Henrywood and Agarwal, Equation (12)

Percentage Accurate: 66.5% → 80.2%

Time: 23.9s

Alternatives: 14

Speedup: 1.4×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 66.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 80.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} [d, h, l, M, D] = \mathsf{sort}([d, h, l, M, D])\\ \\ \begin{array}{l} t_0 := \sqrt{-d}\\ t_1 := \left(0.5 \cdot M\right) \cdot \frac{D}{d}\\ t_2 := 1 - 0.5 \cdot \left(h \cdot \left(t\_1 \cdot \frac{t\_1}{\ell}\right)\right)\\ t_3 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;\ell \leq -1.18 \cdot 10^{+182}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \frac{t\_0}{\sqrt{-\ell}}\right) \cdot t\_2\\ \mathbf{elif}\;\ell \leq -1 \cdot 10^{-311}:\\ \;\;\;\;t\_3 \cdot \left(\frac{t\_0}{\sqrt{-h}} \cdot \left(1 + h \cdot \left({\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2} \cdot \frac{-0.5}{\ell}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2 \cdot \left(t\_3 \cdot \frac{\sqrt{d}}{\sqrt{h}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (- d)))
        (t_1 (* (* 0.5 M) (/ D d)))
        (t_2 (- 1.0 (* 0.5 (* h (* t_1 (/ t_1 l))))))
        (t_3 (sqrt (/ d l))))
   (if (<= l -1.18e+182)
     (* (* (sqrt (/ d h)) (/ t_0 (sqrt (- l)))) t_2)
     (if (<= l -1e-311)
       (*
        t_3
        (*
         (/ t_0 (sqrt (- h)))
         (+ 1.0 (* h (* (pow (* D (/ M (* d 2.0))) 2.0) (/ -0.5 l))))))
       (* t_2 (* t_3 (/ (sqrt d) (sqrt h))))))))

C

assert(d < h && h < l && l < M && M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt(-d);
	double t_1 = (0.5 * M) * (D / d);
	double t_2 = 1.0 - (0.5 * (h * (t_1 * (t_1 / l))));
	double t_3 = sqrt((d / l));
	double tmp;
	if (l <= -1.18e+182) {
		tmp = (sqrt((d / h)) * (t_0 / sqrt(-l))) * t_2;
	} else if (l <= -1e-311) {
		tmp = t_3 * ((t_0 / sqrt(-h)) * (1.0 + (h * (pow((D * (M / (d * 2.0))), 2.0) * (-0.5 / l)))));
	} else {
		tmp = t_2 * (t_3 * (sqrt(d) / sqrt(h)));
	}
	return tmp;
}

Fortran

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
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
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = sqrt(-d)
    t_1 = (0.5d0 * m) * (d_1 / d)
    t_2 = 1.0d0 - (0.5d0 * (h * (t_1 * (t_1 / l))))
    t_3 = sqrt((d / l))
    if (l <= (-1.18d+182)) then
        tmp = (sqrt((d / h)) * (t_0 / sqrt(-l))) * t_2
    else if (l <= (-1d-311)) then
        tmp = t_3 * ((t_0 / sqrt(-h)) * (1.0d0 + (h * (((d_1 * (m / (d * 2.0d0))) ** 2.0d0) * ((-0.5d0) / l)))))
    else
        tmp = t_2 * (t_3 * (sqrt(d) / sqrt(h)))
    end if
    code = tmp
end function

Java

assert d < h && h < l && l < M && M < D;
public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt(-d);
	double t_1 = (0.5 * M) * (D / d);
	double t_2 = 1.0 - (0.5 * (h * (t_1 * (t_1 / l))));
	double t_3 = Math.sqrt((d / l));
	double tmp;
	if (l <= -1.18e+182) {
		tmp = (Math.sqrt((d / h)) * (t_0 / Math.sqrt(-l))) * t_2;
	} else if (l <= -1e-311) {
		tmp = t_3 * ((t_0 / Math.sqrt(-h)) * (1.0 + (h * (Math.pow((D * (M / (d * 2.0))), 2.0) * (-0.5 / l)))));
	} else {
		tmp = t_2 * (t_3 * (Math.sqrt(d) / Math.sqrt(h)));
	}
	return tmp;
}

Python

[d, h, l, M, D] = sort([d, h, l, M, D])
def code(d, h, l, M, D):
	t_0 = math.sqrt(-d)
	t_1 = (0.5 * M) * (D / d)
	t_2 = 1.0 - (0.5 * (h * (t_1 * (t_1 / l))))
	t_3 = math.sqrt((d / l))
	tmp = 0
	if l <= -1.18e+182:
		tmp = (math.sqrt((d / h)) * (t_0 / math.sqrt(-l))) * t_2
	elif l <= -1e-311:
		tmp = t_3 * ((t_0 / math.sqrt(-h)) * (1.0 + (h * (math.pow((D * (M / (d * 2.0))), 2.0) * (-0.5 / l)))))
	else:
		tmp = t_2 * (t_3 * (math.sqrt(d) / math.sqrt(h)))
	return tmp

Julia

d, h, l, M, D = sort([d, h, l, M, D])
function code(d, h, l, M, D)
	t_0 = sqrt(Float64(-d))
	t_1 = Float64(Float64(0.5 * M) * Float64(D / d))
	t_2 = Float64(1.0 - Float64(0.5 * Float64(h * Float64(t_1 * Float64(t_1 / l)))))
	t_3 = sqrt(Float64(d / l))
	tmp = 0.0
	if (l <= -1.18e+182)
		tmp = Float64(Float64(sqrt(Float64(d / h)) * Float64(t_0 / sqrt(Float64(-l)))) * t_2);
	elseif (l <= -1e-311)
		tmp = Float64(t_3 * Float64(Float64(t_0 / sqrt(Float64(-h))) * Float64(1.0 + Float64(h * Float64((Float64(D * Float64(M / Float64(d * 2.0))) ^ 2.0) * Float64(-0.5 / l))))));
	else
		tmp = Float64(t_2 * Float64(t_3 * Float64(sqrt(d) / sqrt(h))));
	end
	return tmp
end

MATLAB

d, h, l, M, D = num2cell(sort([d, h, l, M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt(-d);
	t_1 = (0.5 * M) * (D / d);
	t_2 = 1.0 - (0.5 * (h * (t_1 * (t_1 / l))));
	t_3 = sqrt((d / l));
	tmp = 0.0;
	if (l <= -1.18e+182)
		tmp = (sqrt((d / h)) * (t_0 / sqrt(-l))) * t_2;
	elseif (l <= -1e-311)
		tmp = t_3 * ((t_0 / sqrt(-h)) * (1.0 + (h * (((D * (M / (d * 2.0))) ^ 2.0) * (-0.5 / l)))));
	else
		tmp = t_2 * (t_3 * (sqrt(d) / sqrt(h)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[(-d)], $MachinePrecision]}, Block[{t$95$1 = N[(N[(0.5 * M), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 - N[(0.5 * N[(h * N[(t$95$1 * N[(t$95$1 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -1.18e+182], N[(N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(t$95$0 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[l, -1e-311], N[(t$95$3 * N[(N[(t$95$0 / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(h * N[(N[Power[N[(D * N[(M / N[(d * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(-0.5 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$2 * N[(t$95$3 * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.1799999999999999e182

   1. Initial program 44.1%

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

   3. Simplified 44.1%

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

      1. expm1-log1p-u 43.8%

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

      2. expm1-undefine 43.8%

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

   6. Applied egg-rr 43.8%

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

      1. sub-neg 43.8%

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

      2. metadata-eval 43.8%

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

      3. +-commutative 43.8%

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

      4. log1p-undefine 43.8%

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

      5. rem-exp-log 44.1%

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

      6. associate-+r+ 44.1%

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

      7. metadata-eval 44.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(\color{blue}{0} + \frac{h}{\ell} \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right)\right) \]

      8. metadata-eval 44.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(\color{blue}{\left(0 - 0\right)} + \frac{h}{\ell} \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right)\right) \]

      9. associate--r- 44.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(0 - \left(0 - \frac{h}{\ell} \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right)\right)}\right) \]

      10. neg-sub0 44.1%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0 - \color{blue}{\left(-\frac{h}{\ell} \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right)}\right)\right) \]

      11. associate-*l/ 44.4%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0 - \left(-\color{blue}{\frac{h \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell}}\right)\right)\right) \]

      12. distribute-frac-neg2 44.4%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0 - \color{blue}{\frac{h \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{-\ell}}\right)\right) \]

      13. neg-sub0 44.4%

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

      14. distribute-neg-frac2 44.4%

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

      15. remove-double-neg 44.4%

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

   8. Simplified 47.9%

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

      1. unpow2 47.9%

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

      2. *-un-lft-identity 47.9%

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

      3. times-frac 51.3%

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

      4. associate-*r/ 47.9%

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

      5. frac-times 51.3%

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

      6. *-commutative 51.3%

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

      7. div-inv 51.3%

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

      8. metadata-eval 51.3%

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

      9. associate-*r/ 47.9%

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

      10. frac-times 51.3%

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

      11. *-commutative 51.3%

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

      12. div-inv 51.3%

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

      13. metadata-eval 51.3%

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

   10. Applied egg-rr 51.3%

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

       1. frac-2neg 51.3%

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

       2. sqrt-div 72.3%

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

   12. Applied egg-rr 72.3%

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

   if -1.1799999999999999e182 < l < -9.99999999999948e-312

   1. Initial program 70.9%

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

   3. Simplified 70.9%

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

      1. expm1-log1p-u 40.1%

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

      2. expm1-undefine 40.1%

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

      3. *-commutative 40.1%

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

      4. associate-*l* 40.1%

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

      5. associate-*r/ 40.2%

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

      6. associate-*l/ 39.2%

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

      7. *-commutative 39.2%

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

      8. div-inv 39.2%

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

      9. metadata-eval 39.2%

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

      10. associate-*l* 39.2%

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

   6. Applied egg-rr 39.2%

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

   8. Simplified 74.0%

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

      1. frac-2neg 74.0%

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

      2. sqrt-div 85.9%

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

   10. Applied egg-rr 85.9%

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

   if -9.99999999999948e-312 < l

   1. Initial program 71.8%

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

   3. Simplified 71.1%

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

      1. expm1-log1p-u 70.8%

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

      2. expm1-undefine 70.8%

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

   6. Applied egg-rr 70.8%

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

      1. sub-neg 70.8%

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

      2. metadata-eval 70.8%

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

      3. +-commutative 70.8%

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

      4. log1p-undefine 70.8%

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

      5. rem-exp-log 71.1%

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

      6. associate-+r+ 71.1%

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

      7. metadata-eval 71.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(\color{blue}{0} + \frac{h}{\ell} \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right)\right) \]

