Henrywood and Agarwal, Equation (12)

Percentage Accurate: 66.4% → 81.9%

Time: 40.3s

Alternatives: 23

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 66.4% 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: 81.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := 1 + \frac{h \cdot \left({\left(\left(M \cdot 0.5\right) \cdot \frac{D_m}{d}\right)}^{2} \cdot -0.5\right)}{\ell}\\ \mathbf{if}\;d \leq -9.6 \cdot 10^{-300}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(t_0 \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right)\\ \end{array} \end{array} \]

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 (/ (* h (* (pow (* (* M 0.5) (/ D_m d)) 2.0) -0.5)) l))))
   (if (<= d -9.6e-300)
     (* (/ (sqrt (- d)) (sqrt (- h))) (* (sqrt (/ d l)) t_0))
     (* (/ (sqrt d) (sqrt h)) (* t_0 (/ (sqrt d) (sqrt 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 t_0 = 1.0 + ((h * (pow(((M * 0.5) * (D_m / d)), 2.0) * -0.5)) / l);
	double tmp;
	if (d <= -9.6e-300) {
		tmp = (sqrt(-d) / sqrt(-h)) * (sqrt((d / l)) * t_0);
	} else {
		tmp = (sqrt(d) / sqrt(h)) * (t_0 * (sqrt(d) / sqrt(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) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + ((h * ((((m * 0.5d0) * (d_m / d)) ** 2.0d0) * (-0.5d0))) / l)
    if (d <= (-9.6d-300)) then
        tmp = (sqrt(-d) / sqrt(-h)) * (sqrt((d / l)) * t_0)
    else
        tmp = (sqrt(d) / sqrt(h)) * (t_0 * (sqrt(d) / sqrt(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 t_0 = 1.0 + ((h * (Math.pow(((M * 0.5) * (D_m / d)), 2.0) * -0.5)) / l);
	double tmp;
	if (d <= -9.6e-300) {
		tmp = (Math.sqrt(-d) / Math.sqrt(-h)) * (Math.sqrt((d / l)) * t_0);
	} else {
		tmp = (Math.sqrt(d) / Math.sqrt(h)) * (t_0 * (Math.sqrt(d) / Math.sqrt(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):
	t_0 = 1.0 + ((h * (math.pow(((M * 0.5) * (D_m / d)), 2.0) * -0.5)) / l)
	tmp = 0
	if d <= -9.6e-300:
		tmp = (math.sqrt(-d) / math.sqrt(-h)) * (math.sqrt((d / l)) * t_0)
	else:
		tmp = (math.sqrt(d) / math.sqrt(h)) * (t_0 * (math.sqrt(d) / math.sqrt(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)
	t_0 = Float64(1.0 + Float64(Float64(h * Float64((Float64(Float64(M * 0.5) * Float64(D_m / d)) ^ 2.0) * -0.5)) / l))
	tmp = 0.0
	if (d <= -9.6e-300)
		tmp = Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-h))) * Float64(sqrt(Float64(d / l)) * t_0));
	else
		tmp = Float64(Float64(sqrt(d) / sqrt(h)) * Float64(t_0 * Float64(sqrt(d) / sqrt(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)
	t_0 = 1.0 + ((h * ((((M * 0.5) * (D_m / d)) ^ 2.0) * -0.5)) / l);
	tmp = 0.0;
	if (d <= -9.6e-300)
		tmp = (sqrt(-d) / sqrt(-h)) * (sqrt((d / l)) * t_0);
	else
		tmp = (sqrt(d) / sqrt(h)) * (t_0 * (sqrt(d) / sqrt(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_] := Block[{t$95$0 = N[(1.0 + N[(N[(h * N[(N[Power[N[(N[(M * 0.5), $MachinePrecision] * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -9.6e-300], N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if d < -9.59999999999999998e-300

   1. Initial program 65.3%

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

   3. Simplified 65.2%

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

      1. associate-*l/ 67.9%

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

      2. add-sqr-sqrt 67.9%

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

      3. pow2 67.9%

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

      4. unpow2 67.9%

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

      5. sqrt-prod 46.3%

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

      6. add-sqr-sqrt 67.9%

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

      7. div-inv 67.9%

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

      8. metadata-eval 67.9%

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

   6. Applied egg-rr 67.9%

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

      1. frac-2neg 67.9%

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

      2. sqrt-div 80.8%

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

   8. Applied egg-rr 80.8%

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

   if -9.59999999999999998e-300 < d

   1. Initial program 68.2%

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

   3. Simplified 65.9%

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

      1. associate-*l/ 67.7%

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

      2. add-sqr-sqrt 67.7%

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

      3. pow2 67.7%

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

      4. unpow2 67.7%

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

      5. sqrt-prod 40.0%

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

      6. add-sqr-sqrt 67.7%

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

      7. div-inv 67.7%

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

      8. metadata-eval 67.7%

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

   6. Applied egg-rr 67.7%

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

      1. sqrt-div 71.4%

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

   8. Applied egg-rr 71.4%

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

      1. sqrt-div 88.3%

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

   10. Applied egg-rr 88.3%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -9.6 \cdot 10^{-300}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \frac{h \cdot \left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot -0.5\right)}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\left(1 + \frac{h \cdot \left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot -0.5\right)}{\ell}\right) \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 77.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{M \cdot D_m}{d \cdot 2}\right)}^{2}\right)\right) \leq 10^{+242}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{D_m \cdot \left(0.5 \cdot \frac{M}{d}\right)}{\sqrt{\frac{\ell}{h}}}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \end{array} \end{array} \]

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 (/ (* M D_m) (* d 2.0)) 2.0)))))
      1e+242)
   (*
    (* (sqrt (/ d l)) (sqrt (/ d h)))
    (- 1.0 (* 0.5 (pow (/ (* D_m (* 0.5 (/ M d))) (sqrt (/ l h))) 2.0))))
   (fabs (/ d (sqrt (* 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 (((pow((d / h), 0.5) * pow((d / l), 0.5)) * (1.0 - ((h / l) * (0.5 * pow(((M * D_m) / (d * 2.0)), 2.0))))) <= 1e+242) {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - (0.5 * pow(((D_m * (0.5 * (M / d))) / sqrt((l / h))), 2.0)));
	} else {
		tmp = fabs((d / sqrt((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 (((((d / h) ** 0.5d0) * ((d / l) ** 0.5d0)) * (1.0d0 - ((h / l) * (0.5d0 * (((m * d_m) / (d * 2.0d0)) ** 2.0d0))))) <= 1d+242) then
        tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0d0 - (0.5d0 * (((d_m * (0.5d0 * (m / d))) / sqrt((l / h))) ** 2.0d0)))
    else
        tmp = abs((d / sqrt((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 (((Math.pow((d / h), 0.5) * Math.pow((d / l), 0.5)) * (1.0 - ((h / l) * (0.5 * Math.pow(((M * D_m) / (d * 2.0)), 2.0))))) <= 1e+242) {
		tmp = (Math.sqrt((d / l)) * Math.sqrt((d / h))) * (1.0 - (0.5 * Math.pow(((D_m * (0.5 * (M / d))) / Math.sqrt((l / h))), 2.0)));
	} else {
		tmp = Math.abs((d / Math.sqrt((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 ((math.pow((d / h), 0.5) * math.pow((d / l), 0.5)) * (1.0 - ((h / l) * (0.5 * math.pow(((M * D_m) / (d * 2.0)), 2.0))))) <= 1e+242:
		tmp = (math.sqrt((d / l)) * math.sqrt((d / h))) * (1.0 - (0.5 * math.pow(((D_m * (0.5 * (M / d))) / math.sqrt((l / h))), 2.0)))
	else:
		tmp = math.fabs((d / math.sqrt((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 (Float64(Float64((Float64(d / h) ^ 0.5) * (Float64(d / l) ^ 0.5)) * Float64(1.0 - Float64(Float64(h / l) * Float64(0.5 * (Float64(Float64(M * D_m) / Float64(d * 2.0)) ^ 2.0))))) <= 1e+242)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * Float64(1.0 - Float64(0.5 * (Float64(Float64(D_m * Float64(0.5 * Float64(M / d))) / sqrt(Float64(l / h))) ^ 2.0))));
	else
		tmp = abs(Float64(d / sqrt(Float64(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 (((((d / h) ^ 0.5) * ((d / l) ^ 0.5)) * (1.0 - ((h / l) * (0.5 * (((M * D_m) / (d * 2.0)) ^ 2.0))))) <= 1e+242)
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - (0.5 * (((D_m * (0.5 * (M / d))) / sqrt((l / h))) ^ 2.0)));
	else
		tmp = abs((d / sqrt((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[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[(M * D$95$m), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+242], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[Power[N[(N[(D$95$m * N[(0.5 * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(l / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Abs[N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (-.f64 1 (*.f64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)) (/.f64 h l)))) < 1.00000000000000005e242

   1. Initial program 90.4%

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

   3. Simplified 89.7%

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

      1. *-commutative 89.7%

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

      2. clear-num 89.7%

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

      3. frac-times 90.3%

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

      4. *-un-lft-identity 90.3%

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

   6. Applied egg-rr 90.3%

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

      1. associate-*r/ 89.2%

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

      2. associate-*l/ 89.2%

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

   8. Applied egg-rr 89.2%

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

      1. associate-/l* 90.7%

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

      2. add-sqr-sqrt 90.7%

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

   10. Applied egg-rr 90.1%

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

       1. unpow2 90.1%

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

       2. times-frac 91.3%

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

       3. *-commutative 91.3%

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

       4. associate-/l* 90.7%

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

       5. *-rgt-identity 90.7%

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

       6. *-commutative 90.7%

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

       7. associate-*r/ 90.7%

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

       8. *-commutative 90.7%

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

       9. times-frac 90.7%

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

       10. metadata-eval 90.7%

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

       11. *-commutative 90.7%

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

       12. associate-/r/ 90.6%

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

       13. associate-*l* 90.6%

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

       14. *-commutative 90.6%

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

   12. Simplified 90.6%

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

   if 1.00000000000000005e242 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (-.f64 1 (*.f64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)) (/.f64 h l))))

   1. Initial program 23.1%

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

   3. Simplified 23.1%

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

      1. sqrt-div 18.1%

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

   6. Applied egg-rr 14.3%

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

   7. Taylor expanded in d around inf 33.5%

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

      1. unpow-1 33.5%

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

      2. metadata-eval 33.5%

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

      3. pow-sqr 33.6%

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

      4. rem-sqrt-square 33.6%

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

      5. rem-square-sqrt 33.4%

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

      6. fabs-sqr 33.4%

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

      7. rem-square-sqrt 33.6%

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

   9. Simplified 33.6%

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

       1. add-sqr-sqrt 32.8%

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

       2. sqrt-prod 35.4%

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

       3. rem-sqrt-square 62.8%

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

       4. metadata-eval 62.8%

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

       5. pow-flip 62.8%

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

       6. pow1/2 62.8%

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

       7. div-inv 62.8%

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

   11. Applied egg-rr 62.8%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right)\right) \leq 10^{+242}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{D \cdot \left(0.5 \cdot \frac{M}{d}\right)}{\sqrt{\frac{\ell}{h}}}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{M \cdot D_m}{d \cdot 2}\right)}^{2}\right)\right)\\ \mathbf{if}\;t_0 \leq 10^{+242}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \end{array} \end{array} \]

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
         (*
          (* (pow (/ d h) 0.5) (pow (/ d l) 0.5))
          (- 1.0 (* (/ h l) (* 0.5 (pow (/ (* M D_m) (* d 2.0)) 2.0)))))))
   (if (<= t_0 1e+242) t_0 (fabs (/ d (sqrt (* 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 t_0 = (pow((d / h), 0.5) * pow((d / l), 0.5)) * (1.0 - ((h / l) * (0.5 * pow(((M * D_m) / (d * 2.0)), 2.0))));
	double tmp;
	if (t_0 <= 1e+242) {
		tmp = t_0;
	} else {
		tmp = fabs((d / sqrt((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) :: t_0
    real(8) :: tmp
    t_0 = (((d / h) ** 0.5d0) * ((d / l) ** 0.5d0)) * (1.0d0 - ((h / l) * (0.5d0 * (((m * d_m) / (d * 2.0d0)) ** 2.0d0))))
    if (t_0 <= 1d+242) then
        tmp = t_0
    else
        tmp = abs((d / sqrt((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 t_0 = (Math.pow((d / h), 0.5) * Math.pow((d / l), 0.5)) * (1.0 - ((h / l) * (0.5 * Math.pow(((M * D_m) / (d * 2.0)), 2.0))));
	double tmp;
	if (t_0 <= 1e+242) {
		tmp = t_0;
	} else {
		tmp = Math.abs((d / Math.sqrt((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):
	t_0 = (math.pow((d / h), 0.5) * math.pow((d / l), 0.5)) * (1.0 - ((h / l) * (0.5 * math.pow(((M * D_m) / (d * 2.0)), 2.0))))
	tmp = 0
	if t_0 <= 1e+242:
		tmp = t_0
	else:
		tmp = math.fabs((d / math.sqrt((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)
	t_0 = Float64(Float64((Float64(d / h) ^ 0.5) * (Float64(d / l) ^ 0.5)) * Float64(1.0 - Float64(Float64(h / l) * Float64(0.5 * (Float64(Float64(M * D_m) / Float64(d * 2.0)) ^ 2.0)))))
	tmp = 0.0
	if (t_0 <= 1e+242)
		tmp = t_0;
	else
		tmp = abs(Float64(d / sqrt(Float64(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)
	t_0 = (((d / h) ^ 0.5) * ((d / l) ^ 0.5)) * (1.0 - ((h / l) * (0.5 * (((M * D_m) / (d * 2.0)) ^ 2.0))));
	tmp = 0.0;
	if (t_0 <= 1e+242)
		tmp = t_0;
	else
		tmp = abs((d / sqrt((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_] := Block[{t$95$0 = N[(N[(N[Power[N[(d / h), $MachinePrecision], 0.5], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(0.5 * N[Power[N[(N[(M * D$95$m), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 1e+242], t$95$0, N[Abs[N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (-.f64 1 (*.f64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)) (/.f64 h l)))) < 1.00000000000000005e242

   1. Initial program 90.4%

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

   if 1.00000000000000005e242 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (-.f64 1 (*.f64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)) (/.f64 h l))))

   1. Initial program 23.1%

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

   3. Simplified 23.1%

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

      1. sqrt-div 18.1%

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

   6. Applied egg-rr 14.3%

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

   7. Taylor expanded in d around inf 33.5%

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

      1. unpow-1 33.5%

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

      2. metadata-eval 33.5%

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

      3. pow-sqr 33.6%

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

      4. rem-sqrt-square 33.6%

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

      5. rem-square-sqrt 33.4%

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

      6. fabs-sqr 33.4%

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

      7. rem-square-sqrt 33.6%

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

   9. Simplified 33.6%

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

       1. add-sqr-sqrt 32.8%

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

       2. sqrt-prod 35.4%

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

       3. rem-sqrt-square 62.8%

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

       4. metadata-eval 62.8%

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

       5. pow-flip 62.8%

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

       6. pow1/2 62.8%

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

       7. div-inv 62.8%

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

   11. Applied egg-rr 62.8%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right)\right) \leq 10^{+242}:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -9.6 \cdot 10^{-300}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \frac{h \cdot \left({\left(\left(M \cdot 0.5\right) \cdot \frac{D_m}{d}\right)}^{2} \cdot -0.5\right)}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{D_m}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right)\right)\\ \end{array} \end{array} \]

