Henrywood and Agarwal, Equation (12)

Percentage Accurate: 66.8% → 77.4%

Time: 20.6s

Alternatives: 13

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 66.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 77.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := 1 - \frac{h \cdot \left(0.5 \cdot {\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2}\right)}{\ell}\\ \mathbf{if}\;\ell \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\left({\left({\left(\frac{-1}{h}\right)}^{0.25} \cdot {\left(-d\right)}^{0.25}\right)}^{2} \cdot t_0\right) \cdot t_1\\ \mathbf{elif}\;\ell \leq 7.2 \cdot 10^{+89}:\\ \;\;\;\;t_1 \cdot \left(t_0 \cdot \frac{\sqrt{d}}{\sqrt{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ d l)))
        (t_1 (- 1.0 (/ (* h (* 0.5 (pow (* (/ D d) (* M 0.5)) 2.0))) l))))
   (if (<= l -1e-310)
     (* (* (pow (* (pow (/ -1.0 h) 0.25) (pow (- d) 0.25)) 2.0) t_0) t_1)
     (if (<= l 7.2e+89)
       (* t_1 (* t_0 (/ (sqrt d) (sqrt h))))
       (* 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 = sqrt((d / l));
	double t_1 = 1.0 - ((h * (0.5 * pow(((D / d) * (M * 0.5)), 2.0))) / l);
	double tmp;
	if (l <= -1e-310) {
		tmp = (pow((pow((-1.0 / h), 0.25) * pow(-d, 0.25)), 2.0) * t_0) * t_1;
	} else if (l <= 7.2e+89) {
		tmp = t_1 * (t_0 * (sqrt(d) / sqrt(h)));
	} 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) :: t_1
    real(8) :: tmp
    t_0 = sqrt((d / l))
    t_1 = 1.0d0 - ((h * (0.5d0 * (((d_1 / d) * (m * 0.5d0)) ** 2.0d0))) / l)
    if (l <= (-1d-310)) then
        tmp = ((((((-1.0d0) / h) ** 0.25d0) * (-d ** 0.25d0)) ** 2.0d0) * t_0) * t_1
    else if (l <= 7.2d+89) then
        tmp = t_1 * (t_0 * (sqrt(d) / sqrt(h)))
    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 = Math.sqrt((d / l));
	double t_1 = 1.0 - ((h * (0.5 * Math.pow(((D / d) * (M * 0.5)), 2.0))) / l);
	double tmp;
	if (l <= -1e-310) {
		tmp = (Math.pow((Math.pow((-1.0 / h), 0.25) * Math.pow(-d, 0.25)), 2.0) * t_0) * t_1;
	} else if (l <= 7.2e+89) {
		tmp = t_1 * (t_0 * (Math.sqrt(d) / Math.sqrt(h)));
	} 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 = math.sqrt((d / l))
	t_1 = 1.0 - ((h * (0.5 * math.pow(((D / d) * (M * 0.5)), 2.0))) / l)
	tmp = 0
	if l <= -1e-310:
		tmp = (math.pow((math.pow((-1.0 / h), 0.25) * math.pow(-d, 0.25)), 2.0) * t_0) * t_1
	elif l <= 7.2e+89:
		tmp = t_1 * (t_0 * (math.sqrt(d) / math.sqrt(h)))
	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 = sqrt(Float64(d / l))
	t_1 = Float64(1.0 - Float64(Float64(h * Float64(0.5 * (Float64(Float64(D / d) * Float64(M * 0.5)) ^ 2.0))) / l))
	tmp = 0.0
	if (l <= -1e-310)
		tmp = Float64(Float64((Float64((Float64(-1.0 / h) ^ 0.25) * (Float64(-d) ^ 0.25)) ^ 2.0) * t_0) * t_1);
	elseif (l <= 7.2e+89)
		tmp = Float64(t_1 * Float64(t_0 * Float64(sqrt(d) / sqrt(h))));
	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 = sqrt((d / l));
	t_1 = 1.0 - ((h * (0.5 * (((D / d) * (M * 0.5)) ^ 2.0))) / l);
	tmp = 0.0;
	if (l <= -1e-310)
		tmp = (((((-1.0 / h) ^ 0.25) * (-d ^ 0.25)) ^ 2.0) * t_0) * t_1;
	elseif (l <= 7.2e+89)
		tmp = t_1 * (t_0 * (sqrt(d) / sqrt(h)));
	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[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - N[(N[(h * N[(0.5 * N[Power[N[(N[(D / d), $MachinePrecision] * N[(M * 0.5), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -1e-310], N[(N[(N[Power[N[(N[Power[N[(-1.0 / h), $MachinePrecision], 0.25], $MachinePrecision] * N[Power[(-d), 0.25], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[l, 7.2e+89], N[(t$95$1 * N[(t$95$0 * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $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 < -9.999999999999969e-311

   1. Initial program 63.6%

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

      1. metadata-eval 63.6%

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

      2. unpow1/2 63.6%

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

      3. metadata-eval 63.6%

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

      4. unpow1/2 63.6%

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

      5. *-commutative 63.6%

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

      6. associate-*l* 63.6%

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

      7. times-frac 63.6%

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

      8. metadata-eval 63.6%

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

   3. Simplified 63.6%

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

      1. associate-*r* 63.6%

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

      2. frac-times 63.6%

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

      3. *-commutative 63.6%

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

      4. metadata-eval 63.6%

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

      5. associate-*r/ 66.4%

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

      6. metadata-eval 66.4%

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

      7. *-commutative 66.4%

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

      8. frac-times 66.4%

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

      9. *-commutative 66.4%

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

      10. div-inv 66.4%

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

      11. metadata-eval 66.4%

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

   5. Applied egg-rr 66.4%

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

      1. pow1/2 66.4%

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

      2. sqr-pow 66.3%

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

      3. pow2 66.3%

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

      4. metadata-eval 66.3%

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

   7. Applied egg-rr 66.3%

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

   8. Taylor expanded in h around -inf 75.1%

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

      1. distribute-lft-in 75.1%

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

      2. exp-sum 75.2%

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

      3. *-commutative 75.2%

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

      4. exp-to-pow 75.5%

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

      5. *-commutative 75.5%

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

      6. rem-square-sqrt 0.0%

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

      7. unpow2 0.0%

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

      8. exp-to-pow 0.0%

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

      9. unpow2 0.0%

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

      10. rem-square-sqrt 78.5%

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

      11. mul-1-neg 78.5%

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

   10. Simplified 78.5%

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

   if -9.999999999999969e-311 < l < 7.2e89

   1. Initial program 66.4%

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

      1. metadata-eval 66.4%

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

      2. unpow1/2 66.4%

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

      3. metadata-eval 66.4%

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

      4. unpow1/2 66.4%

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

      5. *-commutative 66.4%

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

      6. associate-*l* 66.4%

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

      7. times-frac 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%

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

      1. associate-*r* 66.5%

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

      2. frac-times 66.4%

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

      3. *-commutative 66.4%

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

      4. metadata-eval 66.4%

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

      5. associate-*r/ 75.7%

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

      6. metadata-eval 75.7%

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

      7. *-commutative 75.7%

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

      8. frac-times 75.7%

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

      9. *-commutative 75.7%

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

      10. div-inv 75.7%

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

      11. metadata-eval 75.7%

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

   5. Applied egg-rr 75.7%

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

      1. sqrt-div 87.9%

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

   7. Applied egg-rr 87.9%

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

   if 7.2e89 < l

   1. Initial program 54.8%

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

      1. metadata-eval 54.8%

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

      2. unpow1/2 54.8%

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

      3. metadata-eval 54.8%

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

      4. unpow1/2 54.8%

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

      5. *-commutative 54.8%

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

      6. associate-*l* 54.8%

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

      7. times-frac 54.7%

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

      8. metadata-eval 54.7%

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

   3. Simplified 54.7%

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

   4. Taylor expanded in d around inf 75.3%

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

      1. *-un-lft-identity 75.3%

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

      2. *-commutative 75.3%

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

   6. Applied egg-rr 75.3%

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

      1. *-lft-identity 75.3%

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

      2. unpow-1 75.3%

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

      3. sqr-pow 75.3%

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

      4. rem-sqrt-square 75.3%

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

      5. sqr-pow 74.9%

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

      6. fabs-sqr 74.9%

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

      7. sqr-pow 75.3%

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

      8. metadata-eval 75.3%

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

   8. Simplified 75.3%

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

      1. unpow-prod-down 80.2%

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

   10. Applied egg-rr 80.2%

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

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\left({\left({\left(\frac{-1}{h}\right)}^{0.25} \cdot {\left(-d\right)}^{0.25}\right)}^{2} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{h \cdot \left(0.5 \cdot {\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2}\right)}{\ell}\right)\\ \mathbf{elif}\;\ell \leq 7.2 \cdot 10^{+89}:\\ \;\;\;\;\left(1 - \frac{h \cdot \left(0.5 \cdot {\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2}\right)}{\ell}\right) \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \frac{\sqrt{d}}{\sqrt{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \]

