Henrywood and Agarwal, Equation (12)

Percentage Accurate: 66.7% → 80.0%

Time: 23.0s

Alternatives: 13

Speedup: 0.6×

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: 66.7% 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: 80.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := 1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\\ t_2 := \frac{\sqrt{d}}{\sqrt{h}}\\ \mathbf{if}\;\ell \leq -1.62 \cdot 10^{-301}:\\ \;\;\;\;{\left({\left(\frac{-1}{h}\right)}^{0.25} \cdot {\left(-d\right)}^{0.25}\right)}^{2} \cdot \left(t_0 \cdot t_1\right)\\ \mathbf{elif}\;\ell \leq 1.36 \cdot 10^{+87}:\\ \;\;\;\;t_2 \cdot \left(t_0 \cdot \left(1 - 0.5 \cdot \frac{h \cdot {\left(\frac{D}{\frac{d}{0.5 \cdot M}}\right)}^{2}}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot \left(t_1 \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ d l)))
        (t_1 (- 1.0 (* 0.5 (* (pow (* (/ M 2.0) (/ D d)) 2.0) (/ h l)))))
        (t_2 (/ (sqrt d) (sqrt h))))
   (if (<= l -1.62e-301)
     (* (pow (* (pow (/ -1.0 h) 0.25) (pow (- d) 0.25)) 2.0) (* t_0 t_1))
     (if (<= l 1.36e+87)
       (*
        t_2
        (* t_0 (- 1.0 (* 0.5 (/ (* h (pow (/ D (/ d (* 0.5 M))) 2.0)) l)))))
       (* t_2 (* t_1 (/ (sqrt d) (sqrt l))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((d / l));
	double t_1 = 1.0 - (0.5 * (pow(((M / 2.0) * (D / d)), 2.0) * (h / l)));
	double t_2 = sqrt(d) / sqrt(h);
	double tmp;
	if (l <= -1.62e-301) {
		tmp = pow((pow((-1.0 / h), 0.25) * pow(-d, 0.25)), 2.0) * (t_0 * t_1);
	} else if (l <= 1.36e+87) {
		tmp = t_2 * (t_0 * (1.0 - (0.5 * ((h * pow((D / (d / (0.5 * M))), 2.0)) / l))));
	} else {
		tmp = t_2 * (t_1 * (sqrt(d) / sqrt(l)));
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = sqrt((d / l))
    t_1 = 1.0d0 - (0.5d0 * ((((m / 2.0d0) * (d_1 / d)) ** 2.0d0) * (h / l)))
    t_2 = sqrt(d) / sqrt(h)
    if (l <= (-1.62d-301)) then
        tmp = (((((-1.0d0) / h) ** 0.25d0) * (-d ** 0.25d0)) ** 2.0d0) * (t_0 * t_1)
    else if (l <= 1.36d+87) then
        tmp = t_2 * (t_0 * (1.0d0 - (0.5d0 * ((h * ((d_1 / (d / (0.5d0 * m))) ** 2.0d0)) / l))))
    else
        tmp = t_2 * (t_1 * (sqrt(d) / sqrt(l)))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt((d / l));
	double t_1 = 1.0 - (0.5 * (Math.pow(((M / 2.0) * (D / d)), 2.0) * (h / l)));
	double t_2 = Math.sqrt(d) / Math.sqrt(h);
	double tmp;
	if (l <= -1.62e-301) {
		tmp = Math.pow((Math.pow((-1.0 / h), 0.25) * Math.pow(-d, 0.25)), 2.0) * (t_0 * t_1);
	} else if (l <= 1.36e+87) {
		tmp = t_2 * (t_0 * (1.0 - (0.5 * ((h * Math.pow((D / (d / (0.5 * M))), 2.0)) / l))));
	} else {
		tmp = t_2 * (t_1 * (Math.sqrt(d) / Math.sqrt(l)));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.sqrt((d / l))
	t_1 = 1.0 - (0.5 * (math.pow(((M / 2.0) * (D / d)), 2.0) * (h / l)))
	t_2 = math.sqrt(d) / math.sqrt(h)
	tmp = 0
	if l <= -1.62e-301:
		tmp = math.pow((math.pow((-1.0 / h), 0.25) * math.pow(-d, 0.25)), 2.0) * (t_0 * t_1)
	elif l <= 1.36e+87:
		tmp = t_2 * (t_0 * (1.0 - (0.5 * ((h * math.pow((D / (d / (0.5 * M))), 2.0)) / l))))
	else:
		tmp = t_2 * (t_1 * (math.sqrt(d) / math.sqrt(l)))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(d / l))
	t_1 = Float64(1.0 - Float64(0.5 * Float64((Float64(Float64(M / 2.0) * Float64(D / d)) ^ 2.0) * Float64(h / l))))
	t_2 = Float64(sqrt(d) / sqrt(h))
	tmp = 0.0
	if (l <= -1.62e-301)
		tmp = Float64((Float64((Float64(-1.0 / h) ^ 0.25) * (Float64(-d) ^ 0.25)) ^ 2.0) * Float64(t_0 * t_1));
	elseif (l <= 1.36e+87)
		tmp = Float64(t_2 * Float64(t_0 * Float64(1.0 - Float64(0.5 * Float64(Float64(h * (Float64(D / Float64(d / Float64(0.5 * M))) ^ 2.0)) / l)))));
	else
		tmp = Float64(t_2 * Float64(t_1 * Float64(sqrt(d) / sqrt(l))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt((d / l));
	t_1 = 1.0 - (0.5 * ((((M / 2.0) * (D / d)) ^ 2.0) * (h / l)));
	t_2 = sqrt(d) / sqrt(h);
	tmp = 0.0;
	if (l <= -1.62e-301)
		tmp = ((((-1.0 / h) ^ 0.25) * (-d ^ 0.25)) ^ 2.0) * (t_0 * t_1);
	elseif (l <= 1.36e+87)
		tmp = t_2 * (t_0 * (1.0 - (0.5 * ((h * ((D / (d / (0.5 * M))) ^ 2.0)) / l))));
	else
		tmp = t_2 * (t_1 * (sqrt(d) / sqrt(l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - N[(0.5 * N[(N[Power[N[(N[(M / 2.0), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -1.62e-301], N[(N[Power[N[(N[Power[N[(-1.0 / h), $MachinePrecision], 0.25], $MachinePrecision] * N[Power[(-d), 0.25], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(t$95$0 * t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.36e+87], N[(t$95$2 * N[(t$95$0 * N[(1.0 - N[(0.5 * N[(N[(h * N[Power[N[(D / N[(d / N[(0.5 * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$2 * N[(t$95$1 * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.61999999999999993e-301

   1. Initial program 62.1%

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

      1. associate-*l* 62.2%

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

      2. metadata-eval 62.2%

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

      3. unpow1/2 62.2%

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

      4. metadata-eval 62.2%

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

      5. unpow1/2 62.2%

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

      6. associate-*l* 62.2%

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

      7. metadata-eval 62.2%

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

      8. times-frac 62.1%

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

   3. Simplified 62.1%

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

      1. pow1/2 62.1%

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

      2. sqr-pow 62.0%

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

      3. pow2 62.0%

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

      4. metadata-eval 62.0%

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

   5. Applied egg-rr 62.0%

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

   6. Taylor expanded in h around -inf 73.9%

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

      1. distribute-lft-in 73.9%

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

      2. exp-sum 74.1%

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

      3. *-commutative 74.1%

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

      4. exp-to-pow 74.4%

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

      5. *-commutative 74.4%

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

      6. rem-square-sqrt 0.0%

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

      7. unpow2 0.0%

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

      8. exp-to-pow 0.0%

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

      9. unpow2 0.0%

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

      10. rem-square-sqrt 76.4%

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

      11. mul-1-neg 76.4%

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

   8. Simplified 76.4%

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

   if -1.61999999999999993e-301 < l < 1.3599999999999999e87

   1. Initial program 80.3%

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

      1. associate-*l* 80.4%

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

      2. metadata-eval 80.4%

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

      3. unpow1/2 80.4%

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

      4. metadata-eval 80.4%

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

      5. unpow1/2 80.4%

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

      6. associate-*l* 80.4%

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

      7. metadata-eval 80.4%

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

      8. times-frac 79.2%

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

   3. Simplified 79.2%

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

      1. associate-*r/ 86.3%

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

      2. div-inv 86.3%

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

      3. metadata-eval 86.3%

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

   5. Applied egg-rr 86.3%

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

   6. Taylor expanded in M around 0 87.5%

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

      1. *-commutative 87.5%

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

      2. *-commutative 87.5%

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

      3. associate-*l/ 87.5%

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

      4. *-commutative 87.5%

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

      5. associate-*r* 87.5%

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

      6. associate-/l* 87.5%

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

      7. *-commutative 87.5%

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

   8. Simplified 87.5%

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

      1. sqrt-div 94.3%

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

   10. Applied egg-rr 94.3%

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

   if 1.3599999999999999e87 < l

   1. Initial program 53.2%

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

      1. associate-*l* 53.2%

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

      2. metadata-eval 53.2%

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

      3. unpow1/2 53.2%

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

      4. metadata-eval 53.2%

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

      5. unpow1/2 53.2%

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

      6. associate-*l* 53.2%

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

      7. metadata-eval 53.2%

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

      8. times-frac 53.3%

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

   3. Simplified 53.3%

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

      1. sqrt-div 65.4%

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

   5. Applied egg-rr 65.4%

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

      1. sqrt-div 68.6%

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

   7. Applied egg-rr 82.5%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.62 \cdot 10^{-301}:\\ \;\;\;\;{\left({\left(\frac{-1}{h}\right)}^{0.25} \cdot {\left(-d\right)}^{0.25}\right)}^{2} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 1.36 \cdot 10^{+87}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \frac{h \cdot {\left(\frac{D}{\frac{d}{0.5 \cdot M}}\right)}^{2}}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right)\\ \end{array} \]

Alternative 2: 77.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right)\right)\\ \mathbf{if}\;t_0 \leq -2 \cdot 10^{-125} \lor \neg \left(t_0 \leq 0\right) \land t_0 \leq 5 \cdot 10^{+163}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{{\left(\frac{D}{\frac{d}{0.5 \cdot M}}\right)}^{2}}{\ell}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left|d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0
         (*
          (* (pow (/ d h) 0.5) (pow (/ d l) 0.5))
          (- 1.0 (* (/ h l) (* 0.5 (pow (/ (* M D) (* d 2.0)) 2.0)))))))
   (if (or (<= t_0 -2e-125) (and (not (<= t_0 0.0)) (<= t_0 5e+163)))
     (*
      (sqrt (/ d h))
      (*
       (sqrt (/ d l))
       (- 1.0 (* 0.5 (* h (/ (pow (/ D (/ d (* 0.5 M))) 2.0) l))))))
     (fabs (* d (pow (* l h) -0.5))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (pow((d / h), 0.5) * pow((d / l), 0.5)) * (1.0 - ((h / l) * (0.5 * pow(((M * D) / (d * 2.0)), 2.0))));
	double tmp;
	if ((t_0 <= -2e-125) || (!(t_0 <= 0.0) && (t_0 <= 5e+163))) {
		tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0 - (0.5 * (h * (pow((D / (d / (0.5 * M))), 2.0) / l)))));
	} else {
		tmp = fabs((d * pow((l * h), -0.5)));
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (((d / h) ** 0.5d0) * ((d / l) ** 0.5d0)) * (1.0d0 - ((h / l) * (0.5d0 * (((m * d_1) / (d * 2.0d0)) ** 2.0d0))))
    if ((t_0 <= (-2d-125)) .or. (.not. (t_0 <= 0.0d0)) .and. (t_0 <= 5d+163)) then
        tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0d0 - (0.5d0 * (h * (((d_1 / (d / (0.5d0 * m))) ** 2.0d0) / l)))))
    else
        tmp = abs((d * ((l * h) ** (-0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = (Math.pow((d / h), 0.5) * Math.pow((d / l), 0.5)) * (1.0 - ((h / l) * (0.5 * Math.pow(((M * D) / (d * 2.0)), 2.0))));
	double tmp;
	if ((t_0 <= -2e-125) || (!(t_0 <= 0.0) && (t_0 <= 5e+163))) {
		tmp = Math.sqrt((d / h)) * (Math.sqrt((d / l)) * (1.0 - (0.5 * (h * (Math.pow((D / (d / (0.5 * M))), 2.0) / l)))));
	} else {
		tmp = Math.abs((d * Math.pow((l * h), -0.5)));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = (math.pow((d / h), 0.5) * math.pow((d / l), 0.5)) * (1.0 - ((h / l) * (0.5 * math.pow(((M * D) / (d * 2.0)), 2.0))))
	tmp = 0
	if (t_0 <= -2e-125) or (not (t_0 <= 0.0) and (t_0 <= 5e+163)):
		tmp = math.sqrt((d / h)) * (math.sqrt((d / l)) * (1.0 - (0.5 * (h * (math.pow((D / (d / (0.5 * M))), 2.0) / l)))))
	else:
		tmp = math.fabs((d * math.pow((l * h), -0.5)))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64((Float64(d / h) ^ 0.5) * (Float64(d / l) ^ 0.5)) * Float64(1.0 - Float64(Float64(h / l) * Float64(0.5 * (Float64(Float64(M * D) / Float64(d * 2.0)) ^ 2.0)))))
	tmp = 0.0
	if ((t_0 <= -2e-125) || (!(t_0 <= 0.0) && (t_0 <= 5e+163)))
		tmp = Float64(sqrt(Float64(d / h)) * Float64(sqrt(Float64(d / l)) * Float64(1.0 - Float64(0.5 * Float64(h * Float64((Float64(D / Float64(d / Float64(0.5 * M))) ^ 2.0) / l))))));
	else
		tmp = abs(Float64(d * (Float64(l * h) ^ -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = (((d / h) ^ 0.5) * ((d / l) ^ 0.5)) * (1.0 - ((h / l) * (0.5 * (((M * D) / (d * 2.0)) ^ 2.0))));
	tmp = 0.0;
	if ((t_0 <= -2e-125) || (~((t_0 <= 0.0)) && (t_0 <= 5e+163)))
		tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0 - (0.5 * (h * (((D / (d / (0.5 * M))) ^ 2.0) / l)))));
	else
		tmp = abs((d * ((l * h) ^ -0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(N[Power[N[(d / h), $MachinePrecision], 0.5], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(0.5 * N[Power[N[(N[(M * D), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -2e-125], And[N[Not[LessEqual[t$95$0, 0.0]], $MachinePrecision], LessEqual[t$95$0, 5e+163]]], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(0.5 * N[(h * N[(N[Power[N[(D / N[(d / N[(0.5 * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Abs[N[(d * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (-.f64 1 (*.f64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)) (/.f64 h l)))) < -2.00000000000000002e-125 or 0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (-.f64 1 (*.f64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)) (/.f64 h l)))) < 5e163