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 -9.6e-300)
   (*
    (/ (sqrt (- d)) (sqrt (- h)))
    (*
     (sqrt (/ d l))
     (+ 1.0 (/ (* h (* (pow (* (* M 0.5) (/ D_m d)) 2.0) -0.5)) l))))
   (*
    (/ (sqrt d) (sqrt h))
    (*
     (/ (sqrt d) (sqrt l))
     (+ 1.0 (* (/ h l) (* -0.5 (pow (* (/ D_m d) (/ M 2.0)) 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 <= -9.6e-300) {
		tmp = (sqrt(-d) / sqrt(-h)) * (sqrt((d / l)) * (1.0 + ((h * (pow(((M * 0.5) * (D_m / d)), 2.0) * -0.5)) / l)));
	} else {
		tmp = (sqrt(d) / sqrt(h)) * ((sqrt(d) / sqrt(l)) * (1.0 + ((h / l) * (-0.5 * pow(((D_m / d) * (M / 2.0)), 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 <= (-9.6d-300)) then
        tmp = (sqrt(-d) / sqrt(-h)) * (sqrt((d / l)) * (1.0d0 + ((h * ((((m * 0.5d0) * (d_m / d)) ** 2.0d0) * (-0.5d0))) / l)))
    else
        tmp = (sqrt(d) / sqrt(h)) * ((sqrt(d) / sqrt(l)) * (1.0d0 + ((h / l) * ((-0.5d0) * (((d_m / d) * (m / 2.0d0)) ** 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 <= -9.6e-300) {
		tmp = (Math.sqrt(-d) / Math.sqrt(-h)) * (Math.sqrt((d / l)) * (1.0 + ((h * (Math.pow(((M * 0.5) * (D_m / d)), 2.0) * -0.5)) / l)));
	} else {
		tmp = (Math.sqrt(d) / Math.sqrt(h)) * ((Math.sqrt(d) / Math.sqrt(l)) * (1.0 + ((h / l) * (-0.5 * Math.pow(((D_m / d) * (M / 2.0)), 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 <= -9.6e-300:
		tmp = (math.sqrt(-d) / math.sqrt(-h)) * (math.sqrt((d / l)) * (1.0 + ((h * (math.pow(((M * 0.5) * (D_m / d)), 2.0) * -0.5)) / l)))
	else:
		tmp = (math.sqrt(d) / math.sqrt(h)) * ((math.sqrt(d) / math.sqrt(l)) * (1.0 + ((h / l) * (-0.5 * math.pow(((D_m / d) * (M / 2.0)), 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 <= -9.6e-300)
		tmp = Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-h))) * Float64(sqrt(Float64(d / l)) * Float64(1.0 + Float64(Float64(h * Float64((Float64(Float64(M * 0.5) * Float64(D_m / d)) ^ 2.0) * -0.5)) / l))));
	else
		tmp = Float64(Float64(sqrt(d) / sqrt(h)) * Float64(Float64(sqrt(d) / sqrt(l)) * Float64(1.0 + Float64(Float64(h / l) * Float64(-0.5 * (Float64(Float64(D_m / d) * Float64(M / 2.0)) ^ 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 <= -9.6e-300)
		tmp = (sqrt(-d) / sqrt(-h)) * (sqrt((d / l)) * (1.0 + ((h * ((((M * 0.5) * (D_m / d)) ^ 2.0) * -0.5)) / l)));
	else
		tmp = (sqrt(d) / sqrt(h)) * ((sqrt(d) / sqrt(l)) * (1.0 + ((h / l) * (-0.5 * (((D_m / d) * (M / 2.0)) ^ 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, -9.6e-300], N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(N[(h * N[(N[Power[N[(N[(M * 0.5), $MachinePrecision] * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(-0.5 * N[Power[N[(N[(D$95$m / d), $MachinePrecision] * N[(M / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -9.59999999999999998e-300

   1. Initial program 65.3%

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

   3. Simplified 65.2%

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

      1. associate-*l/ 67.9%

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

      2. add-sqr-sqrt 67.9%

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

      3. pow2 67.9%

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

      4. unpow2 67.9%

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

      5. sqrt-prod 46.3%

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

      6. add-sqr-sqrt 67.9%

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

      7. div-inv 67.9%

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

      8. metadata-eval 67.9%

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

   6. Applied egg-rr 67.9%

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

      1. frac-2neg 67.9%

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

      2. sqrt-div 80.8%

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

   8. Applied egg-rr 80.8%

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

   if -9.59999999999999998e-300 < d

   1. Initial program 68.2%

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

   3. Simplified 65.9%

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

      1. sqrt-div 71.4%

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

   6. Applied egg-rr 68.7%

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

      1. sqrt-div 88.3%

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

   8. Applied egg-rr 84.1%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -9.6 \cdot 10^{-300}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \frac{h \cdot \left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot -0.5\right)}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := {\left(\left(M \cdot 0.5\right) \cdot \frac{D_m}{d}\right)}^{2}\\ \mathbf{if}\;d \leq -9.6 \cdot 10^{-300}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \frac{h \cdot \left(t_0 \cdot -0.5\right)}{\ell}\right)\right)\\ \mathbf{elif}\;d \leq 2.5 \cdot 10^{-235}:\\ \;\;\;\;\frac{{\left(M \cdot D_m\right)}^{2}}{d} \cdot \left({\left(\frac{\sqrt[3]{\sqrt{h}}}{\sqrt{\ell}}\right)}^{3} \cdot -0.125\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot t_0\right)\right)\\ \end{array} \end{array} \]

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 (pow (* (* M 0.5) (/ D_m d)) 2.0)))
   (if (<= d -9.6e-300)
     (*
      (/ (sqrt (- d)) (sqrt (- h)))
      (* (sqrt (/ d l)) (+ 1.0 (/ (* h (* t_0 -0.5)) l))))
     (if (<= d 2.5e-235)
       (*
        (/ (pow (* M D_m) 2.0) d)
        (* (pow (/ (cbrt (sqrt h)) (sqrt l)) 3.0) -0.125))
       (* (/ d (* (sqrt h) (sqrt l))) (- 1.0 (* (/ h l) (* 0.5 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 = pow(((M * 0.5) * (D_m / d)), 2.0);
	double tmp;
	if (d <= -9.6e-300) {
		tmp = (sqrt(-d) / sqrt(-h)) * (sqrt((d / l)) * (1.0 + ((h * (t_0 * -0.5)) / l)));
	} else if (d <= 2.5e-235) {
		tmp = (pow((M * D_m), 2.0) / d) * (pow((cbrt(sqrt(h)) / sqrt(l)), 3.0) * -0.125);
	} else {
		tmp = (d / (sqrt(h) * sqrt(l))) * (1.0 - ((h / l) * (0.5 * t_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.pow(((M * 0.5) * (D_m / d)), 2.0);
	double tmp;
	if (d <= -9.6e-300) {
		tmp = (Math.sqrt(-d) / Math.sqrt(-h)) * (Math.sqrt((d / l)) * (1.0 + ((h * (t_0 * -0.5)) / l)));
	} else if (d <= 2.5e-235) {
		tmp = (Math.pow((M * D_m), 2.0) / d) * (Math.pow((Math.cbrt(Math.sqrt(h)) / Math.sqrt(l)), 3.0) * -0.125);
	} else {
		tmp = (d / (Math.sqrt(h) * Math.sqrt(l))) * (1.0 - ((h / l) * (0.5 * 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(Float64(M * 0.5) * Float64(D_m / d)) ^ 2.0
	tmp = 0.0
	if (d <= -9.6e-300)
		tmp = Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-h))) * Float64(sqrt(Float64(d / l)) * Float64(1.0 + Float64(Float64(h * Float64(t_0 * -0.5)) / l))));
	elseif (d <= 2.5e-235)
		tmp = Float64(Float64((Float64(M * D_m) ^ 2.0) / d) * Float64((Float64(cbrt(sqrt(h)) / sqrt(l)) ^ 3.0) * -0.125));
	else
		tmp = Float64(Float64(d / Float64(sqrt(h) * sqrt(l))) * Float64(1.0 - Float64(Float64(h / l) * Float64(0.5 * t_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[Power[N[(N[(M * 0.5), $MachinePrecision] * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[d, -9.6e-300], N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(N[(h * N[(t$95$0 * -0.5), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 2.5e-235], N[(N[(N[Power[N[(M * D$95$m), $MachinePrecision], 2.0], $MachinePrecision] / d), $MachinePrecision] * N[(N[Power[N[(N[Power[N[Sqrt[h], $MachinePrecision], 1/3], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision], N[(N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(0.5 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if d < -9.59999999999999998e-300

   1. Initial program 65.3%

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

   3. Simplified 65.2%

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

      1. associate-*l/ 67.9%

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

      2. add-sqr-sqrt 67.9%

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

      3. pow2 67.9%

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

      4. unpow2 67.9%

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

      5. sqrt-prod 46.3%

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

      6. add-sqr-sqrt 67.9%

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

      7. div-inv 67.9%

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

      8. metadata-eval 67.9%

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

   6. Applied egg-rr 67.9%

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

      1. frac-2neg 67.9%

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

      2. sqrt-div 80.8%

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

   8. Applied egg-rr 80.8%

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

   if -9.59999999999999998e-300 < d < 2.4999999999999999e-235

   1. Initial program 39.6%

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

   3. Taylor expanded in d around 0 38.2%

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

      1. *-commutative 38.2%

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

      2. associate-*l* 38.2%

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

      3. *-commutative 38.2%

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

      4. unpow2 38.2%

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

      5. unpow2 38.2%

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

      6. swap-sqr 45.0%

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

      7. unpow2 45.0%

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

      8. *-commutative 45.0%

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

   5. Simplified 45.0%

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

      1. add-cube-cbrt 45.0%

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

      2. pow3 45.0%

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

      3. sqrt-div 44.3%

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

      4. cbrt-div 44.3%

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

      5. unpow3 44.3%

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

      6. sqrt-prod 56.7%

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

      7. sqrt-unprod 62.5%

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

      8. add-cbrt-cube 62.5%

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

   7. Applied egg-rr 62.5%

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

   if 2.4999999999999999e-235 < d

   1. Initial program 72.2%

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

   4. Applied egg-rr 90.3%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -9.6 \cdot 10^{-300}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \frac{h \cdot \left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot -0.5\right)}{\ell}\right)\right)\\ \mathbf{elif}\;d \leq 2.5 \cdot 10^{-235}:\\ \;\;\;\;\frac{{\left(M \cdot D\right)}^{2}}{d} \cdot \left({\left(\frac{\sqrt[3]{\sqrt{h}}}{\sqrt{\ell}}\right)}^{3} \cdot -0.125\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -9.6 \cdot 10^{-300}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{D_m}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right)\right)\\ \mathbf{elif}\;d \leq 5 \cdot 10^{-236}:\\ \;\;\;\;\frac{{\left(M \cdot D_m\right)}^{2} \cdot \left(-0.125 \cdot \frac{\sqrt{h}}{{\ell}^{1.5}}\right)}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\left(M \cdot 0.5\right) \cdot \frac{D_m}{d}\right)}^{2}\right)\right)\\ \end{array} \end{array} \]

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 -9.6e-300)
   (*
    (/ (sqrt (- d)) (sqrt (- h)))
    (*
     (sqrt (/ d l))
     (+ 1.0 (* (/ h l) (* -0.5 (pow (* (/ D_m d) (/ M 2.0)) 2.0))))))
   (if (<= d 5e-236)
     (/ (* (pow (* M D_m) 2.0) (* -0.125 (/ (sqrt h) (pow l 1.5)))) d)
     (*
      (/ d (* (sqrt h) (sqrt l)))
      (- 1.0 (* (/ h l) (* 0.5 (pow (* (* M 0.5) (/ 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 <= -9.6e-300) {
		tmp = (sqrt(-d) / sqrt(-h)) * (sqrt((d / l)) * (1.0 + ((h / l) * (-0.5 * pow(((D_m / d) * (M / 2.0)), 2.0)))));
	} else if (d <= 5e-236) {
		tmp = (pow((M * D_m), 2.0) * (-0.125 * (sqrt(h) / pow(l, 1.5)))) / d;
	} else {
		tmp = (d / (sqrt(h) * sqrt(l))) * (1.0 - ((h / l) * (0.5 * pow(((M * 0.5) * (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 <= (-9.6d-300)) then
        tmp = (sqrt(-d) / sqrt(-h)) * (sqrt((d / l)) * (1.0d0 + ((h / l) * ((-0.5d0) * (((d_m / d) * (m / 2.0d0)) ** 2.0d0)))))
    else if (d <= 5d-236) then
        tmp = (((m * d_m) ** 2.0d0) * ((-0.125d0) * (sqrt(h) / (l ** 1.5d0)))) / d
    else
        tmp = (d / (sqrt(h) * sqrt(l))) * (1.0d0 - ((h / l) * (0.5d0 * (((m * 0.5d0) * (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 <= -9.6e-300) {
		tmp = (Math.sqrt(-d) / Math.sqrt(-h)) * (Math.sqrt((d / l)) * (1.0 + ((h / l) * (-0.5 * Math.pow(((D_m / d) * (M / 2.0)), 2.0)))));
	} else if (d <= 5e-236) {
		tmp = (Math.pow((M * D_m), 2.0) * (-0.125 * (Math.sqrt(h) / Math.pow(l, 1.5)))) / d;
	} else {
		tmp = (d / (Math.sqrt(h) * Math.sqrt(l))) * (1.0 - ((h / l) * (0.5 * Math.pow(((M * 0.5) * (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 <= -9.6e-300:
		tmp = (math.sqrt(-d) / math.sqrt(-h)) * (math.sqrt((d / l)) * (1.0 + ((h / l) * (-0.5 * math.pow(((D_m / d) * (M / 2.0)), 2.0)))))
	elif d <= 5e-236:
		tmp = (math.pow((M * D_m), 2.0) * (-0.125 * (math.sqrt(h) / math.pow(l, 1.5)))) / d
	else:
		tmp = (d / (math.sqrt(h) * math.sqrt(l))) * (1.0 - ((h / l) * (0.5 * math.pow(((M * 0.5) * (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 <= -9.6e-300)
		tmp = Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-h))) * Float64(sqrt(Float64(d / l)) * Float64(1.0 + Float64(Float64(h / l) * Float64(-0.5 * (Float64(Float64(D_m / d) * Float64(M / 2.0)) ^ 2.0))))));
	elseif (d <= 5e-236)
		tmp = Float64(Float64((Float64(M * D_m) ^ 2.0) * Float64(-0.125 * Float64(sqrt(h) / (l ^ 1.5)))) / d);
	else
		tmp = Float64(Float64(d / Float64(sqrt(h) * sqrt(l))) * Float64(1.0 - Float64(Float64(h / l) * Float64(0.5 * (Float64(Float64(M * 0.5) * 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 <= -9.6e-300)
		tmp = (sqrt(-d) / sqrt(-h)) * (sqrt((d / l)) * (1.0 + ((h / l) * (-0.5 * (((D_m / d) * (M / 2.0)) ^ 2.0)))));
	elseif (d <= 5e-236)
		tmp = (((M * D_m) ^ 2.0) * (-0.125 * (sqrt(h) / (l ^ 1.5)))) / d;
	else
		tmp = (d / (sqrt(h) * sqrt(l))) * (1.0 - ((h / l) * (0.5 * (((M * 0.5) * (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, -9.6e-300], N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(-0.5 * N[Power[N[(N[(D$95$m / d), $MachinePrecision] * N[(M / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 5e-236], N[(N[(N[Power[N[(M * D$95$m), $MachinePrecision], 2.0], $MachinePrecision] * N[(-0.125 * N[(N[Sqrt[h], $MachinePrecision] / N[Power[l, 1.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], N[(N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(0.5 * N[Power[N[(N[(M * 0.5), $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 < -9.59999999999999998e-300