Alternative 2: 72.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := 0.5 \cdot \frac{h}{\ell}\\ \mathbf{if}\;\ell \leq -6.8 \cdot 10^{-111}:\\ \;\;\;\;\left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right) \cdot \left(-1 + {\left(M \cdot \frac{D}{\frac{d}{0.5}}\right)}^{2} \cdot t_1\right)\\ \mathbf{elif}\;\ell \leq 1.75 \cdot 10^{-249}:\\ \;\;\;\;\left(t_0 \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - h \cdot \frac{{\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2}}{\frac{\ell}{0.5}}\right)\\ \mathbf{elif}\;\ell \leq 3.2 \cdot 10^{+89}:\\ \;\;\;\;\left(t_0 \cdot \frac{\sqrt{d}}{\sqrt{h}}\right) \cdot \left(1 - t_1 \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ d l))) (t_1 (* 0.5 (/ h l))))
   (if (<= l -6.8e-111)
     (*
      (* d (pow (* l h) -0.5))
      (+ -1.0 (* (pow (* M (/ D (/ d 0.5))) 2.0) t_1)))
     (if (<= l 1.75e-249)
       (*
        (* t_0 (sqrt (/ d h)))
        (- 1.0 (* h (/ (pow (* (/ D d) (* M 0.5)) 2.0) (/ l 0.5)))))
       (if (<= l 3.2e+89)
         (*
          (* t_0 (/ (sqrt d) (sqrt h)))
          (- 1.0 (* t_1 (pow (* (/ D d) (/ M 2.0)) 2.0))))
         (* 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 = sqrt((d / l));
	double t_1 = 0.5 * (h / l);
	double tmp;
	if (l <= -6.8e-111) {
		tmp = (d * pow((l * h), -0.5)) * (-1.0 + (pow((M * (D / (d / 0.5))), 2.0) * t_1));
	} else if (l <= 1.75e-249) {
		tmp = (t_0 * sqrt((d / h))) * (1.0 - (h * (pow(((D / d) * (M * 0.5)), 2.0) / (l / 0.5))));
	} else if (l <= 3.2e+89) {
		tmp = (t_0 * (sqrt(d) / sqrt(h))) * (1.0 - (t_1 * pow(((D / d) * (M / 2.0)), 2.0)));
	} 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) :: t_1
    real(8) :: tmp
    t_0 = sqrt((d / l))
    t_1 = 0.5d0 * (h / l)
    if (l <= (-6.8d-111)) then
        tmp = (d * ((l * h) ** (-0.5d0))) * ((-1.0d0) + (((m * (d_1 / (d / 0.5d0))) ** 2.0d0) * t_1))
    else if (l <= 1.75d-249) then
        tmp = (t_0 * sqrt((d / h))) * (1.0d0 - (h * ((((d_1 / d) * (m * 0.5d0)) ** 2.0d0) / (l / 0.5d0))))
    else if (l <= 3.2d+89) then
        tmp = (t_0 * (sqrt(d) / sqrt(h))) * (1.0d0 - (t_1 * (((d_1 / d) * (m / 2.0d0)) ** 2.0d0)))
    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 = Math.sqrt((d / l));
	double t_1 = 0.5 * (h / l);
	double tmp;
	if (l <= -6.8e-111) {
		tmp = (d * Math.pow((l * h), -0.5)) * (-1.0 + (Math.pow((M * (D / (d / 0.5))), 2.0) * t_1));
	} else if (l <= 1.75e-249) {
		tmp = (t_0 * Math.sqrt((d / h))) * (1.0 - (h * (Math.pow(((D / d) * (M * 0.5)), 2.0) / (l / 0.5))));
	} else if (l <= 3.2e+89) {
		tmp = (t_0 * (Math.sqrt(d) / Math.sqrt(h))) * (1.0 - (t_1 * Math.pow(((D / d) * (M / 2.0)), 2.0)));
	} 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 = math.sqrt((d / l))
	t_1 = 0.5 * (h / l)
	tmp = 0
	if l <= -6.8e-111:
		tmp = (d * math.pow((l * h), -0.5)) * (-1.0 + (math.pow((M * (D / (d / 0.5))), 2.0) * t_1))
	elif l <= 1.75e-249:
		tmp = (t_0 * math.sqrt((d / h))) * (1.0 - (h * (math.pow(((D / d) * (M * 0.5)), 2.0) / (l / 0.5))))
	elif l <= 3.2e+89:
		tmp = (t_0 * (math.sqrt(d) / math.sqrt(h))) * (1.0 - (t_1 * math.pow(((D / d) * (M / 2.0)), 2.0)))
	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 = sqrt(Float64(d / l))
	t_1 = Float64(0.5 * Float64(h / l))
	tmp = 0.0
	if (l <= -6.8e-111)
		tmp = Float64(Float64(d * (Float64(l * h) ^ -0.5)) * Float64(-1.0 + Float64((Float64(M * Float64(D / Float64(d / 0.5))) ^ 2.0) * t_1)));
	elseif (l <= 1.75e-249)
		tmp = Float64(Float64(t_0 * sqrt(Float64(d / h))) * Float64(1.0 - Float64(h * Float64((Float64(Float64(D / d) * Float64(M * 0.5)) ^ 2.0) / Float64(l / 0.5)))));
	elseif (l <= 3.2e+89)
		tmp = Float64(Float64(t_0 * Float64(sqrt(d) / sqrt(h))) * Float64(1.0 - Float64(t_1 * (Float64(Float64(D / d) * Float64(M / 2.0)) ^ 2.0))));
	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 = sqrt((d / l));
	t_1 = 0.5 * (h / l);
	tmp = 0.0;
	if (l <= -6.8e-111)
		tmp = (d * ((l * h) ^ -0.5)) * (-1.0 + (((M * (D / (d / 0.5))) ^ 2.0) * t_1));
	elseif (l <= 1.75e-249)
		tmp = (t_0 * sqrt((d / h))) * (1.0 - (h * ((((D / d) * (M * 0.5)) ^ 2.0) / (l / 0.5))));
	elseif (l <= 3.2e+89)
		tmp = (t_0 * (sqrt(d) / sqrt(h))) * (1.0 - (t_1 * (((D / d) * (M / 2.0)) ^ 2.0)));
	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[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[(h / l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -6.8e-111], N[(N[(d * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] * N[(-1.0 + N[(N[Power[N[(M * N[(D / N[(d / 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.75e-249], N[(N[(t$95$0 * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(h * N[(N[Power[N[(N[(D / d), $MachinePrecision] * N[(M * 0.5), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / N[(l / 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 3.2e+89], N[(N[(t$95$0 * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(t$95$1 * N[Power[N[(N[(D / d), $MachinePrecision] * N[(M / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $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 < -6.79999999999999993e-111

   1. Initial program 62.2%

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

      1. metadata-eval 62.2%

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

      2. unpow1/2 62.2%

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

      3. metadata-eval 62.2%

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

      4. unpow1/2 62.2%

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

      5. *-commutative 62.2%

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

      6. associate-*l* 62.2%

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

      7. times-frac 62.2%

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

      8. metadata-eval 62.2%

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

   3. Simplified 62.2%

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

   5. Applied egg-rr 26.9%

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

      1. expm1-def 36.5%

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

      2. expm1-log1p 47.3%

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

      3. rem-log-exp 43.3%

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

      4. exp-diff 43.3%

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

      5. log-div 43.3%

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

      6. rem-log-exp 43.3%

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

      7. rem-log-exp 47.3%

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

      8. *-commutative 47.3%

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

      9. associate-*l* 47.3%

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

      10. *-commutative 47.3%

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

      11. associate-*l/ 47.3%

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

      12. associate-/l* 47.3%

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

   7. Simplified 47.3%

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

   8. Taylor expanded in d around -inf 74.2%

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

      1. associate-*r* 74.2%

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

      2. mul-1-neg 74.2%

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

      3. *-commutative 74.2%

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

      4. unpow-1 74.2%

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

      5. sqr-pow 74.2%

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

      6. rem-sqrt-square 74.2%

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

      7. sqr-pow 74.0%

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

      8. fabs-sqr 74.0%

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

      9. sqr-pow 74.2%

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

      10. metadata-eval 74.2%

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

   10. Simplified 74.2%

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

   if -6.79999999999999993e-111 < l < 1.75000000000000006e-249

   1. Initial program 62.3%

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

      1. metadata-eval 62.3%

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

      2. unpow1/2 62.3%

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

      3. metadata-eval 62.3%

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

      4. unpow1/2 62.3%

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

      5. *-commutative 62.3%

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

      6. associate-*l* 62.3%

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

      7. times-frac 62.3%

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

      8. metadata-eval 62.3%

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

   3. Simplified 62.3%

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

      1. associate-*r* 62.3%

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

      2. frac-times 62.3%

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

      3. *-commutative 62.3%

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

      4. metadata-eval 62.3%

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

      5. associate-*r/ 78.9%

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

      6. metadata-eval 78.9%

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

      7. *-commutative 78.9%

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

      8. frac-times 78.9%

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

      9. *-commutative 78.9%

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

      10. div-inv 78.9%

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

      11. metadata-eval 78.9%

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

   5. Applied egg-rr 78.9%

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

      1. div-inv 78.9%

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

      2. *-commutative 78.9%

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

   7. Applied egg-rr 78.9%

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

      1. associate-*r/ 78.9%

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

      2. *-rgt-identity 78.9%

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

      3. associate-*l/ 78.9%

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

      4. *-commutative 78.9%

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

      5. *-commutative 78.9%

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

      6. associate-/l* 78.9%

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

   9. Simplified 78.9%

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

   if 1.75000000000000006e-249 < l < 3.19999999999999987e89

   1. Initial program 69.9%

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

      1. metadata-eval 69.9%

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

      2. unpow1/2 69.9%

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

      3. metadata-eval 69.9%

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

      4. unpow1/2 69.9%

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

      5. *-commutative 69.9%

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

      6. associate-*l* 69.9%

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

      7. times-frac 69.9%

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

      8. metadata-eval 69.9%

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

   3. Simplified 69.9%

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

      1. sqrt-div 86.3%

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

   5. Applied egg-rr 81.4%

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

   if 3.19999999999999987e89 < l

   1. Initial program 54.6%

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

      1. metadata-eval 54.6%

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

      2. unpow1/2 54.6%

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

      3. metadata-eval 54.6%

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

      4. unpow1/2 54.6%

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

      5. *-commutative 54.6%

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

      6. associate-*l* 54.6%

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

      7. times-frac 54.4%

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

      8. metadata-eval 54.4%

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

   3. Simplified 54.4%

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

   4. Taylor expanded in d around inf 74.0%

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

      1. *-un-lft-identity 74.0%

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

      2. *-commutative 74.0%

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

   6. Applied egg-rr 74.0%

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

      1. *-lft-identity 74.0%

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

      2. unpow-1 74.0%

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

      3. sqr-pow 74.0%

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

      4. rem-sqrt-square 74.0%

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

      5. sqr-pow 73.7%

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

      6. fabs-sqr 73.7%

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

      7. sqr-pow 74.0%

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

      8. metadata-eval 74.0%

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

   8. Simplified 74.0%

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

      1. unpow-prod-down 78.7%

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

   10. Applied egg-rr 78.7%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -6.8 \cdot 10^{-111}:\\ \;\;\;\;\left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right) \cdot \left(-1 + {\left(M \cdot \frac{D}{\frac{d}{0.5}}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\\ \mathbf{elif}\;\ell \leq 1.75 \cdot 10^{-249}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - h \cdot \frac{{\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2}}{\frac{\ell}{0.5}}\right)\\ \mathbf{elif}\;\ell \leq 3.2 \cdot 10^{+89}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \frac{\sqrt{d}}{\sqrt{h}}\right) \cdot \left(1 - \left(0.5 \cdot \frac{h}{\ell}\right) \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \]