   1. Initial program 93.7%

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

      1. associate-*l* 93.7%

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

      2. metadata-eval 93.7%

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

      3. unpow1/2 93.7%

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

      4. metadata-eval 93.7%

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

      5. unpow1/2 93.7%

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

      6. associate-*l* 93.7%

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

      7. metadata-eval 93.7%

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

      8. times-frac 93.0%

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

   3. Simplified 93.0%

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

      1. associate-*r/ 92.4%

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

      2. div-inv 92.4%

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

      3. metadata-eval 92.4%

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

   5. Applied egg-rr 92.4%

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

      1. metadata-eval 92.4%

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

      2. div-inv 92.4%

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

      3. associate-*r/ 93.0%

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

      4. expm1-log1p-u 92.2%

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

      5. expm1-udef 92.2%

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

      6. *-commutative 92.2%

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

      7. div-inv 92.2%

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

      8. metadata-eval 92.2%

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

   7. Applied egg-rr 92.2%

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

      1. expm1-def 92.2%

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

      2. expm1-log1p 93.0%

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

      3. *-commutative 93.0%

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

      4. associate-*l/ 92.4%

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

      5. *-rgt-identity 92.4%

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

      6. associate-*r/ 92.4%

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

      7. associate-*l* 93.7%

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

      8. associate-*r/ 93.7%

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

      9. *-rgt-identity 93.7%

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

      10. associate-*l/ 94.4%

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

      11. associate-/l* 93.0%

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

      12. *-commutative 93.0%

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

   9. Simplified 93.0%

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

   if -2.00000000000000002e-125 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (-.f64 1 (*.f64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)) (/.f64 h l)))) < 0.0 or 5e163 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (-.f64 1 (*.f64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)) (/.f64 h l))))

   1. Initial program 29.8%

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

      1. metadata-eval 29.8%

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

      2. unpow1/2 29.8%

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

      3. metadata-eval 29.8%

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

      4. unpow1/2 29.8%

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

      5. *-commutative 29.8%

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

      6. associate-*l* 29.8%

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

      7. times-frac 29.7%

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

      8. metadata-eval 29.7%

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

   3. Simplified 29.7%

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

   4. Taylor expanded in d around inf 39.9%

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

      1. expm1-log1p-u 39.1%

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

      2. expm1-udef 29.0%

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

      3. *-commutative 29.0%

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

   6. Applied egg-rr 29.0%

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

      1. expm1-def 39.1%

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

      2. expm1-log1p 39.9%

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

      3. unpow-1 39.9%

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

      4. sqr-pow 39.9%

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

      5. rem-sqrt-square 39.9%

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

      6. sqr-pow 39.7%

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

      7. fabs-sqr 39.7%

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

      8. sqr-pow 39.9%

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

      9. metadata-eval 39.9%

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

   8. Simplified 39.9%

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

      1. add-cbrt-cube 38.9%

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

   10. Applied egg-rr 38.9%

       \[\leadsto \color{blue}{\sqrt[3]{\left({\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}\right) \cdot {\left(h \cdot \ell\right)}^{-0.5}}} \cdot d \]
   11. Step-by-step derivation

       1. associate-*l* 38.9%

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

       2. pow-sqr 38.9%

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

       3. metadata-eval 38.9%

          \[\leadsto \sqrt[3]{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{\color{blue}{-1}}} \cdot d \]

       4. unpow-1 38.9%

          \[\leadsto \sqrt[3]{{\left(h \cdot \ell\right)}^{-0.5} \cdot \color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

   12. Simplified 38.9%

       \[\leadsto \color{blue}{\sqrt[3]{{\left(h \cdot \ell\right)}^{-0.5} \cdot \frac{1}{h \cdot \ell}}} \cdot d \]
   13. Step-by-step derivation

       1. *-commutative 38.9%

          \[\leadsto \color{blue}{d \cdot \sqrt[3]{{\left(h \cdot \ell\right)}^{-0.5} \cdot \frac{1}{h \cdot \ell}}} \]

       2. add-sqr-sqrt 38.8%

          \[\leadsto d \cdot \color{blue}{\left(\sqrt{\sqrt[3]{{\left(h \cdot \ell\right)}^{-0.5} \cdot \frac{1}{h \cdot \ell}}} \cdot \sqrt{\sqrt[3]{{\left(h \cdot \ell\right)}^{-0.5} \cdot \frac{1}{h \cdot \ell}}}\right)} \]

       3. pow2 38.8%

          \[\leadsto d \cdot \color{blue}{{\left(\sqrt{\sqrt[3]{{\left(h \cdot \ell\right)}^{-0.5} \cdot \frac{1}{h \cdot \ell}}}\right)}^{2}} \]

   14. Applied egg-rr 36.1%

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

       1. unpow2 36.1%

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

       2. rem-sqrt-square 65.4%

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

   16. Simplified 65.4%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right)\right) \leq -2 \cdot 10^{-125} \lor \neg \left(\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right)\right) \leq 0\right) \land \left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right)\right) \leq 5 \cdot 10^{+163}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{{\left(\frac{D}{\frac{d}{0.5 \cdot M}}\right)}^{2}}{\ell}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left|d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right|\\ \end{array} \]

Alternative 3: 58.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq -2.9 \cdot 10^{+53}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\\ \mathbf{elif}\;M \leq -1.7 \cdot 10^{-119} \lor \neg \left(M \leq -4.5 \cdot 10^{-176}\right) \land M \leq 2.45 \cdot 10^{-189}:\\ \;\;\;\;\left|d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right|\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{D}{\frac{d}{D}} \cdot \frac{h \cdot \frac{M}{\frac{\ell}{M}}}{d}\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= M -2.9e+53)
   (*
    (sqrt (* (/ d l) (/ d h)))
    (- 1.0 (* (pow (* M (* 0.5 (/ D d))) 2.0) (* 0.5 (/ h l)))))
   (if (or (<= M -1.7e-119) (and (not (<= M -4.5e-176)) (<= M 2.45e-189)))
     (fabs (* d (pow (* l h) -0.5)))
     (*
      (sqrt (/ d h))
      (*
       (sqrt (/ d l))
       (-
        1.0
        (* 0.5 (* 0.25 (* (/ D (/ d D)) (/ (* h (/ M (/ l M))) d))))))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (M <= -2.9e+53) {
		tmp = sqrt(((d / l) * (d / h))) * (1.0 - (pow((M * (0.5 * (D / d))), 2.0) * (0.5 * (h / l))));
	} else if ((M <= -1.7e-119) || (!(M <= -4.5e-176) && (M <= 2.45e-189))) {
		tmp = fabs((d * pow((l * h), -0.5)));
	} else {
		tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0 - (0.5 * (0.25 * ((D / (d / D)) * ((h * (M / (l / M))) / d))))));
	}
	return tmp;
}

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
    real(8) :: tmp
    if (m <= (-2.9d+53)) then
        tmp = sqrt(((d / l) * (d / h))) * (1.0d0 - (((m * (0.5d0 * (d_1 / d))) ** 2.0d0) * (0.5d0 * (h / l))))
    else if ((m <= (-1.7d-119)) .or. (.not. (m <= (-4.5d-176))) .and. (m <= 2.45d-189)) then
        tmp = abs((d * ((l * h) ** (-0.5d0))))
    else
        tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0d0 - (0.5d0 * (0.25d0 * ((d_1 / (d / d_1)) * ((h * (m / (l / m))) / d))))))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (M <= -2.9e+53) {
		tmp = Math.sqrt(((d / l) * (d / h))) * (1.0 - (Math.pow((M * (0.5 * (D / d))), 2.0) * (0.5 * (h / l))));
	} else if ((M <= -1.7e-119) || (!(M <= -4.5e-176) && (M <= 2.45e-189))) {
		tmp = Math.abs((d * Math.pow((l * h), -0.5)));
	} else {
		tmp = Math.sqrt((d / h)) * (Math.sqrt((d / l)) * (1.0 - (0.5 * (0.25 * ((D / (d / D)) * ((h * (M / (l / M))) / d))))));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if M <= -2.9e+53:
		tmp = math.sqrt(((d / l) * (d / h))) * (1.0 - (math.pow((M * (0.5 * (D / d))), 2.0) * (0.5 * (h / l))))
	elif (M <= -1.7e-119) or (not (M <= -4.5e-176) and (M <= 2.45e-189)):
		tmp = math.fabs((d * math.pow((l * h), -0.5)))
	else:
		tmp = math.sqrt((d / h)) * (math.sqrt((d / l)) * (1.0 - (0.5 * (0.25 * ((D / (d / D)) * ((h * (M / (l / M))) / d))))))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (M <= -2.9e+53)
		tmp = Float64(sqrt(Float64(Float64(d / l) * Float64(d / h))) * Float64(1.0 - Float64((Float64(M * Float64(0.5 * Float64(D / d))) ^ 2.0) * Float64(0.5 * Float64(h / l)))));
	elseif ((M <= -1.7e-119) || (!(M <= -4.5e-176) && (M <= 2.45e-189)))
		tmp = abs(Float64(d * (Float64(l * h) ^ -0.5)));
	else
		tmp = Float64(sqrt(Float64(d / h)) * Float64(sqrt(Float64(d / l)) * Float64(1.0 - Float64(0.5 * Float64(0.25 * Float64(Float64(D / Float64(d / D)) * Float64(Float64(h * Float64(M / Float64(l / M))) / d)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (M <= -2.9e+53)
		tmp = sqrt(((d / l) * (d / h))) * (1.0 - (((M * (0.5 * (D / d))) ^ 2.0) * (0.5 * (h / l))));
	elseif ((M <= -1.7e-119) || (~((M <= -4.5e-176)) && (M <= 2.45e-189)))
		tmp = abs((d * ((l * h) ^ -0.5)));
	else
		tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0 - (0.5 * (0.25 * ((D / (d / D)) * ((h * (M / (l / M))) / d))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[M, -2.9e+53], N[(N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(N[Power[N[(M * N[(0.5 * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(0.5 * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[M, -1.7e-119], And[N[Not[LessEqual[M, -4.5e-176]], $MachinePrecision], LessEqual[M, 2.45e-189]]], N[Abs[N[(d * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(0.5 * N[(0.25 * N[(N[(D / N[(d / D), $MachinePrecision]), $MachinePrecision] * N[(N[(h * N[(M / N[(l / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if M < -2.9000000000000002e53

   1. Initial program 74.7%

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

      1. metadata-eval 74.7%

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

      2. unpow1/2 74.7%

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

      3. metadata-eval 74.7%

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

      4. unpow1/2 74.7%

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

      5. *-commutative 74.7%

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

      6. associate-*l* 74.7%

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

      7. times-frac 72.8%

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

      8. metadata-eval 72.8%

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

   3. Simplified 72.8%

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

   5. Applied egg-rr 19.8%

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

      1. expm1-def 21.7%

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

      2. expm1-log1p 63.4%

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

      3. sub-neg 63.4%

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

      4. associate-*r* 63.4%

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

      5. distribute-rgt-neg-in 63.4%

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

      6. *-commutative 63.4%

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

      7. metadata-eval 63.4%

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

      8. associate-*r* 63.4%

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

      9. +-commutative 63.4%

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

      10. fma-def 63.4%

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

      11. fma-def 63.4%

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

   7. Simplified 63.4%

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

   if -2.9000000000000002e53 < M < -1.70000000000000012e-119 or -4.5e-176 < M < 2.4499999999999999e-189

   1. Initial program 63.2%

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

      1. metadata-eval 63.2%

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

      2. unpow1/2 63.2%

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

      3. metadata-eval 63.2%

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

      4. unpow1/2 63.2%

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

      5. *-commutative 63.2%

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

      6. associate-*l* 63.2%

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

      7. times-frac 63.1%

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

      8. metadata-eval 63.1%

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

   3. Simplified 63.1%

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

   4. Taylor expanded in d around inf 45.7%

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

      1. expm1-log1p-u 44.6%

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

      2. expm1-udef 27.3%

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

      3. *-commutative 27.3%

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

   6. Applied egg-rr 27.3%

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

      1. expm1-def 44.6%

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

      2. expm1-log1p 45.7%

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

      3. unpow-1 45.7%

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

      4. sqr-pow 45.7%

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

      5. rem-sqrt-square 46.9%

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

      6. sqr-pow 46.6%

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

      7. fabs-sqr 46.6%

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

      8. sqr-pow 46.9%

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

      9. metadata-eval 46.9%

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

   8. Simplified 46.9%

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

      1. add-cbrt-cube 38.9%

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

   10. Applied egg-rr 38.9%

       \[\leadsto \color{blue}{\sqrt[3]{\left({\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}\right) \cdot {\left(h \cdot \ell\right)}^{-0.5}}} \cdot d \]
   11. Step-by-step derivation

       1. associate-*l* 38.9%

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

       2. pow-sqr 38.9%

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

       3. metadata-eval 38.9%

          \[\leadsto \sqrt[3]{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{\color{blue}{-1}}} \cdot d \]

       4. unpow-1 38.9%

          \[\leadsto \sqrt[3]{{\left(h \cdot \ell\right)}^{-0.5} \cdot \color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

   12. Simplified 38.9%

       \[\leadsto \color{blue}{\sqrt[3]{{\left(h \cdot \ell\right)}^{-0.5} \cdot \frac{1}{h \cdot \ell}}} \cdot d \]
   13. Step-by-step derivation

       1. *-commutative 38.9%

          \[\leadsto \color{blue}{d \cdot \sqrt[3]{{\left(h \cdot \ell\right)}^{-0.5} \cdot \frac{1}{h \cdot \ell}}} \]