   1. Initial program 65.3%

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

   3. Simplified 65.2%

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

      1. frac-2neg 67.9%

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

      2. sqrt-div 80.8%

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

   6. Applied egg-rr 77.4%

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

   if -9.59999999999999998e-300 < d < 4.9999999999999998e-236

   1. Initial program 39.6%

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

   3. Taylor expanded in d around 0 38.2%

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

      1. *-commutative 38.2%

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

      2. associate-*l* 38.2%

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

      3. *-commutative 38.2%

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

      4. unpow2 38.2%

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

      5. unpow2 38.2%

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

      6. swap-sqr 45.0%

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

      7. unpow2 45.0%

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

      8. *-commutative 45.0%

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

   5. Simplified 45.0%

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

      1. associate-*l/ 44.9%

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

      2. sqrt-div 44.2%

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

      3. sqrt-pow1 62.5%

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

      4. metadata-eval 62.5%

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

   7. Applied egg-rr 62.5%

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

   if 4.9999999999999998e-236 < d

   1. Initial program 72.2%

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

   4. Applied egg-rr 90.3%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -9.6 \cdot 10^{-300}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right)\right)\\ \mathbf{elif}\;d \leq 5 \cdot 10^{-236}:\\ \;\;\;\;\frac{{\left(M \cdot D\right)}^{2} \cdot \left(-0.125 \cdot \frac{\sqrt{h}}{{\ell}^{1.5}}\right)}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 79.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := {\left(\left(M \cdot 0.5\right) \cdot \frac{D_m}{d}\right)}^{2}\\ \mathbf{if}\;d \leq -9.6 \cdot 10^{-300}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \frac{h \cdot \left(t_0 \cdot -0.5\right)}{\ell}\right)\right)\\ \mathbf{elif}\;d \leq 1.35 \cdot 10^{-243}:\\ \;\;\;\;\frac{{\left(M \cdot D_m\right)}^{2} \cdot \left(-0.125 \cdot \frac{\sqrt{h}}{{\ell}^{1.5}}\right)}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot t_0\right)\right)\\ \end{array} \end{array} \]

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 (pow (* (* M 0.5) (/ D_m d)) 2.0)))
   (if (<= d -9.6e-300)
     (*
      (/ (sqrt (- d)) (sqrt (- h)))
      (* (sqrt (/ d l)) (+ 1.0 (/ (* h (* t_0 -0.5)) l))))
     (if (<= d 1.35e-243)
       (/ (* (pow (* M D_m) 2.0) (* -0.125 (/ (sqrt h) (pow l 1.5)))) d)
       (* (/ d (* (sqrt h) (sqrt l))) (- 1.0 (* (/ h l) (* 0.5 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 = pow(((M * 0.5) * (D_m / d)), 2.0);
	double tmp;
	if (d <= -9.6e-300) {
		tmp = (sqrt(-d) / sqrt(-h)) * (sqrt((d / l)) * (1.0 + ((h * (t_0 * -0.5)) / l)));
	} else if (d <= 1.35e-243) {
		tmp = (pow((M * D_m), 2.0) * (-0.125 * (sqrt(h) / pow(l, 1.5)))) / d;
	} else {
		tmp = (d / (sqrt(h) * sqrt(l))) * (1.0 - ((h / l) * (0.5 * 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 = ((m * 0.5d0) * (d_m / d)) ** 2.0d0
    if (d <= (-9.6d-300)) then
        tmp = (sqrt(-d) / sqrt(-h)) * (sqrt((d / l)) * (1.0d0 + ((h * (t_0 * (-0.5d0))) / l)))
    else if (d <= 1.35d-243) then
        tmp = (((m * d_m) ** 2.0d0) * ((-0.125d0) * (sqrt(h) / (l ** 1.5d0)))) / d
    else
        tmp = (d / (sqrt(h) * sqrt(l))) * (1.0d0 - ((h / l) * (0.5d0 * 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.pow(((M * 0.5) * (D_m / d)), 2.0);
	double tmp;
	if (d <= -9.6e-300) {
		tmp = (Math.sqrt(-d) / Math.sqrt(-h)) * (Math.sqrt((d / l)) * (1.0 + ((h * (t_0 * -0.5)) / l)));
	} else if (d <= 1.35e-243) {
		tmp = (Math.pow((M * D_m), 2.0) * (-0.125 * (Math.sqrt(h) / Math.pow(l, 1.5)))) / d;
	} else {
		tmp = (d / (Math.sqrt(h) * Math.sqrt(l))) * (1.0 - ((h / l) * (0.5 * 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.pow(((M * 0.5) * (D_m / d)), 2.0)
	tmp = 0
	if d <= -9.6e-300:
		tmp = (math.sqrt(-d) / math.sqrt(-h)) * (math.sqrt((d / l)) * (1.0 + ((h * (t_0 * -0.5)) / l)))
	elif d <= 1.35e-243:
		tmp = (math.pow((M * D_m), 2.0) * (-0.125 * (math.sqrt(h) / math.pow(l, 1.5)))) / d
	else:
		tmp = (d / (math.sqrt(h) * math.sqrt(l))) * (1.0 - ((h / l) * (0.5 * 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(Float64(M * 0.5) * Float64(D_m / d)) ^ 2.0
	tmp = 0.0
	if (d <= -9.6e-300)
		tmp = Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-h))) * Float64(sqrt(Float64(d / l)) * Float64(1.0 + Float64(Float64(h * Float64(t_0 * -0.5)) / l))));
	elseif (d <= 1.35e-243)
		tmp = Float64(Float64((Float64(M * D_m) ^ 2.0) * Float64(-0.125 * Float64(sqrt(h) / (l ^ 1.5)))) / d);
	else
		tmp = Float64(Float64(d / Float64(sqrt(h) * sqrt(l))) * Float64(1.0 - Float64(Float64(h / l) * Float64(0.5 * 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 = ((M * 0.5) * (D_m / d)) ^ 2.0;
	tmp = 0.0;
	if (d <= -9.6e-300)
		tmp = (sqrt(-d) / sqrt(-h)) * (sqrt((d / l)) * (1.0 + ((h * (t_0 * -0.5)) / l)));
	elseif (d <= 1.35e-243)
		tmp = (((M * D_m) ^ 2.0) * (-0.125 * (sqrt(h) / (l ^ 1.5)))) / d;
	else
		tmp = (d / (sqrt(h) * sqrt(l))) * (1.0 - ((h / l) * (0.5 * 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[Power[N[(N[(M * 0.5), $MachinePrecision] * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[d, -9.6e-300], N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(N[(h * N[(t$95$0 * -0.5), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.35e-243], N[(N[(N[Power[N[(M * D$95$m), $MachinePrecision], 2.0], $MachinePrecision] * N[(-0.125 * N[(N[Sqrt[h], $MachinePrecision] / N[Power[l, 1.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], N[(N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(0.5 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if d < -9.59999999999999998e-300

   1. Initial program 65.3%

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

   3. Simplified 65.2%

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

      1. associate-*l/ 67.9%

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

      2. add-sqr-sqrt 67.9%

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

      3. pow2 67.9%

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

      4. unpow2 67.9%

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

      5. sqrt-prod 46.3%

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

      6. add-sqr-sqrt 67.9%

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

      7. div-inv 67.9%

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

      8. metadata-eval 67.9%

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

   6. Applied egg-rr 67.9%

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

      1. frac-2neg 67.9%

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

      2. sqrt-div 80.8%

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

   8. Applied egg-rr 80.8%

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

   if -9.59999999999999998e-300 < d < 1.35000000000000005e-243

   1. Initial program 39.6%

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

   3. Taylor expanded in d around 0 38.2%

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

      1. *-commutative 38.2%

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

      2. associate-*l* 38.2%

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

      3. *-commutative 38.2%

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

      4. unpow2 38.2%

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

      5. unpow2 38.2%

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

      6. swap-sqr 45.0%

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

      7. unpow2 45.0%

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

      8. *-commutative 45.0%

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

   5. Simplified 45.0%

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

      1. associate-*l/ 44.9%

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

      2. sqrt-div 44.2%

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

      3. sqrt-pow1 62.5%

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

      4. metadata-eval 62.5%

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

   7. Applied egg-rr 62.5%

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

   if 1.35000000000000005e-243 < d

   1. Initial program 72.2%

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

   4. Applied egg-rr 90.3%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -9.6 \cdot 10^{-300}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \frac{h \cdot \left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot -0.5\right)}{\ell}\right)\right)\\ \mathbf{elif}\;d \leq 1.35 \cdot 10^{-243}:\\ \;\;\;\;\frac{{\left(M \cdot D\right)}^{2} \cdot \left(-0.125 \cdot \frac{\sqrt{h}}{{\ell}^{1.5}}\right)}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 62.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -2.4 \cdot 10^{-98}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \mathbf{elif}\;d \leq -2 \cdot 10^{-310}:\\ \;\;\;\;d \cdot \sqrt{\log \left(e^{\frac{1}{h \cdot \ell}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D_m \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)\right)\\ \end{array} \end{array} \]

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 -2.4e-98)
   (fabs (/ d (sqrt (* h l))))
   (if (<= d -2e-310)
     (* d (sqrt (log (exp (/ 1.0 (* h l))))))
     (*
      (/ (/ d (sqrt l)) (sqrt h))
      (- 1.0 (* 0.5 (* (/ h l) (pow (* D_m (/ M (* d 2.0))) 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 <= -2.4e-98) {
		tmp = fabs((d / sqrt((h * l))));
	} else if (d <= -2e-310) {
		tmp = d * sqrt(log(exp((1.0 / (h * l)))));
	} else {
		tmp = ((d / sqrt(l)) / sqrt(h)) * (1.0 - (0.5 * ((h / l) * pow((D_m * (M / (d * 2.0))), 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 <= (-2.4d-98)) then
        tmp = abs((d / sqrt((h * l))))
    else if (d <= (-2d-310)) then
        tmp = d * sqrt(log(exp((1.0d0 / (h * l)))))
    else
        tmp = ((d / sqrt(l)) / sqrt(h)) * (1.0d0 - (0.5d0 * ((h / l) * ((d_m * (m / (d * 2.0d0))) ** 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 <= -2.4e-98) {
		tmp = Math.abs((d / Math.sqrt((h * l))));
	} else if (d <= -2e-310) {
		tmp = d * Math.sqrt(Math.log(Math.exp((1.0 / (h * l)))));
	} else {
		tmp = ((d / Math.sqrt(l)) / Math.sqrt(h)) * (1.0 - (0.5 * ((h / l) * Math.pow((D_m * (M / (d * 2.0))), 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 <= -2.4e-98:
		tmp = math.fabs((d / math.sqrt((h * l))))
	elif d <= -2e-310:
		tmp = d * math.sqrt(math.log(math.exp((1.0 / (h * l)))))
	else:
		tmp = ((d / math.sqrt(l)) / math.sqrt(h)) * (1.0 - (0.5 * ((h / l) * math.pow((D_m * (M / (d * 2.0))), 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 <= -2.4e-98)
		tmp = abs(Float64(d / sqrt(Float64(h * l))));
	elseif (d <= -2e-310)
		tmp = Float64(d * sqrt(log(exp(Float64(1.0 / Float64(h * l))))));
	else
		tmp = Float64(Float64(Float64(d / sqrt(l)) / sqrt(h)) * Float64(1.0 - Float64(0.5 * Float64(Float64(h / l) * (Float64(D_m * Float64(M / Float64(d * 2.0))) ^ 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 <= -2.4e-98)
		tmp = abs((d / sqrt((h * l))));
	elseif (d <= -2e-310)
		tmp = d * sqrt(log(exp((1.0 / (h * l)))));
	else
		tmp = ((d / sqrt(l)) / sqrt(h)) * (1.0 - (0.5 * ((h / l) * ((D_m * (M / (d * 2.0))) ^ 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, -2.4e-98], N[Abs[N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[d, -2e-310], N[(d * N[Sqrt[N[Log[N[Exp[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(d / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(D$95$m * N[(M / N[(d * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -2.40000000000000005e-98

   1. Initial program 73.3%

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

   3. Simplified 73.3%

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

      1. sqrt-div 0.0%

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

   6. Applied egg-rr 0.0%

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

   7. Taylor expanded in d around inf 6.5%

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

      1. unpow-1 6.5%

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

      2. metadata-eval 6.5%

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

      3. pow-sqr 6.5%

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

      4. rem-sqrt-square 6.5%

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

      5. rem-square-sqrt 6.5%

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

      6. fabs-sqr 6.5%

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

      7. rem-square-sqrt 6.5%

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

   9. Simplified 6.5%

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

       1. add-sqr-sqrt 0.4%

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

       2. sqrt-prod 36.8%

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

       3. rem-sqrt-square 55.6%

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

       4. metadata-eval 55.6%

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

       5. pow-flip 55.5%

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

       6. pow1/2 55.5%

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

       7. div-inv 55.6%

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

   11. Applied egg-rr 55.6%

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

   if -2.40000000000000005e-98 < d < -1.999999999999994e-310

   1. Initial program 45.0%

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

   3. Taylor expanded in d around inf 12.5%

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

      1. add-log-exp 38.7%

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

   5. Applied egg-rr 38.7%

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

   if -1.999999999999994e-310 < d

   1. Initial program 69.2%

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

   4. Applied egg-rr 42.1%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 - \frac{h}{\ell} \cdot \left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right)\right)\right)} - 1} \]
   5. Step-by-step derivation

      1. expm1-def 55.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 - \frac{h}{\ell} \cdot \left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right)\right)\right)\right)} \]

      2. expm1-log1p 84.5%

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

      3. associate-/r* 84.4%

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

      4. associate-*r* 84.4%

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

      5. *-commutative 84.4%

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

      6. /-rgt-identity 84.4%

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

      7. associate-/l* 84.4%

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

      8. metadata-eval 84.4%

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

      9. times-frac 86.7%

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

      10. associate-*r/ 86.0%

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

      11. *-commutative 86.0%

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

   6. Simplified 86.0%

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

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.4 \cdot 10^{-98}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \mathbf{elif}\;d \leq -2 \cdot 10^{-310}:\\ \;\;\;\;d \cdot \sqrt{\log \left(e^{\frac{1}{h \cdot \ell}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 57.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := \frac{d}{\sqrt{h \cdot \ell}}\\ \mathbf{if}\;d \leq -4.9 \cdot 10^{-101}:\\ \;\;\;\;\left|t_0\right|\\ \mathbf{elif}\;d \leq -2 \cdot 10^{-310}:\\ \;\;\;\;d \cdot \sqrt{\log \left(e^{\frac{1}{h \cdot \ell}}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \mathsf{fma}\left({\left(\frac{M}{\frac{\frac{d}{D_m}}{0.5}}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\\ \end{array} \end{array} \]

