto \sqrt{\frac{\frac{1}{h}}{\ell}} \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right) \]

      5. rem-square-sqrt 36.5%

         \[\leadsto \sqrt{\frac{\frac{1}{h}}{\ell}} \cdot \left(\color{blue}{-1} \cdot d\right) \]

      6. neg-mul-1 36.5%

         \[\leadsto \sqrt{\frac{\frac{1}{h}}{\ell}} \cdot \color{blue}{\left(-d\right)} \]

   7. Simplified 36.5%

      \[\leadsto \color{blue}{\sqrt{\frac{\frac{1}{h}}{\ell}} \cdot \left(-d\right)} \]

   if -5.2e-215 < l

   1. Initial program 71.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 71.9%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 35.3%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. associate-/r* 37.1%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \]

   7. Simplified 37.1%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 36.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5.2 \cdot 10^{-215}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{h}}{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 42.5% accurate, 3.0× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -2.8 \cdot 10^{-215}:\\ \;\;\;\;\left(-d\right) \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (if (<= l -2.8e-215)
   (* (- d) (pow (* l h) -0.5))
   (* d (sqrt (/ (/ 1.0 h) l)))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double tmp;
	if (l <= -2.8e-215) {
		tmp = -d * pow((l * h), -0.5);
	} else {
		tmp = d * sqrt(((1.0 / h) / l));
	}
	return tmp;
}

Fortran

D_m = abs(d)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_m)
    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_m
    real(8) :: tmp
    if (l <= (-2.8d-215)) then
        tmp = -d * ((l * h) ** (-0.5d0))
    else
        tmp = d * sqrt(((1.0d0 / h) / l))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert d < h && h < l && l < M && M < D_m;
public static double code(double d, double h, double l, double M, double D_m) {
	double tmp;
	if (l <= -2.8e-215) {
		tmp = -d * Math.pow((l * h), -0.5);
	} else {
		tmp = d * Math.sqrt(((1.0 / h) / l));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[d, h, l, M, D_m] = sort([d, h, l, M, D_m])
def code(d, h, l, M, D_m):
	tmp = 0
	if l <= -2.8e-215:
		tmp = -d * math.pow((l * h), -0.5)
	else:
		tmp = d * math.sqrt(((1.0 / h) / l))
	return tmp

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	tmp = 0.0
	if (l <= -2.8e-215)
		tmp = Float64(Float64(-d) * (Float64(l * h) ^ -0.5));
	else
		tmp = Float64(d * sqrt(Float64(Float64(1.0 / h) / l)));
	end
	return tmp
end

MATLAB

D_m = abs(D);
d, h, l, M, D_m = num2cell(sort([d, h, l, M, D_m])){:}
function tmp_2 = code(d, h, l, M, D_m)
	tmp = 0.0;
	if (l <= -2.8e-215)
		tmp = -d * ((l * h) ^ -0.5);
	else
		tmp = d * sqrt(((1.0 / h) / l));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := If[LessEqual[l, -2.8e-215], N[((-d) * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], N[(d * N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -2.79999999999999986e-215

   1. Initial program 62.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 62.9%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 6.4%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. *-commutative 6.4%

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

   7. Simplified 6.4%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

   8. Taylor expanded in l around -inf 0.0%

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   9. Step-by-step derivation

      1. associate-*l* 0.0%

         \[\leadsto \color{blue}{d \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. unpow2 0.0%

         \[\leadsto d \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      3. rem-square-sqrt 35.7%

         \[\leadsto d \cdot \left(\color{blue}{-1} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      4. associate-*r* 35.7%

         \[\leadsto \color{blue}{\left(d \cdot -1\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      5. *-commutative 35.7%

         \[\leadsto \color{blue}{\left(-1 \cdot d\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      6. neg-mul-1 35.7%

         \[\leadsto \color{blue}{\left(-d\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      7. distribute-lft-neg-in 35.7%

         \[\leadsto \color{blue}{-d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      8. distribute-rgt-neg-in 35.7%

         \[\leadsto \color{blue}{d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      9. *-commutative 35.7%

         \[\leadsto d \cdot \left(-\sqrt{\frac{1}{\color{blue}{\ell \cdot h}}}\right) \]

      10. unpow-1 35.7%

          \[\leadsto d \cdot \left(-\sqrt{\color{blue}{{\left(\ell \cdot h\right)}^{-1}}}\right) \]

      11. metadata-eval 35.7%

          \[\leadsto d \cdot \left(-\sqrt{{\left(\ell \cdot h\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}}\right) \]

      12. pow-sqr 35.7%

          \[\leadsto d \cdot \left(-\sqrt{\color{blue}{{\left(\ell \cdot h\right)}^{-0.5} \cdot {\left(\ell \cdot h\right)}^{-0.5}}}\right) \]