      8. metadata-eval 71.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(\color{blue}{\left(0 - 0\right)} + \frac{h}{\ell} \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right)\right) \]

      9. associate--r- 71.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(0 - \left(0 - \frac{h}{\ell} \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right)\right)}\right) \]

      10. neg-sub0 71.1%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0 - \color{blue}{\left(-\frac{h}{\ell} \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right)}\right)\right) \]

      11. associate-*l/ 72.9%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0 - \left(-\color{blue}{\frac{h \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell}}\right)\right)\right) \]

      12. distribute-frac-neg2 72.9%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0 - \color{blue}{\frac{h \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{-\ell}}\right)\right) \]

      13. neg-sub0 72.9%

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

      14. distribute-neg-frac2 72.9%

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

      15. remove-double-neg 72.9%

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

   8. Simplified 74.5%

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

      1. unpow2 74.5%

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

      2. *-un-lft-identity 74.5%

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

      3. times-frac 78.4%

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

      4. associate-*r/ 77.7%

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

      5. frac-times 77.6%

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

      6. *-commutative 77.6%

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

      7. div-inv 77.6%

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

      8. metadata-eval 77.6%

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

      9. associate-*r/ 76.9%

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

      10. frac-times 77.6%

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

      11. *-commutative 77.6%

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

      12. div-inv 77.6%

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

      13. metadata-eval 77.6%

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

   10. Applied egg-rr 77.6%

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

       1. sqrt-div 89.5%

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

       2. div-inv 89.5%

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

   12. Applied egg-rr 89.5%

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

       1. associate-*r/ 89.5%

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

       2. *-rgt-identity 89.5%

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

   14. Simplified 89.5%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.18 \cdot 10^{+182}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \left(\left(\left(0.5 \cdot M\right) \cdot \frac{D}{d}\right) \cdot \frac{\left(0.5 \cdot M\right) \cdot \frac{D}{d}}{\ell}\right)\right)\right)\\ \mathbf{elif}\;\ell \leq -1 \cdot 10^{-311}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(1 + h \cdot \left({\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2} \cdot \frac{-0.5}{\ell}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left(h \cdot \left(\left(\left(0.5 \cdot M\right) \cdot \frac{D}{d}\right) \cdot \frac{\left(0.5 \cdot M\right) \cdot \frac{D}{d}}{\ell}\right)\right)\right) \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \frac{\sqrt{d}}{\sqrt{h}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 78.2% accurate, 1.0× speedup

Math

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

FPCore

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* (* 0.5 M) (/ D d)))
        (t_1 (- 1.0 (* 0.5 (* h (* t_0 (/ t_0 l)))))))
   (if (<= l -1e-311)
     (* (* (sqrt (/ d h)) (/ (sqrt (- d)) (sqrt (- l)))) t_1)
     (* t_1 (* (sqrt (/ d l)) (/ (sqrt d) (sqrt h)))))))

C

assert(d < h && h < l && l < M && M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = (0.5 * M) * (D / d);
	double t_1 = 1.0 - (0.5 * (h * (t_0 * (t_0 / l))));
	double tmp;
	if (l <= -1e-311) {
		tmp = (sqrt((d / h)) * (sqrt(-d) / sqrt(-l))) * t_1;
	} else {
		tmp = t_1 * (sqrt((d / l)) * (sqrt(d) / sqrt(h)));
	}
	return tmp;
}

Fortran

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
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
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (0.5d0 * m) * (d_1 / d)
    t_1 = 1.0d0 - (0.5d0 * (h * (t_0 * (t_0 / l))))
    if (l <= (-1d-311)) then
        tmp = (sqrt((d / h)) * (sqrt(-d) / sqrt(-l))) * t_1
    else
        tmp = t_1 * (sqrt((d / l)) * (sqrt(d) / sqrt(h)))
    end if
    code = tmp
end function

Java

assert d < h && h < l && l < M && M < D;
public static double code(double d, double h, double l, double M, double D) {
	double t_0 = (0.5 * M) * (D / d);
	double t_1 = 1.0 - (0.5 * (h * (t_0 * (t_0 / l))));
	double tmp;
	if (l <= -1e-311) {
		tmp = (Math.sqrt((d / h)) * (Math.sqrt(-d) / Math.sqrt(-l))) * t_1;
	} else {
		tmp = t_1 * (Math.sqrt((d / l)) * (Math.sqrt(d) / Math.sqrt(h)));
	}
	return tmp;
}

Python

[d, h, l, M, D] = sort([d, h, l, M, D])
def code(d, h, l, M, D):
	t_0 = (0.5 * M) * (D / d)
	t_1 = 1.0 - (0.5 * (h * (t_0 * (t_0 / l))))
	tmp = 0
	if l <= -1e-311:
		tmp = (math.sqrt((d / h)) * (math.sqrt(-d) / math.sqrt(-l))) * t_1
	else:
		tmp = t_1 * (math.sqrt((d / l)) * (math.sqrt(d) / math.sqrt(h)))
	return tmp

Julia

d, h, l, M, D = sort([d, h, l, M, D])
function code(d, h, l, M, D)
	t_0 = Float64(Float64(0.5 * M) * Float64(D / d))
	t_1 = Float64(1.0 - Float64(0.5 * Float64(h * Float64(t_0 * Float64(t_0 / l)))))
	tmp = 0.0
	if (l <= -1e-311)
		tmp = Float64(Float64(sqrt(Float64(d / h)) * Float64(sqrt(Float64(-d)) / sqrt(Float64(-l)))) * t_1);
	else
		tmp = Float64(t_1 * Float64(sqrt(Float64(d / l)) * Float64(sqrt(d) / sqrt(h))));
	end
	return tmp
end

MATLAB

d, h, l, M, D = num2cell(sort([d, h, l, M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	t_0 = (0.5 * M) * (D / d);
	t_1 = 1.0 - (0.5 * (h * (t_0 * (t_0 / l))));
	tmp = 0.0;
	if (l <= -1e-311)
		tmp = (sqrt((d / h)) * (sqrt(-d) / sqrt(-l))) * t_1;
	else
		tmp = t_1 * (sqrt((d / l)) * (sqrt(d) / sqrt(h)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(0.5 * M), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - N[(0.5 * N[(h * N[(t$95$0 * N[(t$95$0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -1e-311], N[(N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], N[(t$95$1 * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if l < -9.99999999999948e-312

   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 63.8%

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

      1. expm1-log1p-u 63.5%

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

      2. expm1-undefine 63.5%

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

   6. Applied egg-rr 63.5%

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

      1. sub-neg 63.5%

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

      2. metadata-eval 63.5%

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

      3. +-commutative 63.5%

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

      4. log1p-undefine 63.5%

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

      5. rem-exp-log 63.8%

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

      6. associate-+r+ 63.8%

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

      7. metadata-eval 63.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(\color{blue}{0} + \frac{h}{\ell} \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right)\right) \]

      8. metadata-eval 63.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(\color{blue}{\left(0 - 0\right)} + \frac{h}{\ell} \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right)\right) \]

      9. associate--r- 63.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(0 - \left(0 - \frac{h}{\ell} \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right)\right)}\right) \]

      10. neg-sub0 63.8%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0 - \color{blue}{\left(-\frac{h}{\ell} \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right)}\right)\right) \]

      11. associate-*l/ 66.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0 - \left(-\color{blue}{\frac{h \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell}}\right)\right)\right) \]

      12. distribute-frac-neg2 66.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0 - \color{blue}{\frac{h \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{-\ell}}\right)\right) \]

      13. neg-sub0 66.3%

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

      14. distribute-neg-frac2 66.3%

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

      15. remove-double-neg 66.3%

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

   8. Simplified 68.5%

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

      1. unpow2 68.5%

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

      2. *-un-lft-identity 68.5%

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

      3. times-frac 70.0%

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

      4. associate-*r/ 69.3%

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

      5. frac-times 68.5%

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

      6. *-commutative 68.5%

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

      7. div-inv 68.5%

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

      8. metadata-eval 68.5%

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

      9. associate-*r/ 67.8%

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

      10. frac-times 68.5%

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

      11. *-commutative 68.5%

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

      12. div-inv 68.5%

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

      13. metadata-eval 68.5%

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

   10. Applied egg-rr 68.5%

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

       1. frac-2neg 68.5%

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

       2. sqrt-div 77.2%

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

   12. Applied egg-rr 77.2%

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

   if -9.99999999999948e-312 < l

   1. Initial program 71.8%

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

   3. Simplified 71.1%

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

      1. expm1-log1p-u 70.8%

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

      2. expm1-undefine 70.8%

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

   6. Applied egg-rr 70.8%

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

      1. sub-neg 70.8%

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

      2. metadata-eval 70.8%

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

      3. +-commutative 70.8%

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

      4. log1p-undefine 70.8%

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

      5. rem-exp-log 71.1%

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

      6. associate-+r+ 71.1%

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

      7. metadata-eval 71.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(\color{blue}{0} + \frac{h}{\ell} \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right)\right) \]

      8. metadata-eval 71.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(\color{blue}{\left(0 - 0\right)} + \frac{h}{\ell} \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right)\right) \]

      9. associate--r- 71.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(0 - \left(0 - \frac{h}{\ell} \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right)\right)}\right) \]

      10. neg-sub0 71.1%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0 - \color{blue}{\left(-\frac{h}{\ell} \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right)}\right)\right) \]

      11. associate-*l/ 72.9%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0 - \left(-\color{blue}{\frac{h \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell}}\right)\right)\right) \]