       2. add-sqr-sqrt 38.8%

          \[\leadsto d \cdot \color{blue}{\left(\sqrt{\sqrt[3]{{\left(h \cdot \ell\right)}^{-0.5} \cdot \frac{1}{h \cdot \ell}}} \cdot \sqrt{\sqrt[3]{{\left(h \cdot \ell\right)}^{-0.5} \cdot \frac{1}{h \cdot \ell}}}\right)} \]

       3. pow2 38.8%

          \[\leadsto d \cdot \color{blue}{{\left(\sqrt{\sqrt[3]{{\left(h \cdot \ell\right)}^{-0.5} \cdot \frac{1}{h \cdot \ell}}}\right)}^{2}} \]

   14. Applied egg-rr 45.7%

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

       1. unpow2 45.7%

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

       2. rem-sqrt-square 67.2%

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

   16. Simplified 67.2%

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

   if -1.70000000000000012e-119 < M < -4.5e-176 or 2.4499999999999999e-189 < M

   1. Initial program 65.2%

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

      1. associate-*l* 65.2%

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

      2. metadata-eval 65.2%

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

      3. unpow1/2 65.2%

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

      4. metadata-eval 65.2%

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

      5. unpow1/2 65.2%

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

      6. associate-*l* 65.2%

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

      7. metadata-eval 65.2%

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

      8. times-frac 65.2%

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

   3. Simplified 65.2%

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

      1. associate-*r/ 68.2%

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

      2. div-inv 68.2%

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

      3. metadata-eval 68.2%

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

   5. Applied egg-rr 68.2%

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

   6. Taylor expanded in M around 0 68.2%

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

      1. *-commutative 68.2%

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

      2. *-commutative 68.2%

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

      3. associate-*l/ 68.2%

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

      4. *-commutative 68.2%

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

      5. associate-*r* 68.2%

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

      6. associate-/l* 66.6%

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

      7. *-commutative 66.6%

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

   8. Simplified 66.6%

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

   9. Taylor expanded in D around 0 40.4%

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

       1. associate-*r/ 40.4%

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

       2. unpow2 40.4%

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

       3. unpow2 40.4%

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

       4. unpow2 40.4%

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

       5. *-commutative 40.4%

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

       6. associate-*r* 45.4%

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

       7. *-commutative 45.4%

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

       8. associate-*r/ 45.4%

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

       9. times-frac 44.2%

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

       10. associate-*r* 40.9%

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

       11. associate-/l* 39.4%

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

       12. associate-*l/ 41.8%

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

       13. times-frac 50.0%

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

   11. Simplified 59.6%

       \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \color{blue}{\left(0.25 \cdot \left(\frac{D}{\frac{d}{D}} \cdot \frac{h \cdot \frac{M}{\frac{\ell}{M}}}{d}\right)\right)}\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 62.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -2.9 \cdot 10^{+53}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\\ \mathbf{elif}\;M \leq -1.7 \cdot 10^{-119} \lor \neg \left(M \leq -4.5 \cdot 10^{-176}\right) \land M \leq 2.45 \cdot 10^{-189}:\\ \;\;\;\;\left|d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right|\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{D}{\frac{d}{D}} \cdot \frac{h \cdot \frac{M}{\frac{\ell}{M}}}{d}\right)\right)\right)\right)\\ \end{array} \]

Alternative 4: 57.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq -2.9 \cdot 10^{+53} \lor \neg \left(M \leq 4.7 \cdot 10^{-195}\right):\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left|d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (or (<= M -2.9e+53) (not (<= M 4.7e-195)))
   (*
    (sqrt (* (/ d l) (/ d h)))
    (- 1.0 (* (pow (* M (* 0.5 (/ D d))) 2.0) (* 0.5 (/ h l)))))
   (fabs (* d (pow (* l h) -0.5)))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if ((M <= -2.9e+53) || !(M <= 4.7e-195)) {
		tmp = sqrt(((d / l) * (d / h))) * (1.0 - (pow((M * (0.5 * (D / d))), 2.0) * (0.5 * (h / l))));
	} else {
		tmp = fabs((d * pow((l * h), -0.5)));
	}
	return tmp;
}

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
    real(8) :: tmp
    if ((m <= (-2.9d+53)) .or. (.not. (m <= 4.7d-195))) then
        tmp = sqrt(((d / l) * (d / h))) * (1.0d0 - (((m * (0.5d0 * (d_1 / d))) ** 2.0d0) * (0.5d0 * (h / l))))
    else
        tmp = abs((d * ((l * h) ** (-0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if ((M <= -2.9e+53) || !(M <= 4.7e-195)) {
		tmp = Math.sqrt(((d / l) * (d / h))) * (1.0 - (Math.pow((M * (0.5 * (D / d))), 2.0) * (0.5 * (h / l))));
	} else {
		tmp = Math.abs((d * Math.pow((l * h), -0.5)));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if (M <= -2.9e+53) or not (M <= 4.7e-195):
		tmp = math.sqrt(((d / l) * (d / h))) * (1.0 - (math.pow((M * (0.5 * (D / d))), 2.0) * (0.5 * (h / l))))
	else:
		tmp = math.fabs((d * math.pow((l * h), -0.5)))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if ((M <= -2.9e+53) || !(M <= 4.7e-195))
		tmp = Float64(sqrt(Float64(Float64(d / l) * Float64(d / h))) * Float64(1.0 - Float64((Float64(M * Float64(0.5 * Float64(D / d))) ^ 2.0) * Float64(0.5 * Float64(h / l)))));
	else
		tmp = abs(Float64(d * (Float64(l * h) ^ -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if ((M <= -2.9e+53) || ~((M <= 4.7e-195)))
		tmp = sqrt(((d / l) * (d / h))) * (1.0 - (((M * (0.5 * (D / d))) ^ 2.0) * (0.5 * (h / l))));
	else
		tmp = abs((d * ((l * h) ^ -0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[Or[LessEqual[M, -2.9e+53], N[Not[LessEqual[M, 4.7e-195]], $MachinePrecision]], N[(N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(N[Power[N[(M * N[(0.5 * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(0.5 * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Abs[N[(d * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < -2.9000000000000002e53 or 4.7000000000000001e-195 < M

   1. Initial program 67.0%

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

      1. metadata-eval 67.0%

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

      2. unpow1/2 67.0%

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

      3. metadata-eval 67.0%

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

      4. unpow1/2 67.0%

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

      5. *-commutative 67.0%

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

      6. associate-*l* 67.0%

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

      7. times-frac 66.5%

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

      8. metadata-eval 66.5%

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

   3. Simplified 66.5%

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

   5. Applied egg-rr 16.7%

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

      1. expm1-def 21.5%

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

      2. expm1-log1p 58.6%

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

      3. sub-neg 58.6%

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

      4. associate-*r* 58.6%

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

      5. distribute-rgt-neg-in 58.6%

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

      6. *-commutative 58.6%

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

      7. metadata-eval 58.6%

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

      8. associate-*r* 58.6%

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

      9. +-commutative 58.6%

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

      10. fma-def 58.6%

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

      11. fma-def 58.6%

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

   7. Simplified 58.6%

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

   if -2.9000000000000002e53 < M < 4.7000000000000001e-195

   1. Initial program 65.5%

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

      1. metadata-eval 65.5%

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

      2. unpow1/2 65.5%

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

      3. metadata-eval 65.5%

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

      4. unpow1/2 65.5%

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

      5. *-commutative 65.5%

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

      6. associate-*l* 65.5%

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

      7. times-frac 65.4%

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

      8. metadata-eval 65.4%

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

   3. Simplified 65.4%

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

   4. Taylor expanded in d around inf 42.5%

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

      1. expm1-log1p-u 41.4%

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

      2. expm1-udef 25.5%

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

      3. *-commutative 25.5%

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

   6. Applied egg-rr 25.5%

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

      1. expm1-def 41.4%

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

      2. expm1-log1p 42.5%

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

      3. unpow-1 42.5%

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

      4. sqr-pow 42.5%

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

      5. rem-sqrt-square 43.4%

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

      6. sqr-pow 43.2%

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

      7. fabs-sqr 43.2%

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

      8. sqr-pow 43.4%

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

      9. metadata-eval 43.4%

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

   8. Simplified 43.4%

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

      1. add-cbrt-cube 35.6%

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

   10. Applied egg-rr 35.6%

       \[\leadsto \color{blue}{\sqrt[3]{\left({\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}\right) \cdot {\left(h \cdot \ell\right)}^{-0.5}}} \cdot d \]
   11. Step-by-step derivation

       1. associate-*l* 35.6%

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

       2. pow-sqr 35.6%

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

       3. metadata-eval 35.6%

          \[\leadsto \sqrt[3]{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{\color{blue}{-1}}} \cdot d \]

       4. unpow-1 35.6%

          \[\leadsto \sqrt[3]{{\left(h \cdot \ell\right)}^{-0.5} \cdot \color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

   12. Simplified 35.6%

       \[\leadsto \color{blue}{\sqrt[3]{{\left(h \cdot \ell\right)}^{-0.5} \cdot \frac{1}{h \cdot \ell}}} \cdot d \]
   13. Step-by-step derivation

       1. *-commutative 35.6%

          \[\leadsto \color{blue}{d \cdot \sqrt[3]{{\left(h \cdot \ell\right)}^{-0.5} \cdot \frac{1}{h \cdot \ell}}} \]

       2. add-sqr-sqrt 35.5%

          \[\leadsto d \cdot \color{blue}{\left(\sqrt{\sqrt[3]{{\left(h \cdot \ell\right)}^{-0.5} \cdot \frac{1}{h \cdot \ell}}} \cdot \sqrt{\sqrt[3]{{\left(h \cdot \ell\right)}^{-0.5} \cdot \frac{1}{h \cdot \ell}}}\right)} \]

       3. pow2 35.5%

          \[\leadsto d \cdot \color{blue}{{\left(\sqrt{\sqrt[3]{{\left(h \cdot \ell\right)}^{-0.5} \cdot \frac{1}{h \cdot \ell}}}\right)}^{2}} \]

   14. Applied egg-rr 44.7%

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

       1. unpow2 44.7%

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

       2. rem-sqrt-square 64.0%

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

   16. Simplified 64.0%

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

4. Final simplification 60.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -2.9 \cdot 10^{+53} \lor \neg \left(M \leq 4.7 \cdot 10^{-195}\right):\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left|d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right|\\ \end{array} \]

Alternative 5: 47.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\ell \cdot h}\\ \mathbf{if}\;\ell \leq -2.55 \cdot 10^{-222}:\\ \;\;\;\;\left|d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right|\\ \mathbf{elif}\;\ell \leq 2.25 \cdot 10^{-273}:\\ \;\;\;\;d \cdot \sqrt{\sqrt[3]{t_0 \cdot \left(t_0 \cdot t_0\right)}}\\ \mathbf{elif}\;\ell \leq 1.45 \cdot 10^{-236}:\\ \;\;\;\;\frac{D}{\frac{\frac{d}{M \cdot M}}{D}} \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (/ 1.0 (* l h))))
   (if (<= l -2.55e-222)
     (fabs (* d (pow (* l h) -0.5)))
     (if (<= l 2.25e-273)
       (* d (sqrt (cbrt (* t_0 (* t_0 t_0)))))
       (if (<= l 1.45e-236)
         (* (/ D (/ (/ d (* M M)) D)) (* (sqrt (/ h (pow l 3.0))) -0.125))
         (* d (* (pow h -0.5) (pow l -0.5))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = 1.0 / (l * h);
	double tmp;
	if (l <= -2.55e-222) {
		tmp = fabs((d * pow((l * h), -0.5)));
	} else if (l <= 2.25e-273) {
		tmp = d * sqrt(cbrt((t_0 * (t_0 * t_0))));
	} else if (l <= 1.45e-236) {
		tmp = (D / ((d / (M * M)) / D)) * (sqrt((h / pow(l, 3.0))) * -0.125);
	} else {
		tmp = d * (pow(h, -0.5) * pow(l, -0.5));
	}
	return tmp;
}

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = 1.0 / (l * h);
	double tmp;
	if (l <= -2.55e-222) {
		tmp = Math.abs((d * Math.pow((l * h), -0.5)));
	} else if (l <= 2.25e-273) {
		tmp = d * Math.sqrt(Math.cbrt((t_0 * (t_0 * t_0))));
	} else if (l <= 1.45e-236) {
		tmp = (D / ((d / (M * M)) / D)) * (Math.sqrt((h / Math.pow(l, 3.0))) * -0.125);
	} else {
		tmp = d * (Math.pow(h, -0.5) * Math.pow(l, -0.5));
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = Float64(1.0 / Float64(l * h))
	tmp = 0.0
	if (l <= -2.55e-222)
		tmp = abs(Float64(d * (Float64(l * h) ^ -0.5)));
	elseif (l <= 2.25e-273)
		tmp = Float64(d * sqrt(cbrt(Float64(t_0 * Float64(t_0 * t_0)))));
	elseif (l <= 1.45e-236)
		tmp = Float64(Float64(D / Float64(Float64(d / Float64(M * M)) / D)) * Float64(sqrt(Float64(h / (l ^ 3.0))) * -0.125));
	else
		tmp = Float64(d * Float64((h ^ -0.5) * (l ^ -0.5)));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -2.55e-222], N[Abs[N[(d * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 2.25e-273], N[(d * N[Sqrt[N[Power[N[(t$95$0 * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.45e-236], N[(N[(D / N[(N[(d / N[(M * M), $MachinePrecision]), $MachinePrecision] / D), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(h / N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[h, -0.5], $MachinePrecision] * N[Power[l, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -2.5500000000000001e-222

   1. Initial program 61.8%

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

      1. metadata-eval 61.8%

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

      2. unpow1/2 61.8%

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

      3. metadata-eval 61.8%

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

      4. unpow1/2 61.8%

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

      5. *-commutative 61.8%

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

      6. associate-*l* 61.8%

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

      7. times-frac 61.8%

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

      8. metadata-eval 61.8%

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

   3. Simplified 61.8%

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

   4. Taylor expanded in d around inf 10.7%

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

      1. expm1-log1p-u 10.7%

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

      2. expm1-udef 10.6%

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

      3. *-commutative 10.6%

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

   6. Applied egg-rr 10.6%

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

      1. expm1-def 10.7%

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

      2. expm1-log1p 10.7%

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

      3. unpow-1 10.7%

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

      4. sqr-pow 10.7%

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

      5. rem-sqrt-square 10.7%

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

      6. sqr-pow 10.7%

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

      7. fabs-sqr 10.7%

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

      8. sqr-pow 10.7%

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

      9. metadata-eval 10.7%

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

   8. Simplified 10.7%

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

      1. add-cbrt-cube 11.7%

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

   10. Applied egg-rr 11.7%

       \[\leadsto \color{blue}{\sqrt[3]{\left({\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}\right) \cdot {\left(h \cdot \ell\right)}^{-0.5}}} \cdot d \]
   11. Step-by-step derivation