      13. rem-sqrt-square 35.7%

          \[\leadsto d \cdot \left(-\color{blue}{\left|{\left(\ell \cdot h\right)}^{-0.5}\right|}\right) \]

      14. rem-square-sqrt 35.6%

          \[\leadsto d \cdot \left(-\left|\color{blue}{\sqrt{{\left(\ell \cdot h\right)}^{-0.5}} \cdot \sqrt{{\left(\ell \cdot h\right)}^{-0.5}}}\right|\right) \]

      15. fabs-sqr 35.6%

          \[\leadsto d \cdot \left(-\color{blue}{\sqrt{{\left(\ell \cdot h\right)}^{-0.5}} \cdot \sqrt{{\left(\ell \cdot h\right)}^{-0.5}}}\right) \]

      16. rem-square-sqrt 35.7%

          \[\leadsto d \cdot \left(-\color{blue}{{\left(\ell \cdot h\right)}^{-0.5}}\right) \]

   10. Simplified 35.7%

       \[\leadsto \color{blue}{d \cdot \left(-{\left(\ell \cdot h\right)}^{-0.5}\right)} \]

   if -2.79999999999999986e-215 < l

   1. Initial program 71.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 71.9%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 35.3%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. associate-/r* 37.1%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \]

   7. Simplified 37.1%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 36.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.8 \cdot 10^{-215}:\\ \;\;\;\;\left(-d\right) \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 42.5% accurate, 3.0× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -5.2 \cdot 10^{-215}:\\ \;\;\;\;\left(-d\right) \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (if (<= l -5.2e-215) (* (- d) (pow (* l h) -0.5)) (/ d (sqrt (* l h)))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double tmp;
	if (l <= -5.2e-215) {
		tmp = -d * pow((l * h), -0.5);
	} else {
		tmp = d / sqrt((l * h));
	}
	return tmp;
}

Fortran

D_m = abs(d)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_m)
    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_m
    real(8) :: tmp
    if (l <= (-5.2d-215)) then
        tmp = -d * ((l * h) ** (-0.5d0))
    else
        tmp = d / sqrt((l * h))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert d < h && h < l && l < M && M < D_m;
public static double code(double d, double h, double l, double M, double D_m) {
	double tmp;
	if (l <= -5.2e-215) {
		tmp = -d * Math.pow((l * h), -0.5);
	} else {
		tmp = d / Math.sqrt((l * h));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[d, h, l, M, D_m] = sort([d, h, l, M, D_m])
def code(d, h, l, M, D_m):
	tmp = 0
	if l <= -5.2e-215:
		tmp = -d * math.pow((l * h), -0.5)
	else:
		tmp = d / math.sqrt((l * h))
	return tmp

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	tmp = 0.0
	if (l <= -5.2e-215)
		tmp = Float64(Float64(-d) * (Float64(l * h) ^ -0.5));
	else
		tmp = Float64(d / sqrt(Float64(l * h)));
	end
	return tmp
end

MATLAB

D_m = abs(D);
d, h, l, M, D_m = num2cell(sort([d, h, l, M, D_m])){:}
function tmp_2 = code(d, h, l, M, D_m)
	tmp = 0.0;
	if (l <= -5.2e-215)
		tmp = -d * ((l * h) ^ -0.5);
	else
		tmp = d / sqrt((l * h));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := If[LessEqual[l, -5.2e-215], N[((-d) * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -5.2e-215

   1. Initial program 62.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 62.9%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 6.4%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. *-commutative 6.4%

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

   7. Simplified 6.4%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

   8. Taylor expanded in l around -inf 0.0%

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   9. Step-by-step derivation

      1. associate-*l* 0.0%

         \[\leadsto \color{blue}{d \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. unpow2 0.0%

         \[\leadsto d \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      3. rem-square-sqrt 35.7%

         \[\leadsto d \cdot \left(\color{blue}{-1} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      4. associate-*r* 35.7%

         \[\leadsto \color{blue}{\left(d \cdot -1\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      5. *-commutative 35.7%

         \[\leadsto \color{blue}{\left(-1 \cdot d\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      6. neg-mul-1 35.7%

         \[\leadsto \color{blue}{\left(-d\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      7. distribute-lft-neg-in 35.7%

         \[\leadsto \color{blue}{-d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      8. distribute-rgt-neg-in 35.7%

         \[\leadsto \color{blue}{d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      9. *-commutative 35.7%

         \[\leadsto d \cdot \left(-\sqrt{\frac{1}{\color{blue}{\ell \cdot h}}}\right) \]

      10. unpow-1 35.7%

          \[\leadsto d \cdot \left(-\sqrt{\color{blue}{{\left(\ell \cdot h\right)}^{-1}}}\right) \]

      11. metadata-eval 35.7%

          \[\leadsto d \cdot \left(-\sqrt{{\left(\ell \cdot h\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}}\right) \]