      12. distribute-frac-neg2 72.9%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0 - \color{blue}{\frac{h \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{-\ell}}\right)\right) \]

      13. neg-sub0 72.9%

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

      14. distribute-neg-frac2 72.9%

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

      15. remove-double-neg 72.9%

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

   8. Simplified 74.5%

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

      1. unpow2 74.5%

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

      2. *-un-lft-identity 74.5%

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

      3. times-frac 78.4%

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

      4. associate-*r/ 77.7%

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

      5. frac-times 77.6%

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

      6. *-commutative 77.6%

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

      7. div-inv 77.6%

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

      8. metadata-eval 77.6%

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

      9. associate-*r/ 76.9%

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

      10. frac-times 77.6%

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

      11. *-commutative 77.6%

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

      12. div-inv 77.6%

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

      13. metadata-eval 77.6%

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

   10. Applied egg-rr 77.6%

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

       1. sqrt-div 89.5%

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

       2. div-inv 89.5%

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

   12. Applied egg-rr 89.5%

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

       1. associate-*r/ 89.5%

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

       2. *-rgt-identity 89.5%

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

   14. Simplified 89.5%

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

4. Final simplification 83.1%

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

Alternative 3: 79.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} [d, h, l, M, D] = \mathsf{sort}([d, h, l, M, D])\\ \\ \begin{array}{l} t_0 := \left(0.5 \cdot M\right) \cdot \frac{D}{d}\\ t_1 := \sqrt{\frac{d}{\ell}}\\ t_2 := 1 - 0.5 \cdot \left(h \cdot \left(t\_0 \cdot \frac{t\_0}{\ell}\right)\right)\\ \mathbf{if}\;d \leq -4 \cdot 10^{-310}:\\ \;\;\;\;t\_2 \cdot \left(t\_1 \cdot \frac{\sqrt{-d}}{\sqrt{-h}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2 \cdot \left(t\_1 \cdot \frac{\sqrt{d}}{\sqrt{h}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* (* 0.5 M) (/ D d)))
        (t_1 (sqrt (/ d l)))
        (t_2 (- 1.0 (* 0.5 (* h (* t_0 (/ t_0 l)))))))
   (if (<= d -4e-310)
     (* t_2 (* t_1 (/ (sqrt (- d)) (sqrt (- h)))))
     (* t_2 (* t_1 (/ (sqrt d) (sqrt h)))))))

C

assert(d < h && h < l && l < M && M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = (0.5 * M) * (D / d);
	double t_1 = sqrt((d / l));
	double t_2 = 1.0 - (0.5 * (h * (t_0 * (t_0 / l))));
	double tmp;
	if (d <= -4e-310) {
		tmp = t_2 * (t_1 * (sqrt(-d) / sqrt(-h)));
	} else {
		tmp = t_2 * (t_1 * (sqrt(d) / sqrt(h)));
	}
	return tmp;
}

Fortran

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
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
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = (0.5d0 * m) * (d_1 / d)
    t_1 = sqrt((d / l))
    t_2 = 1.0d0 - (0.5d0 * (h * (t_0 * (t_0 / l))))
    if (d <= (-4d-310)) then
        tmp = t_2 * (t_1 * (sqrt(-d) / sqrt(-h)))
    else
        tmp = t_2 * (t_1 * (sqrt(d) / sqrt(h)))
    end if
    code = tmp
end function

Java

assert d < h && h < l && l < M && M < D;
public static double code(double d, double h, double l, double M, double D) {
	double t_0 = (0.5 * M) * (D / d);
	double t_1 = Math.sqrt((d / l));
	double t_2 = 1.0 - (0.5 * (h * (t_0 * (t_0 / l))));
	double tmp;
	if (d <= -4e-310) {
		tmp = t_2 * (t_1 * (Math.sqrt(-d) / Math.sqrt(-h)));
	} else {
		tmp = t_2 * (t_1 * (Math.sqrt(d) / Math.sqrt(h)));
	}
	return tmp;
}

Python

[d, h, l, M, D] = sort([d, h, l, M, D])
def code(d, h, l, M, D):
	t_0 = (0.5 * M) * (D / d)
	t_1 = math.sqrt((d / l))
	t_2 = 1.0 - (0.5 * (h * (t_0 * (t_0 / l))))
	tmp = 0
	if d <= -4e-310:
		tmp = t_2 * (t_1 * (math.sqrt(-d) / math.sqrt(-h)))
	else:
		tmp = t_2 * (t_1 * (math.sqrt(d) / math.sqrt(h)))
	return tmp

Julia

d, h, l, M, D = sort([d, h, l, M, D])
function code(d, h, l, M, D)
	t_0 = Float64(Float64(0.5 * M) * Float64(D / d))
	t_1 = sqrt(Float64(d / l))
	t_2 = Float64(1.0 - Float64(0.5 * Float64(h * Float64(t_0 * Float64(t_0 / l)))))
	tmp = 0.0
	if (d <= -4e-310)
		tmp = Float64(t_2 * Float64(t_1 * Float64(sqrt(Float64(-d)) / sqrt(Float64(-h)))));
	else
		tmp = Float64(t_2 * Float64(t_1 * Float64(sqrt(d) / sqrt(h))));
	end
	return tmp
end

MATLAB

d, h, l, M, D = num2cell(sort([d, h, l, M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	t_0 = (0.5 * M) * (D / d);
	t_1 = sqrt((d / l));
	t_2 = 1.0 - (0.5 * (h * (t_0 * (t_0 / l))));
	tmp = 0.0;
	if (d <= -4e-310)
		tmp = t_2 * (t_1 * (sqrt(-d) / sqrt(-h)));
	else
		tmp = t_2 * (t_1 * (sqrt(d) / sqrt(h)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(0.5 * M), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(1.0 - N[(0.5 * N[(h * N[(t$95$0 * N[(t$95$0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -4e-310], N[(t$95$2 * N[(t$95$1 * N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$2 * N[(t$95$1 * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if d < -3.999999999999988e-310

   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 63.8%

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

      1. expm1-log1p-u 63.5%

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

      2. expm1-undefine 63.5%

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

   6. Applied egg-rr 63.5%

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

      1. sub-neg 63.5%

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

      2. metadata-eval 63.5%

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

      3. +-commutative 63.5%

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

      4. log1p-undefine 63.5%

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

      5. rem-exp-log 63.8%

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

      6. associate-+r+ 63.8%

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

      7. metadata-eval 63.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(\color{blue}{0} + \frac{h}{\ell} \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right)\right) \]

      8. metadata-eval 63.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(\color{blue}{\left(0 - 0\right)} + \frac{h}{\ell} \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right)\right) \]

      9. associate--r- 63.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(0 - \left(0 - \frac{h}{\ell} \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right)\right)}\right) \]

      10. neg-sub0 63.8%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0 - \color{blue}{\left(-\frac{h}{\ell} \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right)}\right)\right) \]

      11. associate-*l/ 66.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0 - \left(-\color{blue}{\frac{h \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell}}\right)\right)\right) \]

      12. distribute-frac-neg2 66.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0 - \color{blue}{\frac{h \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{-\ell}}\right)\right) \]

      13. neg-sub0 66.3%

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

      14. distribute-neg-frac2 66.3%

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

      15. remove-double-neg 66.3%

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

   8. Simplified 68.5%

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

      1. unpow2 68.5%

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

      2. *-un-lft-identity 68.5%

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

      3. times-frac 70.0%

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

      4. associate-*r/ 69.3%

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

      5. frac-times 68.5%

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

      6. *-commutative 68.5%

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

      7. div-inv 68.5%

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

      8. metadata-eval 68.5%

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

      9. associate-*r/ 67.8%

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

      10. frac-times 68.5%

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

      11. *-commutative 68.5%

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

      12. div-inv 68.5%

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

      13. metadata-eval 68.5%

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

   10. Applied egg-rr 68.5%

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

       1. frac-2neg 68.5%

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

       2. sqrt-div 80.0%

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

   12. Applied egg-rr 78.5%

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

   if -3.999999999999988e-310 < d

   1. Initial program 71.8%

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

   3. Simplified 71.1%

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

      1. expm1-log1p-u 70.8%

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

      2. expm1-undefine 70.8%

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

   6. Applied egg-rr 70.8%

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

      1. sub-neg 70.8%

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

      2. metadata-eval 70.8%

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

      3. +-commutative 70.8%

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

      4. log1p-undefine 70.8%

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

      5. rem-exp-log 71.1%

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

      6. associate-+r+ 71.1%

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

      7. metadata-eval 71.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(\color{blue}{0} + \frac{h}{\ell} \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right)\right) \]

      8. metadata-eval 71.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(\color{blue}{\left(0 - 0\right)} + \frac{h}{\ell} \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right)\right) \]

      9. associate--r- 71.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(0 - \left(0 - \frac{h}{\ell} \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right)\right)}\right) \]

      10. neg-sub0 71.1%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0 - \color{blue}{\left(-\frac{h}{\ell} \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right)}\right)\right) \]

      11. associate-*l/ 72.9%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0 - \left(-\color{blue}{\frac{h \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell}}\right)\right)\right) \]

      12. distribute-frac-neg2 72.9%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0 - \color{blue}{\frac{h \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{-\ell}}\right)\right) \]

      13. neg-sub0 72.9%

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

      14. distribute-neg-frac2 72.9%

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

      15. remove-double-neg 72.9%

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

   8. Simplified 74.5%

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

      1. unpow2 74.5%

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

      2. *-un-lft-identity 74.5%

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

      3. times-frac 78.4%

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

      4. associate-*r/ 77.7%

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

      5. frac-times 77.6%

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

      6. *-commutative 77.6%

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

      7. div-inv 77.6%

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

      8. metadata-eval 77.6%

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

      9. associate-*r/ 76.9%

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

      10. frac-times 77.6%

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

      11. *-commutative 77.6%

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

      12. div-inv 77.6%

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

      13. metadata-eval 77.6%

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

   10. Applied egg-rr 77.6%

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

       1. sqrt-div 89.5%

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

       2. div-inv 89.5%

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

   12. Applied egg-rr 89.5%

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

       1. associate-*r/ 89.5%

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

       2. *-rgt-identity 89.5%

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

   14. Simplified 89.5%

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

4. Final simplification 83.8%

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

Alternative 4: 74.1% accurate, 1.0× speedup

Math

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

FPCore

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* (* 0.5 M) (/ D d))) (t_1 (sqrt (/ d l))))
   (if (<= l -1e-311)
     (*
      (* (sqrt (/ d h)) t_1)
      (- 1.0 (* 0.5 (* h (* t_0 (/ (* D (/ (* 0.5 M) d)) l))))))
     (*
      (- 1.0 (* 0.5 (* h (* t_0 (/ t_0 l)))))
      (* t_1 (/ (sqrt d) (sqrt h)))))))