Alternative 3: 72.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := 0.5 \cdot \frac{h}{\ell}\\ \mathbf{if}\;\ell \leq -1.1 \cdot 10^{-110}:\\ \;\;\;\;\left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right) \cdot \left(-1 + {\left(M \cdot \frac{D}{\frac{d}{0.5}}\right)}^{2} \cdot t_1\right)\\ \mathbf{elif}\;\ell \leq 2.05 \cdot 10^{-249}:\\ \;\;\;\;\left(t_0 \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - h \cdot \frac{{\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2}}{\frac{\ell}{0.5}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t_0 \cdot \frac{\sqrt{d}}{\sqrt{h}}\right) \cdot \left(1 - t_1 \cdot {\left(\frac{D}{\frac{d \cdot 2}{M}}\right)}^{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ d l))) (t_1 (* 0.5 (/ h l))))
   (if (<= l -1.1e-110)
     (*
      (* d (pow (* l h) -0.5))
      (+ -1.0 (* (pow (* M (/ D (/ d 0.5))) 2.0) t_1)))
     (if (<= l 2.05e-249)
       (*
        (* t_0 (sqrt (/ d h)))
        (- 1.0 (* h (/ (pow (* (/ D d) (* M 0.5)) 2.0) (/ l 0.5)))))
       (*
        (* t_0 (/ (sqrt d) (sqrt h)))
        (- 1.0 (* t_1 (pow (/ D (/ (* d 2.0) M)) 2.0))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((d / l));
	double t_1 = 0.5 * (h / l);
	double tmp;
	if (l <= -1.1e-110) {
		tmp = (d * pow((l * h), -0.5)) * (-1.0 + (pow((M * (D / (d / 0.5))), 2.0) * t_1));
	} else if (l <= 2.05e-249) {
		tmp = (t_0 * sqrt((d / h))) * (1.0 - (h * (pow(((D / d) * (M * 0.5)), 2.0) / (l / 0.5))));
	} else {
		tmp = (t_0 * (sqrt(d) / sqrt(h))) * (1.0 - (t_1 * pow((D / ((d * 2.0) / M)), 2.0)));
	}
	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) :: tmp
    t_0 = sqrt((d / l))
    t_1 = 0.5d0 * (h / l)
    if (l <= (-1.1d-110)) then
        tmp = (d * ((l * h) ** (-0.5d0))) * ((-1.0d0) + (((m * (d_1 / (d / 0.5d0))) ** 2.0d0) * t_1))
    else if (l <= 2.05d-249) then
        tmp = (t_0 * sqrt((d / h))) * (1.0d0 - (h * ((((d_1 / d) * (m * 0.5d0)) ** 2.0d0) / (l / 0.5d0))))
    else
        tmp = (t_0 * (sqrt(d) / sqrt(h))) * (1.0d0 - (t_1 * ((d_1 / ((d * 2.0d0) / m)) ** 2.0d0)))
    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 = 0.5 * (h / l);
	double tmp;
	if (l <= -1.1e-110) {
		tmp = (d * Math.pow((l * h), -0.5)) * (-1.0 + (Math.pow((M * (D / (d / 0.5))), 2.0) * t_1));
	} else if (l <= 2.05e-249) {
		tmp = (t_0 * Math.sqrt((d / h))) * (1.0 - (h * (Math.pow(((D / d) * (M * 0.5)), 2.0) / (l / 0.5))));
	} else {
		tmp = (t_0 * (Math.sqrt(d) / Math.sqrt(h))) * (1.0 - (t_1 * Math.pow((D / ((d * 2.0) / M)), 2.0)));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.sqrt((d / l))
	t_1 = 0.5 * (h / l)
	tmp = 0
	if l <= -1.1e-110:
		tmp = (d * math.pow((l * h), -0.5)) * (-1.0 + (math.pow((M * (D / (d / 0.5))), 2.0) * t_1))
	elif l <= 2.05e-249:
		tmp = (t_0 * math.sqrt((d / h))) * (1.0 - (h * (math.pow(((D / d) * (M * 0.5)), 2.0) / (l / 0.5))))
	else:
		tmp = (t_0 * (math.sqrt(d) / math.sqrt(h))) * (1.0 - (t_1 * math.pow((D / ((d * 2.0) / M)), 2.0)))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(d / l))
	t_1 = Float64(0.5 * Float64(h / l))
	tmp = 0.0
	if (l <= -1.1e-110)
		tmp = Float64(Float64(d * (Float64(l * h) ^ -0.5)) * Float64(-1.0 + Float64((Float64(M * Float64(D / Float64(d / 0.5))) ^ 2.0) * t_1)));
	elseif (l <= 2.05e-249)
		tmp = Float64(Float64(t_0 * sqrt(Float64(d / h))) * Float64(1.0 - Float64(h * Float64((Float64(Float64(D / d) * Float64(M * 0.5)) ^ 2.0) / Float64(l / 0.5)))));
	else
		tmp = Float64(Float64(t_0 * Float64(sqrt(d) / sqrt(h))) * Float64(1.0 - Float64(t_1 * (Float64(D / Float64(Float64(d * 2.0) / M)) ^ 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt((d / l));
	t_1 = 0.5 * (h / l);
	tmp = 0.0;
	if (l <= -1.1e-110)
		tmp = (d * ((l * h) ^ -0.5)) * (-1.0 + (((M * (D / (d / 0.5))) ^ 2.0) * t_1));
	elseif (l <= 2.05e-249)
		tmp = (t_0 * sqrt((d / h))) * (1.0 - (h * ((((D / d) * (M * 0.5)) ^ 2.0) / (l / 0.5))));
	else
		tmp = (t_0 * (sqrt(d) / sqrt(h))) * (1.0 - (t_1 * ((D / ((d * 2.0) / M)) ^ 2.0)));
	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[(0.5 * N[(h / l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -1.1e-110], N[(N[(d * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] * N[(-1.0 + N[(N[Power[N[(M * N[(D / N[(d / 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 2.05e-249], N[(N[(t$95$0 * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(h * N[(N[Power[N[(N[(D / d), $MachinePrecision] * N[(M * 0.5), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / N[(l / 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(t$95$1 * N[Power[N[(D / N[(N[(d * 2.0), $MachinePrecision] / M), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.1e-110

   1. Initial program 62.2%

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

      1. metadata-eval 62.2%

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

      2. unpow1/2 62.2%

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

      3. metadata-eval 62.2%

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

      4. unpow1/2 62.2%

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

      5. *-commutative 62.2%

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

      6. associate-*l* 62.2%

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

      7. times-frac 62.2%

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

      8. metadata-eval 62.2%

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

   3. Simplified 62.2%

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

   5. Applied egg-rr 26.9%

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

      1. expm1-def 36.5%

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

      2. expm1-log1p 47.3%

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

      3. rem-log-exp 43.3%

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

      4. exp-diff 43.3%

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

      5. log-div 43.3%

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

      6. rem-log-exp 43.3%

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

      7. rem-log-exp 47.3%

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

      8. *-commutative 47.3%

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

      9. associate-*l* 47.3%

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

      10. *-commutative 47.3%

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

      11. associate-*l/ 47.3%

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

      12. associate-/l* 47.3%

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

   7. Simplified 47.3%

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

   8. Taylor expanded in d around -inf 74.2%

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

      1. associate-*r* 74.2%

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

      2. mul-1-neg 74.2%

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

      3. *-commutative 74.2%

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

      4. unpow-1 74.2%

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

      5. sqr-pow 74.2%

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

      6. rem-sqrt-square 74.2%

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

      7. sqr-pow 74.0%

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

      8. fabs-sqr 74.0%

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

      9. sqr-pow 74.2%

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

      10. metadata-eval 74.2%

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

   10. Simplified 74.2%

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

   if -1.1e-110 < l < 2.05000000000000002e-249

   1. Initial program 62.3%

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

      1. metadata-eval 62.3%

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

      2. unpow1/2 62.3%

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

      3. metadata-eval 62.3%

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

      4. unpow1/2 62.3%

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

      5. *-commutative 62.3%

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

      6. associate-*l* 62.3%

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

      7. times-frac 62.3%

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

      8. metadata-eval 62.3%

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

   3. Simplified 62.3%

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

      1. associate-*r* 62.3%

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

      2. frac-times 62.3%

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

      3. *-commutative 62.3%

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

      4. metadata-eval 62.3%

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

      5. associate-*r/ 78.9%

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

      6. metadata-eval 78.9%

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

      7. *-commutative 78.9%

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

      8. frac-times 78.9%

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

      9. *-commutative 78.9%

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

      10. div-inv 78.9%

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

      11. metadata-eval 78.9%

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

   5. Applied egg-rr 78.9%

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

      1. div-inv 78.9%

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

      2. *-commutative 78.9%

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

   7. Applied egg-rr 78.9%

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

      1. associate-*r/ 78.9%

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

      2. *-rgt-identity 78.9%

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

      3. associate-*l/ 78.9%

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

      4. *-commutative 78.9%

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

      5. *-commutative 78.9%

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

      6. associate-/l* 78.9%

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

   9. Simplified 78.9%

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

   if 2.05000000000000002e-249 < l

   1. Initial program 64.8%

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

      1. metadata-eval 64.8%

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

      2. unpow1/2 64.8%

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

      3. metadata-eval 64.8%

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

      4. unpow1/2 64.8%

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

      5. *-commutative 64.8%

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

      6. associate-*l* 64.8%

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

      7. times-frac 64.7%

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

      8. metadata-eval 64.7%

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

   3. Simplified 64.7%

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

      1. sqrt-div 81.3%

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

   5. Applied egg-rr 77.2%

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

      1. frac-times 76.9%

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

   7. Applied egg-rr 76.9%

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

      1. *-commutative 76.9%

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

      2. associate-/l* 77.2%

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

   9. Simplified 77.2%

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

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.1 \cdot 10^{-110}:\\ \;\;\;\;\left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right) \cdot \left(-1 + {\left(M \cdot \frac{D}{\frac{d}{0.5}}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\\ \mathbf{elif}\;\ell \leq 2.05 \cdot 10^{-249}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - h \cdot \frac{{\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2}}{\frac{\ell}{0.5}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \frac{\sqrt{d}}{\sqrt{h}}\right) \cdot \left(1 - \left(0.5 \cdot \frac{h}{\ell}\right) \cdot {\left(\frac{D}{\frac{d \cdot 2}{M}}\right)}^{2}\right)\\ \end{array} \]

Alternative 4: 73.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;h \leq -5.2 \cdot 10^{-277}:\\ \;\;\;\;\left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right) \cdot \left(-1 + {\left(M \cdot \frac{D}{\frac{d}{0.5}}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\\ \mathbf{elif}\;h \leq 9 \cdot 10^{-293}:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \frac{h \cdot \left(0.5 \cdot {\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2}\right)}{\ell}\right) \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \frac{\sqrt{d}}{\sqrt{h}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= h -5.2e-277)
   (*
    (* d (pow (* l h) -0.5))
    (+ -1.0 (* (pow (* M (/ D (/ d 0.5))) 2.0) (* 0.5 (/ h l)))))
   (if (<= h 9e-293)
     (*
      (* (pow (/ d h) 0.5) (pow (/ d l) 0.5))
      (- 1.0 (* (/ h l) (* 0.5 (pow (/ (* D M) (* d 2.0)) 2.0)))))
     (*
      (- 1.0 (/ (* h (* 0.5 (pow (* (/ D d) (* M 0.5)) 2.0))) l))
      (* (sqrt (/ d l)) (/ (sqrt d) (sqrt h)))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (h <= -5.2e-277) {
		tmp = (d * pow((l * h), -0.5)) * (-1.0 + (pow((M * (D / (d / 0.5))), 2.0) * (0.5 * (h / l))));
	} else if (h <= 9e-293) {
		tmp = (pow((d / h), 0.5) * pow((d / l), 0.5)) * (1.0 - ((h / l) * (0.5 * pow(((D * M) / (d * 2.0)), 2.0))));
	} else {
		tmp = (1.0 - ((h * (0.5 * pow(((D / d) * (M * 0.5)), 2.0))) / l)) * (sqrt((d / l)) * (sqrt(d) / sqrt(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) :: tmp
    if (h <= (-5.2d-277)) then
        tmp = (d * ((l * h) ** (-0.5d0))) * ((-1.0d0) + (((m * (d_1 / (d / 0.5d0))) ** 2.0d0) * (0.5d0 * (h / l))))
    else if (h <= 9d-293) then
        tmp = (((d / h) ** 0.5d0) * ((d / l) ** 0.5d0)) * (1.0d0 - ((h / l) * (0.5d0 * (((d_1 * m) / (d * 2.0d0)) ** 2.0d0))))
    else
        tmp = (1.0d0 - ((h * (0.5d0 * (((d_1 / d) * (m * 0.5d0)) ** 2.0d0))) / l)) * (sqrt((d / l)) * (sqrt(d) / sqrt(h)))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (h <= -5.2e-277) {
		tmp = (d * Math.pow((l * h), -0.5)) * (-1.0 + (Math.pow((M * (D / (d / 0.5))), 2.0) * (0.5 * (h / l))));
	} else if (h <= 9e-293) {
		tmp = (Math.pow((d / h), 0.5) * Math.pow((d / l), 0.5)) * (1.0 - ((h / l) * (0.5 * Math.pow(((D * M) / (d * 2.0)), 2.0))));
	} else {
		tmp = (1.0 - ((h * (0.5 * Math.pow(((D / d) * (M * 0.5)), 2.0))) / l)) * (Math.sqrt((d / l)) * (Math.sqrt(d) / Math.sqrt(h)));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if h <= -5.2e-277:
		tmp = (d * math.pow((l * h), -0.5)) * (-1.0 + (math.pow((M * (D / (d / 0.5))), 2.0) * (0.5 * (h / l))))
	elif h <= 9e-293:
		tmp = (math.pow((d / h), 0.5) * math.pow((d / l), 0.5)) * (1.0 - ((h / l) * (0.5 * math.pow(((D * M) / (d * 2.0)), 2.0))))
	else:
		tmp = (1.0 - ((h * (0.5 * math.pow(((D / d) * (M * 0.5)), 2.0))) / l)) * (math.sqrt((d / l)) * (math.sqrt(d) / math.sqrt(h)))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (h <= -5.2e-277)
		tmp = Float64(Float64(d * (Float64(l * h) ^ -0.5)) * Float64(-1.0 + Float64((Float64(M * Float64(D / Float64(d / 0.5))) ^ 2.0) * Float64(0.5 * Float64(h / l)))));
	elseif (h <= 9e-293)
		tmp = Float64(Float64((Float64(d / h) ^ 0.5) * (Float64(d / l) ^ 0.5)) * Float64(1.0 - Float64(Float64(h / l) * Float64(0.5 * (Float64(Float64(D * M) / Float64(d * 2.0)) ^ 2.0)))));
	else
		tmp = Float64(Float64(1.0 - Float64(Float64(h * Float64(0.5 * (Float64(Float64(D / d) * Float64(M * 0.5)) ^ 2.0))) / l)) * Float64(sqrt(Float64(d / l)) * Float64(sqrt(d) / sqrt(h))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (h <= -5.2e-277)
		tmp = (d * ((l * h) ^ -0.5)) * (-1.0 + (((M * (D / (d / 0.5))) ^ 2.0) * (0.5 * (h / l))));
	elseif (h <= 9e-293)
		tmp = (((d / h) ^ 0.5) * ((d / l) ^ 0.5)) * (1.0 - ((h / l) * (0.5 * (((D * M) / (d * 2.0)) ^ 2.0))));
	else
		tmp = (1.0 - ((h * (0.5 * (((D / d) * (M * 0.5)) ^ 2.0))) / l)) * (sqrt((d / l)) * (sqrt(d) / sqrt(h)));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[h, -5.2e-277], N[(N[(d * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] * N[(-1.0 + N[(N[Power[N[(M * N[(D / N[(d / 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(0.5 * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, 9e-293], N[(N[(N[Power[N[(d / h), $MachinePrecision], 0.5], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(0.5 * N[Power[N[(N[(D * M), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[(N[(h * N[(0.5 * N[Power[N[(N[(D / d), $MachinePrecision] * N[(M * 0.5), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if h < -5.2e-277