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 (/ d (sqrt (* h l)))))
   (if (<= d -4.9e-101)
     (fabs t_0)
     (if (<= d -2e-310)
       (* d (sqrt (log (exp (/ 1.0 (* h l))))))
       (* t_0 (fma (pow (/ M (/ (/ d D_m) 0.5)) 2.0) (* -0.5 (/ h l)) 1.0))))))

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 = d / sqrt((h * l));
	double tmp;
	if (d <= -4.9e-101) {
		tmp = fabs(t_0);
	} else if (d <= -2e-310) {
		tmp = d * sqrt(log(exp((1.0 / (h * l)))));
	} else {
		tmp = t_0 * fma(pow((M / ((d / D_m) / 0.5)), 2.0), (-0.5 * (h / l)), 1.0);
	}
	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(d / sqrt(Float64(h * l)))
	tmp = 0.0
	if (d <= -4.9e-101)
		tmp = abs(t_0);
	elseif (d <= -2e-310)
		tmp = Float64(d * sqrt(log(exp(Float64(1.0 / Float64(h * l))))));
	else
		tmp = Float64(t_0 * fma((Float64(M / Float64(Float64(d / D_m) / 0.5)) ^ 2.0), Float64(-0.5 * Float64(h / l)), 1.0));
	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[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -4.9e-101], N[Abs[t$95$0], $MachinePrecision], If[LessEqual[d, -2e-310], N[(d * N[Sqrt[N[Log[N[Exp[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(N[Power[N[(M / N[(N[(d / D$95$m), $MachinePrecision] / 0.5), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(-0.5 * N[(h / l), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if d < -4.9e-101

   1. Initial program 73.3%

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

   3. Simplified 73.3%

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

      1. sqrt-div 0.0%

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

   6. Applied egg-rr 0.0%

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

   7. Taylor expanded in d around inf 6.5%

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

      1. unpow-1 6.5%

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

      2. metadata-eval 6.5%

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

      3. pow-sqr 6.5%

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

      4. rem-sqrt-square 6.5%

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

      5. rem-square-sqrt 6.5%

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

      6. fabs-sqr 6.5%

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

      7. rem-square-sqrt 6.5%

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

   9. Simplified 6.5%

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

       1. add-sqr-sqrt 0.4%

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

       2. sqrt-prod 36.8%

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

       3. rem-sqrt-square 55.6%

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

       4. metadata-eval 55.6%

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

       5. pow-flip 55.5%

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

       6. pow1/2 55.5%

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

       7. div-inv 55.6%

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

   11. Applied egg-rr 55.6%

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

   if -4.9e-101 < d < -1.999999999999994e-310

   1. Initial program 45.0%

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

   3. Taylor expanded in d around inf 12.5%

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

      1. add-log-exp 38.7%

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

   5. Applied egg-rr 38.7%

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

   if -1.999999999999994e-310 < d

   1. Initial program 69.2%

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

   3. Simplified 66.9%

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

      1. sqrt-div 72.5%

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

   6. Applied egg-rr 69.8%

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

      1. expm1-log1p-u 45.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h}} \cdot \left(\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot -0.5\right)\right)\right)\right)\right)} \]

      2. expm1-udef 32.2%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h}} \cdot \left(\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot -0.5\right)\right)\right)\right)} - 1} \]

   8. Applied egg-rr 39.8%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{d}{\sqrt{h \cdot \ell}} \cdot \mathsf{fma}\left(\frac{h}{\ell}, -0.5 \cdot {\left(\frac{M}{\frac{d \cdot 2}{D}}\right)}^{2}, 1\right)\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 47.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{d}{\sqrt{h \cdot \ell}} \cdot \mathsf{fma}\left(\frac{h}{\ell}, -0.5 \cdot {\left(\frac{M}{\frac{d \cdot 2}{D}}\right)}^{2}, 1\right)\right)\right)} \]

      2. expm1-log1p 74.6%

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

      3. *-commutative 74.6%

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

   10. Simplified 74.6%

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

4. Final simplification 62.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -4.9 \cdot 10^{-101}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \mathbf{elif}\;d \leq -2 \cdot 10^{-310}:\\ \;\;\;\;d \cdot \sqrt{\log \left(e^{\frac{1}{h \cdot \ell}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}} \cdot \mathsf{fma}\left({\left(\frac{M}{\frac{\frac{d}{D}}{0.5}}\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 71.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -4.6 \cdot 10^{-296}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{D_m}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right)\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D_m \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)\right)\\ \end{array} \end{array} \]

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 (<= l -4.6e-296)
   (*
    (*
     (sqrt (/ d l))
     (+ 1.0 (* (/ h l) (* -0.5 (pow (* (/ D_m d) (/ M 2.0)) 2.0)))))
    (sqrt (/ d h)))
   (*
    (/ (/ d (sqrt l)) (sqrt h))
    (- 1.0 (* 0.5 (* (/ h l) (pow (* D_m (/ M (* d 2.0))) 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 (l <= -4.6e-296) {
		tmp = (sqrt((d / l)) * (1.0 + ((h / l) * (-0.5 * pow(((D_m / d) * (M / 2.0)), 2.0))))) * sqrt((d / h));
	} else {
		tmp = ((d / sqrt(l)) / sqrt(h)) * (1.0 - (0.5 * ((h / l) * pow((D_m * (M / (d * 2.0))), 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 (l <= (-4.6d-296)) then
        tmp = (sqrt((d / l)) * (1.0d0 + ((h / l) * ((-0.5d0) * (((d_m / d) * (m / 2.0d0)) ** 2.0d0))))) * sqrt((d / h))
    else
        tmp = ((d / sqrt(l)) / sqrt(h)) * (1.0d0 - (0.5d0 * ((h / l) * ((d_m * (m / (d * 2.0d0))) ** 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 (l <= -4.6e-296) {
		tmp = (Math.sqrt((d / l)) * (1.0 + ((h / l) * (-0.5 * Math.pow(((D_m / d) * (M / 2.0)), 2.0))))) * Math.sqrt((d / h));
	} else {
		tmp = ((d / Math.sqrt(l)) / Math.sqrt(h)) * (1.0 - (0.5 * ((h / l) * Math.pow((D_m * (M / (d * 2.0))), 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 l <= -4.6e-296:
		tmp = (math.sqrt((d / l)) * (1.0 + ((h / l) * (-0.5 * math.pow(((D_m / d) * (M / 2.0)), 2.0))))) * math.sqrt((d / h))
	else:
		tmp = ((d / math.sqrt(l)) / math.sqrt(h)) * (1.0 - (0.5 * ((h / l) * math.pow((D_m * (M / (d * 2.0))), 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 (l <= -4.6e-296)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * Float64(1.0 + Float64(Float64(h / l) * Float64(-0.5 * (Float64(Float64(D_m / d) * Float64(M / 2.0)) ^ 2.0))))) * sqrt(Float64(d / h)));
	else
		tmp = Float64(Float64(Float64(d / sqrt(l)) / sqrt(h)) * Float64(1.0 - Float64(0.5 * Float64(Float64(h / l) * (Float64(D_m * Float64(M / Float64(d * 2.0))) ^ 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 (l <= -4.6e-296)
		tmp = (sqrt((d / l)) * (1.0 + ((h / l) * (-0.5 * (((D_m / d) * (M / 2.0)) ^ 2.0))))) * sqrt((d / h));
	else
		tmp = ((d / sqrt(l)) / sqrt(h)) * (1.0 - (0.5 * ((h / l) * ((D_m * (M / (d * 2.0))) ^ 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[l, -4.6e-296], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(-0.5 * N[Power[N[(N[(D$95$m / d), $MachinePrecision] * N[(M / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(d / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(D$95$m * N[(M / N[(d * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -4.60000000000000008e-296

   1. Initial program 64.8%

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

   3. Simplified 64.8%

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

   if -4.60000000000000008e-296 < l

   1. Initial program 68.6%

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

   4. Applied egg-rr 41.8%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 - \frac{h}{\ell} \cdot \left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right)\right)\right)} - 1} \]
   5. Step-by-step derivation

      1. expm1-def 54.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 - \frac{h}{\ell} \cdot \left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right)\right)\right)\right)} \]

      2. expm1-log1p 83.8%

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

      3. associate-/r* 83.8%

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

      4. associate-*r* 83.8%

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

      5. *-commutative 83.8%

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

      6. /-rgt-identity 83.8%

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

      7. associate-/l* 83.8%

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

      8. metadata-eval 83.8%

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

      9. times-frac 86.0%

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

      10. associate-*r/ 85.3%

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

      11. *-commutative 85.3%

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

   6. Simplified 85.3%

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

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4.6 \cdot 10^{-296}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right)\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 72.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 4.6 \cdot 10^{-148}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \frac{h \cdot \left({\left(\left(M \cdot 0.5\right) \cdot \frac{D_m}{d}\right)}^{2} \cdot -0.5\right)}{\ell}\right)\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D_m \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)\right)\\ \end{array} \end{array} \]

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 (<= l 4.6e-148)
   (*
    (*
     (sqrt (/ d l))
     (+ 1.0 (/ (* h (* (pow (* (* M 0.5) (/ D_m d)) 2.0) -0.5)) l)))
    (sqrt (/ d h)))
   (*
    (/ (/ d (sqrt l)) (sqrt h))
    (- 1.0 (* 0.5 (* (/ h l) (pow (* D_m (/ M (* d 2.0))) 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 (l <= 4.6e-148) {
		tmp = (sqrt((d / l)) * (1.0 + ((h * (pow(((M * 0.5) * (D_m / d)), 2.0) * -0.5)) / l))) * sqrt((d / h));
	} else {
		tmp = ((d / sqrt(l)) / sqrt(h)) * (1.0 - (0.5 * ((h / l) * pow((D_m * (M / (d * 2.0))), 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 (l <= 4.6d-148) then
        tmp = (sqrt((d / l)) * (1.0d0 + ((h * ((((m * 0.5d0) * (d_m / d)) ** 2.0d0) * (-0.5d0))) / l))) * sqrt((d / h))
    else
        tmp = ((d / sqrt(l)) / sqrt(h)) * (1.0d0 - (0.5d0 * ((h / l) * ((d_m * (m / (d * 2.0d0))) ** 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 (l <= 4.6e-148) {
		tmp = (Math.sqrt((d / l)) * (1.0 + ((h * (Math.pow(((M * 0.5) * (D_m / d)), 2.0) * -0.5)) / l))) * Math.sqrt((d / h));
	} else {
		tmp = ((d / Math.sqrt(l)) / Math.sqrt(h)) * (1.0 - (0.5 * ((h / l) * Math.pow((D_m * (M / (d * 2.0))), 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 l <= 4.6e-148:
		tmp = (math.sqrt((d / l)) * (1.0 + ((h * (math.pow(((M * 0.5) * (D_m / d)), 2.0) * -0.5)) / l))) * math.sqrt((d / h))
	else:
		tmp = ((d / math.sqrt(l)) / math.sqrt(h)) * (1.0 - (0.5 * ((h / l) * math.pow((D_m * (M / (d * 2.0))), 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 (l <= 4.6e-148)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * Float64(1.0 + Float64(Float64(h * Float64((Float64(Float64(M * 0.5) * Float64(D_m / d)) ^ 2.0) * -0.5)) / l))) * sqrt(Float64(d / h)));
	else
		tmp = Float64(Float64(Float64(d / sqrt(l)) / sqrt(h)) * Float64(1.0 - Float64(0.5 * Float64(Float64(h / l) * (Float64(D_m * Float64(M / Float64(d * 2.0))) ^ 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 (l <= 4.6e-148)
		tmp = (sqrt((d / l)) * (1.0 + ((h * ((((M * 0.5) * (D_m / d)) ^ 2.0) * -0.5)) / l))) * sqrt((d / h));
	else
		tmp = ((d / sqrt(l)) / sqrt(h)) * (1.0 - (0.5 * ((h / l) * ((D_m * (M / (d * 2.0))) ^ 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[l, 4.6e-148], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(N[(h * N[(N[Power[N[(N[(M * 0.5), $MachinePrecision] * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(d / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(D$95$m * N[(M / N[(d * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 4.59999999999999995e-148

   1. Initial program 64.8%

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

   3. Simplified 64.8%

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

      1. associate-*l/ 68.7%

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

      2. add-sqr-sqrt 68.7%

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

      3. pow2 68.7%

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

      4. unpow2 68.7%

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

      5. sqrt-prod 42.7%

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

      6. add-sqr-sqrt 68.7%

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

      7. div-inv 68.7%

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

      8. metadata-eval 68.7%

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

   6. Applied egg-rr 68.7%

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

   if 4.59999999999999995e-148 < l

   1. Initial program 70.1%

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

   4. Applied egg-rr 43.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 - \frac{h}{\ell} \cdot \left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right)\right)\right)} - 1} \]
   5. Step-by-step derivation

      1. expm1-def 61.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 - \frac{h}{\ell} \cdot \left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right)\right)\right)\right)} \]

      2. expm1-log1p 87.1%

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

      3. associate-/r* 87.0%

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

      4. associate-*r* 87.0%

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

      5. *-commutative 87.0%

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

      6. /-rgt-identity 87.0%

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

      7. associate-/l* 87.0%

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

      8. metadata-eval 87.0%

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

      9. times-frac 89.1%

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

      10. associate-*r/ 89.1%

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

      11. *-commutative 89.1%

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

   6. Simplified 89.1%

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

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 4.6 \cdot 10^{-148}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \frac{h \cdot \left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot -0.5\right)}{\ell}\right)\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 72.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.95 \cdot 10^{-150}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - 0.5 \cdot \frac{h \cdot {\left(\frac{M}{\frac{d \cdot 2}{D_m}}\right)}^{2}}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D_m \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)\right)\\ \end{array} \end{array} \]

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 (<= l 1.95e-150)
   (*
    (* (sqrt (/ d l)) (sqrt (/ d h)))
    (- 1.0 (* 0.5 (/ (* h (pow (/ M (/ (* d 2.0) D_m)) 2.0)) l))))
   (*
    (/ (/ d (sqrt l)) (sqrt h))
    (- 1.0 (* 0.5 (* (/ h l) (pow (* D_m (/ M (* d 2.0))) 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 (l <= 1.95e-150) {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - (0.5 * ((h * pow((M / ((d * 2.0) / D_m)), 2.0)) / l)));
	} else {
		tmp = ((d / sqrt(l)) / sqrt(h)) * (1.0 - (0.5 * ((h / l) * pow((D_m * (M / (d * 2.0))), 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 (l <= 1.95d-150) then
        tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0d0 - (0.5d0 * ((h * ((m / ((d * 2.0d0) / d_m)) ** 2.0d0)) / l)))
    else
        tmp = ((d / sqrt(l)) / sqrt(h)) * (1.0d0 - (0.5d0 * ((h / l) * ((d_m * (m / (d * 2.0d0))) ** 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 (l <= 1.95e-150) {
		tmp = (Math.sqrt((d / l)) * Math.sqrt((d / h))) * (1.0 - (0.5 * ((h * Math.pow((M / ((d * 2.0) / D_m)), 2.0)) / l)));
	} else {
		tmp = ((d / Math.sqrt(l)) / Math.sqrt(h)) * (1.0 - (0.5 * ((h / l) * Math.pow((D_m * (M / (d * 2.0))), 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 l <= 1.95e-150:
		tmp = (math.sqrt((d / l)) * math.sqrt((d / h))) * (1.0 - (0.5 * ((h * math.pow((M / ((d * 2.0) / D_m)), 2.0)) / l)))
	else:
		tmp = ((d / math.sqrt(l)) / math.sqrt(h)) * (1.0 - (0.5 * ((h / l) * math.pow((D_m * (M / (d * 2.0))), 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 (l <= 1.95e-150)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * Float64(1.0 - Float64(0.5 * Float64(Float64(h * (Float64(M / Float64(Float64(d * 2.0) / D_m)) ^ 2.0)) / l))));
	else
		tmp = Float64(Float64(Float64(d / sqrt(l)) / sqrt(h)) * Float64(1.0 - Float64(0.5 * Float64(Float64(h / l) * (Float64(D_m * Float64(M / Float64(d * 2.0))) ^ 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 (l <= 1.95e-150)
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - (0.5 * ((h * ((M / ((d * 2.0) / D_m)) ^ 2.0)) / l)));
	else
		tmp = ((d / sqrt(l)) / sqrt(h)) * (1.0 - (0.5 * ((h / l) * ((D_m * (M / (d * 2.0))) ^ 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[l, 1.95e-150], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(h * N[Power[N[(M / N[(N[(d * 2.0), $MachinePrecision] / D$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(d / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(D$95$m * N[(M / N[(d * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.9500000000000001e-150