       1. associate-*l* 11.7%

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

       2. pow-sqr 11.7%

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

       3. metadata-eval 11.7%

          \[\leadsto \sqrt[3]{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{\color{blue}{-1}}} \cdot d \]

       4. unpow-1 11.7%

          \[\leadsto \sqrt[3]{{\left(h \cdot \ell\right)}^{-0.5} \cdot \color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

   12. Simplified 11.7%

       \[\leadsto \color{blue}{\sqrt[3]{{\left(h \cdot \ell\right)}^{-0.5} \cdot \frac{1}{h \cdot \ell}}} \cdot d \]
   13. Step-by-step derivation

       1. *-commutative 11.7%

          \[\leadsto \color{blue}{d \cdot \sqrt[3]{{\left(h \cdot \ell\right)}^{-0.5} \cdot \frac{1}{h \cdot \ell}}} \]

       2. add-sqr-sqrt 11.7%

          \[\leadsto d \cdot \color{blue}{\left(\sqrt{\sqrt[3]{{\left(h \cdot \ell\right)}^{-0.5} \cdot \frac{1}{h \cdot \ell}}} \cdot \sqrt{\sqrt[3]{{\left(h \cdot \ell\right)}^{-0.5} \cdot \frac{1}{h \cdot \ell}}}\right)} \]

       3. pow2 11.7%

          \[\leadsto d \cdot \color{blue}{{\left(\sqrt{\sqrt[3]{{\left(h \cdot \ell\right)}^{-0.5} \cdot \frac{1}{h \cdot \ell}}}\right)}^{2}} \]

   14. Applied egg-rr 26.9%

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

       1. unpow2 26.9%

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

       2. rem-sqrt-square 43.6%

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

   16. Simplified 43.6%

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

   if -2.5500000000000001e-222 < l < 2.2499999999999998e-273

   1. Initial program 57.9%

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

      1. metadata-eval 57.9%

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

      2. unpow1/2 57.9%

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

      3. metadata-eval 57.9%

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

      4. unpow1/2 57.9%

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

      5. *-commutative 57.9%

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

      6. associate-*l* 57.9%

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

      7. times-frac 57.9%

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

      8. metadata-eval 57.9%

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

   3. Simplified 57.9%

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

   4. Taylor expanded in d around inf 41.4%

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

      1. add-cbrt-cube 58.8%

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

      2. *-commutative 58.8%

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

      3. *-commutative 58.8%

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

      4. *-commutative 58.8%

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

   6. Applied egg-rr 58.8%

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

   if 2.2499999999999998e-273 < l < 1.45e-236

   1. Initial program 80.4%

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

      1. associate-*l* 80.4%

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

      2. metadata-eval 80.4%

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

      3. unpow1/2 80.4%

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

      4. metadata-eval 80.4%

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

      5. unpow1/2 80.4%

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

      6. associate-*l* 80.4%

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

      7. metadata-eval 80.4%

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

      8. times-frac 73.8%

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

   3. Simplified 73.8%

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

      1. sqrt-div 73.9%

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

   5. Applied egg-rr 73.9%

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

   6. Taylor expanded in d around 0 47.0%

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

      1. *-commutative 47.0%

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

      2. associate-*l* 47.0%

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

      3. associate-/l* 46.7%

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

      4. unpow2 46.7%

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

      5. associate-/l* 60.0%

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

      6. unpow2 60.0%

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

   8. Simplified 60.0%

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

   if 1.45e-236 < l

   1. Initial program 70.5%

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

      1. metadata-eval 70.5%

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

      2. unpow1/2 70.5%

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

      3. metadata-eval 70.5%

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

      4. unpow1/2 70.5%

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

      5. *-commutative 70.5%

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

      6. associate-*l* 70.5%

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

      7. times-frac 70.5%

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

      8. metadata-eval 70.5%

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

   3. Simplified 70.5%

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

   4. Taylor expanded in d around inf 54.5%

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

      1. expm1-log1p-u 53.3%

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

      2. expm1-udef 30.5%

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

      3. *-commutative 30.5%

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

   6. Applied egg-rr 30.5%

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

      1. expm1-def 53.3%

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

      2. expm1-log1p 54.5%

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

      3. unpow-1 54.5%

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

      4. sqr-pow 54.5%

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

      5. rem-sqrt-square 55.3%

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

      6. sqr-pow 55.1%

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

      7. fabs-sqr 55.1%

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

      8. sqr-pow 55.3%

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

      9. metadata-eval 55.3%

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

   8. Simplified 55.3%

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

      1. unpow-prod-down 60.8%

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

   10. Applied egg-rr 60.8%

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

4. Final simplification 54.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.55 \cdot 10^{-222}:\\ \;\;\;\;\left|d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right|\\ \mathbf{elif}\;\ell \leq 2.25 \cdot 10^{-273}:\\ \;\;\;\;d \cdot \sqrt{\sqrt[3]{\frac{1}{\ell \cdot h} \cdot \left(\frac{1}{\ell \cdot h} \cdot \frac{1}{\ell \cdot h}\right)}}\\ \mathbf{elif}\;\ell \leq 1.45 \cdot 10^{-236}:\\ \;\;\;\;\frac{D}{\frac{\frac{d}{M \cdot M}}{D}} \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \]

Alternative 6: 46.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -6.3 \cdot 10^{-223}:\\ \;\;\;\;\left|d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right|\\ \mathbf{elif}\;\ell \leq 1.36 \cdot 10^{-275}:\\ \;\;\;\;d \cdot \sqrt[3]{{\left(\frac{1}{\ell \cdot h}\right)}^{1.5}}\\ \mathbf{elif}\;\ell \leq 1.5 \cdot 10^{-236}:\\ \;\;\;\;\frac{D}{\frac{\frac{d}{M \cdot M}}{D}} \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= l -6.3e-223)
   (fabs (* d (pow (* l h) -0.5)))
   (if (<= l 1.36e-275)
     (* d (cbrt (pow (/ 1.0 (* l h)) 1.5)))
     (if (<= l 1.5e-236)
       (* (/ D (/ (/ d (* M M)) D)) (* (sqrt (/ h (pow l 3.0))) -0.125))
       (* d (* (pow h -0.5) (pow l -0.5)))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -6.3e-223) {
		tmp = fabs((d * pow((l * h), -0.5)));
	} else if (l <= 1.36e-275) {
		tmp = d * cbrt(pow((1.0 / (l * h)), 1.5));
	} else if (l <= 1.5e-236) {
		tmp = (D / ((d / (M * M)) / D)) * (sqrt((h / pow(l, 3.0))) * -0.125);
	} else {
		tmp = d * (pow(h, -0.5) * pow(l, -0.5));
	}
	return tmp;
}

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -6.3e-223) {
		tmp = Math.abs((d * Math.pow((l * h), -0.5)));
	} else if (l <= 1.36e-275) {
		tmp = d * Math.cbrt(Math.pow((1.0 / (l * h)), 1.5));
	} else if (l <= 1.5e-236) {
		tmp = (D / ((d / (M * M)) / D)) * (Math.sqrt((h / Math.pow(l, 3.0))) * -0.125);
	} else {
		tmp = d * (Math.pow(h, -0.5) * Math.pow(l, -0.5));
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= -6.3e-223)
		tmp = abs(Float64(d * (Float64(l * h) ^ -0.5)));
	elseif (l <= 1.36e-275)
		tmp = Float64(d * cbrt((Float64(1.0 / Float64(l * h)) ^ 1.5)));
	elseif (l <= 1.5e-236)
		tmp = Float64(Float64(D / Float64(Float64(d / Float64(M * M)) / D)) * Float64(sqrt(Float64(h / (l ^ 3.0))) * -0.125));
	else
		tmp = Float64(d * Float64((h ^ -0.5) * (l ^ -0.5)));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[l, -6.3e-223], N[Abs[N[(d * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 1.36e-275], N[(d * N[Power[N[Power[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision], 1.5], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.5e-236], N[(N[(D / N[(N[(d / N[(M * M), $MachinePrecision]), $MachinePrecision] / D), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(h / N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[h, -0.5], $MachinePrecision] * N[Power[l, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if l < -6.29999999999999986e-223

   1. Initial program 61.8%

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

      1. metadata-eval 61.8%

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

      2. unpow1/2 61.8%

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

      3. metadata-eval 61.8%

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

      4. unpow1/2 61.8%

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

      5. *-commutative 61.8%

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

      6. associate-*l* 61.8%

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

      7. times-frac 61.8%

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

      8. metadata-eval 61.8%

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

   3. Simplified 61.8%

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

   4. Taylor expanded in d around inf 10.7%

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

      1. expm1-log1p-u 10.7%

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

      2. expm1-udef 10.6%

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

      3. *-commutative 10.6%

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

   6. Applied egg-rr 10.6%

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

      1. expm1-def 10.7%

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

      2. expm1-log1p 10.7%

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

      3. unpow-1 10.7%

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

      4. sqr-pow 10.7%

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

      5. rem-sqrt-square 10.7%

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

      6. sqr-pow 10.7%

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

      7. fabs-sqr 10.7%

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

      8. sqr-pow 10.7%

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

      9. metadata-eval 10.7%

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

   8. Simplified 10.7%

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

      1. add-cbrt-cube 11.7%

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

   10. Applied egg-rr 11.7%

       \[\leadsto \color{blue}{\sqrt[3]{\left({\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}\right) \cdot {\left(h \cdot \ell\right)}^{-0.5}}} \cdot d \]
   11. Step-by-step derivation

       1. associate-*l* 11.7%

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

       2. pow-sqr 11.7%

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

       3. metadata-eval 11.7%

          \[\leadsto \sqrt[3]{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{\color{blue}{-1}}} \cdot d \]

       4. unpow-1 11.7%

          \[\leadsto \sqrt[3]{{\left(h \cdot \ell\right)}^{-0.5} \cdot \color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

   12. Simplified 11.7%

       \[\leadsto \color{blue}{\sqrt[3]{{\left(h \cdot \ell\right)}^{-0.5} \cdot \frac{1}{h \cdot \ell}}} \cdot d \]
   13. Step-by-step derivation

       1. *-commutative 11.7%

          \[\leadsto \color{blue}{d \cdot \sqrt[3]{{\left(h \cdot \ell\right)}^{-0.5} \cdot \frac{1}{h \cdot \ell}}} \]

       2. add-sqr-sqrt 11.7%

          \[\leadsto d \cdot \color{blue}{\left(\sqrt{\sqrt[3]{{\left(h \cdot \ell\right)}^{-0.5} \cdot \frac{1}{h \cdot \ell}}} \cdot \sqrt{\sqrt[3]{{\left(h \cdot \ell\right)}^{-0.5} \cdot \frac{1}{h \cdot \ell}}}\right)} \]

       3. pow2 11.7%

          \[\leadsto d \cdot \color{blue}{{\left(\sqrt{\sqrt[3]{{\left(h \cdot \ell\right)}^{-0.5} \cdot \frac{1}{h \cdot \ell}}}\right)}^{2}} \]

   14. Applied egg-rr 26.9%

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

       1. unpow2 26.9%

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

       2. rem-sqrt-square 43.6%

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

   16. Simplified 43.6%

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

   if -6.29999999999999986e-223 < l < 1.35999999999999997e-275

   1. Initial program 57.9%

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

      1. metadata-eval 57.9%

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

      2. unpow1/2 57.9%

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

      3. metadata-eval 57.9%

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

      4. unpow1/2 57.9%

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

      5. *-commutative 57.9%

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

      6. associate-*l* 57.9%

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

      7. times-frac 57.9%

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

      8. metadata-eval 57.9%

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

   3. Simplified 57.9%

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

   4. Taylor expanded in d around inf 41.4%

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

      1. expm1-log1p-u 41.3%

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

      2. expm1-udef 41.3%

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

      3. *-commutative 41.3%

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

   6. Applied egg-rr 41.3%

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

      1. add-cbrt-cube 48.2%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\sqrt{e^{\mathsf{log1p}\left(\frac{1}{h \cdot \ell}\right)} - 1} \cdot \sqrt{e^{\mathsf{log1p}\left(\frac{1}{h \cdot \ell}\right)} - 1}\right) \cdot \sqrt{e^{\mathsf{log1p}\left(\frac{1}{h \cdot \ell}\right)} - 1}}} \cdot d \]

      2. add-sqr-sqrt 48.2%

         \[\leadsto \sqrt[3]{\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{1}{h \cdot \ell}\right)} - 1\right)} \cdot \sqrt{e^{\mathsf{log1p}\left(\frac{1}{h \cdot \ell}\right)} - 1}} \cdot d \]

      3. expm1-def 48.2%

         \[\leadsto \sqrt[3]{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{h \cdot \ell}\right)\right)} \cdot \sqrt{e^{\mathsf{log1p}\left(\frac{1}{h \cdot \ell}\right)} - 1}} \cdot d \]

      4. expm1-log1p-u 48.3%

         \[\leadsto \sqrt[3]{\color{blue}{\frac{1}{h \cdot \ell}} \cdot \sqrt{e^{\mathsf{log1p}\left(\frac{1}{h \cdot \ell}\right)} - 1}} \cdot d \]

      5. associate-/r* 48.3%

         \[\leadsto \sqrt[3]{\color{blue}{\frac{\frac{1}{h}}{\ell}} \cdot \sqrt{e^{\mathsf{log1p}\left(\frac{1}{h \cdot \ell}\right)} - 1}} \cdot d \]

      6. expm1-def 48.3%

         \[\leadsto \sqrt[3]{\frac{\frac{1}{h}}{\ell} \cdot \sqrt{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{h \cdot \ell}\right)\right)}}} \cdot d \]

      7. expm1-log1p-u 48.3%

         \[\leadsto \sqrt[3]{\frac{\frac{1}{h}}{\ell} \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}}} \cdot d \]