   1. Initial program 62.2%

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

      1. metadata-eval 62.2%

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

      2. unpow1/2 62.2%

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

      3. metadata-eval 62.2%

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

      4. unpow1/2 62.2%

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

      5. *-commutative 62.2%

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

      6. associate-*l* 62.2%

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

      7. times-frac 62.2%

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

      8. metadata-eval 62.2%

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

   3. Simplified 62.2%

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

   5. Applied egg-rr 27.6%

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

      1. expm1-def 35.5%

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

      2. expm1-log1p 51.3%

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

      3. rem-log-exp 48.4%

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

      4. exp-diff 48.4%

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

      5. log-div 48.4%

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

      6. rem-log-exp 48.4%

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

      7. rem-log-exp 51.3%

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

      8. *-commutative 51.3%

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

      9. associate-*l* 51.3%

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

      10. *-commutative 51.3%

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

      11. associate-*l/ 51.3%

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

      12. associate-/l* 51.3%

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

   7. Simplified 51.3%

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

   8. Taylor expanded in d around -inf 71.9%

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

      1. associate-*r* 71.9%

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

      2. mul-1-neg 71.9%

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

      3. *-commutative 71.9%

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

      4. unpow-1 71.9%

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

      5. sqr-pow 72.0%

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

      6. rem-sqrt-square 72.4%

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

      7. sqr-pow 72.2%

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

      8. fabs-sqr 72.2%

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

      9. sqr-pow 72.4%

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

      10. metadata-eval 72.4%

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

   10. Simplified 72.4%

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

   if -5.2e-277 < h < 9.0000000000000005e-293

   1. Initial program 83.7%

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

   if 9.0000000000000005e-293 < h

   1. Initial program 62.6%

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

      1. metadata-eval 62.6%

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

      2. unpow1/2 62.6%

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

      3. metadata-eval 62.6%

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

      4. unpow1/2 62.6%

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

      5. *-commutative 62.6%

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

      6. associate-*l* 62.6%

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

      7. times-frac 62.5%

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

      8. metadata-eval 62.5%

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

   3. Simplified 62.5%

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

      1. associate-*r* 62.5%

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

      2. frac-times 62.6%

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

      3. *-commutative 62.6%

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

      4. metadata-eval 62.6%

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

      5. associate-*r/ 69.5%

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

      6. metadata-eval 69.5%

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

      7. *-commutative 69.5%

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

      8. frac-times 69.5%

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

      9. *-commutative 69.5%

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

      10. div-inv 69.5%

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

      11. metadata-eval 69.5%

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

   5. Applied egg-rr 69.5%

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

      1. sqrt-div 83.0%

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

   7. Applied egg-rr 83.0%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -5.2 \cdot 10^{-277}:\\ \;\;\;\;\left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right) \cdot \left(-1 + {\left(M \cdot \frac{D}{\frac{d}{0.5}}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\\ \mathbf{elif}\;h \leq 9 \cdot 10^{-293}:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \frac{h \cdot \left(0.5 \cdot {\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2}\right)}{\ell}\right) \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \frac{\sqrt{d}}{\sqrt{h}}\right)\\ \end{array} \]

Alternative 5: 69.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -5 \cdot 10^{-111}:\\ \;\;\;\;\left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right) \cdot \left(-1 + {\left(M \cdot \frac{D}{\frac{d}{0.5}}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\\ \mathbf{elif}\;\ell \leq 6.6 \cdot 10^{+89}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - h \cdot \frac{{\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2}}{\frac{\ell}{0.5}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= l -5e-111)
   (*
    (* d (pow (* l h) -0.5))
    (+ -1.0 (* (pow (* M (/ D (/ d 0.5))) 2.0) (* 0.5 (/ h l)))))
   (if (<= l 6.6e+89)
     (*
      (* (sqrt (/ d l)) (sqrt (/ d h)))
      (- 1.0 (* h (/ (pow (* (/ D d) (* M 0.5)) 2.0) (/ l 0.5)))))
     (* 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 <= -5e-111) {
		tmp = (d * pow((l * h), -0.5)) * (-1.0 + (pow((M * (D / (d / 0.5))), 2.0) * (0.5 * (h / l))));
	} else if (l <= 6.6e+89) {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - (h * (pow(((D / d) * (M * 0.5)), 2.0) / (l / 0.5))));
	} 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 <= (-5d-111)) then
        tmp = (d * ((l * h) ** (-0.5d0))) * ((-1.0d0) + (((m * (d_1 / (d / 0.5d0))) ** 2.0d0) * (0.5d0 * (h / l))))
    else if (l <= 6.6d+89) then
        tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0d0 - (h * ((((d_1 / d) * (m * 0.5d0)) ** 2.0d0) / (l / 0.5d0))))
    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 <= -5e-111) {
		tmp = (d * Math.pow((l * h), -0.5)) * (-1.0 + (Math.pow((M * (D / (d / 0.5))), 2.0) * (0.5 * (h / l))));
	} else if (l <= 6.6e+89) {
		tmp = (Math.sqrt((d / l)) * Math.sqrt((d / h))) * (1.0 - (h * (Math.pow(((D / d) * (M * 0.5)), 2.0) / (l / 0.5))));
	} 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 <= -5e-111:
		tmp = (d * math.pow((l * h), -0.5)) * (-1.0 + (math.pow((M * (D / (d / 0.5))), 2.0) * (0.5 * (h / l))))
	elif l <= 6.6e+89:
		tmp = (math.sqrt((d / l)) * math.sqrt((d / h))) * (1.0 - (h * (math.pow(((D / d) * (M * 0.5)), 2.0) / (l / 0.5))))
	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 <= -5e-111)
		tmp = Float64(Float64(d * (Float64(l * h) ^ -0.5)) * Float64(-1.0 + Float64((Float64(M * Float64(D / Float64(d / 0.5))) ^ 2.0) * Float64(0.5 * Float64(h / l)))));
	elseif (l <= 6.6e+89)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * Float64(1.0 - Float64(h * Float64((Float64(Float64(D / d) * Float64(M * 0.5)) ^ 2.0) / Float64(l / 0.5)))));
	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 <= -5e-111)
		tmp = (d * ((l * h) ^ -0.5)) * (-1.0 + (((M * (D / (d / 0.5))) ^ 2.0) * (0.5 * (h / l))));
	elseif (l <= 6.6e+89)
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - (h * ((((D / d) * (M * 0.5)) ^ 2.0) / (l / 0.5))));
	else
		tmp = d * ((h ^ -0.5) * (l ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[l, -5e-111], N[(N[(d * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] * N[(-1.0 + N[(N[Power[N[(M * N[(D / N[(d / 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(0.5 * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 6.6e+89], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(h * N[(N[Power[N[(N[(D / d), $MachinePrecision] * N[(M * 0.5), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / N[(l / 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 < -5.0000000000000003e-111

   1. Initial program 62.2%

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

      1. metadata-eval 62.2%

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

      2. unpow1/2 62.2%

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

      3. metadata-eval 62.2%

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

      4. unpow1/2 62.2%

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

      5. *-commutative 62.2%

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

      6. associate-*l* 62.2%

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

      7. times-frac 62.2%

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

      8. metadata-eval 62.2%

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

   3. Simplified 62.2%

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

   5. Applied egg-rr 26.9%

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

      1. expm1-def 36.5%

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

      2. expm1-log1p 47.3%

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

      3. rem-log-exp 43.3%

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

      4. exp-diff 43.3%

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

      5. log-div 43.3%

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

      6. rem-log-exp 43.3%

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

      7. rem-log-exp 47.3%

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

      8. *-commutative 47.3%

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

      9. associate-*l* 47.3%

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

      10. *-commutative 47.3%

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

      11. associate-*l/ 47.3%

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

      12. associate-/l* 47.3%

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

   7. Simplified 47.3%

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

   8. Taylor expanded in d around -inf 74.2%

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

      1. associate-*r* 74.2%

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

      2. mul-1-neg 74.2%

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

      3. *-commutative 74.2%

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

      4. unpow-1 74.2%

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

      5. sqr-pow 74.2%

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

      6. rem-sqrt-square 74.2%

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

      7. sqr-pow 74.0%

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

      8. fabs-sqr 74.0%

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

      9. sqr-pow 74.2%

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

      10. metadata-eval 74.2%

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

   10. Simplified 74.2%

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

   if -5.0000000000000003e-111 < l < 6.59999999999999948e89

   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 66.5%

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

      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%

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

      1. associate-*r* 66.5%

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

      2. frac-times 66.5%

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

      3. *-commutative 66.5%

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

      4. metadata-eval 66.5%

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

      5. associate-*r/ 75.5%

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

      6. metadata-eval 75.5%

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

      7. *-commutative 75.5%

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

      8. frac-times 75.5%

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

      9. *-commutative 75.5%

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

      10. div-inv 75.5%

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

      11. metadata-eval 75.5%

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

   5. Applied egg-rr 75.5%

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

      1. div-inv 75.5%

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

      2. *-commutative 75.5%

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

   7. Applied egg-rr 75.5%

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

      1. associate-*r/ 75.5%

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

      2. *-rgt-identity 75.5%

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

      3. associate-*l/ 74.8%

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

      4. *-commutative 74.8%

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

      5. *-commutative 74.8%

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

      6. associate-/l* 74.8%

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

   9. Simplified 74.8%

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

   if 6.59999999999999948e89 < l

   1. Initial program 54.8%

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

      1. metadata-eval 54.8%

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

      2. unpow1/2 54.8%

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

      3. metadata-eval 54.8%

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

      4. unpow1/2 54.8%

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

      5. *-commutative 54.8%

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

      6. associate-*l* 54.8%

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

      7. times-frac 54.7%

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

      8. metadata-eval 54.7%

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

   3. Simplified 54.7%

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

   4. Taylor expanded in d around inf 75.3%

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

      1. *-un-lft-identity 75.3%

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

      2. *-commutative 75.3%

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

   6. Applied egg-rr 75.3%

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

      1. *-lft-identity 75.3%

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

      2. unpow-1 75.3%

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

      3. sqr-pow 75.3%

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

      4. rem-sqrt-square 75.3%

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

      5. sqr-pow 74.9%

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

      6. fabs-sqr 74.9%

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

      7. sqr-pow 75.3%

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

      8. metadata-eval 75.3%

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

   8. Simplified 75.3%

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

      1. unpow-prod-down 80.2%

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

   10. Applied egg-rr 80.2%

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

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5 \cdot 10^{-111}:\\ \;\;\;\;\left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right) \cdot \left(-1 + {\left(M \cdot \frac{D}{\frac{d}{0.5}}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\\ \mathbf{elif}\;\ell \leq 6.6 \cdot 10^{+89}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - h \cdot \frac{{\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2}}{\frac{\ell}{0.5}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \]