   1. Initial program 64.8%

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

   3. Simplified 65.4%

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

      1. *-commutative 65.4%

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

      2. clear-num 65.3%

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

      3. frac-times 65.4%

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

      4. *-un-lft-identity 65.4%

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

   6. Applied egg-rr 65.4%

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

      1. associate-*r/ 69.3%

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

      2. associate-*l/ 69.3%

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

   8. Applied egg-rr 69.3%

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

   if 1.9500000000000001e-150 < l

   1. Initial program 70.1%

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

   4. Applied egg-rr 43.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 - \frac{h}{\ell} \cdot \left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right)\right)\right)} - 1} \]
   5. Step-by-step derivation

      1. expm1-def 61.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 - \frac{h}{\ell} \cdot \left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right)\right)\right)\right)} \]

      2. expm1-log1p 87.1%

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

      3. associate-/r* 87.0%

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

      4. associate-*r* 87.0%

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

      5. *-commutative 87.0%

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

      6. /-rgt-identity 87.0%

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

      7. associate-/l* 87.0%

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

      8. metadata-eval 87.0%

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

      9. times-frac 89.1%

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

      10. associate-*r/ 89.1%

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

      11. *-commutative 89.1%

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

   6. Simplified 89.1%

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

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.95 \cdot 10^{-150}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - 0.5 \cdot \frac{h \cdot {\left(\frac{M}{\frac{d \cdot 2}{D}}\right)}^{2}}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 58.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := \frac{d}{\sqrt{h \cdot \ell}}\\ \mathbf{if}\;d \leq -1.45 \cdot 10^{-93}:\\ \;\;\;\;\left|t_0\right|\\ \mathbf{elif}\;d \leq -2 \cdot 10^{-310}:\\ \;\;\;\;d \cdot \sqrt{\log \left(e^{\frac{1}{h \cdot \ell}}\right)}\\ \mathbf{elif}\;d \leq 2.15 \cdot 10^{-238}:\\ \;\;\;\;\frac{{\left(M \cdot D_m\right)}^{2} \cdot \left(-0.125 \cdot \frac{\sqrt{h}}{{\ell}^{1.5}}\right)}{d}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(1 + {\left(\frac{D_m}{d} \cdot \frac{M}{2}\right)}^{2} \cdot \left(-0.5 \cdot \frac{h}{\ell}\right)\right)\\ \end{array} \end{array} \]

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 (/ d (sqrt (* h l)))))
   (if (<= d -1.45e-93)
     (fabs t_0)
     (if (<= d -2e-310)
       (* d (sqrt (log (exp (/ 1.0 (* h l))))))
       (if (<= d 2.15e-238)
         (/ (* (pow (* M D_m) 2.0) (* -0.125 (/ (sqrt h) (pow l 1.5)))) d)
         (*
          t_0
          (+ 1.0 (* (pow (* (/ D_m d) (/ M 2.0)) 2.0) (* -0.5 (/ 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 t_0 = d / sqrt((h * l));
	double tmp;
	if (d <= -1.45e-93) {
		tmp = fabs(t_0);
	} else if (d <= -2e-310) {
		tmp = d * sqrt(log(exp((1.0 / (h * l)))));
	} else if (d <= 2.15e-238) {
		tmp = (pow((M * D_m), 2.0) * (-0.125 * (sqrt(h) / pow(l, 1.5)))) / d;
	} else {
		tmp = t_0 * (1.0 + (pow(((D_m / d) * (M / 2.0)), 2.0) * (-0.5 * (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) :: t_0
    real(8) :: tmp
    t_0 = d / sqrt((h * l))
    if (d <= (-1.45d-93)) then
        tmp = abs(t_0)
    else if (d <= (-2d-310)) then
        tmp = d * sqrt(log(exp((1.0d0 / (h * l)))))
    else if (d <= 2.15d-238) then
        tmp = (((m * d_m) ** 2.0d0) * ((-0.125d0) * (sqrt(h) / (l ** 1.5d0)))) / d
    else
        tmp = t_0 * (1.0d0 + ((((d_m / d) * (m / 2.0d0)) ** 2.0d0) * ((-0.5d0) * (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 t_0 = d / Math.sqrt((h * l));
	double tmp;
	if (d <= -1.45e-93) {
		tmp = Math.abs(t_0);
	} else if (d <= -2e-310) {
		tmp = d * Math.sqrt(Math.log(Math.exp((1.0 / (h * l)))));
	} else if (d <= 2.15e-238) {
		tmp = (Math.pow((M * D_m), 2.0) * (-0.125 * (Math.sqrt(h) / Math.pow(l, 1.5)))) / d;
	} else {
		tmp = t_0 * (1.0 + (Math.pow(((D_m / d) * (M / 2.0)), 2.0) * (-0.5 * (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):
	t_0 = d / math.sqrt((h * l))
	tmp = 0
	if d <= -1.45e-93:
		tmp = math.fabs(t_0)
	elif d <= -2e-310:
		tmp = d * math.sqrt(math.log(math.exp((1.0 / (h * l)))))
	elif d <= 2.15e-238:
		tmp = (math.pow((M * D_m), 2.0) * (-0.125 * (math.sqrt(h) / math.pow(l, 1.5)))) / d
	else:
		tmp = t_0 * (1.0 + (math.pow(((D_m / d) * (M / 2.0)), 2.0) * (-0.5 * (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)
	t_0 = Float64(d / sqrt(Float64(h * l)))
	tmp = 0.0
	if (d <= -1.45e-93)
		tmp = abs(t_0);
	elseif (d <= -2e-310)
		tmp = Float64(d * sqrt(log(exp(Float64(1.0 / Float64(h * l))))));
	elseif (d <= 2.15e-238)
		tmp = Float64(Float64((Float64(M * D_m) ^ 2.0) * Float64(-0.125 * Float64(sqrt(h) / (l ^ 1.5)))) / d);
	else
		tmp = Float64(t_0 * Float64(1.0 + Float64((Float64(Float64(D_m / d) * Float64(M / 2.0)) ^ 2.0) * Float64(-0.5 * Float64(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)
	t_0 = d / sqrt((h * l));
	tmp = 0.0;
	if (d <= -1.45e-93)
		tmp = abs(t_0);
	elseif (d <= -2e-310)
		tmp = d * sqrt(log(exp((1.0 / (h * l)))));
	elseif (d <= 2.15e-238)
		tmp = (((M * D_m) ^ 2.0) * (-0.125 * (sqrt(h) / (l ^ 1.5)))) / d;
	else
		tmp = t_0 * (1.0 + ((((D_m / d) * (M / 2.0)) ^ 2.0) * (-0.5 * (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_] := Block[{t$95$0 = N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -1.45e-93], N[Abs[t$95$0], $MachinePrecision], If[LessEqual[d, -2e-310], N[(d * N[Sqrt[N[Log[N[Exp[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 2.15e-238], N[(N[(N[Power[N[(M * D$95$m), $MachinePrecision], 2.0], $MachinePrecision] * N[(-0.125 * N[(N[Sqrt[h], $MachinePrecision] / N[Power[l, 1.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], N[(t$95$0 * N[(1.0 + N[(N[Power[N[(N[(D$95$m / d), $MachinePrecision] * N[(M / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(-0.5 * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -1.4499999999999999e-93

   1. Initial program 73.3%

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

   3. Simplified 73.3%

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

      1. sqrt-div 0.0%

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

   6. Applied egg-rr 0.0%

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

   7. Taylor expanded in d around inf 6.5%

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

      1. unpow-1 6.5%

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

      2. metadata-eval 6.5%

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

      3. pow-sqr 6.5%

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

      4. rem-sqrt-square 6.5%

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

      5. rem-square-sqrt 6.5%

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

      6. fabs-sqr 6.5%

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

      7. rem-square-sqrt 6.5%

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

   9. Simplified 6.5%

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

       1. add-sqr-sqrt 0.4%

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

       2. sqrt-prod 36.8%

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

       3. rem-sqrt-square 55.6%

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

       4. metadata-eval 55.6%

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

       5. pow-flip 55.5%

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

       6. pow1/2 55.5%

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

       7. div-inv 55.6%

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

   11. Applied egg-rr 55.6%

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

   if -1.4499999999999999e-93 < d < -1.999999999999994e-310

   1. Initial program 45.0%

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

   3. Taylor expanded in d around inf 12.5%

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

      1. add-log-exp 38.7%

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

   5. Applied egg-rr 38.7%

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

   if -1.999999999999994e-310 < d < 2.14999999999999984e-238

   1. Initial program 44.4%

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

   3. Taylor expanded in d around 0 42.9%

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

      1. *-commutative 42.9%

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

      2. associate-*l* 42.9%

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

      3. *-commutative 42.9%

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

      4. unpow2 42.9%

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

      5. unpow2 42.9%

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

      6. swap-sqr 50.6%

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

      7. unpow2 50.6%

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

      8. *-commutative 50.6%

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

   5. Simplified 50.6%

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

      1. associate-*l/ 50.5%

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

      2. sqrt-div 50.5%

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

      3. sqrt-pow1 71.4%

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

      4. metadata-eval 71.4%

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

   7. Applied egg-rr 71.4%

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

   if 2.14999999999999984e-238 < d

   1. Initial program 72.2%

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

   3. Simplified 71.4%

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

      1. sqrt-div 76.8%

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

   6. Applied egg-rr 74.6%

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

      1. expm1-log1p-u 49.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h}} \cdot \left(\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot -0.5\right)\right)\right)\right)\right)} \]

      2. expm1-udef 35.3%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h}} \cdot \left(\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot -0.5\right)\right)\right)\right)} - 1} \]

   8. Applied egg-rr 42.9%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{d}{\sqrt{h \cdot \ell}} \cdot \mathsf{fma}\left(\frac{h}{\ell}, -0.5 \cdot {\left(\frac{M}{\frac{d \cdot 2}{D}}\right)}^{2}, 1\right)\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 51.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{d}{\sqrt{h \cdot \ell}} \cdot \mathsf{fma}\left(\frac{h}{\ell}, -0.5 \cdot {\left(\frac{M}{\frac{d \cdot 2}{D}}\right)}^{2}, 1\right)\right)\right)} \]

      2. expm1-log1p 78.3%

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

      3. *-commutative 78.3%

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

   10. Simplified 78.3%

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

       1. fma-udef 78.3%

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

       2. *-un-lft-identity 78.3%

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

       3. div-inv 78.3%

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

       4. metadata-eval 78.3%

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

       5. times-frac 78.3%

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

       6. clear-num 78.3%

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

       7. *-commutative 78.3%

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

   12. Applied egg-rr 78.3%

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

4. Final simplification 63.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.45 \cdot 10^{-93}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \mathbf{elif}\;d \leq -2 \cdot 10^{-310}:\\ \;\;\;\;d \cdot \sqrt{\log \left(e^{\frac{1}{h \cdot \ell}}\right)}\\ \mathbf{elif}\;d \leq 2.15 \cdot 10^{-238}:\\ \;\;\;\;\frac{{\left(M \cdot D\right)}^{2} \cdot \left(-0.125 \cdot \frac{\sqrt{h}}{{\ell}^{1.5}}\right)}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 + {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2} \cdot \left(-0.5 \cdot \frac{h}{\ell}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 56.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := \frac{d}{\sqrt{h \cdot \ell}}\\ \mathbf{if}\;d \leq -3.7 \cdot 10^{-92}:\\ \;\;\;\;\left|t_0\right|\\ \mathbf{elif}\;d \leq 3 \cdot 10^{-297}:\\ \;\;\;\;d \cdot \sqrt{\log \left(e^{\frac{1}{h \cdot \ell}}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(1 + {\left(\frac{D_m}{d} \cdot \frac{M}{2}\right)}^{2} \cdot \left(-0.5 \cdot \frac{h}{\ell}\right)\right)\\ \end{array} \end{array} \]

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 (/ d (sqrt (* h l)))))
   (if (<= d -3.7e-92)
     (fabs t_0)
     (if (<= d 3e-297)
       (* d (sqrt (log (exp (/ 1.0 (* h l))))))
       (*
        t_0
        (+ 1.0 (* (pow (* (/ D_m d) (/ M 2.0)) 2.0) (* -0.5 (/ 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 t_0 = d / sqrt((h * l));
	double tmp;
	if (d <= -3.7e-92) {
		tmp = fabs(t_0);
	} else if (d <= 3e-297) {
		tmp = d * sqrt(log(exp((1.0 / (h * l)))));
	} else {
		tmp = t_0 * (1.0 + (pow(((D_m / d) * (M / 2.0)), 2.0) * (-0.5 * (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) :: t_0
    real(8) :: tmp
    t_0 = d / sqrt((h * l))
    if (d <= (-3.7d-92)) then
        tmp = abs(t_0)
    else if (d <= 3d-297) then
        tmp = d * sqrt(log(exp((1.0d0 / (h * l)))))
    else
        tmp = t_0 * (1.0d0 + ((((d_m / d) * (m / 2.0d0)) ** 2.0d0) * ((-0.5d0) * (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 t_0 = d / Math.sqrt((h * l));
	double tmp;
	if (d <= -3.7e-92) {
		tmp = Math.abs(t_0);
	} else if (d <= 3e-297) {
		tmp = d * Math.sqrt(Math.log(Math.exp((1.0 / (h * l)))));
	} else {
		tmp = t_0 * (1.0 + (Math.pow(((D_m / d) * (M / 2.0)), 2.0) * (-0.5 * (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):
	t_0 = d / math.sqrt((h * l))
	tmp = 0
	if d <= -3.7e-92:
		tmp = math.fabs(t_0)
	elif d <= 3e-297:
		tmp = d * math.sqrt(math.log(math.exp((1.0 / (h * l)))))
	else:
		tmp = t_0 * (1.0 + (math.pow(((D_m / d) * (M / 2.0)), 2.0) * (-0.5 * (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)
	t_0 = Float64(d / sqrt(Float64(h * l)))
	tmp = 0.0
	if (d <= -3.7e-92)
		tmp = abs(t_0);
	elseif (d <= 3e-297)
		tmp = Float64(d * sqrt(log(exp(Float64(1.0 / Float64(h * l))))));
	else
		tmp = Float64(t_0 * Float64(1.0 + Float64((Float64(Float64(D_m / d) * Float64(M / 2.0)) ^ 2.0) * Float64(-0.5 * Float64(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)
	t_0 = d / sqrt((h * l));
	tmp = 0.0;
	if (d <= -3.7e-92)
		tmp = abs(t_0);
	elseif (d <= 3e-297)
		tmp = d * sqrt(log(exp((1.0 / (h * l)))));
	else
		tmp = t_0 * (1.0 + ((((D_m / d) * (M / 2.0)) ^ 2.0) * (-0.5 * (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_] := Block[{t$95$0 = N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -3.7e-92], N[Abs[t$95$0], $MachinePrecision], If[LessEqual[d, 3e-297], N[(d * N[Sqrt[N[Log[N[Exp[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(1.0 + N[(N[Power[N[(N[(D$95$m / d), $MachinePrecision] * N[(M / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(-0.5 * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if d < -3.69999999999999977e-92