C

assert(d < h && h < l && l < M && M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = (0.5 * M) * (D / d);
	double t_1 = sqrt((d / l));
	double tmp;
	if (l <= -1e-311) {
		tmp = (sqrt((d / h)) * t_1) * (1.0 - (0.5 * (h * (t_0 * ((D * ((0.5 * M) / d)) / l)))));
	} else {
		tmp = (1.0 - (0.5 * (h * (t_0 * (t_0 / l))))) * (t_1 * (sqrt(d) / sqrt(h)));
	}
	return tmp;
}

Fortran

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
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
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (0.5d0 * m) * (d_1 / d)
    t_1 = sqrt((d / l))
    if (l <= (-1d-311)) then
        tmp = (sqrt((d / h)) * t_1) * (1.0d0 - (0.5d0 * (h * (t_0 * ((d_1 * ((0.5d0 * m) / d)) / l)))))
    else
        tmp = (1.0d0 - (0.5d0 * (h * (t_0 * (t_0 / l))))) * (t_1 * (sqrt(d) / sqrt(h)))
    end if
    code = tmp
end function

Java

assert d < h && h < l && l < M && M < D;
public static double code(double d, double h, double l, double M, double D) {
	double t_0 = (0.5 * M) * (D / d);
	double t_1 = Math.sqrt((d / l));
	double tmp;
	if (l <= -1e-311) {
		tmp = (Math.sqrt((d / h)) * t_1) * (1.0 - (0.5 * (h * (t_0 * ((D * ((0.5 * M) / d)) / l)))));
	} else {
		tmp = (1.0 - (0.5 * (h * (t_0 * (t_0 / l))))) * (t_1 * (Math.sqrt(d) / Math.sqrt(h)));
	}
	return tmp;
}

Python

[d, h, l, M, D] = sort([d, h, l, M, D])
def code(d, h, l, M, D):
	t_0 = (0.5 * M) * (D / d)
	t_1 = math.sqrt((d / l))
	tmp = 0
	if l <= -1e-311:
		tmp = (math.sqrt((d / h)) * t_1) * (1.0 - (0.5 * (h * (t_0 * ((D * ((0.5 * M) / d)) / l)))))
	else:
		tmp = (1.0 - (0.5 * (h * (t_0 * (t_0 / l))))) * (t_1 * (math.sqrt(d) / math.sqrt(h)))
	return tmp

Julia

d, h, l, M, D = sort([d, h, l, M, D])
function code(d, h, l, M, D)
	t_0 = Float64(Float64(0.5 * M) * Float64(D / d))
	t_1 = sqrt(Float64(d / l))
	tmp = 0.0
	if (l <= -1e-311)
		tmp = Float64(Float64(sqrt(Float64(d / h)) * t_1) * Float64(1.0 - Float64(0.5 * Float64(h * Float64(t_0 * Float64(Float64(D * Float64(Float64(0.5 * M) / d)) / l))))));
	else
		tmp = Float64(Float64(1.0 - Float64(0.5 * Float64(h * Float64(t_0 * Float64(t_0 / l))))) * Float64(t_1 * Float64(sqrt(d) / sqrt(h))));
	end
	return tmp
end

MATLAB

d, h, l, M, D = num2cell(sort([d, h, l, M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	t_0 = (0.5 * M) * (D / d);
	t_1 = sqrt((d / l));
	tmp = 0.0;
	if (l <= -1e-311)
		tmp = (sqrt((d / h)) * t_1) * (1.0 - (0.5 * (h * (t_0 * ((D * ((0.5 * M) / d)) / l)))));
	else
		tmp = (1.0 - (0.5 * (h * (t_0 * (t_0 / l))))) * (t_1 * (sqrt(d) / sqrt(h)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(0.5 * M), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -1e-311], N[(N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(h * N[(t$95$0 * N[(N[(D * N[(N[(0.5 * M), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[(0.5 * N[(h * N[(t$95$0 * N[(t$95$0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if l < -9.99999999999948e-312

   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 63.8%

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

      1. expm1-log1p-u 63.5%

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

      2. expm1-undefine 63.5%

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

   6. Applied egg-rr 63.5%

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

      1. sub-neg 63.5%

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

      2. metadata-eval 63.5%

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

      3. +-commutative 63.5%

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

      4. log1p-undefine 63.5%

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

      5. rem-exp-log 63.8%

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

      6. associate-+r+ 63.8%

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

      7. metadata-eval 63.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(\color{blue}{0} + \frac{h}{\ell} \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right)\right) \]

      8. metadata-eval 63.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(\color{blue}{\left(0 - 0\right)} + \frac{h}{\ell} \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right)\right) \]

      9. associate--r- 63.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(0 - \left(0 - \frac{h}{\ell} \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right)\right)}\right) \]

      10. neg-sub0 63.8%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0 - \color{blue}{\left(-\frac{h}{\ell} \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right)}\right)\right) \]

      11. associate-*l/ 66.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0 - \left(-\color{blue}{\frac{h \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell}}\right)\right)\right) \]

      12. distribute-frac-neg2 66.3%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0 - \color{blue}{\frac{h \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{-\ell}}\right)\right) \]

      13. neg-sub0 66.3%

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

      14. distribute-neg-frac2 66.3%

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

      15. remove-double-neg 66.3%

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

   8. Simplified 68.5%

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

      1. unpow2 68.5%

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

      2. *-un-lft-identity 68.5%

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

      3. times-frac 70.0%

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

      4. associate-*r/ 69.3%

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

      5. frac-times 68.5%

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

      6. *-commutative 68.5%

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

      7. div-inv 68.5%

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

      8. metadata-eval 68.5%

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

      9. associate-*r/ 67.8%

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

      10. frac-times 68.5%

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

      11. *-commutative 68.5%

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

      12. div-inv 68.5%

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

      13. metadata-eval 68.5%

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

   10. Applied egg-rr 68.5%

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

   11. Taylor expanded in M around 0 67.8%

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

       1. *-commutative 67.8%

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

       2. associate-/l* 68.5%

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

       3. associate-*l* 68.5%

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

       4. associate-*l/ 68.5%

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

   13. Simplified 68.5%

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

   if -9.99999999999948e-312 < l

   1. Initial program 71.8%

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

   3. Simplified 71.1%

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

      1. expm1-log1p-u 70.8%

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

      2. expm1-undefine 70.8%

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

   6. Applied egg-rr 70.8%

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

      1. sub-neg 70.8%

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

      2. metadata-eval 70.8%

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

      3. +-commutative 70.8%

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

      4. log1p-undefine 70.8%

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

      5. rem-exp-log 71.1%

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

      6. associate-+r+ 71.1%

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

      7. metadata-eval 71.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(\color{blue}{0} + \frac{h}{\ell} \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right)\right) \]

      8. metadata-eval 71.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(\color{blue}{\left(0 - 0\right)} + \frac{h}{\ell} \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right)\right) \]

      9. associate--r- 71.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(0 - \left(0 - \frac{h}{\ell} \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right)\right)}\right) \]

      10. neg-sub0 71.1%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0 - \color{blue}{\left(-\frac{h}{\ell} \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right)}\right)\right) \]

      11. associate-*l/ 72.9%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0 - \left(-\color{blue}{\frac{h \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell}}\right)\right)\right) \]

      12. distribute-frac-neg2 72.9%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0 - \color{blue}{\frac{h \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{-\ell}}\right)\right) \]

      13. neg-sub0 72.9%

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

      14. distribute-neg-frac2 72.9%

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

      15. remove-double-neg 72.9%

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

   8. Simplified 74.5%

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

      1. unpow2 74.5%

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

      2. *-un-lft-identity 74.5%

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

      3. times-frac 78.4%

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

      4. associate-*r/ 77.7%

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

      5. frac-times 77.6%

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

      6. *-commutative 77.6%

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

      7. div-inv 77.6%

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

      8. metadata-eval 77.6%

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

      9. associate-*r/ 76.9%

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

      10. frac-times 77.6%

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

      11. *-commutative 77.6%

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

      12. div-inv 77.6%

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

      13. metadata-eval 77.6%

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

   10. Applied egg-rr 77.6%

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

       1. sqrt-div 89.5%

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

       2. div-inv 89.5%

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

   12. Applied egg-rr 89.5%

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

       1. associate-*r/ 89.5%

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

       2. *-rgt-identity 89.5%

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

   14. Simplified 89.5%

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

4. Final simplification 78.6%

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

Alternative 5: 58.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} [d, h, l, M, D] = \mathsf{sort}([d, h, l, M, D])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -6.2 \cdot 10^{+75}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{elif}\;\ell \leq 7 \cdot 10^{-6}:\\ \;\;\;\;\sqrt{\frac{d}{\ell \cdot \frac{h}{d}}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\left(0.5 \cdot M\right) \cdot \frac{D}{d}\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (if (<= l -6.2e+75)
   (* (- d) (sqrt (/ (/ 1.0 l) h)))
   (if (<= l 7e-6)
     (*
      (sqrt (/ d (* l (/ h d))))
      (+ 1.0 (* -0.5 (* (/ h l) (pow (* (* 0.5 M) (/ D d)) 2.0)))))
     (* d (/ (sqrt (/ 1.0 l)) (sqrt h))))))