Alternative 6: 70.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -4 \cdot 10^{-111}:\\ \;\;\;\;\left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right) \cdot \left(-1 + {\left(M \cdot \frac{D}{\frac{d}{0.5}}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\\ \mathbf{elif}\;\ell \leq 6.3 \cdot 10^{+89}:\\ \;\;\;\;\left(1 - \frac{h \cdot \left(0.5 \cdot {\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2}\right)}{\ell}\right) \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= l -4e-111)
   (*
    (* d (pow (* l h) -0.5))
    (+ -1.0 (* (pow (* M (/ D (/ d 0.5))) 2.0) (* 0.5 (/ h l)))))
   (if (<= l 6.3e+89)
     (*
      (- 1.0 (/ (* h (* 0.5 (pow (* (/ D d) (* M 0.5)) 2.0))) l))
      (* (sqrt (/ d l)) (sqrt (/ d h))))
     (* 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 <= -4e-111) {
		tmp = (d * pow((l * h), -0.5)) * (-1.0 + (pow((M * (D / (d / 0.5))), 2.0) * (0.5 * (h / l))));
	} else if (l <= 6.3e+89) {
		tmp = (1.0 - ((h * (0.5 * pow(((D / d) * (M * 0.5)), 2.0))) / l)) * (sqrt((d / l)) * sqrt((d / h)));
	} 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 <= (-4d-111)) then
        tmp = (d * ((l * h) ** (-0.5d0))) * ((-1.0d0) + (((m * (d_1 / (d / 0.5d0))) ** 2.0d0) * (0.5d0 * (h / l))))
    else if (l <= 6.3d+89) then
        tmp = (1.0d0 - ((h * (0.5d0 * (((d_1 / d) * (m * 0.5d0)) ** 2.0d0))) / l)) * (sqrt((d / l)) * sqrt((d / h)))
    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 <= -4e-111) {
		tmp = (d * Math.pow((l * h), -0.5)) * (-1.0 + (Math.pow((M * (D / (d / 0.5))), 2.0) * (0.5 * (h / l))));
	} else if (l <= 6.3e+89) {
		tmp = (1.0 - ((h * (0.5 * Math.pow(((D / d) * (M * 0.5)), 2.0))) / l)) * (Math.sqrt((d / l)) * Math.sqrt((d / h)));
	} 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 <= -4e-111:
		tmp = (d * math.pow((l * h), -0.5)) * (-1.0 + (math.pow((M * (D / (d / 0.5))), 2.0) * (0.5 * (h / l))))
	elif l <= 6.3e+89:
		tmp = (1.0 - ((h * (0.5 * math.pow(((D / d) * (M * 0.5)), 2.0))) / l)) * (math.sqrt((d / l)) * math.sqrt((d / h)))
	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 <= -4e-111)
		tmp = Float64(Float64(d * (Float64(l * h) ^ -0.5)) * Float64(-1.0 + Float64((Float64(M * Float64(D / Float64(d / 0.5))) ^ 2.0) * Float64(0.5 * Float64(h / l)))));
	elseif (l <= 6.3e+89)
		tmp = Float64(Float64(1.0 - Float64(Float64(h * Float64(0.5 * (Float64(Float64(D / d) * Float64(M * 0.5)) ^ 2.0))) / l)) * Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))));
	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 <= -4e-111)
		tmp = (d * ((l * h) ^ -0.5)) * (-1.0 + (((M * (D / (d / 0.5))) ^ 2.0) * (0.5 * (h / l))));
	elseif (l <= 6.3e+89)
		tmp = (1.0 - ((h * (0.5 * (((D / d) * (M * 0.5)) ^ 2.0))) / l)) * (sqrt((d / l)) * sqrt((d / h)));
	else
		tmp = d * ((h ^ -0.5) * (l ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[l, -4e-111], N[(N[(d * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] * N[(-1.0 + N[(N[Power[N[(M * N[(D / N[(d / 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(0.5 * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 6.3e+89], N[(N[(1.0 - N[(N[(h * N[(0.5 * N[Power[N[(N[(D / d), $MachinePrecision] * N[(M * 0.5), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $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 < -4.00000000000000035e-111

   1. Initial program 62.2%

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

      1. metadata-eval 62.2%

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

      2. unpow1/2 62.2%

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

      3. metadata-eval 62.2%

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

      4. unpow1/2 62.2%

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

      5. *-commutative 62.2%

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

      6. associate-*l* 62.2%

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

      7. times-frac 62.2%

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

      8. metadata-eval 62.2%

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

   3. Simplified 62.2%

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

   5. Applied egg-rr 26.9%

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

      1. expm1-def 36.5%

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

      2. expm1-log1p 47.3%

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

      3. rem-log-exp 43.3%

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

      4. exp-diff 43.3%

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

      5. log-div 43.3%

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

      6. rem-log-exp 43.3%

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

      7. rem-log-exp 47.3%

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

      8. *-commutative 47.3%

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

      9. associate-*l* 47.3%

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

      10. *-commutative 47.3%

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

      11. associate-*l/ 47.3%

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

      12. associate-/l* 47.3%

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

   7. Simplified 47.3%

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

   8. Taylor expanded in d around -inf 74.2%

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

      1. associate-*r* 74.2%

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

      2. mul-1-neg 74.2%

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

      3. *-commutative 74.2%

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

      4. unpow-1 74.2%

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

      5. sqr-pow 74.2%

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

      6. rem-sqrt-square 74.2%

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

      7. sqr-pow 74.0%

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

      8. fabs-sqr 74.0%

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

      9. sqr-pow 74.2%

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

      10. metadata-eval 74.2%

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

   10. Simplified 74.2%

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

   if -4.00000000000000035e-111 < l < 6.3e89

   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 66.5%

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

      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%

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

      1. associate-*r* 66.5%

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

      2. frac-times 66.5%

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

      3. *-commutative 66.5%

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

      4. metadata-eval 66.5%

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

      5. associate-*r/ 75.5%

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

      6. metadata-eval 75.5%

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

      7. *-commutative 75.5%

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

      8. frac-times 75.5%

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

      9. *-commutative 75.5%

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

      10. div-inv 75.5%

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

      11. metadata-eval 75.5%

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

   5. Applied egg-rr 75.5%

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

   if 6.3e89 < l

   1. Initial program 54.8%

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

      1. metadata-eval 54.8%

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

      2. unpow1/2 54.8%

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

      3. metadata-eval 54.8%

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

      4. unpow1/2 54.8%

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

      5. *-commutative 54.8%

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

      6. associate-*l* 54.8%

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

      7. times-frac 54.7%

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

      8. metadata-eval 54.7%

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

   3. Simplified 54.7%

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

   4. Taylor expanded in d around inf 75.3%

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

      1. *-un-lft-identity 75.3%

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

      2. *-commutative 75.3%

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

   6. Applied egg-rr 75.3%

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

      1. *-lft-identity 75.3%

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

      2. unpow-1 75.3%

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

      3. sqr-pow 75.3%

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

      4. rem-sqrt-square 75.3%

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

      5. sqr-pow 74.9%

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

      6. fabs-sqr 74.9%

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

      7. sqr-pow 75.3%

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

      8. metadata-eval 75.3%

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

   8. Simplified 75.3%

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

      1. unpow-prod-down 80.2%

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

   10. Applied egg-rr 80.2%

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

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4 \cdot 10^{-111}:\\ \;\;\;\;\left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right) \cdot \left(-1 + {\left(M \cdot \frac{D}{\frac{d}{0.5}}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\\ \mathbf{elif}\;\ell \leq 6.3 \cdot 10^{+89}:\\ \;\;\;\;\left(1 - \frac{h \cdot \left(0.5 \cdot {\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2}\right)}{\ell}\right) \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \]

Alternative 7: 64.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -4.5 \cdot 10^{-111}:\\ \;\;\;\;\left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right) \cdot \left(-1 + {\left(M \cdot \frac{D}{\frac{d}{0.5}}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\\ \mathbf{elif}\;\ell \leq 1850000000:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - h \cdot \frac{0.5 \cdot {\left(0.5 \cdot \left(\frac{D}{d} \cdot M\right)\right)}^{2}}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= l -4.5e-111)
   (*
    (* d (pow (* l h) -0.5))
    (+ -1.0 (* (pow (* M (/ D (/ d 0.5))) 2.0) (* 0.5 (/ h l)))))
   (if (<= l 1850000000.0)
     (*
      (sqrt (* (/ d l) (/ d h)))
      (- 1.0 (* h (/ (* 0.5 (pow (* 0.5 (* (/ D d) M)) 2.0)) l))))
     (* 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 <= -4.5e-111) {
		tmp = (d * pow((l * h), -0.5)) * (-1.0 + (pow((M * (D / (d / 0.5))), 2.0) * (0.5 * (h / l))));
	} else if (l <= 1850000000.0) {
		tmp = sqrt(((d / l) * (d / h))) * (1.0 - (h * ((0.5 * pow((0.5 * ((D / d) * M)), 2.0)) / l)));
	} 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 <= (-4.5d-111)) then
        tmp = (d * ((l * h) ** (-0.5d0))) * ((-1.0d0) + (((m * (d_1 / (d / 0.5d0))) ** 2.0d0) * (0.5d0 * (h / l))))
    else if (l <= 1850000000.0d0) then
        tmp = sqrt(((d / l) * (d / h))) * (1.0d0 - (h * ((0.5d0 * ((0.5d0 * ((d_1 / d) * m)) ** 2.0d0)) / l)))
    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 <= -4.5e-111) {
		tmp = (d * Math.pow((l * h), -0.5)) * (-1.0 + (Math.pow((M * (D / (d / 0.5))), 2.0) * (0.5 * (h / l))));
	} else if (l <= 1850000000.0) {
		tmp = Math.sqrt(((d / l) * (d / h))) * (1.0 - (h * ((0.5 * Math.pow((0.5 * ((D / d) * M)), 2.0)) / l)));
	} 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 <= -4.5e-111:
		tmp = (d * math.pow((l * h), -0.5)) * (-1.0 + (math.pow((M * (D / (d / 0.5))), 2.0) * (0.5 * (h / l))))
	elif l <= 1850000000.0:
		tmp = math.sqrt(((d / l) * (d / h))) * (1.0 - (h * ((0.5 * math.pow((0.5 * ((D / d) * M)), 2.0)) / l)))
	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 <= -4.5e-111)
		tmp = Float64(Float64(d * (Float64(l * h) ^ -0.5)) * Float64(-1.0 + Float64((Float64(M * Float64(D / Float64(d / 0.5))) ^ 2.0) * Float64(0.5 * Float64(h / l)))));
	elseif (l <= 1850000000.0)
		tmp = Float64(sqrt(Float64(Float64(d / l) * Float64(d / h))) * Float64(1.0 - Float64(h * Float64(Float64(0.5 * (Float64(0.5 * Float64(Float64(D / d) * M)) ^ 2.0)) / l))));
	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 <= -4.5e-111)
		tmp = (d * ((l * h) ^ -0.5)) * (-1.0 + (((M * (D / (d / 0.5))) ^ 2.0) * (0.5 * (h / l))));
	elseif (l <= 1850000000.0)
		tmp = sqrt(((d / l) * (d / h))) * (1.0 - (h * ((0.5 * ((0.5 * ((D / d) * M)) ^ 2.0)) / l)));
	else
		tmp = d * ((h ^ -0.5) * (l ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[l, -4.5e-111], N[(N[(d * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] * N[(-1.0 + N[(N[Power[N[(M * N[(D / N[(d / 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(0.5 * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1850000000.0], N[(N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(h * N[(N[(0.5 * N[Power[N[(0.5 * N[(N[(D / d), $MachinePrecision] * M), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $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 < -4.49999999999999994e-111