      8. associate-/r* 48.3%

         \[\leadsto \sqrt[3]{\frac{\frac{1}{h}}{\ell} \cdot \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}}} \cdot d \]

   8. Applied egg-rr 48.3%

      \[\leadsto \color{blue}{\sqrt[3]{\frac{\frac{1}{h}}{\ell} \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}}} \cdot d \]
   9. Step-by-step derivation

      1. *-commutative 48.3%

         \[\leadsto \sqrt[3]{\color{blue}{\sqrt{\frac{\frac{1}{h}}{\ell}} \cdot \frac{\frac{1}{h}}{\ell}}} \cdot d \]

      2. unpow1/2 48.3%

         \[\leadsto \sqrt[3]{\color{blue}{{\left(\frac{\frac{1}{h}}{\ell}\right)}^{0.5}} \cdot \frac{\frac{1}{h}}{\ell}} \cdot d \]

      3. pow-plus 48.3%

         \[\leadsto \sqrt[3]{\color{blue}{{\left(\frac{\frac{1}{h}}{\ell}\right)}^{\left(0.5 + 1\right)}}} \cdot d \]

      4. associate-/r* 48.3%

         \[\leadsto \sqrt[3]{{\color{blue}{\left(\frac{1}{h \cdot \ell}\right)}}^{\left(0.5 + 1\right)}} \cdot d \]

      5. metadata-eval 48.3%

         \[\leadsto \sqrt[3]{{\left(\frac{1}{h \cdot \ell}\right)}^{\color{blue}{1.5}}} \cdot d \]

   10. Simplified 48.3%

       \[\leadsto \color{blue}{\sqrt[3]{{\left(\frac{1}{h \cdot \ell}\right)}^{1.5}}} \cdot d \]

   if 1.35999999999999997e-275 < l < 1.50000000000000007e-236

   1. Initial program 80.4%

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

      1. associate-*l* 80.4%

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

      2. metadata-eval 80.4%

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

      3. unpow1/2 80.4%

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

      4. metadata-eval 80.4%

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

      5. unpow1/2 80.4%

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

      6. associate-*l* 80.4%

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

      7. metadata-eval 80.4%

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

      8. times-frac 73.8%

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

   3. Simplified 73.8%

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

      1. sqrt-div 73.9%

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

   5. Applied egg-rr 73.9%

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

   6. Taylor expanded in d around 0 47.0%

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

      1. *-commutative 47.0%

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

      2. associate-*l* 47.0%

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

      3. associate-/l* 46.7%

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

      4. unpow2 46.7%

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

      5. associate-/l* 60.0%

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

      6. unpow2 60.0%

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

   8. Simplified 60.0%

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

   if 1.50000000000000007e-236 < l

   1. Initial program 70.5%

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

      1. metadata-eval 70.5%

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

      2. unpow1/2 70.5%

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

      3. metadata-eval 70.5%

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

      4. unpow1/2 70.5%

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

      5. *-commutative 70.5%

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

      6. associate-*l* 70.5%

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

      7. times-frac 70.5%

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

      8. metadata-eval 70.5%

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

   3. Simplified 70.5%

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

   4. Taylor expanded in d around inf 54.5%

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

      1. expm1-log1p-u 53.3%

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

      2. expm1-udef 30.5%

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

      3. *-commutative 30.5%

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

   6. Applied egg-rr 30.5%

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

      1. expm1-def 53.3%

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

      2. expm1-log1p 54.5%

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

      3. unpow-1 54.5%

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

      4. sqr-pow 54.5%

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

      5. rem-sqrt-square 55.3%

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

      6. sqr-pow 55.1%

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

      7. fabs-sqr 55.1%

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

      8. sqr-pow 55.3%

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

      9. metadata-eval 55.3%

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

   8. Simplified 55.3%

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

      1. unpow-prod-down 60.8%

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

   10. Applied egg-rr 60.8%

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

4. Final simplification 53.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -6.3 \cdot 10^{-223}:\\ \;\;\;\;\left|d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right|\\ \mathbf{elif}\;\ell \leq 1.36 \cdot 10^{-275}:\\ \;\;\;\;d \cdot \sqrt[3]{{\left(\frac{1}{\ell \cdot h}\right)}^{1.5}}\\ \mathbf{elif}\;\ell \leq 1.5 \cdot 10^{-236}:\\ \;\;\;\;\frac{D}{\frac{\frac{d}{M \cdot M}}{D}} \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot -0.125\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \]

Alternative 7: 46.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := d \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \mathbf{if}\;\ell \leq -9.8 \cdot 10^{-222}:\\ \;\;\;\;\left|t_0\right|\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\sqrt[3]{\left(\frac{1}{\ell \cdot h} \cdot \left(d \cdot d\right)\right) \cdot t_0}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* d (pow (* l h) -0.5))))
   (if (<= l -9.8e-222)
     (fabs t_0)
     (if (<= l -5e-310)
       (cbrt (* (* (/ 1.0 (* l h)) (* d d)) t_0))
       (* d (* (pow h -0.5) (pow l -0.5)))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = d * pow((l * h), -0.5);
	double tmp;
	if (l <= -9.8e-222) {
		tmp = fabs(t_0);
	} else if (l <= -5e-310) {
		tmp = cbrt((((1.0 / (l * h)) * (d * d)) * t_0));
	} else {
		tmp = d * (pow(h, -0.5) * pow(l, -0.5));
	}
	return tmp;
}

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = d * Math.pow((l * h), -0.5);
	double tmp;
	if (l <= -9.8e-222) {
		tmp = Math.abs(t_0);
	} else if (l <= -5e-310) {
		tmp = Math.cbrt((((1.0 / (l * h)) * (d * d)) * t_0));
	} else {
		tmp = d * (Math.pow(h, -0.5) * Math.pow(l, -0.5));
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = Float64(d * (Float64(l * h) ^ -0.5))
	tmp = 0.0
	if (l <= -9.8e-222)
		tmp = abs(t_0);
	elseif (l <= -5e-310)
		tmp = cbrt(Float64(Float64(Float64(1.0 / Float64(l * h)) * Float64(d * d)) * t_0));
	else
		tmp = Float64(d * Float64((h ^ -0.5) * (l ^ -0.5)));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(d * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -9.8e-222], N[Abs[t$95$0], $MachinePrecision], If[LessEqual[l, -5e-310], N[Power[N[(N[(N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision] * N[(d * d), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], 1/3], $MachinePrecision], N[(d * N[(N[Power[h, -0.5], $MachinePrecision] * N[Power[l, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < -9.7999999999999999e-222

   1. Initial program 61.8%

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

      1. metadata-eval 61.8%

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

      2. unpow1/2 61.8%

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

      3. metadata-eval 61.8%

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

      4. unpow1/2 61.8%

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

      5. *-commutative 61.8%

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

      6. associate-*l* 61.8%

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

      7. times-frac 61.8%

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

      8. metadata-eval 61.8%

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

   3. Simplified 61.8%

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

   4. Taylor expanded in d around inf 10.7%

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

      1. expm1-log1p-u 10.7%

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

      2. expm1-udef 10.6%

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

      3. *-commutative 10.6%

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

   6. Applied egg-rr 10.6%

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

      1. expm1-def 10.7%

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

      2. expm1-log1p 10.7%

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

      3. unpow-1 10.7%

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

      4. sqr-pow 10.7%

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

      5. rem-sqrt-square 10.7%

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

      6. sqr-pow 10.7%

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

      7. fabs-sqr 10.7%

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

      8. sqr-pow 10.7%

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

      9. metadata-eval 10.7%

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

   8. Simplified 10.7%

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

      1. add-cbrt-cube 11.7%

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

   10. Applied egg-rr 11.7%

       \[\leadsto \color{blue}{\sqrt[3]{\left({\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}\right) \cdot {\left(h \cdot \ell\right)}^{-0.5}}} \cdot d \]
   11. Step-by-step derivation

       1. associate-*l* 11.7%

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

       2. pow-sqr 11.7%

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

       3. metadata-eval 11.7%

          \[\leadsto \sqrt[3]{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{\color{blue}{-1}}} \cdot d \]

       4. unpow-1 11.7%

          \[\leadsto \sqrt[3]{{\left(h \cdot \ell\right)}^{-0.5} \cdot \color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

   12. Simplified 11.7%

       \[\leadsto \color{blue}{\sqrt[3]{{\left(h \cdot \ell\right)}^{-0.5} \cdot \frac{1}{h \cdot \ell}}} \cdot d \]
   13. Step-by-step derivation

       1. *-commutative 11.7%

          \[\leadsto \color{blue}{d \cdot \sqrt[3]{{\left(h \cdot \ell\right)}^{-0.5} \cdot \frac{1}{h \cdot \ell}}} \]

       2. add-sqr-sqrt 11.7%

          \[\leadsto d \cdot \color{blue}{\left(\sqrt{\sqrt[3]{{\left(h \cdot \ell\right)}^{-0.5} \cdot \frac{1}{h \cdot \ell}}} \cdot \sqrt{\sqrt[3]{{\left(h \cdot \ell\right)}^{-0.5} \cdot \frac{1}{h \cdot \ell}}}\right)} \]

       3. pow2 11.7%

          \[\leadsto d \cdot \color{blue}{{\left(\sqrt{\sqrt[3]{{\left(h \cdot \ell\right)}^{-0.5} \cdot \frac{1}{h \cdot \ell}}}\right)}^{2}} \]

   14. Applied egg-rr 26.9%

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

       1. unpow2 26.9%

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

       2. rem-sqrt-square 43.6%

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

   16. Simplified 43.6%

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

   if -9.7999999999999999e-222 < l < -4.999999999999985e-310

   1. Initial program 60.8%

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

      1. metadata-eval 60.8%

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

      2. unpow1/2 60.8%

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

      3. metadata-eval 60.8%

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

      4. unpow1/2 60.8%

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

      5. *-commutative 60.8%

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

      6. associate-*l* 60.8%

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

      7. times-frac 60.8%

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

      8. metadata-eval 60.8%

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

   3. Simplified 60.8%

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

   4. Taylor expanded in d around inf 34.4%

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

      1. expm1-log1p-u 34.4%

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

      2. expm1-udef 34.4%

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

      3. *-commutative 34.4%

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

   6. Applied egg-rr 34.4%

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

      1. expm1-def 34.4%

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

      2. expm1-log1p 34.4%

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

      3. unpow-1 34.4%

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

      4. sqr-pow 34.4%

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

      5. rem-sqrt-square 34.4%

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

      6. sqr-pow 34.4%

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

      7. fabs-sqr 34.4%

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

      8. sqr-pow 34.4%

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

      9. metadata-eval 34.4%

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

   8. Simplified 34.4%

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

      1. add-cbrt-cube 49.3%

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

      2. *-commutative 49.3%

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

      3. *-commutative 49.3%

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

      4. *-commutative 49.3%

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

   10. Applied egg-rr 49.3%

       \[\leadsto \color{blue}{\sqrt[3]{\left(\left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right) \cdot \left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right)\right) \cdot \left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right)}} \]
   11. Step-by-step derivation

       1. associate-*l* 49.3%

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

       2. swap-sqr 45.3%

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

       3. pow-sqr 45.3%

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

       4. metadata-eval 45.3%

          \[\leadsto \sqrt[3]{\left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right) \cdot \left(\left(d \cdot d\right) \cdot {\left(h \cdot \ell\right)}^{\color{blue}{-1}}\right)} \]

       5. unpow-1 45.3%

          \[\leadsto \sqrt[3]{\left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right) \cdot \left(\left(d \cdot d\right) \cdot \color{blue}{\frac{1}{h \cdot \ell}}\right)} \]

   12. Simplified 45.3%

       \[\leadsto \color{blue}{\sqrt[3]{\left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right) \cdot \left(\left(d \cdot d\right) \cdot \frac{1}{h \cdot \ell}\right)}} \]

   if -4.999999999999985e-310 < l

   1. Initial program 70.7%

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

      1. metadata-eval 70.7%

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

      2. unpow1/2 70.7%

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

      3. metadata-eval 70.7%

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

      4. unpow1/2 70.7%

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

      5. *-commutative 70.7%

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

      6. associate-*l* 70.7%

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

      7. times-frac 70.0%

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

      8. metadata-eval 70.0%

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

   3. Simplified 70.0%

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

   4. Taylor expanded in d around inf 51.0%

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

      1. expm1-log1p-u 49.9%

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

      2. expm1-udef 30.2%

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

      3. *-commutative 30.2%

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

   6. Applied egg-rr 30.2%

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

      1. expm1-def 49.9%

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

      2. expm1-log1p 51.0%

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

      3. unpow-1 51.0%

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

      4. sqr-pow 51.1%

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

      5. rem-sqrt-square 51.7%

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

      6. sqr-pow 51.5%

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

      7. fabs-sqr 51.5%

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

      8. sqr-pow 51.7%

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

      9. metadata-eval 51.7%

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

   8. Simplified 51.7%

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

      1. unpow-prod-down 56.5%

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

   10. Applied egg-rr 56.5%

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

4. Final simplification 50.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -9.8 \cdot 10^{-222}:\\ \;\;\;\;\left|d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right|\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\sqrt[3]{\left(\frac{1}{\ell \cdot h} \cdot \left(d \cdot d\right)\right) \cdot \left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right)}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \]

Alternative 8: 39.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := d \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \mathbf{if}\;M \leq -2.1 \cdot 10^{+216}:\\ \;\;\;\;\sqrt[3]{\left(\frac{1}{\ell \cdot h} \cdot \left(d \cdot d\right)\right) \cdot t_0}\\ \mathbf{elif}\;M \leq 3.9 \cdot 10^{-63}:\\ \;\;\;\;\left|t_0\right|\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \frac{D \cdot D}{\frac{d}{M \cdot M}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* d (pow (* l h) -0.5))))
   (if (<= M -2.1e+216)
     (cbrt (* (* (/ 1.0 (* l h)) (* d d)) t_0))
     (if (<= M 3.9e-63)
       (fabs t_0)
       (* -0.125 (* (sqrt (/ h (pow l 3.0))) (/ (* D D) (/ d (* M M)))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = d * pow((l * h), -0.5);
	double tmp;
	if (M <= -2.1e+216) {
		tmp = cbrt((((1.0 / (l * h)) * (d * d)) * t_0));
	} else if (M <= 3.9e-63) {
		tmp = fabs(t_0);
	} else {
		tmp = -0.125 * (sqrt((h / pow(l, 3.0))) * ((D * D) / (d / (M * M))));
	}
	return tmp;
}