C

assert(d < h && h < l && l < M && M < D);
double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -6.2e+75) {
		tmp = -d * sqrt(((1.0 / l) / h));
	} else if (l <= 7e-6) {
		tmp = sqrt((d / (l * (h / d)))) * (1.0 + (-0.5 * ((h / l) * pow(((0.5 * M) * (D / d)), 2.0))));
	} else {
		tmp = d * (sqrt((1.0 / l)) / sqrt(h));
	}
	return tmp;
}

Fortran

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
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
    real(8) :: tmp
    if (l <= (-6.2d+75)) then
        tmp = -d * sqrt(((1.0d0 / l) / h))
    else if (l <= 7d-6) then
        tmp = sqrt((d / (l * (h / d)))) * (1.0d0 + ((-0.5d0) * ((h / l) * (((0.5d0 * m) * (d_1 / d)) ** 2.0d0))))
    else
        tmp = d * (sqrt((1.0d0 / l)) / sqrt(h))
    end if
    code = tmp
end function

Java

assert d < h && h < l && l < M && M < D;
public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -6.2e+75) {
		tmp = -d * Math.sqrt(((1.0 / l) / h));
	} else if (l <= 7e-6) {
		tmp = Math.sqrt((d / (l * (h / d)))) * (1.0 + (-0.5 * ((h / l) * Math.pow(((0.5 * M) * (D / d)), 2.0))));
	} else {
		tmp = d * (Math.sqrt((1.0 / l)) / Math.sqrt(h));
	}
	return tmp;
}

Python

[d, h, l, M, D] = sort([d, h, l, M, D])
def code(d, h, l, M, D):
	tmp = 0
	if l <= -6.2e+75:
		tmp = -d * math.sqrt(((1.0 / l) / h))
	elif l <= 7e-6:
		tmp = math.sqrt((d / (l * (h / d)))) * (1.0 + (-0.5 * ((h / l) * math.pow(((0.5 * M) * (D / d)), 2.0))))
	else:
		tmp = d * (math.sqrt((1.0 / l)) / math.sqrt(h))
	return tmp

Julia

d, h, l, M, D = sort([d, h, l, M, D])
function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= -6.2e+75)
		tmp = Float64(Float64(-d) * sqrt(Float64(Float64(1.0 / l) / h)));
	elseif (l <= 7e-6)
		tmp = Float64(sqrt(Float64(d / Float64(l * Float64(h / d)))) * Float64(1.0 + Float64(-0.5 * Float64(Float64(h / l) * (Float64(Float64(0.5 * M) * Float64(D / d)) ^ 2.0)))));
	else
		tmp = Float64(d * Float64(sqrt(Float64(1.0 / l)) / sqrt(h)));
	end
	return tmp
end

MATLAB

d, h, l, M, D = num2cell(sort([d, h, l, M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (l <= -6.2e+75)
		tmp = -d * sqrt(((1.0 / l) / h));
	elseif (l <= 7e-6)
		tmp = sqrt((d / (l * (h / d)))) * (1.0 + (-0.5 * ((h / l) * (((0.5 * M) * (D / d)) ^ 2.0))));
	else
		tmp = d * (sqrt((1.0 / l)) / sqrt(h));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := If[LessEqual[l, -6.2e+75], N[((-d) * N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 7e-6], N[(N[Sqrt[N[(d / N[(l * N[(h / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(-0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(0.5 * M), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Sqrt[N[(1.0 / l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -6.2000000000000002e75

   1. Initial program 53.8%

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

   3. Simplified 52.0%

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

      1. sqrt-div 0.0%

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

      2. clear-num 0.0%

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

   6. Applied egg-rr 0.0%

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

   7. Taylor expanded in l around -inf 0.0%

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

      1. *-commutative 0.0%

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

      2. *-commutative 0.0%

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

      3. associate-/r* 0.0%

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

      4. *-commutative 0.0%

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

      5. unpow2 0.0%

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

      6. rem-square-sqrt 60.1%

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

      7. neg-mul-1 60.1%

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

   9. Simplified 60.1%

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

   if -6.2000000000000002e75 < l < 6.99999999999999989e-6

   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 75.3%

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

      1. sqrt-div 35.0%

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

      2. clear-num 35.0%

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

   6. Applied egg-rr 35.0%

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

      1. pow1 35.0%

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

   8. Applied egg-rr 62.3%

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

      1. unpow1 62.3%

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

      2. associate-/l/ 61.6%

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

      3. *-commutative 61.6%

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

   10. Simplified 61.6%

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

   if 6.99999999999999989e-6 < l

   1. Initial program 64.2%

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

   3. Simplified 64.2%

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

      1. sqrt-div 76.9%

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

      2. clear-num 76.9%

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

   6. Applied egg-rr 76.9%

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

   7. Taylor expanded in h around 0 56.8%

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

      1. *-commutative 56.8%

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

   9. Simplified 56.8%

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

       1. associate-/r* 58.4%

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

       2. sqrt-div 69.6%

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

   11. Applied egg-rr 69.6%

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

4. Final simplification 63.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -6.2 \cdot 10^{+75}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{elif}\;\ell \leq 7 \cdot 10^{-6}:\\ \;\;\;\;\sqrt{\frac{d}{\ell \cdot \frac{h}{d}}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\left(0.5 \cdot M\right) \cdot \frac{D}{d}\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 69.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} [d, h, l, M, D] = \mathsf{sort}([d, h, l, M, D])\\ \\ \begin{array}{l} t_0 := \left(0.5 \cdot M\right) \cdot \frac{D}{d}\\ \left(1 - 0.5 \cdot \left(h \cdot \left(t\_0 \cdot \frac{t\_0}{\ell}\right)\right)\right) \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \frac{1}{\sqrt{\frac{h}{d}}}\right) \end{array} \end{array} \]

FPCore

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* (* 0.5 M) (/ D d))))
   (*
    (- 1.0 (* 0.5 (* h (* t_0 (/ t_0 l)))))
    (* (sqrt (/ d l)) (/ 1.0 (sqrt (/ h d)))))))

C

assert(d < h && h < l && l < M && M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = (0.5 * M) * (D / d);
	return (1.0 - (0.5 * (h * (t_0 * (t_0 / l))))) * (sqrt((d / l)) * (1.0 / sqrt((h / d))));
}

Fortran

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
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
    real(8) :: t_0
    t_0 = (0.5d0 * m) * (d_1 / d)
    code = (1.0d0 - (0.5d0 * (h * (t_0 * (t_0 / l))))) * (sqrt((d / l)) * (1.0d0 / sqrt((h / d))))
end function

Java

assert d < h && h < l && l < M && M < D;
public static double code(double d, double h, double l, double M, double D) {
	double t_0 = (0.5 * M) * (D / d);
	return (1.0 - (0.5 * (h * (t_0 * (t_0 / l))))) * (Math.sqrt((d / l)) * (1.0 / Math.sqrt((h / d))));
}

Python

[d, h, l, M, D] = sort([d, h, l, M, D])
def code(d, h, l, M, D):
	t_0 = (0.5 * M) * (D / d)
	return (1.0 - (0.5 * (h * (t_0 * (t_0 / l))))) * (math.sqrt((d / l)) * (1.0 / math.sqrt((h / d))))

Julia

d, h, l, M, D = sort([d, h, l, M, D])
function code(d, h, l, M, D)
	t_0 = Float64(Float64(0.5 * M) * Float64(D / d))
	return Float64(Float64(1.0 - Float64(0.5 * Float64(h * Float64(t_0 * Float64(t_0 / l))))) * Float64(sqrt(Float64(d / l)) * Float64(1.0 / sqrt(Float64(h / d)))))
end

MATLAB

d, h, l, M, D = num2cell(sort([d, h, l, M, D])){:}
function tmp = code(d, h, l, M, D)
	t_0 = (0.5 * M) * (D / d);
	tmp = (1.0 - (0.5 * (h * (t_0 * (t_0 / l))))) * (sqrt((d / l)) * (1.0 / sqrt((h / d))));
end

Wolfram

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(0.5 * M), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision]}, N[(N[(1.0 - N[(0.5 * N[(h * N[(t$95$0 * N[(t$95$0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(1.0 / N[Sqrt[N[(h / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 68.4%

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

3. Simplified 67.3%

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

   1. expm1-log1p-u 67.0%

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

   2. expm1-undefine 67.0%

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

6. Applied egg-rr 67.0%

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

   1. sub-neg 67.0%

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

   2. metadata-eval 67.0%

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

   3. +-commutative 67.0%

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

   4. log1p-undefine 67.0%

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

   5. rem-exp-log 67.3%

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

   6. associate-+r+ 67.3%

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

   7. metadata-eval 67.3%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(\color{blue}{0} + \frac{h}{\ell} \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right)\right) \]

   8. metadata-eval 67.3%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(\color{blue}{\left(0 - 0\right)} + \frac{h}{\ell} \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right)\right) \]

   9. associate--r- 67.3%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(0 - \left(0 - \frac{h}{\ell} \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right)\right)}\right) \]

   10. neg-sub0 67.3%

       \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0 - \color{blue}{\left(-\frac{h}{\ell} \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right)}\right)\right) \]

   11. associate-*l/ 69.5%

       \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0 - \left(-\color{blue}{\frac{h \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell}}\right)\right)\right) \]

   12. distribute-frac-neg2 69.5%

       \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0 - \color{blue}{\frac{h \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{-\ell}}\right)\right) \]

   13. neg-sub0 69.5%

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

   14. distribute-neg-frac2 69.5%

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

   15. remove-double-neg 69.5%

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

8. Simplified 71.4%

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

   1. unpow2 71.4%

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

   2. *-un-lft-identity 71.4%

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

   3. times-frac 74.0%

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

   4. associate-*r/ 73.3%

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

   5. frac-times 72.9%

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

   6. *-commutative 72.9%

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

   7. div-inv 72.9%

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

   8. metadata-eval 72.9%

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

   9. associate-*r/ 72.2%

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

   10. frac-times 72.9%

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

   11. *-commutative 72.9%

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

   12. div-inv 72.9%

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

   13. metadata-eval 72.9%

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

10. Applied egg-rr 72.9%

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

    1. clear-num 72.4%

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

    2. sqrt-div 73.0%

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

    3. metadata-eval 73.0%

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

12. Applied egg-rr 73.0%

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

13. Final simplification 73.0%

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

Alternative 7: 69.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} [d, h, l, M, D] = \mathsf{sort}([d, h, l, M, D])\\ \\ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \left(\left(\left(0.5 \cdot M\right) \cdot \frac{D}{d}\right) \cdot \frac{D \cdot \left(M \cdot \frac{0.5}{d}\right)}{\ell}\right)\right)\right) \end{array} \]