   1. Initial program 73.3%

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

   3. Simplified 73.3%

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

      1. sqrt-div 0.0%

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

   6. Applied egg-rr 0.0%

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

   7. Taylor expanded in d around inf 6.5%

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

      1. unpow-1 6.5%

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

      2. metadata-eval 6.5%

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

      3. pow-sqr 6.5%

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

      4. rem-sqrt-square 6.5%

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

      5. rem-square-sqrt 6.5%

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

      6. fabs-sqr 6.5%

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

      7. rem-square-sqrt 6.5%

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

   9. Simplified 6.5%

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

       1. add-sqr-sqrt 0.4%

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

       2. sqrt-prod 36.8%

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

       3. rem-sqrt-square 55.6%

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

       4. metadata-eval 55.6%

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

       5. pow-flip 55.5%

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

       6. pow1/2 55.5%

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

       7. div-inv 55.6%

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

   11. Applied egg-rr 55.6%

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

   if -3.69999999999999977e-92 < d < 2.99999999999999995e-297

   1. Initial program 46.7%

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

   3. Taylor expanded in d around inf 17.7%

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

      1. add-log-exp 38.9%

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

   5. Applied egg-rr 38.9%

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

   if 2.99999999999999995e-297 < d

   1. Initial program 69.5%

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

   3. Simplified 67.9%

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

      1. sqrt-div 73.8%

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

   6. Applied egg-rr 71.0%

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

      1. expm1-log1p-u 45.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h}} \cdot \left(\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot -0.5\right)\right)\right)\right)\right)} \]

      2. expm1-udef 32.7%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h}} \cdot \left(\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot -0.5\right)\right)\right)\right)} - 1} \]

   8. Applied egg-rr 39.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{d}{\sqrt{h \cdot \ell}} \cdot \mathsf{fma}\left(\frac{h}{\ell}, -0.5 \cdot {\left(\frac{M}{\frac{d \cdot 2}{D}}\right)}^{2}, 1\right)\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 47.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{d}{\sqrt{h \cdot \ell}} \cdot \mathsf{fma}\left(\frac{h}{\ell}, -0.5 \cdot {\left(\frac{M}{\frac{d \cdot 2}{D}}\right)}^{2}, 1\right)\right)\right)} \]

      2. expm1-log1p 75.2%

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

      3. *-commutative 75.2%

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

   10. Simplified 75.2%

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

       1. fma-udef 75.2%

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

       2. *-un-lft-identity 75.2%

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

       3. div-inv 75.2%

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

       4. metadata-eval 75.2%

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

       5. times-frac 75.2%

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

       6. clear-num 75.2%

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

       7. *-commutative 75.2%

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

   12. Applied egg-rr 75.2%

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

4. Final simplification 61.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -3.7 \cdot 10^{-92}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \mathbf{elif}\;d \leq 3 \cdot 10^{-297}:\\ \;\;\;\;d \cdot \sqrt{\log \left(e^{\frac{1}{h \cdot \ell}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 + {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2} \cdot \left(-0.5 \cdot \frac{h}{\ell}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 56.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := \frac{d}{\sqrt{h \cdot \ell}}\\ \mathbf{if}\;d \leq -6.2 \cdot 10^{-97}:\\ \;\;\;\;\left|t_0\right|\\ \mathbf{elif}\;d \leq 3 \cdot 10^{-297}:\\ \;\;\;\;d \cdot \log \left(e^{{\left(h \cdot \ell\right)}^{-0.5}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(1 + {\left(\frac{D_m}{d} \cdot \frac{M}{2}\right)}^{2} \cdot \left(-0.5 \cdot \frac{h}{\ell}\right)\right)\\ \end{array} \end{array} \]

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 (/ d (sqrt (* h l)))))
   (if (<= d -6.2e-97)
     (fabs t_0)
     (if (<= d 3e-297)
       (* d (log (exp (pow (* h l) -0.5))))
       (*
        t_0
        (+ 1.0 (* (pow (* (/ D_m d) (/ M 2.0)) 2.0) (* -0.5 (/ 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 t_0 = d / sqrt((h * l));
	double tmp;
	if (d <= -6.2e-97) {
		tmp = fabs(t_0);
	} else if (d <= 3e-297) {
		tmp = d * log(exp(pow((h * l), -0.5)));
	} else {
		tmp = t_0 * (1.0 + (pow(((D_m / d) * (M / 2.0)), 2.0) * (-0.5 * (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) :: t_0
    real(8) :: tmp
    t_0 = d / sqrt((h * l))
    if (d <= (-6.2d-97)) then
        tmp = abs(t_0)
    else if (d <= 3d-297) then
        tmp = d * log(exp(((h * l) ** (-0.5d0))))
    else
        tmp = t_0 * (1.0d0 + ((((d_m / d) * (m / 2.0d0)) ** 2.0d0) * ((-0.5d0) * (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 t_0 = d / Math.sqrt((h * l));
	double tmp;
	if (d <= -6.2e-97) {
		tmp = Math.abs(t_0);
	} else if (d <= 3e-297) {
		tmp = d * Math.log(Math.exp(Math.pow((h * l), -0.5)));
	} else {
		tmp = t_0 * (1.0 + (Math.pow(((D_m / d) * (M / 2.0)), 2.0) * (-0.5 * (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):
	t_0 = d / math.sqrt((h * l))
	tmp = 0
	if d <= -6.2e-97:
		tmp = math.fabs(t_0)
	elif d <= 3e-297:
		tmp = d * math.log(math.exp(math.pow((h * l), -0.5)))
	else:
		tmp = t_0 * (1.0 + (math.pow(((D_m / d) * (M / 2.0)), 2.0) * (-0.5 * (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)
	t_0 = Float64(d / sqrt(Float64(h * l)))
	tmp = 0.0
	if (d <= -6.2e-97)
		tmp = abs(t_0);
	elseif (d <= 3e-297)
		tmp = Float64(d * log(exp((Float64(h * l) ^ -0.5))));
	else
		tmp = Float64(t_0 * Float64(1.0 + Float64((Float64(Float64(D_m / d) * Float64(M / 2.0)) ^ 2.0) * Float64(-0.5 * Float64(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)
	t_0 = d / sqrt((h * l));
	tmp = 0.0;
	if (d <= -6.2e-97)
		tmp = abs(t_0);
	elseif (d <= 3e-297)
		tmp = d * log(exp(((h * l) ^ -0.5)));
	else
		tmp = t_0 * (1.0 + ((((D_m / d) * (M / 2.0)) ^ 2.0) * (-0.5 * (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_] := Block[{t$95$0 = N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -6.2e-97], N[Abs[t$95$0], $MachinePrecision], If[LessEqual[d, 3e-297], N[(d * N[Log[N[Exp[N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(1.0 + N[(N[Power[N[(N[(D$95$m / d), $MachinePrecision] * N[(M / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(-0.5 * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if d < -6.20000000000000004e-97

   1. Initial program 73.3%

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

   3. Simplified 73.3%

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

      1. sqrt-div 0.0%

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

   6. Applied egg-rr 0.0%

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

   7. Taylor expanded in d around inf 6.5%

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

      1. unpow-1 6.5%

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

      2. metadata-eval 6.5%

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

      3. pow-sqr 6.5%

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

      4. rem-sqrt-square 6.5%

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

      5. rem-square-sqrt 6.5%

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

      6. fabs-sqr 6.5%

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

      7. rem-square-sqrt 6.5%

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

   9. Simplified 6.5%

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

       1. add-sqr-sqrt 0.4%

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

       2. sqrt-prod 36.8%

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

       3. rem-sqrt-square 55.6%

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

       4. metadata-eval 55.6%

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

       5. pow-flip 55.5%

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

       6. pow1/2 55.5%

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

       7. div-inv 55.6%

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

   11. Applied egg-rr 55.6%

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

   if -6.20000000000000004e-97 < d < 2.99999999999999995e-297

   1. Initial program 46.7%

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

   3. Taylor expanded in d around inf 17.7%

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

      1. add-log-exp 38.9%

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

      2. inv-pow 38.9%

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

      3. sqrt-pow1 38.9%

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

      4. metadata-eval 38.9%

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

   5. Applied egg-rr 38.9%

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

   if 2.99999999999999995e-297 < d

   1. Initial program 69.5%

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

   3. Simplified 67.9%

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

      1. sqrt-div 73.8%

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

   6. Applied egg-rr 71.0%

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

      1. expm1-log1p-u 45.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h}} \cdot \left(\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot -0.5\right)\right)\right)\right)\right)} \]

      2. expm1-udef 32.7%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h}} \cdot \left(\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot -0.5\right)\right)\right)\right)} - 1} \]

   8. Applied egg-rr 39.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{d}{\sqrt{h \cdot \ell}} \cdot \mathsf{fma}\left(\frac{h}{\ell}, -0.5 \cdot {\left(\frac{M}{\frac{d \cdot 2}{D}}\right)}^{2}, 1\right)\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 47.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{d}{\sqrt{h \cdot \ell}} \cdot \mathsf{fma}\left(\frac{h}{\ell}, -0.5 \cdot {\left(\frac{M}{\frac{d \cdot 2}{D}}\right)}^{2}, 1\right)\right)\right)} \]

      2. expm1-log1p 75.2%

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

      3. *-commutative 75.2%

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

   10. Simplified 75.2%

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

       1. fma-udef 75.2%

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

       2. *-un-lft-identity 75.2%

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

       3. div-inv 75.2%

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

       4. metadata-eval 75.2%

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

       5. times-frac 75.2%

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

       6. clear-num 75.2%

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

       7. *-commutative 75.2%

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

   12. Applied egg-rr 75.2%

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

4. Final simplification 61.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -6.2 \cdot 10^{-97}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \mathbf{elif}\;d \leq 3 \cdot 10^{-297}:\\ \;\;\;\;d \cdot \log \left(e^{{\left(h \cdot \ell\right)}^{-0.5}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 + {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2} \cdot \left(-0.5 \cdot \frac{h}{\ell}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 58.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{h \cdot \ell}\\ t_1 := \frac{d}{t_0}\\ \mathbf{if}\;\ell \leq -1.65 \cdot 10^{-234}:\\ \;\;\;\;\left|t_1\right|\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{d}{\log \left(e^{t_0}\right)}\\ \mathbf{elif}\;\ell \leq 4 \cdot 10^{+136}:\\ \;\;\;\;t_1 \cdot \left(1 + {\left(\frac{D_m}{d} \cdot \frac{M}{2}\right)}^{2} \cdot \left(-0.5 \cdot \frac{h}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

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
 (let* ((t_0 (sqrt (* h l))) (t_1 (/ d t_0)))
   (if (<= l -1.65e-234)
     (fabs t_1)
     (if (<= l -5e-310)
       (/ d (log (exp t_0)))
       (if (<= l 4e+136)
         (* t_1 (+ 1.0 (* (pow (* (/ D_m d) (/ M 2.0)) 2.0) (* -0.5 (/ h l)))))
         (/ d (* (sqrt h) (sqrt 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 t_0 = sqrt((h * l));
	double t_1 = d / t_0;
	double tmp;
	if (l <= -1.65e-234) {
		tmp = fabs(t_1);
	} else if (l <= -5e-310) {
		tmp = d / log(exp(t_0));
	} else if (l <= 4e+136) {
		tmp = t_1 * (1.0 + (pow(((D_m / d) * (M / 2.0)), 2.0) * (-0.5 * (h / l))));
	} else {
		tmp = d / (sqrt(h) * sqrt(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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sqrt((h * l))
    t_1 = d / t_0
    if (l <= (-1.65d-234)) then
        tmp = abs(t_1)
    else if (l <= (-5d-310)) then
        tmp = d / log(exp(t_0))
    else if (l <= 4d+136) then
        tmp = t_1 * (1.0d0 + ((((d_m / d) * (m / 2.0d0)) ** 2.0d0) * ((-0.5d0) * (h / l))))
    else
        tmp = d / (sqrt(h) * sqrt(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 t_0 = Math.sqrt((h * l));
	double t_1 = d / t_0;
	double tmp;
	if (l <= -1.65e-234) {
		tmp = Math.abs(t_1);
	} else if (l <= -5e-310) {
		tmp = d / Math.log(Math.exp(t_0));
	} else if (l <= 4e+136) {
		tmp = t_1 * (1.0 + (Math.pow(((D_m / d) * (M / 2.0)), 2.0) * (-0.5 * (h / l))));
	} else {
		tmp = d / (Math.sqrt(h) * Math.sqrt(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):
	t_0 = math.sqrt((h * l))
	t_1 = d / t_0
	tmp = 0
	if l <= -1.65e-234:
		tmp = math.fabs(t_1)
	elif l <= -5e-310:
		tmp = d / math.log(math.exp(t_0))
	elif l <= 4e+136:
		tmp = t_1 * (1.0 + (math.pow(((D_m / d) * (M / 2.0)), 2.0) * (-0.5 * (h / l))))
	else:
		tmp = d / (math.sqrt(h) * math.sqrt(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)
	t_0 = sqrt(Float64(h * l))
	t_1 = Float64(d / t_0)
	tmp = 0.0
	if (l <= -1.65e-234)
		tmp = abs(t_1);
	elseif (l <= -5e-310)
		tmp = Float64(d / log(exp(t_0)));
	elseif (l <= 4e+136)
		tmp = Float64(t_1 * Float64(1.0 + Float64((Float64(Float64(D_m / d) * Float64(M / 2.0)) ^ 2.0) * Float64(-0.5 * Float64(h / l)))));
	else
		tmp = Float64(d / Float64(sqrt(h) * sqrt(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)
	t_0 = sqrt((h * l));
	t_1 = d / t_0;
	tmp = 0.0;
	if (l <= -1.65e-234)
		tmp = abs(t_1);
	elseif (l <= -5e-310)
		tmp = d / log(exp(t_0));
	elseif (l <= 4e+136)
		tmp = t_1 * (1.0 + ((((D_m / d) * (M / 2.0)) ^ 2.0) * (-0.5 * (h / l))));
	else
		tmp = d / (sqrt(h) * sqrt(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_] := Block[{t$95$0 = N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(d / t$95$0), $MachinePrecision]}, If[LessEqual[l, -1.65e-234], N[Abs[t$95$1], $MachinePrecision], If[LessEqual[l, -5e-310], N[(d / N[Log[N[Exp[t$95$0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 4e+136], N[(t$95$1 * N[(1.0 + N[(N[Power[N[(N[(D$95$m / d), $MachinePrecision] * N[(M / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(-0.5 * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -1.65000000000000007e-234