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = d * Math.pow((l * h), -0.5);
	double tmp;
	if (M <= -2.1e+216) {
		tmp = Math.cbrt((((1.0 / (l * h)) * (d * d)) * t_0));
	} else if (M <= 3.9e-63) {
		tmp = Math.abs(t_0);
	} else {
		tmp = -0.125 * (Math.sqrt((h / Math.pow(l, 3.0))) * ((D * D) / (d / (M * M))));
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = Float64(d * (Float64(l * h) ^ -0.5))
	tmp = 0.0
	if (M <= -2.1e+216)
		tmp = cbrt(Float64(Float64(Float64(1.0 / Float64(l * h)) * Float64(d * d)) * t_0));
	elseif (M <= 3.9e-63)
		tmp = abs(t_0);
	else
		tmp = Float64(-0.125 * Float64(sqrt(Float64(h / (l ^ 3.0))) * Float64(Float64(D * D) / Float64(d / Float64(M * M)))));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(d * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[M, -2.1e+216], N[Power[N[(N[(N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision] * N[(d * d), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], 1/3], $MachinePrecision], If[LessEqual[M, 3.9e-63], N[Abs[t$95$0], $MachinePrecision], N[(-0.125 * N[(N[Sqrt[N[(h / N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(D * D), $MachinePrecision] / N[(d / N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if M < -2.10000000000000001e216

   1. Initial program 72.9%

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

      1. metadata-eval 72.9%

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

      2. unpow1/2 72.9%

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

      3. metadata-eval 72.9%

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

      4. unpow1/2 72.9%

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

      5. *-commutative 72.9%

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

      6. associate-*l* 72.9%

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

      7. times-frac 68.3%

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

      8. metadata-eval 68.3%

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

   3. Simplified 68.3%

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

   4. Taylor expanded in d around inf 30.8%

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

      1. expm1-log1p-u 30.6%

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

      2. expm1-udef 21.7%

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

      3. *-commutative 21.7%

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

   6. Applied egg-rr 21.7%

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

      1. expm1-def 30.6%

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

      2. expm1-log1p 30.8%

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

      3. unpow-1 30.8%

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

      4. sqr-pow 30.8%

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

      5. rem-sqrt-square 30.8%

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

      6. sqr-pow 30.7%

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

      7. fabs-sqr 30.7%

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

      8. sqr-pow 30.8%

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

      9. metadata-eval 30.8%

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

   8. Simplified 30.8%

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

      1. add-cbrt-cube 39.2%

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

      2. *-commutative 39.2%

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

      3. *-commutative 39.2%

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

      4. *-commutative 39.2%

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

   10. Applied egg-rr 39.2%

       \[\leadsto \color{blue}{\sqrt[3]{\left(\left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right) \cdot \left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right)\right) \cdot \left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right)}} \]
   11. Step-by-step derivation

       1. associate-*l* 39.2%

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

       2. swap-sqr 34.7%

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

       3. pow-sqr 34.7%

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

       4. metadata-eval 34.7%

          \[\leadsto \sqrt[3]{\left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right) \cdot \left(\left(d \cdot d\right) \cdot {\left(h \cdot \ell\right)}^{\color{blue}{-1}}\right)} \]

       5. unpow-1 34.7%

          \[\leadsto \sqrt[3]{\left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right) \cdot \left(\left(d \cdot d\right) \cdot \color{blue}{\frac{1}{h \cdot \ell}}\right)} \]

   12. Simplified 34.7%

       \[\leadsto \color{blue}{\sqrt[3]{\left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right) \cdot \left(\left(d \cdot d\right) \cdot \frac{1}{h \cdot \ell}\right)}} \]

   if -2.10000000000000001e216 < M < 3.90000000000000022e-63

   1. Initial program 67.4%

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

      1. metadata-eval 67.4%

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

      2. unpow1/2 67.4%

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

      3. metadata-eval 67.4%

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

      4. unpow1/2 67.4%

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

      5. *-commutative 67.4%

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

      6. associate-*l* 67.4%

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

      7. times-frac 67.4%

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

      8. metadata-eval 67.4%

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

   3. Simplified 67.4%

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

   4. Taylor expanded in d around inf 37.6%

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

      1. expm1-log1p-u 36.8%

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

      2. expm1-udef 23.2%

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

      3. *-commutative 23.2%

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

   6. Applied egg-rr 23.2%

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

      1. expm1-def 36.8%

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

      2. expm1-log1p 37.6%

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

      3. unpow-1 37.6%

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

      4. sqr-pow 37.6%

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

      5. rem-sqrt-square 38.2%

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

      6. sqr-pow 38.1%

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

      7. fabs-sqr 38.1%

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

      8. sqr-pow 38.2%

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

      9. metadata-eval 38.2%

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

   8. Simplified 38.2%

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

      1. add-cbrt-cube 32.4%

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

   10. Applied egg-rr 32.4%

       \[\leadsto \color{blue}{\sqrt[3]{\left({\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}\right) \cdot {\left(h \cdot \ell\right)}^{-0.5}}} \cdot d \]
   11. Step-by-step derivation

       1. associate-*l* 32.4%

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

       2. pow-sqr 32.4%

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

       3. metadata-eval 32.4%

          \[\leadsto \sqrt[3]{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{\color{blue}{-1}}} \cdot d \]

       4. unpow-1 32.4%

          \[\leadsto \sqrt[3]{{\left(h \cdot \ell\right)}^{-0.5} \cdot \color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

   12. Simplified 32.4%

       \[\leadsto \color{blue}{\sqrt[3]{{\left(h \cdot \ell\right)}^{-0.5} \cdot \frac{1}{h \cdot \ell}}} \cdot d \]
   13. Step-by-step derivation

       1. *-commutative 32.4%

          \[\leadsto \color{blue}{d \cdot \sqrt[3]{{\left(h \cdot \ell\right)}^{-0.5} \cdot \frac{1}{h \cdot \ell}}} \]

       2. add-sqr-sqrt 32.3%

          \[\leadsto d \cdot \color{blue}{\left(\sqrt{\sqrt[3]{{\left(h \cdot \ell\right)}^{-0.5} \cdot \frac{1}{h \cdot \ell}}} \cdot \sqrt{\sqrt[3]{{\left(h \cdot \ell\right)}^{-0.5} \cdot \frac{1}{h \cdot \ell}}}\right)} \]

       3. pow2 32.3%

          \[\leadsto d \cdot \color{blue}{{\left(\sqrt{\sqrt[3]{{\left(h \cdot \ell\right)}^{-0.5} \cdot \frac{1}{h \cdot \ell}}}\right)}^{2}} \]

   14. Applied egg-rr 39.8%

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

       1. unpow2 39.8%

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

       2. rem-sqrt-square 57.4%

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

   16. Simplified 57.4%

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

   if 3.90000000000000022e-63 < M

   1. Initial program 63.0%

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

      1. metadata-eval 63.0%

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

      2. unpow1/2 63.0%

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

      3. metadata-eval 63.0%

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

      4. unpow1/2 63.0%

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

      5. *-commutative 63.0%

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

      6. associate-*l* 63.0%

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

      7. times-frac 63.0%

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

      8. metadata-eval 63.0%

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

   3. Simplified 63.0%

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

   4. Taylor expanded in d around 0 20.2%

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

      1. associate-/l* 20.0%

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

      2. unpow2 20.0%

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

      3. unpow2 20.0%

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

   6. Simplified 20.0%

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

4. Final simplification 43.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -2.1 \cdot 10^{+216}:\\ \;\;\;\;\sqrt[3]{\left(\frac{1}{\ell \cdot h} \cdot \left(d \cdot d\right)\right) \cdot \left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right)}\\ \mathbf{elif}\;M \leq 3.9 \cdot 10^{-63}:\\ \;\;\;\;\left|d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right|\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \frac{D \cdot D}{\frac{d}{M \cdot M}}\right)\\ \end{array} \]

Alternative 9: 45.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := d \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \mathbf{if}\;\ell \leq -7.2 \cdot 10^{-219}:\\ \;\;\;\;\left|t_0\right|\\ \mathbf{elif}\;\ell \leq 7.5 \cdot 10^{-307}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* d (pow (* l h) -0.5))))
   (if (<= l -7.2e-219)
     (fabs t_0)
     (if (<= l 7.5e-307) t_0 (* d (* (pow h -0.5) (pow l -0.5)))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = d * pow((l * h), -0.5);
	double tmp;
	if (l <= -7.2e-219) {
		tmp = fabs(t_0);
	} else if (l <= 7.5e-307) {
		tmp = t_0;
	} else {
		tmp = d * (pow(h, -0.5) * pow(l, -0.5));
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = d * ((l * h) ** (-0.5d0))
    if (l <= (-7.2d-219)) then
        tmp = abs(t_0)
    else if (l <= 7.5d-307) then
        tmp = t_0
    else
        tmp = d * ((h ** (-0.5d0)) * (l ** (-0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = d * Math.pow((l * h), -0.5);
	double tmp;
	if (l <= -7.2e-219) {
		tmp = Math.abs(t_0);
	} else if (l <= 7.5e-307) {
		tmp = t_0;
	} else {
		tmp = d * (Math.pow(h, -0.5) * Math.pow(l, -0.5));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = d * math.pow((l * h), -0.5)
	tmp = 0
	if l <= -7.2e-219:
		tmp = math.fabs(t_0)
	elif l <= 7.5e-307:
		tmp = t_0
	else:
		tmp = d * (math.pow(h, -0.5) * math.pow(l, -0.5))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(d * (Float64(l * h) ^ -0.5))
	tmp = 0.0
	if (l <= -7.2e-219)
		tmp = abs(t_0);
	elseif (l <= 7.5e-307)
		tmp = t_0;
	else
		tmp = Float64(d * Float64((h ^ -0.5) * (l ^ -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = d * ((l * h) ^ -0.5);
	tmp = 0.0;
	if (l <= -7.2e-219)
		tmp = abs(t_0);
	elseif (l <= 7.5e-307)
		tmp = t_0;
	else
		tmp = d * ((h ^ -0.5) * (l ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(d * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -7.2e-219], N[Abs[t$95$0], $MachinePrecision], If[LessEqual[l, 7.5e-307], t$95$0, N[(d * N[(N[Power[h, -0.5], $MachinePrecision] * N[Power[l, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < -7.19999999999999947e-219

   1. Initial program 61.8%

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

      1. metadata-eval 61.8%

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

      2. unpow1/2 61.8%

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

      3. metadata-eval 61.8%

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

      4. unpow1/2 61.8%

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

      5. *-commutative 61.8%

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

      6. associate-*l* 61.8%

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

      7. times-frac 61.8%

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

      8. metadata-eval 61.8%

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

   3. Simplified 61.8%

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

   4. Taylor expanded in d around inf 10.7%

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

      1. expm1-log1p-u 10.7%

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

      2. expm1-udef 10.6%

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

      3. *-commutative 10.6%

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

   6. Applied egg-rr 10.6%

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

      1. expm1-def 10.7%

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

      2. expm1-log1p 10.7%

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

      3. unpow-1 10.7%

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

      4. sqr-pow 10.7%

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

      5. rem-sqrt-square 10.7%

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

      6. sqr-pow 10.7%

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

      7. fabs-sqr 10.7%

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

      8. sqr-pow 10.7%

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

      9. metadata-eval 10.7%

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

   8. Simplified 10.7%

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

      1. add-cbrt-cube 11.7%

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

   10. Applied egg-rr 11.7%

       \[\leadsto \color{blue}{\sqrt[3]{\left({\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}\right) \cdot {\left(h \cdot \ell\right)}^{-0.5}}} \cdot d \]
   11. Step-by-step derivation

       1. associate-*l* 11.7%

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

       2. pow-sqr 11.7%

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

       3. metadata-eval 11.7%

          \[\leadsto \sqrt[3]{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{\color{blue}{-1}}} \cdot d \]

       4. unpow-1 11.7%

          \[\leadsto \sqrt[3]{{\left(h \cdot \ell\right)}^{-0.5} \cdot \color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

   12. Simplified 11.7%

       \[\leadsto \color{blue}{\sqrt[3]{{\left(h \cdot \ell\right)}^{-0.5} \cdot \frac{1}{h \cdot \ell}}} \cdot d \]
   13. Step-by-step derivation

       1. *-commutative 11.7%

          \[\leadsto \color{blue}{d \cdot \sqrt[3]{{\left(h \cdot \ell\right)}^{-0.5} \cdot \frac{1}{h \cdot \ell}}} \]

       2. add-sqr-sqrt 11.7%

          \[\leadsto d \cdot \color{blue}{\left(\sqrt{\sqrt[3]{{\left(h \cdot \ell\right)}^{-0.5} \cdot \frac{1}{h \cdot \ell}}} \cdot \sqrt{\sqrt[3]{{\left(h \cdot \ell\right)}^{-0.5} \cdot \frac{1}{h \cdot \ell}}}\right)} \]

       3. pow2 11.7%

          \[\leadsto d \cdot \color{blue}{{\left(\sqrt{\sqrt[3]{{\left(h \cdot \ell\right)}^{-0.5} \cdot \frac{1}{h \cdot \ell}}}\right)}^{2}} \]