FPCore

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (*
  (* (sqrt (/ d h)) (sqrt (/ d l)))
  (- 1.0 (* 0.5 (* h (* (* (* 0.5 M) (/ D d)) (/ (* D (* M (/ 0.5 d))) l)))))))

C

assert(d < h && h < l && l < M && M < D);
double code(double d, double h, double l, double M, double D) {
	return (sqrt((d / h)) * sqrt((d / l))) * (1.0 - (0.5 * (h * (((0.5 * M) * (D / d)) * ((D * (M * (0.5 / d))) / l)))));
}

Fortran

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
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 = (sqrt((d / h)) * sqrt((d / l))) * (1.0d0 - (0.5d0 * (h * (((0.5d0 * m) * (d_1 / d)) * ((d_1 * (m * (0.5d0 / d))) / l)))))
end function

Java

assert d < h && h < l && l < M && M < D;
public static double code(double d, double h, double l, double M, double D) {
	return (Math.sqrt((d / h)) * Math.sqrt((d / l))) * (1.0 - (0.5 * (h * (((0.5 * M) * (D / d)) * ((D * (M * (0.5 / d))) / l)))));
}

Python

[d, h, l, M, D] = sort([d, h, l, M, D])
def code(d, h, l, M, D):
	return (math.sqrt((d / h)) * math.sqrt((d / l))) * (1.0 - (0.5 * (h * (((0.5 * M) * (D / d)) * ((D * (M * (0.5 / d))) / l)))))

Julia

d, h, l, M, D = sort([d, h, l, M, D])
function code(d, h, l, M, D)
	return Float64(Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l))) * Float64(1.0 - Float64(0.5 * Float64(h * Float64(Float64(Float64(0.5 * M) * Float64(D / d)) * Float64(Float64(D * Float64(M * Float64(0.5 / d))) / l))))))
end

MATLAB

d, h, l, M, D = num2cell(sort([d, h, l, M, D])){:}
function tmp = code(d, h, l, M, D)
	tmp = (sqrt((d / h)) * sqrt((d / l))) * (1.0 - (0.5 * (h * (((0.5 * M) * (D / d)) * ((D * (M * (0.5 / d))) / l)))));
end

Wolfram

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := N[(N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(h * N[(N[(N[(0.5 * M), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision] * N[(N[(D * N[(M * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 68.4%

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

3. Simplified 67.3%

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

   1. expm1-log1p-u 67.0%

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

   2. expm1-undefine 67.0%

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

6. Applied egg-rr 67.0%

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

   1. sub-neg 67.0%

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

   2. metadata-eval 67.0%

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

   3. +-commutative 67.0%

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

   4. log1p-undefine 67.0%

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

   5. rem-exp-log 67.3%

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

   6. associate-+r+ 67.3%

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

   7. metadata-eval 67.3%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(\color{blue}{0} + \frac{h}{\ell} \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right)\right) \]

   8. metadata-eval 67.3%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(\color{blue}{\left(0 - 0\right)} + \frac{h}{\ell} \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right)\right) \]

   9. associate--r- 67.3%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(0 - \left(0 - \frac{h}{\ell} \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right)\right)}\right) \]

   10. neg-sub0 67.3%

       \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0 - \color{blue}{\left(-\frac{h}{\ell} \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right)}\right)\right) \]

   11. associate-*l/ 69.5%

       \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0 - \left(-\color{blue}{\frac{h \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell}}\right)\right)\right) \]

   12. distribute-frac-neg2 69.5%

       \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(0 - \color{blue}{\frac{h \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{-\ell}}\right)\right) \]

   13. neg-sub0 69.5%

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

   14. distribute-neg-frac2 69.5%

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

   15. remove-double-neg 69.5%

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

8. Simplified 71.4%

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

   1. unpow2 71.4%

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

   2. *-un-lft-identity 71.4%

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

   3. times-frac 74.0%

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

   4. associate-*r/ 73.3%

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

   5. frac-times 72.9%

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

   6. *-commutative 72.9%

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

   7. div-inv 72.9%

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

   8. metadata-eval 72.9%

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

   9. associate-*r/ 72.2%

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

   10. frac-times 72.9%

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

   11. *-commutative 72.9%

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

   12. div-inv 72.9%

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

   13. metadata-eval 72.9%

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

10. Applied egg-rr 72.9%

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

11. Taylor expanded in M around 0 72.2%

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

    1. *-commutative 72.2%

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

    2. associate-/l* 72.9%

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

    3. associate-*l* 72.9%

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

    4. associate-*l/ 72.9%

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

    5. associate-/l* 72.9%

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

13. Simplified 72.9%

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

14. Final simplification 72.9%

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

Alternative 8: 46.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} [d, h, l, M, D] = \mathsf{sort}([d, h, l, M, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 2.85 \cdot 10^{-257}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (if (<= d 2.85e-257)
   (* (- d) (sqrt (/ (/ 1.0 l) h)))
   (* d (/ (sqrt (/ 1.0 l)) (sqrt h)))))

C

assert(d < h && h < l && l < M && M < D);
double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= 2.85e-257) {
		tmp = -d * sqrt(((1.0 / l) / h));
	} else {
		tmp = d * (sqrt((1.0 / l)) / sqrt(h));
	}
	return tmp;
}

Fortran

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
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
    real(8) :: tmp
    if (d <= 2.85d-257) then
        tmp = -d * sqrt(((1.0d0 / l) / h))
    else
        tmp = d * (sqrt((1.0d0 / l)) / sqrt(h))
    end if
    code = tmp
end function

Java

assert d < h && h < l && l < M && M < D;
public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= 2.85e-257) {
		tmp = -d * Math.sqrt(((1.0 / l) / h));
	} else {
		tmp = d * (Math.sqrt((1.0 / l)) / Math.sqrt(h));
	}
	return tmp;
}

Python

[d, h, l, M, D] = sort([d, h, l, M, D])
def code(d, h, l, M, D):
	tmp = 0
	if d <= 2.85e-257:
		tmp = -d * math.sqrt(((1.0 / l) / h))
	else:
		tmp = d * (math.sqrt((1.0 / l)) / math.sqrt(h))
	return tmp

Julia

d, h, l, M, D = sort([d, h, l, M, D])
function code(d, h, l, M, D)
	tmp = 0.0
	if (d <= 2.85e-257)
		tmp = Float64(Float64(-d) * sqrt(Float64(Float64(1.0 / l) / h)));
	else
		tmp = Float64(d * Float64(sqrt(Float64(1.0 / l)) / sqrt(h)));
	end
	return tmp
end

MATLAB

d, h, l, M, D = num2cell(sort([d, h, l, M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (d <= 2.85e-257)
		tmp = -d * sqrt(((1.0 / l) / h));
	else
		tmp = d * (sqrt((1.0 / l)) / sqrt(h));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := If[LessEqual[d, 2.85e-257], N[((-d) * N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Sqrt[N[(1.0 / l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 2.8499999999999999e-257

   1. Initial program 65.3%

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

   3. Simplified 64.0%

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

      1. sqrt-div 5.0%

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

      2. clear-num 5.0%

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

   6. Applied egg-rr 5.0%

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

   7. Taylor expanded in l around -inf 0.0%

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

      1. *-commutative 0.0%

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

      2. *-commutative 0.0%

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

      3. associate-/r* 0.0%

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

      4. *-commutative 0.0%

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

      5. unpow2 0.0%

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

      6. rem-square-sqrt 49.7%

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

      7. neg-mul-1 49.7%

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

   9. Simplified 49.7%

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

   if 2.8499999999999999e-257 < 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%

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

      1. sqrt-div 80.9%

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

      2. clear-num 80.9%

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

   6. Applied egg-rr 80.9%

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

   7. Taylor expanded in h around 0 47.0%

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

      1. *-commutative 47.0%

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

   9. Simplified 47.0%

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

       1. associate-/r* 48.0%

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

       2. sqrt-div 58.9%

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

   11. Applied egg-rr 58.9%

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

4. Final simplification 53.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 2.85 \cdot 10^{-257}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 46.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} [d, h, l, M, D] = \mathsf{sort}([d, h, l, M, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 1.95 \cdot 10^{-257}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{\ell}^{-0.5}}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (if (<= d 1.95e-257)
   (* (- d) (sqrt (/ (/ 1.0 l) h)))
   (* d (/ (pow l -0.5) (sqrt h)))))

C

assert(d < h && h < l && l < M && M < D);
double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= 1.95e-257) {
		tmp = -d * sqrt(((1.0 / l) / h));
	} else {
		tmp = d * (pow(l, -0.5) / sqrt(h));
	}
	return tmp;
}

Fortran

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
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
    real(8) :: tmp
    if (d <= 1.95d-257) then
        tmp = -d * sqrt(((1.0d0 / l) / h))
    else
        tmp = d * ((l ** (-0.5d0)) / sqrt(h))
    end if
    code = tmp
end function

Java

assert d < h && h < l && l < M && M < D;
public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= 1.95e-257) {
		tmp = -d * Math.sqrt(((1.0 / l) / h));
	} else {
		tmp = d * (Math.pow(l, -0.5) / Math.sqrt(h));
	}
	return tmp;
}

Python

[d, h, l, M, D] = sort([d, h, l, M, D])
def code(d, h, l, M, D):
	tmp = 0
	if d <= 1.95e-257:
		tmp = -d * math.sqrt(((1.0 / l) / h))
	else:
		tmp = d * (math.pow(l, -0.5) / math.sqrt(h))
	return tmp