   1. Initial program 65.0%

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

   3. Simplified 65.0%

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

      1. sqrt-div 0.0%

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

   6. Applied egg-rr 0.0%

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

   7. Taylor expanded in d around inf 7.9%

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

      1. unpow-1 7.9%

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

      2. metadata-eval 7.9%

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

      3. pow-sqr 7.9%

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

      4. rem-sqrt-square 8.0%

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

      5. rem-square-sqrt 8.0%

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

      6. fabs-sqr 8.0%

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

      7. rem-square-sqrt 8.0%

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

   9. Simplified 8.0%

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

       1. add-sqr-sqrt 3.0%

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

       2. sqrt-prod 31.6%

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

       3. rem-sqrt-square 47.8%

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

       4. metadata-eval 47.8%

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

       5. pow-flip 47.7%

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

       6. pow1/2 47.7%

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

       7. div-inv 47.7%

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

   11. Applied egg-rr 47.7%

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

   if -1.65000000000000007e-234 < l < -4.999999999999985e-310

   1. Initial program 55.6%

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

   3. Simplified 55.6%

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

      1. sqrt-div 0.0%

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

   6. Applied egg-rr 0.0%

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

   7. Taylor expanded in d around inf 14.1%

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

      1. unpow-1 14.1%

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

      2. metadata-eval 14.1%

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

      3. pow-sqr 14.1%

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

      4. rem-sqrt-square 14.1%

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

      5. rem-square-sqrt 14.1%

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

      6. fabs-sqr 14.1%

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

      7. rem-square-sqrt 14.1%

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

   9. Simplified 14.1%

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

       1. metadata-eval 14.1%

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

       2. pow-flip 14.1%

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

       3. pow1/2 14.1%

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

       4. div-inv 14.1%

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

       5. expm1-log1p-u 1.2%

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

       6. expm1-udef 0.9%

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

   11. Applied egg-rr 0.9%

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

       1. expm1-def 1.2%

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

       2. expm1-log1p 14.1%

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

   13. Simplified 14.1%

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

       1. add-log-exp 56.3%

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

   15. Applied egg-rr 56.3%

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

   if -4.999999999999985e-310 < l < 4.00000000000000023e136

   1. Initial program 76.7%

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

   3. Simplified 74.7%

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

      1. sqrt-div 79.3%

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

   6. Applied egg-rr 75.6%

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

      1. expm1-log1p-u 43.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h}} \cdot \left(\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot -0.5\right)\right)\right)\right)\right)} \]

      2. expm1-udef 34.9%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h}} \cdot \left(\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot -0.5\right)\right)\right)\right)} - 1} \]

   8. Applied egg-rr 33.8%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{d}{\sqrt{h \cdot \ell}} \cdot \mathsf{fma}\left(\frac{h}{\ell}, -0.5 \cdot {\left(\frac{M}{\frac{d \cdot 2}{D}}\right)}^{2}, 1\right)\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 42.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{d}{\sqrt{h \cdot \ell}} \cdot \mathsf{fma}\left(\frac{h}{\ell}, -0.5 \cdot {\left(\frac{M}{\frac{d \cdot 2}{D}}\right)}^{2}, 1\right)\right)\right)} \]

      2. expm1-log1p 76.8%

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

      3. *-commutative 76.8%

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

   10. Simplified 76.8%

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

       1. fma-udef 76.8%

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

       2. *-un-lft-identity 76.8%

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

       3. div-inv 76.8%

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

       4. metadata-eval 76.8%

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

       5. times-frac 76.8%

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

       6. clear-num 76.8%

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

       7. *-commutative 76.8%

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

   12. Applied egg-rr 76.8%

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

   if 4.00000000000000023e136 < l

   1. Initial program 47.9%

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

   3. Simplified 44.9%

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

      1. sqrt-div 53.4%

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

   6. Applied egg-rr 53.2%

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

   7. Taylor expanded in d around inf 67.7%

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

      1. unpow-1 67.7%

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

      2. metadata-eval 67.7%

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

      3. pow-sqr 67.8%

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

      4. rem-sqrt-square 67.8%

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

      5. rem-square-sqrt 67.5%

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

      6. fabs-sqr 67.5%

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

      7. rem-square-sqrt 67.8%

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

   9. Simplified 67.8%

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

       1. metadata-eval 67.8%

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

       2. pow-flip 67.8%

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

       3. pow1/2 67.8%

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

       4. div-inv 67.8%

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

       5. expm1-log1p-u 62.5%

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

       6. expm1-udef 56.8%

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

   11. Applied egg-rr 56.8%

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

       1. expm1-def 62.5%

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

       2. expm1-log1p 67.8%

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

   13. Simplified 67.8%

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

       1. *-un-lft-identity 67.8%

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

       2. sqrt-prod 88.0%

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

       3. times-frac 87.9%

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

   15. Applied egg-rr 87.9%

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

       1. *-commutative 87.9%

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

       2. times-frac 88.0%

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

       3. *-rgt-identity 88.0%

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

   17. Simplified 88.0%

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

4. Final simplification 63.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.65 \cdot 10^{-234}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{d}{\log \left(e^{\sqrt{h \cdot \ell}}\right)}\\ \mathbf{elif}\;\ell \leq 4 \cdot 10^{+136}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 + {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2} \cdot \left(-0.5 \cdot \frac{h}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 57.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := \frac{d}{\sqrt{h \cdot \ell}}\\ \mathbf{if}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left|t_0\right|\\ \mathbf{elif}\;\ell \leq 5 \cdot 10^{+136}:\\ \;\;\;\;t_0 \cdot \left(1 + {\left(\frac{D_m}{d} \cdot \frac{M}{2}\right)}^{2} \cdot \left(-0.5 \cdot \frac{h}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

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
 (let* ((t_0 (/ d (sqrt (* h l)))))
   (if (<= l -5e-310)
     (fabs t_0)
     (if (<= l 5e+136)
       (* t_0 (+ 1.0 (* (pow (* (/ D_m d) (/ M 2.0)) 2.0) (* -0.5 (/ h l)))))
       (/ d (* (sqrt h) (sqrt 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 t_0 = d / sqrt((h * l));
	double tmp;
	if (l <= -5e-310) {
		tmp = fabs(t_0);
	} else if (l <= 5e+136) {
		tmp = t_0 * (1.0 + (pow(((D_m / d) * (M / 2.0)), 2.0) * (-0.5 * (h / l))));
	} else {
		tmp = d / (sqrt(h) * sqrt(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) :: t_0
    real(8) :: tmp
    t_0 = d / sqrt((h * l))
    if (l <= (-5d-310)) then
        tmp = abs(t_0)
    else if (l <= 5d+136) then
        tmp = t_0 * (1.0d0 + ((((d_m / d) * (m / 2.0d0)) ** 2.0d0) * ((-0.5d0) * (h / l))))
    else
        tmp = d / (sqrt(h) * sqrt(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 t_0 = d / Math.sqrt((h * l));
	double tmp;
	if (l <= -5e-310) {
		tmp = Math.abs(t_0);
	} else if (l <= 5e+136) {
		tmp = t_0 * (1.0 + (Math.pow(((D_m / d) * (M / 2.0)), 2.0) * (-0.5 * (h / l))));
	} else {
		tmp = d / (Math.sqrt(h) * Math.sqrt(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):
	t_0 = d / math.sqrt((h * l))
	tmp = 0
	if l <= -5e-310:
		tmp = math.fabs(t_0)
	elif l <= 5e+136:
		tmp = t_0 * (1.0 + (math.pow(((D_m / d) * (M / 2.0)), 2.0) * (-0.5 * (h / l))))
	else:
		tmp = d / (math.sqrt(h) * math.sqrt(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)
	t_0 = Float64(d / sqrt(Float64(h * l)))
	tmp = 0.0
	if (l <= -5e-310)
		tmp = abs(t_0);
	elseif (l <= 5e+136)
		tmp = Float64(t_0 * Float64(1.0 + Float64((Float64(Float64(D_m / d) * Float64(M / 2.0)) ^ 2.0) * Float64(-0.5 * Float64(h / l)))));
	else
		tmp = Float64(d / Float64(sqrt(h) * sqrt(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)
	t_0 = d / sqrt((h * l));
	tmp = 0.0;
	if (l <= -5e-310)
		tmp = abs(t_0);
	elseif (l <= 5e+136)
		tmp = t_0 * (1.0 + ((((D_m / d) * (M / 2.0)) ^ 2.0) * (-0.5 * (h / l))));
	else
		tmp = d / (sqrt(h) * sqrt(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_] := Block[{t$95$0 = N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -5e-310], N[Abs[t$95$0], $MachinePrecision], If[LessEqual[l, 5e+136], N[(t$95$0 * N[(1.0 + N[(N[Power[N[(N[(D$95$m / d), $MachinePrecision] * N[(M / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(-0.5 * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < -4.999999999999985e-310

   1. Initial program 64.3%

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

   3. Simplified 64.3%

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

      1. sqrt-div 0.0%

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

   6. Applied egg-rr 0.0%

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

   7. Taylor expanded in d around inf 8.4%

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

      1. unpow-1 8.4%

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

      2. metadata-eval 8.4%

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

      3. pow-sqr 8.4%

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

      4. rem-sqrt-square 8.4%

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

      5. rem-square-sqrt 8.4%

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

      6. fabs-sqr 8.4%

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

      7. rem-square-sqrt 8.4%

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

   9. Simplified 8.4%

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

       1. add-sqr-sqrt 2.8%

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

       2. sqrt-prod 29.5%

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

       3. rem-sqrt-square 45.3%

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

       4. metadata-eval 45.3%

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

       5. pow-flip 45.2%

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

       6. pow1/2 45.2%

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

       7. div-inv 45.2%

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

   11. Applied egg-rr 45.2%

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

   if -4.999999999999985e-310 < l < 5.0000000000000002e136

   1. Initial program 76.7%

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

   3. Simplified 74.7%

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

      1. sqrt-div 79.3%

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

   6. Applied egg-rr 75.6%

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

      1. expm1-log1p-u 43.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h}} \cdot \left(\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot -0.5\right)\right)\right)\right)\right)} \]

      2. expm1-udef 34.9%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h}} \cdot \left(\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot -0.5\right)\right)\right)\right)} - 1} \]

   8. Applied egg-rr 33.8%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{d}{\sqrt{h \cdot \ell}} \cdot \mathsf{fma}\left(\frac{h}{\ell}, -0.5 \cdot {\left(\frac{M}{\frac{d \cdot 2}{D}}\right)}^{2}, 1\right)\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 42.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{d}{\sqrt{h \cdot \ell}} \cdot \mathsf{fma}\left(\frac{h}{\ell}, -0.5 \cdot {\left(\frac{M}{\frac{d \cdot 2}{D}}\right)}^{2}, 1\right)\right)\right)} \]

      2. expm1-log1p 76.8%

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

      3. *-commutative 76.8%

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

   10. Simplified 76.8%

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

       1. fma-udef 76.8%

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

       2. *-un-lft-identity 76.8%

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

       3. div-inv 76.8%

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

       4. metadata-eval 76.8%

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

       5. times-frac 76.8%

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

       6. clear-num 76.8%

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

       7. *-commutative 76.8%

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

   12. Applied egg-rr 76.8%

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

   if 5.0000000000000002e136 < l

   1. Initial program 47.9%

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

   3. Simplified 44.9%

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

      1. sqrt-div 53.4%

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

   6. Applied egg-rr 53.2%

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

   7. Taylor expanded in d around inf 67.7%

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

      1. unpow-1 67.7%

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

      2. metadata-eval 67.7%

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

      3. pow-sqr 67.8%

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

      4. rem-sqrt-square 67.8%

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

      5. rem-square-sqrt 67.5%

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

      6. fabs-sqr 67.5%

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

      7. rem-square-sqrt 67.8%

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

   9. Simplified 67.8%

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

       1. metadata-eval 67.8%

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

       2. pow-flip 67.8%

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

       3. pow1/2 67.8%

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

       4. div-inv 67.8%

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

       5. expm1-log1p-u 62.5%

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

       6. expm1-udef 56.8%

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

   11. Applied egg-rr 56.8%

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

       1. expm1-def 62.5%

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

       2. expm1-log1p 67.8%

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

   13. Simplified 67.8%

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

       1. *-un-lft-identity 67.8%

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

       2. sqrt-prod 88.0%

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

       3. times-frac 87.9%

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

   15. Applied egg-rr 87.9%

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

       1. *-commutative 87.9%

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

       2. times-frac 88.0%

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

       3. *-rgt-identity 88.0%

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

   17. Simplified 88.0%

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

4. Final simplification 62.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \mathbf{elif}\;\ell \leq 5 \cdot 10^{+136}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 + {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2} \cdot \left(-0.5 \cdot \frac{h}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 50.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \mathbf{elif}\;d \leq 1.65 \cdot 10^{-161}:\\ \;\;\;\;\left(\left(M \cdot D_m\right) \cdot \left(\left(M \cdot D_m\right) \cdot \frac{1}{d}\right)\right) \cdot \left(-0.125 \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

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 (<= d -2e-310)
   (fabs (/ d (sqrt (* h l))))
   (if (<= d 1.65e-161)
     (*
      (* (* M D_m) (* (* M D_m) (/ 1.0 d)))
      (* -0.125 (sqrt (/ h (pow l 3.0)))))
     (/ d (* (sqrt h) (sqrt 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 (d <= -2e-310) {
		tmp = fabs((d / sqrt((h * l))));
	} else if (d <= 1.65e-161) {
		tmp = ((M * D_m) * ((M * D_m) * (1.0 / d))) * (-0.125 * sqrt((h / pow(l, 3.0))));
	} else {
		tmp = d / (sqrt(h) * sqrt(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 (d <= (-2d-310)) then
        tmp = abs((d / sqrt((h * l))))
    else if (d <= 1.65d-161) then
        tmp = ((m * d_m) * ((m * d_m) * (1.0d0 / d))) * ((-0.125d0) * sqrt((h / (l ** 3.0d0))))
    else
        tmp = d / (sqrt(h) * sqrt(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 (d <= -2e-310) {
		tmp = Math.abs((d / Math.sqrt((h * l))));
	} else if (d <= 1.65e-161) {
		tmp = ((M * D_m) * ((M * D_m) * (1.0 / d))) * (-0.125 * Math.sqrt((h / Math.pow(l, 3.0))));
	} else {
		tmp = d / (Math.sqrt(h) * Math.sqrt(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 d <= -2e-310:
		tmp = math.fabs((d / math.sqrt((h * l))))
	elif d <= 1.65e-161:
		tmp = ((M * D_m) * ((M * D_m) * (1.0 / d))) * (-0.125 * math.sqrt((h / math.pow(l, 3.0))))
	else:
		tmp = d / (math.sqrt(h) * math.sqrt(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 (d <= -2e-310)
		tmp = abs(Float64(d / sqrt(Float64(h * l))));
	elseif (d <= 1.65e-161)
		tmp = Float64(Float64(Float64(M * D_m) * Float64(Float64(M * D_m) * Float64(1.0 / d))) * Float64(-0.125 * sqrt(Float64(h / (l ^ 3.0)))));
	else
		tmp = Float64(d / Float64(sqrt(h) * sqrt(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 (d <= -2e-310)
		tmp = abs((d / sqrt((h * l))));
	elseif (d <= 1.65e-161)
		tmp = ((M * D_m) * ((M * D_m) * (1.0 / d))) * (-0.125 * sqrt((h / (l ^ 3.0))));
	else
		tmp = d / (sqrt(h) * sqrt(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[d, -2e-310], N[Abs[N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[d, 1.65e-161], N[(N[(N[(M * D$95$m), $MachinePrecision] * N[(N[(M * D$95$m), $MachinePrecision] * N[(1.0 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-0.125 * N[Sqrt[N[(h / N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.999999999999994e-310