   14. Applied egg-rr 26.9%

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

       1. unpow2 26.9%

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

       2. rem-sqrt-square 43.6%

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

   16. Simplified 43.6%

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

   if -7.19999999999999947e-219 < l < 7.5000000000000006e-307

   1. Initial program 60.8%

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

      1. metadata-eval 60.8%

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

      2. unpow1/2 60.8%

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

      3. metadata-eval 60.8%

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

      4. unpow1/2 60.8%

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

      5. *-commutative 60.8%

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

      6. associate-*l* 60.8%

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

      7. times-frac 60.8%

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

      8. metadata-eval 60.8%

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

   3. Simplified 60.8%

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

   4. Taylor expanded in d around inf 34.4%

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

      1. expm1-log1p-u 34.4%

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

      2. expm1-udef 34.4%

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

      3. *-commutative 34.4%

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

   6. Applied egg-rr 34.4%

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

      1. expm1-def 34.4%

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

      2. expm1-log1p 34.4%

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

      3. unpow-1 34.4%

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

      4. sqr-pow 34.4%

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

      5. rem-sqrt-square 34.4%

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

      6. sqr-pow 34.4%

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

      7. fabs-sqr 34.4%

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

      8. sqr-pow 34.4%

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

      9. metadata-eval 34.4%

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

   8. Simplified 34.4%

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

   if 7.5000000000000006e-307 < l

   1. Initial program 70.7%

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

      1. metadata-eval 70.7%

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

      2. unpow1/2 70.7%

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

      3. metadata-eval 70.7%

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

      4. unpow1/2 70.7%

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

      5. *-commutative 70.7%

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

      6. associate-*l* 70.7%

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

      7. times-frac 70.0%

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

      8. metadata-eval 70.0%

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

   3. Simplified 70.0%

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

   4. Taylor expanded in d around inf 51.0%

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

      1. expm1-log1p-u 49.9%

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

      2. expm1-udef 30.2%

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

      3. *-commutative 30.2%

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

   6. Applied egg-rr 30.2%

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

      1. expm1-def 49.9%

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

      2. expm1-log1p 51.0%

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

      3. unpow-1 51.0%

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

      4. sqr-pow 51.1%

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

      5. rem-sqrt-square 51.7%

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

      6. sqr-pow 51.5%

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

      7. fabs-sqr 51.5%

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

      8. sqr-pow 51.7%

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

      9. metadata-eval 51.7%

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

   8. Simplified 51.7%

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

      1. unpow-prod-down 56.5%

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

   10. Applied egg-rr 56.5%

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

4. Final simplification 49.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -7.2 \cdot 10^{-219}:\\ \;\;\;\;\left|d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right|\\ \mathbf{elif}\;\ell \leq 7.5 \cdot 10^{-307}:\\ \;\;\;\;d \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \]

Alternative 10: 46.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.18 \cdot 10^{-226}:\\ \;\;\;\;\left|d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right|\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;d \cdot {\left({\left(\ell \cdot h\right)}^{-1.5}\right)}^{0.3333333333333333}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= l -1.18e-226)
   (fabs (* d (pow (* l h) -0.5)))
   (if (<= l -5e-310)
     (* d (pow (pow (* l h) -1.5) 0.3333333333333333))
     (* d (* (pow h -0.5) (pow l -0.5))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -1.18e-226) {
		tmp = fabs((d * pow((l * h), -0.5)));
	} else if (l <= -5e-310) {
		tmp = d * pow(pow((l * h), -1.5), 0.3333333333333333);
	} else {
		tmp = d * (pow(h, -0.5) * pow(l, -0.5));
	}
	return tmp;
}

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
    real(8) :: tmp
    if (l <= (-1.18d-226)) then
        tmp = abs((d * ((l * h) ** (-0.5d0))))
    else if (l <= (-5d-310)) then
        tmp = d * (((l * h) ** (-1.5d0)) ** 0.3333333333333333d0)
    else
        tmp = d * ((h ** (-0.5d0)) * (l ** (-0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -1.18e-226) {
		tmp = Math.abs((d * Math.pow((l * h), -0.5)));
	} else if (l <= -5e-310) {
		tmp = d * Math.pow(Math.pow((l * h), -1.5), 0.3333333333333333);
	} else {
		tmp = d * (Math.pow(h, -0.5) * Math.pow(l, -0.5));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if l <= -1.18e-226:
		tmp = math.fabs((d * math.pow((l * h), -0.5)))
	elif l <= -5e-310:
		tmp = d * math.pow(math.pow((l * h), -1.5), 0.3333333333333333)
	else:
		tmp = d * (math.pow(h, -0.5) * math.pow(l, -0.5))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= -1.18e-226)
		tmp = abs(Float64(d * (Float64(l * h) ^ -0.5)));
	elseif (l <= -5e-310)
		tmp = Float64(d * ((Float64(l * h) ^ -1.5) ^ 0.3333333333333333));
	else
		tmp = Float64(d * Float64((h ^ -0.5) * (l ^ -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (l <= -1.18e-226)
		tmp = abs((d * ((l * h) ^ -0.5)));
	elseif (l <= -5e-310)
		tmp = d * (((l * h) ^ -1.5) ^ 0.3333333333333333);
	else
		tmp = d * ((h ^ -0.5) * (l ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[l, -1.18e-226], N[Abs[N[(d * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, -5e-310], N[(d * N[Power[N[Power[N[(l * h), $MachinePrecision], -1.5], $MachinePrecision], 0.3333333333333333], $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[h, -0.5], $MachinePrecision] * N[Power[l, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.1799999999999999e-226

   1. Initial program 61.8%

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

      1. metadata-eval 61.8%

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

      2. unpow1/2 61.8%

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

      3. metadata-eval 61.8%

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

      4. unpow1/2 61.8%

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

      5. *-commutative 61.8%

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

      6. associate-*l* 61.8%

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

      7. times-frac 61.8%

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

      8. metadata-eval 61.8%

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

   3. Simplified 61.8%

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

   4. Taylor expanded in d around inf 10.7%

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

      1. expm1-log1p-u 10.7%

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

      2. expm1-udef 10.6%

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

      3. *-commutative 10.6%

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

   6. Applied egg-rr 10.6%

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

      1. expm1-def 10.7%

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

      2. expm1-log1p 10.7%

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

      3. unpow-1 10.7%

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

      4. sqr-pow 10.7%

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

      5. rem-sqrt-square 10.7%

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

      6. sqr-pow 10.7%

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

      7. fabs-sqr 10.7%

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

      8. sqr-pow 10.7%

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

      9. metadata-eval 10.7%

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

   8. Simplified 10.7%

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

      1. add-cbrt-cube 11.7%

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

   10. Applied egg-rr 11.7%

       \[\leadsto \color{blue}{\sqrt[3]{\left({\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}\right) \cdot {\left(h \cdot \ell\right)}^{-0.5}}} \cdot d \]
   11. Step-by-step derivation

       1. associate-*l* 11.7%

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

       2. pow-sqr 11.7%

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

       3. metadata-eval 11.7%

          \[\leadsto \sqrt[3]{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{\color{blue}{-1}}} \cdot d \]

       4. unpow-1 11.7%

          \[\leadsto \sqrt[3]{{\left(h \cdot \ell\right)}^{-0.5} \cdot \color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

   12. Simplified 11.7%

       \[\leadsto \color{blue}{\sqrt[3]{{\left(h \cdot \ell\right)}^{-0.5} \cdot \frac{1}{h \cdot \ell}}} \cdot d \]
   13. Step-by-step derivation

       1. *-commutative 11.7%

          \[\leadsto \color{blue}{d \cdot \sqrt[3]{{\left(h \cdot \ell\right)}^{-0.5} \cdot \frac{1}{h \cdot \ell}}} \]

       2. add-sqr-sqrt 11.7%

          \[\leadsto d \cdot \color{blue}{\left(\sqrt{\sqrt[3]{{\left(h \cdot \ell\right)}^{-0.5} \cdot \frac{1}{h \cdot \ell}}} \cdot \sqrt{\sqrt[3]{{\left(h \cdot \ell\right)}^{-0.5} \cdot \frac{1}{h \cdot \ell}}}\right)} \]

       3. pow2 11.7%

          \[\leadsto d \cdot \color{blue}{{\left(\sqrt{\sqrt[3]{{\left(h \cdot \ell\right)}^{-0.5} \cdot \frac{1}{h \cdot \ell}}}\right)}^{2}} \]

   14. Applied egg-rr 26.9%

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

       1. unpow2 26.9%

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

       2. rem-sqrt-square 43.6%

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

   16. Simplified 43.6%

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

   if -1.1799999999999999e-226 < l < -4.999999999999985e-310

   1. Initial program 60.8%

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

      1. metadata-eval 60.8%

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

      2. unpow1/2 60.8%

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

      3. metadata-eval 60.8%

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

      4. unpow1/2 60.8%

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

      5. *-commutative 60.8%

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

      6. associate-*l* 60.8%

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

      7. times-frac 60.8%

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

      8. metadata-eval 60.8%

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

   3. Simplified 60.8%

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

   4. Taylor expanded in d around inf 34.4%

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

      1. expm1-log1p-u 34.4%

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

      2. expm1-udef 34.4%

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

      3. *-commutative 34.4%

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

   6. Applied egg-rr 34.4%

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

      1. expm1-def 34.4%

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

      2. expm1-log1p 34.4%

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

      3. unpow-1 34.4%

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

      4. sqr-pow 34.4%

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

      5. rem-sqrt-square 34.4%

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

      6. sqr-pow 34.4%

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

      7. fabs-sqr 34.4%

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

      8. sqr-pow 34.4%

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

      9. metadata-eval 34.4%

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

   8. Simplified 34.4%

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

      1. add-cbrt-cube 42.1%

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

   10. Applied egg-rr 42.1%

       \[\leadsto \color{blue}{\sqrt[3]{\left({\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}\right) \cdot {\left(h \cdot \ell\right)}^{-0.5}}} \cdot d \]
   11. Step-by-step derivation

       1. associate-*l* 42.1%

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

       2. pow-sqr 42.1%

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

       3. metadata-eval 42.1%

          \[\leadsto \sqrt[3]{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{\color{blue}{-1}}} \cdot d \]

       4. unpow-1 42.1%

          \[\leadsto \sqrt[3]{{\left(h \cdot \ell\right)}^{-0.5} \cdot \color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

   12. Simplified 42.1%

       \[\leadsto \color{blue}{\sqrt[3]{{\left(h \cdot \ell\right)}^{-0.5} \cdot \frac{1}{h \cdot \ell}}} \cdot d \]
   13. Step-by-step derivation

       1. pow1/3 42.1%

          \[\leadsto \color{blue}{{\left({\left(h \cdot \ell\right)}^{-0.5} \cdot \frac{1}{h \cdot \ell}\right)}^{0.3333333333333333}} \cdot d \]

       2. inv-pow 42.1%

          \[\leadsto {\left({\left(h \cdot \ell\right)}^{-0.5} \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-1}}\right)}^{0.3333333333333333} \cdot d \]

       3. pow-prod-up 42.1%

          \[\leadsto {\color{blue}{\left({\left(h \cdot \ell\right)}^{\left(-0.5 + -1\right)}\right)}}^{0.3333333333333333} \cdot d \]

       4. metadata-eval 42.1%

          \[\leadsto {\left({\left(h \cdot \ell\right)}^{\color{blue}{-1.5}}\right)}^{0.3333333333333333} \cdot d \]

   14. Applied egg-rr 42.1%

       \[\leadsto \color{blue}{{\left({\left(h \cdot \ell\right)}^{-1.5}\right)}^{0.3333333333333333}} \cdot d \]

   if -4.999999999999985e-310 < l

   1. Initial program 70.7%

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

      1. metadata-eval 70.7%

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

      2. unpow1/2 70.7%

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

      3. metadata-eval 70.7%

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

      4. unpow1/2 70.7%

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

      5. *-commutative 70.7%

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

      6. associate-*l* 70.7%

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

      7. times-frac 70.0%

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

      8. metadata-eval 70.0%

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

   3. Simplified 70.0%

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

   4. Taylor expanded in d around inf 51.0%

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

      1. expm1-log1p-u 49.9%

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

      2. expm1-udef 30.2%

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

      3. *-commutative 30.2%

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

   6. Applied egg-rr 30.2%

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

      1. expm1-def 49.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{1}{h \cdot \ell}}\right)\right)} \cdot d \]

      2. expm1-log1p 51.0%

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      3. unpow-1 51.0%

         \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \cdot d \]

      4. sqr-pow 51.1%

         \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}}} \cdot d \]

      5. rem-sqrt-square 51.7%

         \[\leadsto \color{blue}{\left|{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}\right|} \cdot d \]

      6. sqr-pow 51.5%

         \[\leadsto \left|\color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)}}\right| \cdot d \]

      7. fabs-sqr 51.5%

         \[\leadsto \color{blue}{\left({\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)}\right)} \cdot d \]

      8. sqr-pow 51.7%

         \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}} \cdot d \]

      9. metadata-eval 51.7%

         \[\leadsto {\left(h \cdot \ell\right)}^{\color{blue}{-0.5}} \cdot d \]

   8. Simplified 51.7%

      \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \cdot d \]
   9. Step-by-step derivation

      1. unpow-prod-down 56.5%

         \[\leadsto \color{blue}{\left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)} \cdot d \]

   10. Applied egg-rr 56.5%

       \[\leadsto \color{blue}{\left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)} \cdot d \]
3. Recombined 3 regimes into one program.