      12. pow-sqr 35.7%

          \[\leadsto d \cdot \left(-\sqrt{\color{blue}{{\left(\ell \cdot h\right)}^{-0.5} \cdot {\left(\ell \cdot h\right)}^{-0.5}}}\right) \]

      13. rem-sqrt-square 35.7%

          \[\leadsto d \cdot \left(-\color{blue}{\left|{\left(\ell \cdot h\right)}^{-0.5}\right|}\right) \]

      14. rem-square-sqrt 35.6%

          \[\leadsto d \cdot \left(-\left|\color{blue}{\sqrt{{\left(\ell \cdot h\right)}^{-0.5}} \cdot \sqrt{{\left(\ell \cdot h\right)}^{-0.5}}}\right|\right) \]

      15. fabs-sqr 35.6%

          \[\leadsto d \cdot \left(-\color{blue}{\sqrt{{\left(\ell \cdot h\right)}^{-0.5}} \cdot \sqrt{{\left(\ell \cdot h\right)}^{-0.5}}}\right) \]

      16. rem-square-sqrt 35.7%

          \[\leadsto d \cdot \left(-\color{blue}{{\left(\ell \cdot h\right)}^{-0.5}}\right) \]

   10. Simplified 35.7%

       \[\leadsto \color{blue}{d \cdot \left(-{\left(\ell \cdot h\right)}^{-0.5}\right)} \]

   if -5.2e-215 < l

   1. Initial program 71.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 71.9%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in M around 0 32.6%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{1}\right) \]
   6. Step-by-step derivation

      1. sqrt-div 36.8%

         \[\leadsto \color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot 1\right) \]

      2. *-rgt-identity 36.8%

         \[\leadsto \frac{\sqrt{d}}{\sqrt{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]

      3. sqrt-div 39.9%

         \[\leadsto \frac{\sqrt{d}}{\sqrt{h}} \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}} \]

      4. frac-times 39.8%

         \[\leadsto \color{blue}{\frac{\sqrt{d} \cdot \sqrt{d}}{\sqrt{h} \cdot \sqrt{\ell}}} \]

      5. add-sqr-sqrt 40.0%

         \[\leadsto \frac{\color{blue}{d}}{\sqrt{h} \cdot \sqrt{\ell}} \]

      6. sqrt-prod 35.4%

         \[\leadsto \frac{d}{\color{blue}{\sqrt{h \cdot \ell}}} \]

   7. Applied egg-rr 35.4%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 35.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5.2 \cdot 10^{-215}:\\ \;\;\;\;\left(-d\right) \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 26.7% accurate, 3.2× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \frac{d}{\sqrt{\ell \cdot h}} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m) :precision binary64 (/ d (sqrt (* l h))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	return d / sqrt((l * h));
}

Fortran

D_m = abs(d)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_m)
    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_m
    code = d / sqrt((l * h))
end function

Java

D_m = Math.abs(D);
assert d < h && h < l && l < M && M < D_m;
public static double code(double d, double h, double l, double M, double D_m) {
	return d / Math.sqrt((l * h));
}

Python

D_m = math.fabs(D)
[d, h, l, M, D_m] = sort([d, h, l, M, D_m])
def code(d, h, l, M, D_m):
	return d / math.sqrt((l * h))

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	return Float64(d / sqrt(Float64(l * h)))
end

MATLAB

D_m = abs(D);
d, h, l, M, D_m = num2cell(sort([d, h, l, M, D_m])){:}
function tmp = code(d, h, l, M, D_m)
	tmp = d / sqrt((l * h));
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 67.3%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

3. Simplified 68.1%

   \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in M around 0 33.1%

   \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{1}\right) \]
6. Step-by-step derivation

   1. sqrt-div 21.0%

      \[\leadsto \color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot 1\right) \]

   2. *-rgt-identity 21.0%

      \[\leadsto \frac{\sqrt{d}}{\sqrt{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]

   3. sqrt-div 22.7%

      \[\leadsto \frac{\sqrt{d}}{\sqrt{h}} \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}} \]

   4. frac-times 22.7%

      \[\leadsto \color{blue}{\frac{\sqrt{d} \cdot \sqrt{d}}{\sqrt{h} \cdot \sqrt{\ell}}} \]

   5. add-sqr-sqrt 22.8%

      \[\leadsto \frac{\color{blue}{d}}{\sqrt{h} \cdot \sqrt{\ell}} \]

   6. sqrt-prod 22.9%

      \[\leadsto \frac{d}{\color{blue}{\sqrt{h \cdot \ell}}} \]

7. Applied egg-rr 22.9%

   \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]

8. Final simplification 22.9%

   \[\leadsto \frac{d}{\sqrt{\ell \cdot h}} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024131 
(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)))))