Julia

d, h, l, M, D = sort([d, h, l, M, D])
function code(d, h, l, M, D)
	tmp = 0.0
	if (d <= 1.95e-257)
		tmp = Float64(Float64(-d) * sqrt(Float64(Float64(1.0 / l) / h)));
	else
		tmp = Float64(d * Float64((l ^ -0.5) / sqrt(h)));
	end
	return tmp
end

MATLAB

d, h, l, M, D = num2cell(sort([d, h, l, M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (d <= 1.95e-257)
		tmp = -d * sqrt(((1.0 / l) / h));
	else
		tmp = d * ((l ^ -0.5) / sqrt(h));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := If[LessEqual[d, 1.95e-257], N[((-d) * N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[l, -0.5], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 1.9500000000000001e-257

   1. Initial program 65.3%

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

   3. Simplified 64.0%

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

      1. sqrt-div 5.0%

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

      2. clear-num 5.0%

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

   6. Applied egg-rr 5.0%

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

   7. Taylor expanded in l around -inf 0.0%

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

      1. *-commutative 0.0%

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

      2. *-commutative 0.0%

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

      3. associate-/r* 0.0%

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

      4. *-commutative 0.0%

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

      5. unpow2 0.0%

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

      6. rem-square-sqrt 49.7%

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

      7. neg-mul-1 49.7%

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

   9. Simplified 49.7%

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

   if 1.9500000000000001e-257 < 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%

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

      1. sqrt-div 80.9%

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

      2. clear-num 80.9%

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

   6. Applied egg-rr 80.9%

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

   7. Taylor expanded in h around 0 47.0%

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

      1. *-commutative 47.0%

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

   9. Simplified 47.0%

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

   10. Taylor expanded in l around 0 47.0%

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

       1. unpow-1 47.0%

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

       2. metadata-eval 47.0%

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

       3. pow-sqr 47.0%

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

       4. rem-sqrt-square 47.0%

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

       5. rem-square-sqrt 47.0%

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

       6. fabs-sqr 47.0%

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

       7. rem-square-sqrt 47.0%

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

   12. Simplified 47.0%

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

       1. add-sqr-sqrt 47.0%

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

       2. sqrt-unprod 47.0%

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

       3. pow-prod-up 47.0%

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

       4. metadata-eval 47.0%

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

       5. inv-pow 47.0%

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

       6. associate-/l/ 48.0%

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

       7. sqrt-div 58.9%

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

       8. inv-pow 58.9%

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

       9. sqrt-pow1 58.9%

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

       10. metadata-eval 58.9%

           \[\leadsto d \cdot \frac{{\ell}^{\color{blue}{-0.5}}}{\sqrt{h}} \]

   14. Applied egg-rr 58.9%

       \[\leadsto d \cdot \color{blue}{\frac{{\ell}^{-0.5}}{\sqrt{h}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 53.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 1.95 \cdot 10^{-257}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{\ell}^{-0.5}}{\sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 42.6% accurate, 2.9× speedup

Math

\[\begin{array}{l} [d, h, l, M, D] = \mathsf{sort}([d, h, l, M, D])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{if}\;\ell \leq -5.9 \cdot 10^{-274}:\\ \;\;\;\;\left(-d\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;d \cdot t\_0\\ \end{array} \end{array} \]

FPCore

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ (/ 1.0 l) h))))
   (if (<= l -5.9e-274) (* (- d) t_0) (* d t_0))))

C

assert(d < h && h < l && l < M && M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt(((1.0 / l) / h));
	double tmp;
	if (l <= -5.9e-274) {
		tmp = -d * t_0;
	} else {
		tmp = d * t_0;
	}
	return tmp;
}

Fortran

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(((1.0d0 / l) / h))
    if (l <= (-5.9d-274)) then
        tmp = -d * t_0
    else
        tmp = d * t_0
    end if
    code = tmp
end function

Java

assert d < h && h < l && l < M && M < D;
public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt(((1.0 / l) / h));
	double tmp;
	if (l <= -5.9e-274) {
		tmp = -d * t_0;
	} else {
		tmp = d * t_0;
	}
	return tmp;
}

Python

[d, h, l, M, D] = sort([d, h, l, M, D])
def code(d, h, l, M, D):
	t_0 = math.sqrt(((1.0 / l) / h))
	tmp = 0
	if l <= -5.9e-274:
		tmp = -d * t_0
	else:
		tmp = d * t_0
	return tmp

Julia

d, h, l, M, D = sort([d, h, l, M, D])
function code(d, h, l, M, D)
	t_0 = sqrt(Float64(Float64(1.0 / l) / h))
	tmp = 0.0
	if (l <= -5.9e-274)
		tmp = Float64(Float64(-d) * t_0);
	else
		tmp = Float64(d * t_0);
	end
	return tmp
end

MATLAB

d, h, l, M, D = num2cell(sort([d, h, l, M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt(((1.0 / l) / h));
	tmp = 0.0;
	if (l <= -5.9e-274)
		tmp = -d * t_0;
	else
		tmp = d * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -5.9e-274], N[((-d) * t$95$0), $MachinePrecision], N[(d * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < -5.89999999999999953e-274

   1. Initial program 64.0%

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

   3. Simplified 62.4%

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

      1. sqrt-div 0.0%

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

      2. clear-num 0.0%

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

   6. Applied egg-rr 0.0%

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

   7. Taylor expanded in l around -inf 0.0%

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

      1. *-commutative 0.0%

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

      2. *-commutative 0.0%

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

      3. associate-/r* 0.0%

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

      4. *-commutative 0.0%

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

      5. unpow2 0.0%

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

      6. rem-square-sqrt 51.5%

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

      7. neg-mul-1 51.5%

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

   9. Simplified 51.5%

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

   if -5.89999999999999953e-274 < l

   1. Initial program 72.5%

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

   3. Simplified 71.8%

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

      1. sqrt-div 74.6%

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

      2. clear-num 74.6%

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

   6. Applied egg-rr 74.6%

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

   7. Taylor expanded in h around 0 46.4%

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

      1. *-commutative 46.4%

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

      2. associate-/r* 47.3%

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

   9. Simplified 47.3%

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

4. Final simplification 49.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5.9 \cdot 10^{-274}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 42.5% accurate, 3.0× speedup

Math

\[\begin{array}{l} [d, h, l, M, D] = \mathsf{sort}([d, h, l, M, D])\\ \\ \begin{array}{l} t_0 := {\left(\ell \cdot h\right)}^{-0.5}\\ \mathbf{if}\;\ell \leq -4.3 \cdot 10^{-271}:\\ \;\;\;\;\left(-d\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;d \cdot t\_0\\ \end{array} \end{array} \]

FPCore

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (pow (* l h) -0.5)))
   (if (<= l -4.3e-271) (* (- d) t_0) (* d t_0))))

C

assert(d < h && h < l && l < M && M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = pow((l * h), -0.5);
	double tmp;
	if (l <= -4.3e-271) {
		tmp = -d * t_0;
	} else {
		tmp = d * t_0;
	}
	return tmp;
}

Fortran

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (l * h) ** (-0.5d0)
    if (l <= (-4.3d-271)) then
        tmp = -d * t_0
    else
        tmp = d * t_0
    end if
    code = tmp
end function

Java

assert d < h && h < l && l < M && M < D;
public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.pow((l * h), -0.5);
	double tmp;
	if (l <= -4.3e-271) {
		tmp = -d * t_0;
	} else {
		tmp = d * t_0;
	}
	return tmp;
}

Python

[d, h, l, M, D] = sort([d, h, l, M, D])
def code(d, h, l, M, D):
	t_0 = math.pow((l * h), -0.5)
	tmp = 0
	if l <= -4.3e-271:
		tmp = -d * t_0
	else:
		tmp = d * t_0
	return tmp

Julia

d, h, l, M, D = sort([d, h, l, M, D])
function code(d, h, l, M, D)
	t_0 = Float64(l * h) ^ -0.5
	tmp = 0.0
	if (l <= -4.3e-271)
		tmp = Float64(Float64(-d) * t_0);
	else
		tmp = Float64(d * t_0);
	end
	return tmp
end

MATLAB

d, h, l, M, D = num2cell(sort([d, h, l, M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	t_0 = (l * h) ^ -0.5;
	tmp = 0.0;
	if (l <= -4.3e-271)
		tmp = -d * t_0;
	else
		tmp = d * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]}, If[LessEqual[l, -4.3e-271], N[((-d) * t$95$0), $MachinePrecision], N[(d * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < -4.3e-271

   1. Initial program 64.0%

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

   3. Simplified 62.4%

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

      1. sqrt-div 0.0%

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

      2. clear-num 0.0%

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

   6. Applied egg-rr 0.0%

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

   7. Taylor expanded in h around 0 8.5%

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

      1. *-commutative 8.5%

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

   9. Simplified 8.5%

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

   10. Taylor expanded in l around -inf 0.0%

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

       1. *-commutative 0.0%

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

       2. unpow2 0.0%

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

       3. rem-square-sqrt 51.2%

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

       4. unpow-1 51.2%

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

       5. metadata-eval 51.2%

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

       6. pow-sqr 51.2%

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

       7. rem-sqrt-square 51.2%

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

       8. rem-square-sqrt 51.1%

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

       9. fabs-sqr 51.1%

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

       10. rem-square-sqrt 51.2%

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

       11. mul-1-neg 51.2%

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

   12. Simplified 51.2%

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

   if -4.3e-271 < l

   1. Initial program 72.5%

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

   3. Simplified 71.8%

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

      1. sqrt-div 74.6%

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

      2. clear-num 74.6%

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

   6. Applied egg-rr 74.6%

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

   7. Taylor expanded in h around 0 46.4%

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

      1. *-commutative 46.4%

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

   9. Simplified 46.4%

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

   10. Taylor expanded in l around 0 46.4%

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

       1. unpow-1 46.4%

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

       2. metadata-eval 46.4%

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

       3. pow-sqr 46.4%

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

       4. rem-sqrt-square 46.4%

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

       5. rem-square-sqrt 46.4%

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

       6. fabs-sqr 46.4%

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

       7. rem-square-sqrt 46.4%

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

   12. Simplified 46.4%

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

4. Final simplification 48.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4.3 \cdot 10^{-271}:\\ \;\;\;\;\left(-d\right) \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;d \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 42.5% accurate, 3.0× speedup