   1. Initial program 64.3%

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

   3. Simplified 64.3%

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

      1. sqrt-div 0.0%

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

   6. Applied egg-rr 0.0%

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

   7. Taylor expanded in d around inf 8.4%

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

      1. unpow-1 8.4%

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

      2. metadata-eval 8.4%

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

      3. pow-sqr 8.4%

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

      4. rem-sqrt-square 8.4%

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

      5. rem-square-sqrt 8.4%

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

      6. fabs-sqr 8.4%

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

      7. rem-square-sqrt 8.4%

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

   9. Simplified 8.4%

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

       1. add-sqr-sqrt 2.8%

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

       2. sqrt-prod 29.5%

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

       3. rem-sqrt-square 45.3%

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

       4. metadata-eval 45.3%

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

       5. pow-flip 45.2%

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

       6. pow1/2 45.2%

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

       7. div-inv 45.2%

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

   11. Applied egg-rr 45.2%

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

   if -1.999999999999994e-310 < d < 1.6499999999999999e-161

   1. Initial program 62.1%

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

   3. Taylor expanded in d around 0 52.0%

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

      1. *-commutative 52.0%

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

      2. associate-*l* 52.0%

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

      3. *-commutative 52.0%

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

      4. unpow2 52.0%

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

      5. unpow2 52.0%

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

      6. swap-sqr 58.9%

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

      7. unpow2 58.9%

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

      8. *-commutative 58.9%

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

   5. Simplified 58.9%

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

      1. div-inv 58.9%

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

      2. unpow2 58.9%

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

      3. associate-*l* 62.8%

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

   7. Applied egg-rr 62.8%

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

   if 1.6499999999999999e-161 < 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 70.4%

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

      1. sqrt-div 75.8%

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

   6. Applied egg-rr 73.3%

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

   7. Taylor expanded in d around inf 63.9%

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

      1. unpow-1 63.9%

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

      2. metadata-eval 63.9%

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

      3. pow-sqr 63.9%

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

      4. rem-sqrt-square 63.9%

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

      5. rem-square-sqrt 63.7%

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

      6. fabs-sqr 63.7%

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

      7. rem-square-sqrt 63.9%

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

   9. Simplified 63.9%

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

       1. metadata-eval 63.9%

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

       2. pow-flip 63.9%

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

       3. pow1/2 63.9%

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

       4. div-inv 63.9%

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

       5. expm1-log1p-u 60.1%

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

       6. expm1-udef 51.9%

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

   11. Applied egg-rr 51.9%

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

       1. expm1-def 60.1%

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

       2. expm1-log1p 63.9%

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

   13. Simplified 63.9%

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

       1. *-un-lft-identity 63.9%

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

       2. sqrt-prod 74.9%

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

       3. times-frac 74.9%

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

   15. Applied egg-rr 74.9%

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

       1. *-commutative 74.9%

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

       2. times-frac 74.9%

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

       3. *-rgt-identity 74.9%

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

   17. Simplified 74.9%

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

4. Final simplification 58.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \mathbf{elif}\;d \leq 1.65 \cdot 10^{-161}:\\ \;\;\;\;\left(\left(M \cdot D\right) \cdot \left(\left(M \cdot D\right) \cdot \frac{1}{d}\right)\right) \cdot \left(-0.125 \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 46.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 10^{-204}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

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 (<= d 1e-204)
   (* (- d) (sqrt (/ 1.0 (* h l))))
   (/ d (* (sqrt h) (sqrt 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 (d <= 1e-204) {
		tmp = -d * sqrt((1.0 / (h * l)));
	} else {
		tmp = d / (sqrt(h) * sqrt(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 (d <= 1d-204) then
        tmp = -d * sqrt((1.0d0 / (h * l)))
    else
        tmp = d / (sqrt(h) * sqrt(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 (d <= 1e-204) {
		tmp = -d * Math.sqrt((1.0 / (h * l)));
	} else {
		tmp = d / (Math.sqrt(h) * Math.sqrt(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 d <= 1e-204:
		tmp = -d * math.sqrt((1.0 / (h * l)))
	else:
		tmp = d / (math.sqrt(h) * math.sqrt(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 (d <= 1e-204)
		tmp = Float64(Float64(-d) * sqrt(Float64(1.0 / Float64(h * l))));
	else
		tmp = Float64(d / Float64(sqrt(h) * sqrt(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 (d <= 1e-204)
		tmp = -d * sqrt((1.0 / (h * l)));
	else
		tmp = d / (sqrt(h) * sqrt(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[d, 1e-204], N[((-d) * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 1e-204

   1. Initial program 62.8%

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

   3. Taylor expanded in d around inf 9.3%

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

      1. add-sqr-sqrt 4.5%

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

      2. sqrt-unprod 26.4%

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

      3. pow2 26.4%

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

      4. inv-pow 26.4%

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

      5. sqrt-pow1 26.9%

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

      6. metadata-eval 26.9%

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

   5. Applied egg-rr 26.9%

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

   6. Taylor expanded in d around -inf 41.4%

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

      1. mul-1-neg 41.4%

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

      2. *-commutative 41.4%

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

      3. distribute-rgt-neg-in 41.4%

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

   8. Simplified 41.4%

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

   if 1e-204 < d

   1. Initial program 72.3%

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

   3. Simplified 71.4%

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

      1. sqrt-div 77.1%

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

   6. Applied egg-rr 74.9%

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

   7. Taylor expanded in d around inf 60.8%

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

      1. unpow-1 60.8%

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

      2. metadata-eval 60.8%

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

      3. pow-sqr 60.9%

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

      4. rem-sqrt-square 60.9%

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

      5. rem-square-sqrt 60.6%

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

      6. fabs-sqr 60.6%

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

      7. rem-square-sqrt 60.9%

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

   9. Simplified 60.9%

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

       1. metadata-eval 60.9%

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

       2. pow-flip 60.8%

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

       3. pow1/2 60.8%

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

       4. div-inv 60.8%

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

       5. expm1-log1p-u 57.4%

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

       6. expm1-udef 48.2%

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

   11. Applied egg-rr 48.2%

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

       1. expm1-def 57.4%

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

       2. expm1-log1p 60.8%

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

   13. Simplified 60.8%

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

       1. *-un-lft-identity 60.8%

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

       2. sqrt-prod 70.8%

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

       3. times-frac 70.7%

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

   15. Applied egg-rr 70.7%

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

       1. *-commutative 70.7%

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

       2. times-frac 70.8%

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

       3. *-rgt-identity 70.8%

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

   17. Simplified 70.8%

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

4. Final simplification 53.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 10^{-204}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 42.8% accurate, 1.6× speedup

Math

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

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 (fabs (/ d (sqrt (* 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) {
	return fabs((d / sqrt((h * l))));
}

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 = abs((d / sqrt((h * l))))
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 Math.abs((d / Math.sqrt((h * l))));
}

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 math.fabs((d / math.sqrt((h * l))))

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 abs(Float64(d / sqrt(Float64(h * l))))
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 = abs((d / sqrt((h * l))));
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[Abs[N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 66.7%

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

3. Simplified 65.6%

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

   1. sqrt-div 35.7%

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

6. Applied egg-rr 34.3%

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

7. Taylor expanded in d around inf 30.5%

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

   1. unpow-1 30.5%

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

   2. metadata-eval 30.5%

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

   3. pow-sqr 30.5%

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

   4. rem-sqrt-square 30.5%

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

   5. rem-square-sqrt 30.4%

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

   6. fabs-sqr 30.4%

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

   7. rem-square-sqrt 30.5%

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

9. Simplified 30.5%

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

    1. add-sqr-sqrt 27.5%

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

    2. sqrt-prod 33.6%

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

    3. rem-sqrt-square 49.2%

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

    4. metadata-eval 49.2%

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

    5. pow-flip 49.1%

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

    6. pow1/2 49.1%

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

    7. div-inv 49.1%

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

11. Applied egg-rr 49.1%

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

12. Final simplification 49.1%

    \[\leadsto \left|\frac{d}{\sqrt{h \cdot \ell}}\right| \]
13. Add Preprocessing

Alternative 21: 42.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 1.42 \cdot 10^{-204}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \end{array} \end{array} \]

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 1.42e-204)
   (* (- d) (sqrt (/ 1.0 (* h l))))
   (* d (sqrt (/ (/ 1.0 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 <= 1.42e-204) {
		tmp = -d * sqrt((1.0 / (h * l)));
	} else {
		tmp = d * sqrt(((1.0 / 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 <= 1.42d-204) then
        tmp = -d * sqrt((1.0d0 / (h * l)))
    else
        tmp = d * sqrt(((1.0d0 / 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 <= 1.42e-204) {
		tmp = -d * Math.sqrt((1.0 / (h * l)));
	} else {
		tmp = d * Math.sqrt(((1.0 / 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 <= 1.42e-204:
		tmp = -d * math.sqrt((1.0 / (h * l)))
	else:
		tmp = d * math.sqrt(((1.0 / 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 <= 1.42e-204)
		tmp = Float64(Float64(-d) * sqrt(Float64(1.0 / Float64(h * l))));
	else
		tmp = Float64(d * sqrt(Float64(Float64(1.0 / 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 <= 1.42e-204)
		tmp = -d * sqrt((1.0 / (h * l)));
	else
		tmp = d * sqrt(((1.0 / 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, 1.42e-204], N[((-d) * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d * N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 1.4200000000000001e-204

   1. Initial program 62.8%

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

   3. Taylor expanded in d around inf 9.3%

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

      1. add-sqr-sqrt 4.5%

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

      2. sqrt-unprod 26.4%

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

      3. pow2 26.4%

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

      4. inv-pow 26.4%

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

      5. sqrt-pow1 26.9%

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

      6. metadata-eval 26.9%

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

   5. Applied egg-rr 26.9%

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

   6. Taylor expanded in d around -inf 41.4%

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

      1. mul-1-neg 41.4%

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

      2. *-commutative 41.4%

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

      3. distribute-rgt-neg-in 41.4%

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

   8. Simplified 41.4%

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

   if 1.4200000000000001e-204 < d

   1. Initial program 72.3%

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

   3. Taylor expanded in d around inf 60.8%

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

      1. *-commutative 60.8%

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

      2. associate-/r* 60.9%

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

   5. Simplified 60.9%

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

4. Final simplification 49.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 1.42 \cdot 10^{-204}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 26.4% accurate, 3.1× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ d \cdot {\left(h \cdot \ell\right)}^{-0.5} \end{array} \]

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 (pow (* 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) {
	return d * pow((h * l), -0.5);
}

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 * ((h * l) ** (-0.5d0))
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.pow((h * l), -0.5);
}

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.pow((h * l), -0.5)

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 * (Float64(h * l) ^ -0.5))
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 * ((h * l) ^ -0.5);
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[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 66.7%

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

3. Simplified 65.6%

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

   1. sqrt-div 35.7%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}} \cdot \left(1 + \frac{h \cdot \left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot -0.5\right)}{\ell}\right)\right) \]

6. Applied egg-rr 34.3%

   \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot -0.5\right)\right)\right) \]

7. Taylor expanded in d around inf 30.5%

   \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
8. Step-by-step derivation

   1. unpow-1 30.5%

      \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \]

   2. metadata-eval 30.5%

      \[\leadsto d \cdot \sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \]

   3. pow-sqr 30.5%

      \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}} \]

   4. rem-sqrt-square 30.5%

      \[\leadsto d \cdot \color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|} \]

   5. rem-square-sqrt 30.4%

      \[\leadsto d \cdot \left|\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right| \]

   6. fabs-sqr 30.4%

      \[\leadsto d \cdot \color{blue}{\left(\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}\right)} \]

   7. rem-square-sqrt 30.5%

      \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

9. Simplified 30.5%

   \[\leadsto \color{blue}{d \cdot {\left(h \cdot \ell\right)}^{-0.5}} \]

10. Final simplification 30.5%

    \[\leadsto d \cdot {\left(h \cdot \ell\right)}^{-0.5} \]
11. Add Preprocessing

Alternative 23: 26.4% accurate, 3.2× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \frac{d}{\sqrt{h \cdot \ell}} \end{array} \]

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 (* 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) {
	return d / sqrt((h * l));
}

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((h * l))
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((h * l));
}

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((h * l))

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(h * l)))
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((h * l));
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[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 66.7%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

3. Simplified 65.6%

   \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot -0.5\right)\right)\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. sqrt-div 35.7%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}} \cdot \left(1 + \frac{h \cdot \left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot -0.5\right)}{\ell}\right)\right) \]

6. Applied egg-rr 34.3%

   \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot -0.5\right)\right)\right) \]

7. Taylor expanded in d around inf 30.5%

   \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
8. Step-by-step derivation

   1. unpow-1 30.5%

      \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \]

   2. metadata-eval 30.5%

      \[\leadsto d \cdot \sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \]

   3. pow-sqr 30.5%

      \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}} \]

   4. rem-sqrt-square 30.5%

      \[\leadsto d \cdot \color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|} \]

   5. rem-square-sqrt 30.4%

      \[\leadsto d \cdot \left|\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right| \]

   6. fabs-sqr 30.4%

      \[\leadsto d \cdot \color{blue}{\left(\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}\right)} \]

   7. rem-square-sqrt 30.5%

      \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

9. Simplified 30.5%

   \[\leadsto \color{blue}{d \cdot {\left(h \cdot \ell\right)}^{-0.5}} \]
10. Step-by-step derivation

    1. metadata-eval 30.5%

       \[\leadsto d \cdot {\left(h \cdot \ell\right)}^{\color{blue}{\left(-0.5\right)}} \]

    2. pow-flip 30.5%

       \[\leadsto d \cdot \color{blue}{\frac{1}{{\left(h \cdot \ell\right)}^{0.5}}} \]

    3. pow1/2 30.5%

       \[\leadsto d \cdot \frac{1}{\color{blue}{\sqrt{h \cdot \ell}}} \]

    4. div-inv 30.4%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]

    5. expm1-log1p-u 26.7%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{d}{\sqrt{h \cdot \ell}}\right)\right)} \]

    6. expm1-udef 22.6%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{d}{\sqrt{h \cdot \ell}}\right)} - 1} \]

11. Applied egg-rr 22.6%

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{d}{\sqrt{h \cdot \ell}}\right)} - 1} \]
12. Step-by-step derivation

    1. expm1-def 26.7%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{d}{\sqrt{h \cdot \ell}}\right)\right)} \]

    2. expm1-log1p 30.4%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]

13. Simplified 30.4%

    \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]

14. Final simplification 30.4%

    \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024014 
(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)))))