   1. Initial program 62.2%

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

      1. metadata-eval 62.2%

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

      2. unpow1/2 62.2%

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

      3. metadata-eval 62.2%

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

      4. unpow1/2 62.2%

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

      5. *-commutative 62.2%

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

      6. associate-*l* 62.2%

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

      7. times-frac 62.2%

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

      8. metadata-eval 62.2%

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

   3. Simplified 62.2%

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

   5. Applied egg-rr 26.9%

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

      1. expm1-def 36.5%

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

      2. expm1-log1p 47.3%

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

      3. rem-log-exp 43.3%

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

      4. exp-diff 43.3%

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

      5. log-div 43.3%

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

      6. rem-log-exp 43.3%

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

      7. rem-log-exp 47.3%

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

      8. *-commutative 47.3%

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

      9. associate-*l* 47.3%

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

      10. *-commutative 47.3%

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

      11. associate-*l/ 47.3%

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

      12. associate-/l* 47.3%

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

   7. Simplified 47.3%

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

   8. Taylor expanded in d around -inf 74.2%

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

      1. associate-*r* 74.2%

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

      2. mul-1-neg 74.2%

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

      3. *-commutative 74.2%

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

      4. unpow-1 74.2%

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

      5. sqr-pow 74.2%

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

      6. rem-sqrt-square 74.2%

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

      7. sqr-pow 74.0%

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

      8. fabs-sqr 74.0%

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

      9. sqr-pow 74.2%

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

      10. metadata-eval 74.2%

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

   10. Simplified 74.2%

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

   if -4.49999999999999994e-111 < l < 1.85e9

   1. Initial program 64.0%

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

      1. metadata-eval 64.0%

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

      2. unpow1/2 64.0%

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

      3. metadata-eval 64.0%

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

      4. unpow1/2 64.0%

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

      5. *-commutative 64.0%

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

      6. associate-*l* 64.0%

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

      7. times-frac 64.1%

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

      8. metadata-eval 64.1%

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

   3. Simplified 64.1%

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

      1. associate-*r* 64.1%

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

      2. frac-times 64.0%

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

      3. *-commutative 64.0%

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

      4. metadata-eval 64.0%

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

      5. associate-*r/ 74.0%

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

      6. metadata-eval 74.0%

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

      7. *-commutative 74.0%

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

      8. frac-times 74.0%

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

      9. *-commutative 74.0%

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

      10. div-inv 74.0%

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

      11. metadata-eval 74.0%

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

   5. Applied egg-rr 74.0%

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

      1. pow1 74.0%

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

      2. sqrt-unprod 70.6%

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

      3. associate-/l* 61.6%

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

      4. *-commutative 61.6%

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

   7. Applied egg-rr 61.6%

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

      1. unpow1 61.6%

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

      2. sub-neg 61.6%

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

      3. +-commutative 61.6%

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

      4. +-commutative 61.6%

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

      5. sub-neg 61.6%

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

      6. *-lft-identity 61.6%

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

      7. *-lft-identity 61.6%

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

      8. associate-/r/ 69.8%

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

   9. Simplified 69.8%

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

   if 1.85e9 < l

   1. Initial program 63.9%

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

      1. metadata-eval 63.9%

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

      2. unpow1/2 63.9%

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

      3. metadata-eval 63.9%

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

      4. unpow1/2 63.9%

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

      5. *-commutative 63.9%

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

      6. associate-*l* 63.9%

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

      7. times-frac 63.8%

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

      8. metadata-eval 63.8%

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

   3. Simplified 63.8%

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

   4. Taylor expanded in d around inf 65.0%

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

      1. *-un-lft-identity 65.0%

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

      2. *-commutative 65.0%

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

   6. Applied egg-rr 65.0%

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

      1. *-lft-identity 65.0%

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

      2. unpow-1 65.0%

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

      3. sqr-pow 65.0%

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

      4. rem-sqrt-square 65.0%

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

      5. sqr-pow 64.7%

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

      6. fabs-sqr 64.7%

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

      7. sqr-pow 65.0%

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

      8. metadata-eval 65.0%

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

   8. Simplified 65.0%

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

      1. unpow-prod-down 71.1%

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

   10. Applied egg-rr 71.1%

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

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4.5 \cdot 10^{-111}:\\ \;\;\;\;\left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right) \cdot \left(-1 + {\left(M \cdot \frac{D}{\frac{d}{0.5}}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\\ \mathbf{elif}\;\ell \leq 1850000000:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - h \cdot \frac{0.5 \cdot {\left(0.5 \cdot \left(\frac{D}{d} \cdot M\right)\right)}^{2}}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \]

Alternative 8: 60.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 135000000:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - h \cdot \frac{0.5 \cdot {\left(0.5 \cdot \left(\frac{D}{d} \cdot M\right)\right)}^{2}}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= l 135000000.0)
   (*
    (sqrt (* (/ d l) (/ d h)))
    (- 1.0 (* h (/ (* 0.5 (pow (* 0.5 (* (/ D d) M)) 2.0)) l))))
   (* 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 <= 135000000.0) {
		tmp = sqrt(((d / l) * (d / h))) * (1.0 - (h * ((0.5 * pow((0.5 * ((D / d) * M)), 2.0)) / l)));
	} 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 <= 135000000.0d0) then
        tmp = sqrt(((d / l) * (d / h))) * (1.0d0 - (h * ((0.5d0 * ((0.5d0 * ((d_1 / d) * m)) ** 2.0d0)) / l)))
    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 <= 135000000.0) {
		tmp = Math.sqrt(((d / l) * (d / h))) * (1.0 - (h * ((0.5 * Math.pow((0.5 * ((D / d) * M)), 2.0)) / l)));
	} 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 <= 135000000.0:
		tmp = math.sqrt(((d / l) * (d / h))) * (1.0 - (h * ((0.5 * math.pow((0.5 * ((D / d) * M)), 2.0)) / l)))
	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 <= 135000000.0)
		tmp = Float64(sqrt(Float64(Float64(d / l) * Float64(d / h))) * Float64(1.0 - Float64(h * Float64(Float64(0.5 * (Float64(0.5 * Float64(Float64(D / d) * M)) ^ 2.0)) / l))));
	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 <= 135000000.0)
		tmp = sqrt(((d / l) * (d / h))) * (1.0 - (h * ((0.5 * ((0.5 * ((D / d) * M)) ^ 2.0)) / l)));
	else
		tmp = d * ((h ^ -0.5) * (l ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[l, 135000000.0], N[(N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(h * N[(N[(0.5 * N[Power[N[(0.5 * N[(N[(D / d), $MachinePrecision] * M), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[h, -0.5], $MachinePrecision] * N[Power[l, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.35e8

   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 63.3%

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

      8. metadata-eval 63.3%

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

   3. Simplified 63.3%

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

      1. associate-*r* 63.3%

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

      2. frac-times 63.2%

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

      3. *-commutative 63.2%

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

      4. metadata-eval 63.2%

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

      5. associate-*r/ 69.1%

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

      6. metadata-eval 69.1%

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

      7. *-commutative 69.1%

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

      8. frac-times 69.1%

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

      9. *-commutative 69.1%

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

      10. div-inv 69.1%

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

      11. metadata-eval 69.1%

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

   5. Applied egg-rr 69.1%

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

      1. pow1 69.1%

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

      2. sqrt-unprod 60.7%

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

      3. associate-/l* 55.4%

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

      4. *-commutative 55.4%

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

   7. Applied egg-rr 55.4%

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

      1. unpow1 55.4%

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

      2. sub-neg 55.4%

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

      3. +-commutative 55.4%

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

      4. +-commutative 55.4%

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

      5. sub-neg 55.4%

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

      6. *-lft-identity 55.4%

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

      7. *-lft-identity 55.4%

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

      8. associate-/r/ 60.5%

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

   9. Simplified 60.5%

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

   if 1.35e8 < l

   1. Initial program 63.9%

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

      1. metadata-eval 63.9%

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

      2. unpow1/2 63.9%

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

      3. metadata-eval 63.9%

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

      4. unpow1/2 63.9%

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

      5. *-commutative 63.9%

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

      6. associate-*l* 63.9%

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

      7. times-frac 63.8%

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

      8. metadata-eval 63.8%

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

   3. Simplified 63.8%

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

   4. Taylor expanded in d around inf 65.0%

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

      1. *-un-lft-identity 65.0%

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

      2. *-commutative 65.0%

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

   6. Applied egg-rr 65.0%

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

      1. *-lft-identity 65.0%

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

      2. unpow-1 65.0%

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

      3. sqr-pow 65.0%

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

      4. rem-sqrt-square 65.0%

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

      5. sqr-pow 64.7%

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

      6. fabs-sqr 64.7%

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

      7. sqr-pow 65.0%

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

      8. metadata-eval 65.0%

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

   8. Simplified 65.0%

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

      1. unpow-prod-down 71.1%

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

   10. Applied egg-rr 71.1%

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

4. Final simplification 63.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 135000000:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - h \cdot \frac{0.5 \cdot {\left(0.5 \cdot \left(\frac{D}{d} \cdot M\right)\right)}^{2}}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \]