4. Final simplification 50.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.18 \cdot 10^{-226}:\\ \;\;\;\;\left|d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right|\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;d \cdot {\left({\left(\ell \cdot h\right)}^{-1.5}\right)}^{0.3333333333333333}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \]

Alternative 11: 38.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := d \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \mathbf{if}\;M \leq -2.35 \cdot 10^{+216}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;M \leq 3.9 \cdot 10^{-63}:\\ \;\;\;\;\left|t_0\right|\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{-1 + \left(1 + \frac{1}{\ell \cdot h}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* d (pow (* l h) -0.5))))
   (if (<= M -2.35e+216)
     t_0
     (if (<= M 3.9e-63)
       (fabs t_0)
       (* d (sqrt (+ -1.0 (+ 1.0 (/ 1.0 (* l h))))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = d * pow((l * h), -0.5);
	double tmp;
	if (M <= -2.35e+216) {
		tmp = t_0;
	} else if (M <= 3.9e-63) {
		tmp = fabs(t_0);
	} else {
		tmp = d * sqrt((-1.0 + (1.0 + (1.0 / (l * h)))));
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = d * ((l * h) ** (-0.5d0))
    if (m <= (-2.35d+216)) then
        tmp = t_0
    else if (m <= 3.9d-63) then
        tmp = abs(t_0)
    else
        tmp = d * sqrt(((-1.0d0) + (1.0d0 + (1.0d0 / (l * h)))))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = d * Math.pow((l * h), -0.5);
	double tmp;
	if (M <= -2.35e+216) {
		tmp = t_0;
	} else if (M <= 3.9e-63) {
		tmp = Math.abs(t_0);
	} else {
		tmp = d * Math.sqrt((-1.0 + (1.0 + (1.0 / (l * h)))));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = d * math.pow((l * h), -0.5)
	tmp = 0
	if M <= -2.35e+216:
		tmp = t_0
	elif M <= 3.9e-63:
		tmp = math.fabs(t_0)
	else:
		tmp = d * math.sqrt((-1.0 + (1.0 + (1.0 / (l * h)))))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(d * (Float64(l * h) ^ -0.5))
	tmp = 0.0
	if (M <= -2.35e+216)
		tmp = t_0;
	elseif (M <= 3.9e-63)
		tmp = abs(t_0);
	else
		tmp = Float64(d * sqrt(Float64(-1.0 + Float64(1.0 + Float64(1.0 / Float64(l * h))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = d * ((l * h) ^ -0.5);
	tmp = 0.0;
	if (M <= -2.35e+216)
		tmp = t_0;
	elseif (M <= 3.9e-63)
		tmp = abs(t_0);
	else
		tmp = d * sqrt((-1.0 + (1.0 + (1.0 / (l * h)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(d * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[M, -2.35e+216], t$95$0, If[LessEqual[M, 3.9e-63], N[Abs[t$95$0], $MachinePrecision], N[(d * N[Sqrt[N[(-1.0 + N[(1.0 + N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if M < -2.3500000000000001e216

   1. Initial program 72.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 72.9%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 72.9%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 72.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 72.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 72.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 72.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 68.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 68.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 68.3%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]

   4. Taylor expanded in d around inf 30.8%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   5. Step-by-step derivation

      1. expm1-log1p-u 30.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{1}{\ell \cdot h}}\right)\right)} \cdot d \]

      2. expm1-udef 21.7%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{1}{\ell \cdot h}}\right)} - 1\right)} \cdot d \]

      3. *-commutative 21.7%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\sqrt{\frac{1}{\color{blue}{h \cdot \ell}}}\right)} - 1\right) \cdot d \]

   6. Applied egg-rr 21.7%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{1}{h \cdot \ell}}\right)} - 1\right)} \cdot d \]
   7. Step-by-step derivation

      1. expm1-def 30.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{1}{h \cdot \ell}}\right)\right)} \cdot d \]

      2. expm1-log1p 30.8%

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      3. unpow-1 30.8%

         \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \cdot d \]

      4. sqr-pow 30.8%

         \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}}} \cdot d \]

      5. rem-sqrt-square 30.8%

         \[\leadsto \color{blue}{\left|{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}\right|} \cdot d \]

      6. sqr-pow 30.7%

         \[\leadsto \left|\color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)}}\right| \cdot d \]

      7. fabs-sqr 30.7%

         \[\leadsto \color{blue}{\left({\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)}\right)} \cdot d \]

      8. sqr-pow 30.8%

         \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}} \cdot d \]

      9. metadata-eval 30.8%

         \[\leadsto {\left(h \cdot \ell\right)}^{\color{blue}{-0.5}} \cdot d \]

   8. Simplified 30.8%

      \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \cdot d \]

   if -2.3500000000000001e216 < M < 3.90000000000000022e-63

   1. Initial program 67.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 67.4%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 67.4%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 67.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 67.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 67.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 67.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 67.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 67.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 67.4%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]

   4. Taylor expanded in d around inf 37.6%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   5. Step-by-step derivation

      1. expm1-log1p-u 36.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{1}{\ell \cdot h}}\right)\right)} \cdot d \]

      2. expm1-udef 23.2%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{1}{\ell \cdot h}}\right)} - 1\right)} \cdot d \]

      3. *-commutative 23.2%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\sqrt{\frac{1}{\color{blue}{h \cdot \ell}}}\right)} - 1\right) \cdot d \]

   6. Applied egg-rr 23.2%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{1}{h \cdot \ell}}\right)} - 1\right)} \cdot d \]
   7. Step-by-step derivation

      1. expm1-def 36.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{1}{h \cdot \ell}}\right)\right)} \cdot d \]

      2. expm1-log1p 37.6%

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      3. unpow-1 37.6%

         \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \cdot d \]

      4. sqr-pow 37.6%

         \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}}} \cdot d \]

      5. rem-sqrt-square 38.2%

         \[\leadsto \color{blue}{\left|{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}\right|} \cdot d \]

      6. sqr-pow 38.1%

         \[\leadsto \left|\color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)}}\right| \cdot d \]

      7. fabs-sqr 38.1%

         \[\leadsto \color{blue}{\left({\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)}\right)} \cdot d \]

      8. sqr-pow 38.2%

         \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}} \cdot d \]

      9. metadata-eval 38.2%

         \[\leadsto {\left(h \cdot \ell\right)}^{\color{blue}{-0.5}} \cdot d \]

   8. Simplified 38.2%

      \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \cdot d \]
   9. Step-by-step derivation

      1. add-cbrt-cube 32.4%

         \[\leadsto \color{blue}{\sqrt[3]{\left({\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}\right) \cdot {\left(h \cdot \ell\right)}^{-0.5}}} \cdot d \]

   10. Applied egg-rr 32.4%

       \[\leadsto \color{blue}{\sqrt[3]{\left({\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}\right) \cdot {\left(h \cdot \ell\right)}^{-0.5}}} \cdot d \]
   11. Step-by-step derivation

       1. associate-*l* 32.4%

          \[\leadsto \sqrt[3]{\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot \left({\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}\right)}} \cdot d \]

       2. pow-sqr 32.4%

          \[\leadsto \sqrt[3]{{\left(h \cdot \ell\right)}^{-0.5} \cdot \color{blue}{{\left(h \cdot \ell\right)}^{\left(2 \cdot -0.5\right)}}} \cdot d \]

       3. metadata-eval 32.4%

          \[\leadsto \sqrt[3]{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{\color{blue}{-1}}} \cdot d \]

       4. unpow-1 32.4%

          \[\leadsto \sqrt[3]{{\left(h \cdot \ell\right)}^{-0.5} \cdot \color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

   12. Simplified 32.4%

       \[\leadsto \color{blue}{\sqrt[3]{{\left(h \cdot \ell\right)}^{-0.5} \cdot \frac{1}{h \cdot \ell}}} \cdot d \]
   13. Step-by-step derivation

       1. *-commutative 32.4%

          \[\leadsto \color{blue}{d \cdot \sqrt[3]{{\left(h \cdot \ell\right)}^{-0.5} \cdot \frac{1}{h \cdot \ell}}} \]

       2. add-sqr-sqrt 32.3%

          \[\leadsto d \cdot \color{blue}{\left(\sqrt{\sqrt[3]{{\left(h \cdot \ell\right)}^{-0.5} \cdot \frac{1}{h \cdot \ell}}} \cdot \sqrt{\sqrt[3]{{\left(h \cdot \ell\right)}^{-0.5} \cdot \frac{1}{h \cdot \ell}}}\right)} \]

       3. pow2 32.3%

          \[\leadsto d \cdot \color{blue}{{\left(\sqrt{\sqrt[3]{{\left(h \cdot \ell\right)}^{-0.5} \cdot \frac{1}{h \cdot \ell}}}\right)}^{2}} \]

   14. Applied egg-rr 39.8%

       \[\leadsto \color{blue}{\sqrt{{\left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right)}^{2}}} \]
   15. Step-by-step derivation

       1. unpow2 39.8%

          \[\leadsto \sqrt{\color{blue}{\left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right) \cdot \left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right)}} \]

       2. rem-sqrt-square 57.4%

          \[\leadsto \color{blue}{\left|d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right|} \]

   16. Simplified 57.4%

       \[\leadsto \color{blue}{\left|d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right|} \]

   if 3.90000000000000022e-63 < M

   1. Initial program 63.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 63.0%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 63.0%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 63.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 63.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 63.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 63.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 63.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 63.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 63.0%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]

   4. Taylor expanded in d around inf 29.5%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   5. Step-by-step derivation

      1. expm1-log1p-u 29.2%

         \[\leadsto \sqrt{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\ell \cdot h}\right)\right)}} \cdot d \]

      2. expm1-udef 23.4%

         \[\leadsto \sqrt{\color{blue}{e^{\mathsf{log1p}\left(\frac{1}{\ell \cdot h}\right)} - 1}} \cdot d \]

      3. *-commutative 23.4%

         \[\leadsto \sqrt{e^{\mathsf{log1p}\left(\frac{1}{\color{blue}{h \cdot \ell}}\right)} - 1} \cdot d \]

   6. Applied egg-rr 23.4%

      \[\leadsto \sqrt{\color{blue}{e^{\mathsf{log1p}\left(\frac{1}{h \cdot \ell}\right)} - 1}} \cdot d \]

   7. Taylor expanded in h around inf 23.8%

      \[\leadsto \sqrt{\color{blue}{\left(\frac{1}{\ell \cdot h} + 1\right)} - 1} \cdot d \]
3. Recombined 3 regimes into one program.

4. Final simplification 44.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -2.35 \cdot 10^{+216}:\\ \;\;\;\;d \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \mathbf{elif}\;M \leq 3.9 \cdot 10^{-63}:\\ \;\;\;\;\left|d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right|\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{-1 + \left(1 + \frac{1}{\ell \cdot h}\right)}\\ \end{array} \]

Alternative 12: 26.3% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ d \cdot {\left(\ell \cdot h\right)}^{-0.5} \end{array} \]

FPCore

(FPCore (d h l M D) :precision binary64 (* d (pow (* l h) -0.5)))

C

double code(double d, double h, double l, double M, double D) {
	return d * pow((l * h), -0.5);
}

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 * ((l * h) ** (-0.5d0))
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	return d * Math.pow((l * h), -0.5);
}

Python

def code(d, h, l, M, D):
	return d * math.pow((l * h), -0.5)

Julia

function code(d, h, l, M, D)
	return Float64(d * (Float64(l * h) ^ -0.5))
end

MATLAB

function tmp = code(d, h, l, M, D)
	tmp = d * ((l * h) ^ -0.5);
end

Wolfram

code[d_, h_, l_, M_, D_] := N[(d * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 66.5%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
2. Step-by-step derivation

   1. metadata-eval 66.5%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. unpow1/2 66.5%

      \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. metadata-eval 66.5%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. unpow1/2 66.5%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   5. *-commutative 66.5%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

   6. associate-*l* 66.5%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

   7. times-frac 66.1%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

   8. metadata-eval 66.1%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

3. Simplified 66.1%

   \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]

4. Taylor expanded in d around inf 34.4%

   \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
5. Step-by-step derivation

   1. expm1-log1p-u 33.8%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{1}{\ell \cdot h}}\right)\right)} \cdot d \]

   2. expm1-udef 23.3%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{1}{\ell \cdot h}}\right)} - 1\right)} \cdot d \]

   3. *-commutative 23.3%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\sqrt{\frac{1}{\color{blue}{h \cdot \ell}}}\right)} - 1\right) \cdot d \]

6. Applied egg-rr 23.3%

   \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{1}{h \cdot \ell}}\right)} - 1\right)} \cdot d \]
7. Step-by-step derivation

   1. expm1-def 33.8%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{1}{h \cdot \ell}}\right)\right)} \cdot d \]

   2. expm1-log1p 34.4%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

   3. unpow-1 34.4%

      \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \cdot d \]

   4. sqr-pow 34.4%

      \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}}} \cdot d \]

   5. rem-sqrt-square 34.8%

      \[\leadsto \color{blue}{\left|{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}\right|} \cdot d \]

   6. sqr-pow 34.7%

      \[\leadsto \left|\color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)}}\right| \cdot d \]

   7. fabs-sqr 34.7%

      \[\leadsto \color{blue}{\left({\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)}\right)} \cdot d \]

   8. sqr-pow 34.8%

      \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}} \cdot d \]

   9. metadata-eval 34.8%

      \[\leadsto {\left(h \cdot \ell\right)}^{\color{blue}{-0.5}} \cdot d \]

8. Simplified 34.8%

   \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \cdot d \]

9. Final simplification 34.8%

   \[\leadsto d \cdot {\left(\ell \cdot h\right)}^{-0.5} \]

Alternative 13: 4.4% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ d \cdot \sqrt{0} \end{array} \]

FPCore

(FPCore (d h l M D) :precision binary64 (* d (sqrt 0.0)))

C

double code(double d, double h, double l, double M, double D) {
	return d * sqrt(0.0);
}

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 * sqrt(0.0d0)
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	return d * Math.sqrt(0.0);
}

Python

def code(d, h, l, M, D):
	return d * math.sqrt(0.0)

Julia

function code(d, h, l, M, D)
	return Float64(d * sqrt(0.0))
end

MATLAB

function tmp = code(d, h, l, M, D)
	tmp = d * sqrt(0.0);
end

Wolfram

code[d_, h_, l_, M_, D_] := N[(d * N[Sqrt[0.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 66.5%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
2. Step-by-step derivation

   1. metadata-eval 66.5%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. unpow1/2 66.5%

      \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. metadata-eval 66.5%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. unpow1/2 66.5%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   5. *-commutative 66.5%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

   6. associate-*l* 66.5%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

   7. times-frac 66.1%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

   8. metadata-eval 66.1%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

3. Simplified 66.1%

   \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]

4. Taylor expanded in d around inf 34.4%

   \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
5. Step-by-step derivation

   1. expm1-log1p-u 33.8%

      \[\leadsto \sqrt{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\ell \cdot h}\right)\right)}} \cdot d \]

   2. expm1-udef 23.1%

      \[\leadsto \sqrt{\color{blue}{e^{\mathsf{log1p}\left(\frac{1}{\ell \cdot h}\right)} - 1}} \cdot d \]

   3. *-commutative 23.1%

      \[\leadsto \sqrt{e^{\mathsf{log1p}\left(\frac{1}{\color{blue}{h \cdot \ell}}\right)} - 1} \cdot d \]

6. Applied egg-rr 23.1%

   \[\leadsto \sqrt{\color{blue}{e^{\mathsf{log1p}\left(\frac{1}{h \cdot \ell}\right)} - 1}} \cdot d \]

7. Taylor expanded in h around inf 4.8%

   \[\leadsto \sqrt{\color{blue}{1} - 1} \cdot d \]

8. Final simplification 4.8%

   \[\leadsto d \cdot \sqrt{0} \]

Reproduce

herbie shell --seed 2023187 
(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)))))