Math

\[\begin{array}{l} [d, h, l, M, D] = \mathsf{sort}([d, h, l, M, D])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -7.7 \cdot 10^{-265}:\\ \;\;\;\;\left(-d\right) \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (if (<= l -7.7e-265)
   (* (- d) (pow (* l h) -0.5))
   (* d (sqrt (/ (/ 1.0 h) l)))))

C

assert(d < h && h < l && l < M && M < D);
double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -7.7e-265) {
		tmp = -d * pow((l * h), -0.5);
	} else {
		tmp = d * sqrt(((1.0 / h) / l));
	}
	return tmp;
}

Fortran

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
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
    real(8) :: tmp
    if (l <= (-7.7d-265)) then
        tmp = -d * ((l * h) ** (-0.5d0))
    else
        tmp = d * sqrt(((1.0d0 / h) / l))
    end if
    code = tmp
end function

Java

assert d < h && h < l && l < M && M < D;
public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -7.7e-265) {
		tmp = -d * Math.pow((l * h), -0.5);
	} else {
		tmp = d * Math.sqrt(((1.0 / h) / l));
	}
	return tmp;
}

Python

[d, h, l, M, D] = sort([d, h, l, M, D])
def code(d, h, l, M, D):
	tmp = 0
	if l <= -7.7e-265:
		tmp = -d * math.pow((l * h), -0.5)
	else:
		tmp = d * math.sqrt(((1.0 / h) / l))
	return tmp

Julia

d, h, l, M, D = sort([d, h, l, M, D])
function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= -7.7e-265)
		tmp = Float64(Float64(-d) * (Float64(l * h) ^ -0.5));
	else
		tmp = Float64(d * sqrt(Float64(Float64(1.0 / h) / l)));
	end
	return tmp
end

MATLAB

d, h, l, M, D = num2cell(sort([d, h, l, M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (l <= -7.7e-265)
		tmp = -d * ((l * h) ^ -0.5);
	else
		tmp = d * sqrt(((1.0 / h) / l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := If[LessEqual[l, -7.7e-265], N[((-d) * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], N[(d * N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -7.70000000000000023e-265

   1. Initial program 64.0%

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

   3. Simplified 62.4%

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

      1. sqrt-div 0.0%

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

      2. clear-num 0.0%

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

   6. Applied egg-rr 0.0%

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

   7. Taylor expanded in h around 0 8.5%

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

      1. *-commutative 8.5%

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

   9. Simplified 8.5%

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

   10. Taylor expanded in l around -inf 0.0%

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

       1. *-commutative 0.0%

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

       2. unpow2 0.0%

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

       3. rem-square-sqrt 51.2%

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

       4. unpow-1 51.2%

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

       5. metadata-eval 51.2%

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

       6. pow-sqr 51.2%

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

       7. rem-sqrt-square 51.2%

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

       8. rem-square-sqrt 51.1%

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

       9. fabs-sqr 51.1%

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

       10. rem-square-sqrt 51.2%

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

       11. mul-1-neg 51.2%

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

   12. Simplified 51.2%

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

   if -7.70000000000000023e-265 < l

   1. Initial program 72.5%

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

   3. Simplified 71.8%

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

      1. sqrt-div 74.6%

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

      2. clear-num 74.6%

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

   6. Applied egg-rr 74.6%

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

   7. Taylor expanded in h around 0 46.4%

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

      1. *-commutative 46.4%

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

   9. Simplified 46.4%

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

   10. Taylor expanded in l around 0 46.4%

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

       1. unpow-1 46.4%

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

       2. metadata-eval 46.4%

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

       3. pow-sqr 46.4%

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

       4. rem-sqrt-square 46.4%

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

       5. rem-square-sqrt 46.4%

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

       6. fabs-sqr 46.4%

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

       7. rem-square-sqrt 46.4%

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

   12. Simplified 46.4%

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

   13. Taylor expanded in h around 0 46.4%

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

       1. associate-/r* 47.3%

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

   15. Simplified 47.3%

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

4. Final simplification 49.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -7.7 \cdot 10^{-265}:\\ \;\;\;\;\left(-d\right) \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 42.5% accurate, 3.0× speedup

Math

\[\begin{array}{l} [d, h, l, M, D] = \mathsf{sort}([d, h, l, M, D])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -2.3 \cdot 10^{-267}:\\ \;\;\;\;\left(-d\right) \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \end{array} \end{array} \]

FPCore

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (if (<= l -2.3e-267)
   (* (- d) (pow (* l h) -0.5))
   (* d (sqrt (/ (/ 1.0 l) h)))))

C

assert(d < h && h < l && l < M && M < D);
double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -2.3e-267) {
		tmp = -d * pow((l * h), -0.5);
	} else {
		tmp = d * sqrt(((1.0 / l) / h));
	}
	return tmp;
}

Fortran

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
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
    real(8) :: tmp
    if (l <= (-2.3d-267)) then
        tmp = -d * ((l * h) ** (-0.5d0))
    else
        tmp = d * sqrt(((1.0d0 / l) / h))
    end if
    code = tmp
end function

Java

assert d < h && h < l && l < M && M < D;
public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -2.3e-267) {
		tmp = -d * Math.pow((l * h), -0.5);
	} else {
		tmp = d * Math.sqrt(((1.0 / l) / h));
	}
	return tmp;
}

Python

[d, h, l, M, D] = sort([d, h, l, M, D])
def code(d, h, l, M, D):
	tmp = 0
	if l <= -2.3e-267:
		tmp = -d * math.pow((l * h), -0.5)
	else:
		tmp = d * math.sqrt(((1.0 / l) / h))
	return tmp

Julia

d, h, l, M, D = sort([d, h, l, M, D])
function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= -2.3e-267)
		tmp = Float64(Float64(-d) * (Float64(l * h) ^ -0.5));
	else
		tmp = Float64(d * sqrt(Float64(Float64(1.0 / l) / h)));
	end
	return tmp
end

MATLAB

d, h, l, M, D = num2cell(sort([d, h, l, M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (l <= -2.3e-267)
		tmp = -d * ((l * h) ^ -0.5);
	else
		tmp = d * sqrt(((1.0 / l) / h));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := If[LessEqual[l, -2.3e-267], N[((-d) * N[Power[N[(l * h), $MachinePrecision], -0.5], $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 l < -2.30000000000000005e-267

   1. Initial program 64.0%

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

   3. Simplified 62.4%

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

      1. sqrt-div 0.0%

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

      2. clear-num 0.0%

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

   6. Applied egg-rr 0.0%

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

   7. Taylor expanded in h around 0 8.5%

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

      1. *-commutative 8.5%

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

   9. Simplified 8.5%

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

   10. Taylor expanded in l around -inf 0.0%

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

       1. *-commutative 0.0%

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

       2. unpow2 0.0%

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

       3. rem-square-sqrt 51.2%

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

       4. unpow-1 51.2%

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

       5. metadata-eval 51.2%

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

       6. pow-sqr 51.2%

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

       7. rem-sqrt-square 51.2%

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

       8. rem-square-sqrt 51.1%

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

       9. fabs-sqr 51.1%

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

       10. rem-square-sqrt 51.2%

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

       11. mul-1-neg 51.2%

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

   12. Simplified 51.2%

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

   if -2.30000000000000005e-267 < l

   1. Initial program 72.5%

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

   3. Simplified 71.8%

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

      1. sqrt-div 74.6%

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

      2. clear-num 74.6%

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

   6. Applied egg-rr 74.6%

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

   7. Taylor expanded in h around 0 46.4%

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

      1. *-commutative 46.4%

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

      2. associate-/r* 47.3%

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

   9. Simplified 47.3%

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

4. Final simplification 49.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.3 \cdot 10^{-267}:\\ \;\;\;\;\left(-d\right) \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 26.2% accurate, 3.1× speedup

Math

\[\begin{array}{l} [d, h, l, M, D] = \mathsf{sort}([d, h, l, M, D])\\ \\ d \cdot {\left(\ell \cdot h\right)}^{-0.5} \end{array} \]

FPCore

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D) :precision binary64 (* d (pow (* l h) -0.5)))

C

assert(d < h && h < l && l < M && M < D);
double code(double d, double h, double l, double M, double D) {
	return d * pow((l * h), -0.5);
}

Fortran

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
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 * ((l * h) ** (-0.5d0))
end function

Java

assert d < h && h < l && l < M && M < D;
public static double code(double d, double h, double l, double M, double D) {
	return d * Math.pow((l * h), -0.5);
}

Python

[d, h, l, M, D] = sort([d, h, l, M, D])
def code(d, h, l, M, D):
	return d * math.pow((l * h), -0.5)

Julia

d, h, l, M, D = sort([d, h, l, M, D])
function code(d, h, l, M, D)
	return Float64(d * (Float64(l * h) ^ -0.5))
end

MATLAB

d, h, l, M, D = num2cell(sort([d, h, l, M, D])){:}
function tmp = code(d, h, l, M, D)
	tmp = d * ((l * h) ^ -0.5);
end

Wolfram

NOTE: d, h, l, M, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := N[(d * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 68.4%

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

3. Simplified 67.3%

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

   1. sqrt-div 38.8%

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

   2. clear-num 38.8%

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

6. Applied egg-rr 38.8%

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

7. Taylor expanded in h around 0 28.2%

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

   1. *-commutative 28.2%

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

9. Simplified 28.2%

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

10. Taylor expanded in l around 0 28.2%

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

    1. unpow-1 28.2%

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

    2. metadata-eval 28.2%

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

    3. pow-sqr 28.2%

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

    4. rem-sqrt-square 28.2%

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

    5. rem-square-sqrt 28.2%

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

    6. fabs-sqr 28.2%

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

    7. rem-square-sqrt 28.2%

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

12. Simplified 28.2%

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

13. Final simplification 28.2%

    \[\leadsto d \cdot {\left(\ell \cdot h\right)}^{-0.5} \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024084 
(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)))))