Alternative 9: 53.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.46 \cdot 10^{-285} \lor \neg \left(\ell \leq 9.5 \cdot 10^{-254}\right) \land \ell \leq 75000000:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \left(\frac{D}{d} \cdot \frac{D \cdot \left(M \cdot \left(h \cdot M\right)\right)}{\ell \cdot d}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (or (<= l 1.46e-285) (and (not (<= l 9.5e-254)) (<= l 75000000.0)))
   (*
    (sqrt (* (/ d l) (/ d h)))
    (- 1.0 (* 0.125 (* (/ D d) (/ (* D (* M (* h M))) (* l d))))))
   (* d (/ (sqrt (/ 1.0 l)) (sqrt h)))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if ((l <= 1.46e-285) || (!(l <= 9.5e-254) && (l <= 75000000.0))) {
		tmp = sqrt(((d / l) * (d / h))) * (1.0 - (0.125 * ((D / d) * ((D * (M * (h * M))) / (l * d)))));
	} else {
		tmp = d * (sqrt((1.0 / l)) / sqrt(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) :: tmp
    if ((l <= 1.46d-285) .or. (.not. (l <= 9.5d-254)) .and. (l <= 75000000.0d0)) then
        tmp = sqrt(((d / l) * (d / h))) * (1.0d0 - (0.125d0 * ((d_1 / d) * ((d_1 * (m * (h * m))) / (l * d)))))
    else
        tmp = d * (sqrt((1.0d0 / l)) / sqrt(h))
    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 <= 1.46e-285) || (!(l <= 9.5e-254) && (l <= 75000000.0))) {
		tmp = Math.sqrt(((d / l) * (d / h))) * (1.0 - (0.125 * ((D / d) * ((D * (M * (h * M))) / (l * d)))));
	} else {
		tmp = d * (Math.sqrt((1.0 / l)) / Math.sqrt(h));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if (l <= 1.46e-285) or (not (l <= 9.5e-254) and (l <= 75000000.0)):
		tmp = math.sqrt(((d / l) * (d / h))) * (1.0 - (0.125 * ((D / d) * ((D * (M * (h * M))) / (l * d)))))
	else:
		tmp = d * (math.sqrt((1.0 / l)) / math.sqrt(h))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if ((l <= 1.46e-285) || (!(l <= 9.5e-254) && (l <= 75000000.0)))
		tmp = Float64(sqrt(Float64(Float64(d / l) * Float64(d / h))) * Float64(1.0 - Float64(0.125 * Float64(Float64(D / d) * Float64(Float64(D * Float64(M * Float64(h * M))) / Float64(l * d))))));
	else
		tmp = Float64(d * Float64(sqrt(Float64(1.0 / l)) / sqrt(h)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if ((l <= 1.46e-285) || (~((l <= 9.5e-254)) && (l <= 75000000.0)))
		tmp = sqrt(((d / l) * (d / h))) * (1.0 - (0.125 * ((D / d) * ((D * (M * (h * M))) / (l * d)))));
	else
		tmp = d * (sqrt((1.0 / l)) / sqrt(h));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[Or[LessEqual[l, 1.46e-285], And[N[Not[LessEqual[l, 9.5e-254]], $MachinePrecision], LessEqual[l, 75000000.0]]], N[(N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(0.125 * N[(N[(D / d), $MachinePrecision] * N[(N[(D * N[(M * N[(h * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Sqrt[N[(1.0 / l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.45999999999999992e-285 or 9.5000000000000003e-254 < l < 7.5e7

   1. Initial program 64.2%

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

      1. metadata-eval 64.2%

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

      2. unpow1/2 64.2%

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

      3. metadata-eval 64.2%

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

      4. unpow1/2 64.2%

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

      5. *-commutative 64.2%

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

      6. associate-*l* 64.2%

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

      7. times-frac 64.2%

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

      8. metadata-eval 64.2%

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

   3. Simplified 64.2%

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

   5. Applied egg-rr 27.1%

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

      1. expm1-def 34.2%

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

      2. expm1-log1p 55.9%

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

      3. rem-log-exp 52.0%

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

      4. exp-diff 52.0%

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

      5. log-div 52.0%

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

      6. rem-log-exp 52.0%

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

      7. rem-log-exp 55.9%

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

      8. *-commutative 55.9%

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

      9. associate-*l* 55.9%

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

      10. *-commutative 55.9%

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

      11. associate-*l/ 55.9%

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

      12. associate-/l* 55.9%

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

   7. Simplified 55.9%

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

   8. Taylor expanded in M around 0 41.6%

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

      1. associate-*r/ 41.6%

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

      2. *-commutative 41.6%

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

      3. associate-*r/ 41.6%

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

      4. unpow2 41.6%

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

      5. *-commutative 41.6%

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

      6. associate-*l* 44.0%

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

      7. *-commutative 44.0%

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

      8. unpow2 44.0%

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

      9. associate-*l* 46.0%

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

      10. times-frac 51.4%

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

      11. *-commutative 51.4%

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

      12. unpow2 51.4%

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

      13. associate-*r* 53.6%

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

   10. Simplified 53.6%

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

   if 1.45999999999999992e-285 < l < 9.5000000000000003e-254 or 7.5e7 < l

   1. Initial program 61.2%

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

      1. metadata-eval 61.2%

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

      2. unpow1/2 61.2%

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

      3. metadata-eval 61.2%

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

      4. unpow1/2 61.2%

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

      5. *-commutative 61.2%

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

      6. associate-*l* 61.2%

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

      7. times-frac 61.1%

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

      8. metadata-eval 61.1%

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

   3. Simplified 61.1%

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

   4. Taylor expanded in d around inf 63.6%

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

      1. *-commutative 63.6%

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

      2. associate-/r* 63.5%

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

   6. Simplified 63.5%

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

      1. sqrt-div 72.7%

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

   8. Applied egg-rr 72.7%

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

4. Final simplification 58.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.46 \cdot 10^{-285} \lor \neg \left(\ell \leq 9.5 \cdot 10^{-254}\right) \land \ell \leq 75000000:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \left(\frac{D}{d} \cdot \frac{D \cdot \left(M \cdot \left(h \cdot M\right)\right)}{\ell \cdot d}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}\\ \end{array} \]

Alternative 10: 53.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \left(\frac{D}{d} \cdot \frac{D \cdot \left(M \cdot \left(h \cdot M\right)\right)}{\ell \cdot d}\right)\right)\\ \mathbf{if}\;\ell \leq 1.32 \cdot 10^{-285}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq 1.95 \cdot 10^{-253}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}\\ \mathbf{elif}\;\ell \leq 2800000:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0
         (*
          (sqrt (* (/ d l) (/ d h)))
          (- 1.0 (* 0.125 (* (/ D d) (/ (* D (* M (* h M))) (* l d))))))))
   (if (<= l 1.32e-285)
     t_0
     (if (<= l 1.95e-253)
       (* d (/ (sqrt (/ 1.0 l)) (sqrt h)))
       (if (<= l 2800000.0) 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 = sqrt(((d / l) * (d / h))) * (1.0 - (0.125 * ((D / d) * ((D * (M * (h * M))) / (l * d)))));
	double tmp;
	if (l <= 1.32e-285) {
		tmp = t_0;
	} else if (l <= 1.95e-253) {
		tmp = d * (sqrt((1.0 / l)) / sqrt(h));
	} else if (l <= 2800000.0) {
		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 = sqrt(((d / l) * (d / h))) * (1.0d0 - (0.125d0 * ((d_1 / d) * ((d_1 * (m * (h * m))) / (l * d)))))
    if (l <= 1.32d-285) then
        tmp = t_0
    else if (l <= 1.95d-253) then
        tmp = d * (sqrt((1.0d0 / l)) / sqrt(h))
    else if (l <= 2800000.0d0) 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 = Math.sqrt(((d / l) * (d / h))) * (1.0 - (0.125 * ((D / d) * ((D * (M * (h * M))) / (l * d)))));
	double tmp;
	if (l <= 1.32e-285) {
		tmp = t_0;
	} else if (l <= 1.95e-253) {
		tmp = d * (Math.sqrt((1.0 / l)) / Math.sqrt(h));
	} else if (l <= 2800000.0) {
		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 = math.sqrt(((d / l) * (d / h))) * (1.0 - (0.125 * ((D / d) * ((D * (M * (h * M))) / (l * d)))))
	tmp = 0
	if l <= 1.32e-285:
		tmp = t_0
	elif l <= 1.95e-253:
		tmp = d * (math.sqrt((1.0 / l)) / math.sqrt(h))
	elif l <= 2800000.0:
		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(sqrt(Float64(Float64(d / l) * Float64(d / h))) * Float64(1.0 - Float64(0.125 * Float64(Float64(D / d) * Float64(Float64(D * Float64(M * Float64(h * M))) / Float64(l * d))))))
	tmp = 0.0
	if (l <= 1.32e-285)
		tmp = t_0;
	elseif (l <= 1.95e-253)
		tmp = Float64(d * Float64(sqrt(Float64(1.0 / l)) / sqrt(h)));
	elseif (l <= 2800000.0)
		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 = sqrt(((d / l) * (d / h))) * (1.0 - (0.125 * ((D / d) * ((D * (M * (h * M))) / (l * d)))));
	tmp = 0.0;
	if (l <= 1.32e-285)
		tmp = t_0;
	elseif (l <= 1.95e-253)
		tmp = d * (sqrt((1.0 / l)) / sqrt(h));
	elseif (l <= 2800000.0)
		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[(N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(0.125 * N[(N[(D / d), $MachinePrecision] * N[(N[(D * N[(M * N[(h * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, 1.32e-285], t$95$0, If[LessEqual[l, 1.95e-253], N[(d * N[(N[Sqrt[N[(1.0 / l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 2800000.0], 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 < 1.3199999999999999e-285 or 1.9499999999999999e-253 < l < 2.8e6

   1. Initial program 64.2%

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

      1. metadata-eval 64.2%

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

      2. unpow1/2 64.2%

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

      3. metadata-eval 64.2%

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

      4. unpow1/2 64.2%

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

      5. *-commutative 64.2%

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

      6. associate-*l* 64.2%

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

      7. times-frac 64.2%

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

      8. metadata-eval 64.2%

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

   3. Simplified 64.2%

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

   5. Applied egg-rr 27.1%

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

      1. expm1-def 34.2%

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

      2. expm1-log1p 55.9%

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

      3. rem-log-exp 52.0%

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

      4. exp-diff 52.0%

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

      5. log-div 52.0%

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

      6. rem-log-exp 52.0%

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

      7. rem-log-exp 55.9%

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

      8. *-commutative 55.9%

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

      9. associate-*l* 55.9%

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

      10. *-commutative 55.9%

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

      11. associate-*l/ 55.9%

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

      12. associate-/l* 55.9%

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

   7. Simplified 55.9%

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

   8. Taylor expanded in M around 0 41.6%

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

      1. associate-*r/ 41.6%

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

      2. *-commutative 41.6%

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

      3. associate-*r/ 41.6%

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

      4. unpow2 41.6%

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

      5. *-commutative 41.6%

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

      6. associate-*l* 44.0%

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

      7. *-commutative 44.0%

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

      8. unpow2 44.0%

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

      9. associate-*l* 46.0%

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

      10. times-frac 51.4%

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

      11. *-commutative 51.4%

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

      12. unpow2 51.4%

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

      13. associate-*r* 53.6%

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

   10. Simplified 53.6%

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

   if 1.3199999999999999e-285 < l < 1.9499999999999999e-253

   1. Initial program 46.8%

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

      1. metadata-eval 46.8%

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

      2. unpow1/2 46.8%

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

      3. metadata-eval 46.8%

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

      4. unpow1/2 46.8%

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

      5. *-commutative 46.8%

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

      6. associate-*l* 46.8%

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

      7. times-frac 46.8%

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

      8. metadata-eval 46.8%

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

   3. Simplified 46.8%

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

   4. Taylor expanded in d around inf 55.8%

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

      1. *-commutative 55.8%

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

      2. associate-/r* 56.0%

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

   6. Simplified 56.0%

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

      1. sqrt-div 81.4%

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

   8. Applied egg-rr 81.4%

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

   if 2.8e6 < l

   1. Initial program 63.9%

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

      1. metadata-eval 63.9%

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

      2. unpow1/2 63.9%

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

      3. metadata-eval 63.9%

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

      4. unpow1/2 63.9%

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

      5. *-commutative 63.9%

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

      6. associate-*l* 63.9%

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

      7. times-frac 63.8%

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

      8. metadata-eval 63.8%

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

   3. Simplified 63.8%

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

   4. Taylor expanded in d around inf 65.0%

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

      1. *-un-lft-identity 65.0%

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

      2. *-commutative 65.0%

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

   6. Applied egg-rr 65.0%

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

      1. *-lft-identity 65.0%

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

      2. unpow-1 65.0%

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

      3. sqr-pow 65.0%

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

      4. rem-sqrt-square 65.0%

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

      5. sqr-pow 64.7%

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

      6. fabs-sqr 64.7%

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

      7. sqr-pow 65.0%

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

      8. metadata-eval 65.0%

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

   8. Simplified 65.0%

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

      1. unpow-prod-down 71.1%

         \[\leadsto \color{blue}{\left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)} \cdot d \]

   10. Applied egg-rr 71.1%

       \[\leadsto \color{blue}{\left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)} \cdot d \]
3. Recombined 3 regimes into one program.

4. Final simplification 58.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.32 \cdot 10^{-285}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \left(\frac{D}{d} \cdot \frac{D \cdot \left(M \cdot \left(h \cdot M\right)\right)}{\ell \cdot d}\right)\right)\\ \mathbf{elif}\;\ell \leq 1.95 \cdot 10^{-253}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}\\ \mathbf{elif}\;\ell \leq 2800000:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \left(\frac{D}{d} \cdot \frac{D \cdot \left(M \cdot \left(h \cdot M\right)\right)}{\ell \cdot d}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({h}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \end{array} \]

Alternative 11: 46.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq 4.8 \cdot 10^{-166}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \left(\frac{D}{d} \cdot \frac{D \cdot \left(M \cdot \left(h \cdot M\right)\right)}{\ell \cdot d}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= M 4.8e-166)
   (* (sqrt (/ d l)) (sqrt (/ d h)))
   (*
    (sqrt (* (/ d l) (/ d h)))
    (- 1.0 (* 0.125 (* (/ D d) (/ (* D (* M (* h M))) (* l d))))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (M <= 4.8e-166) {
		tmp = sqrt((d / l)) * sqrt((d / h));
	} else {
		tmp = sqrt(((d / l) * (d / h))) * (1.0 - (0.125 * ((D / d) * ((D * (M * (h * M))) / (l * 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 <= 4.8d-166) then
        tmp = sqrt((d / l)) * sqrt((d / h))
    else
        tmp = sqrt(((d / l) * (d / h))) * (1.0d0 - (0.125d0 * ((d_1 / d) * ((d_1 * (m * (h * m))) / (l * 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 <= 4.8e-166) {
		tmp = Math.sqrt((d / l)) * Math.sqrt((d / h));
	} else {
		tmp = Math.sqrt(((d / l) * (d / h))) * (1.0 - (0.125 * ((D / d) * ((D * (M * (h * M))) / (l * d)))));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if M <= 4.8e-166:
		tmp = math.sqrt((d / l)) * math.sqrt((d / h))
	else:
		tmp = math.sqrt(((d / l) * (d / h))) * (1.0 - (0.125 * ((D / d) * ((D * (M * (h * M))) / (l * d)))))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (M <= 4.8e-166)
		tmp = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)));
	else
		tmp = Float64(sqrt(Float64(Float64(d / l) * Float64(d / h))) * Float64(1.0 - Float64(0.125 * Float64(Float64(D / d) * Float64(Float64(D * Float64(M * Float64(h * M))) / Float64(l * d))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (M <= 4.8e-166)
		tmp = sqrt((d / l)) * sqrt((d / h));
	else
		tmp = sqrt(((d / l) * (d / h))) * (1.0 - (0.125 * ((D / d) * ((D * (M * (h * M))) / (l * d)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[M, 4.8e-166], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(0.125 * N[(N[(D / d), $MachinePrecision] * N[(N[(D * N[(M * N[(h * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 4.7999999999999997e-166

   1. Initial program 59.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. associate-*l* 58.9%

         \[\leadsto \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]

      2. metadata-eval 58.9%

         \[\leadsto {\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      3. unpow1/2 58.9%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h}}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      4. metadata-eval 58.9%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      5. unpow1/2 58.9%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]

      6. sub-neg 58.9%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(1 + \left(-\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)}\right) \]

      7. +-commutative 58.9%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(\left(-\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) + 1\right)}\right) \]

      8. *-commutative 58.9%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\left(-\color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) + 1\right)\right) \]

      9. associate-*l* 58.9%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\left(-\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) + 1\right)\right) \]

      10. distribute-rgt-neg-in 58.9%

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(-\frac{1}{2} \cdot \frac{h}{\ell}\right)} + 1\right)\right) \]

      11. *-commutative 58.9%

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(-\color{blue}{\frac{h}{\ell} \cdot \frac{1}{2}}\right) + 1\right)\right) \]

   3. Simplified 58.9%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(M \cdot \frac{\frac{D}{d}}{2}\right)}^{2}, \frac{-0.5}{\frac{\ell}{h}}, 1\right)\right)} \]

   4. Taylor expanded in M around 0 46.7%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{1}\right) \]

   if 4.7999999999999997e-166 < M

   1. Initial program 72.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 72.3%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 72.3%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 72.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 72.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 72.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 72.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 72.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 72.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 72.3%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]

   5. Applied egg-rr 28.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 35.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\right)\right)} \]

      2. expm1-log1p 62.6%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]

      3. rem-log-exp 56.8%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \color{blue}{\log \left(e^{1 - {\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)}\right)} \]

      4. exp-diff 56.8%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \log \color{blue}{\left(\frac{e^{1}}{e^{{\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)}}\right)} \]

      5. log-div 56.8%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \color{blue}{\left(\log \left(e^{1}\right) - \log \left(e^{{\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)}\right)\right)} \]

      6. rem-log-exp 56.8%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(\color{blue}{1} - \log \left(e^{{\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)}\right)\right) \]

      7. rem-log-exp 62.6%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \color{blue}{{\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)}\right) \]

      8. *-commutative 62.6%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\color{blue}{\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

      9. associate-*l* 62.6%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\color{blue}{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

      10. *-commutative 62.6%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \color{blue}{\left(\frac{D}{d} \cdot 0.5\right)}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

      11. associate-*l/ 62.6%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \color{blue}{\frac{D \cdot 0.5}{d}}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

      12. associate-/l* 62.6%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \color{blue}{\frac{D}{\frac{d}{0.5}}}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

   7. Simplified 62.6%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{\frac{d}{0.5}}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]

   8. Taylor expanded in M around 0 46.0%

      \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \color{blue}{0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}}\right) \]
   9. Step-by-step derivation

      1. associate-*r/ 46.0%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \color{blue}{\frac{0.125 \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\ell \cdot {d}^{2}}}\right) \]

      2. *-commutative 46.0%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{0.125 \cdot \left({D}^{2} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)}{\ell \cdot {d}^{2}}\right) \]

      3. associate-*r/ 46.0%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \color{blue}{0.125 \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{\ell \cdot {d}^{2}}}\right) \]

      4. unpow2 46.0%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - 0.125 \cdot \frac{\color{blue}{\left(D \cdot D\right)} \cdot \left(h \cdot {M}^{2}\right)}{\ell \cdot {d}^{2}}\right) \]

      5. *-commutative 46.0%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - 0.125 \cdot \frac{\left(D \cdot D\right) \cdot \color{blue}{\left({M}^{2} \cdot h\right)}}{\ell \cdot {d}^{2}}\right) \]

      6. associate-*l* 48.7%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - 0.125 \cdot \frac{\color{blue}{D \cdot \left(D \cdot \left({M}^{2} \cdot h\right)\right)}}{\ell \cdot {d}^{2}}\right) \]

      7. *-commutative 48.7%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - 0.125 \cdot \frac{D \cdot \left(D \cdot \left({M}^{2} \cdot h\right)\right)}{\color{blue}{{d}^{2} \cdot \ell}}\right) \]

      8. unpow2 48.7%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - 0.125 \cdot \frac{D \cdot \left(D \cdot \left({M}^{2} \cdot h\right)\right)}{\color{blue}{\left(d \cdot d\right)} \cdot \ell}\right) \]

      9. associate-*l* 48.8%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - 0.125 \cdot \frac{D \cdot \left(D \cdot \left({M}^{2} \cdot h\right)\right)}{\color{blue}{d \cdot \left(d \cdot \ell\right)}}\right) \]

      10. times-frac 54.9%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - 0.125 \cdot \color{blue}{\left(\frac{D}{d} \cdot \frac{D \cdot \left({M}^{2} \cdot h\right)}{d \cdot \ell}\right)}\right) \]

      11. *-commutative 54.9%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - 0.125 \cdot \left(\frac{D}{d} \cdot \frac{D \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}}{d \cdot \ell}\right)\right) \]

      12. unpow2 54.9%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - 0.125 \cdot \left(\frac{D}{d} \cdot \frac{D \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)}{d \cdot \ell}\right)\right) \]

      13. associate-*r* 57.5%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - 0.125 \cdot \left(\frac{D}{d} \cdot \frac{D \cdot \color{blue}{\left(\left(h \cdot M\right) \cdot M\right)}}{d \cdot \ell}\right)\right) \]

   10. Simplified 57.5%

       \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \color{blue}{0.125 \cdot \left(\frac{D}{d} \cdot \frac{D \cdot \left(\left(h \cdot M\right) \cdot M\right)}{d \cdot \ell}\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 50.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 4.8 \cdot 10^{-166}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \left(\frac{D}{d} \cdot \frac{D \cdot \left(M \cdot \left(h \cdot M\right)\right)}{\ell \cdot d}\right)\right)\\ \end{array} \]

Alternative 12: 51.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 26000:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \left(\frac{D}{d} \cdot \frac{D \cdot \left(M \cdot \left(h \cdot M\right)\right)}{\ell \cdot d}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= l 26000.0)
   (*
    (sqrt (* (/ d l) (/ d h)))
    (- 1.0 (* 0.125 (* (/ D d) (/ (* D (* M (* h M))) (* l d))))))
   (* d (pow (* l h) -0.5))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= 26000.0) {
		tmp = sqrt(((d / l) * (d / h))) * (1.0 - (0.125 * ((D / d) * ((D * (M * (h * M))) / (l * d)))));
	} else {
		tmp = 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 (l <= 26000.0d0) then
        tmp = sqrt(((d / l) * (d / h))) * (1.0d0 - (0.125d0 * ((d_1 / d) * ((d_1 * (m * (h * m))) / (l * d)))))
    else
        tmp = 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 (l <= 26000.0) {
		tmp = Math.sqrt(((d / l) * (d / h))) * (1.0 - (0.125 * ((D / d) * ((D * (M * (h * M))) / (l * d)))));
	} else {
		tmp = d * Math.pow((l * h), -0.5);
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if l <= 26000.0:
		tmp = math.sqrt(((d / l) * (d / h))) * (1.0 - (0.125 * ((D / d) * ((D * (M * (h * M))) / (l * d)))))
	else:
		tmp = d * math.pow((l * h), -0.5)
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= 26000.0)
		tmp = Float64(sqrt(Float64(Float64(d / l) * Float64(d / h))) * Float64(1.0 - Float64(0.125 * Float64(Float64(D / d) * Float64(Float64(D * Float64(M * Float64(h * M))) / Float64(l * d))))));
	else
		tmp = 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 (l <= 26000.0)
		tmp = sqrt(((d / l) * (d / h))) * (1.0 - (0.125 * ((D / d) * ((D * (M * (h * M))) / (l * d)))));
	else
		tmp = d * ((l * h) ^ -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[l, 26000.0], N[(N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(0.125 * N[(N[(D / d), $MachinePrecision] * N[(N[(D * N[(M * N[(h * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 26000

   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 63.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 63.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 63.3%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]

   5. Applied egg-rr 26.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 33.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\right)\right)} \]

      2. expm1-log1p 54.9%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]

      3. rem-log-exp 51.2%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \color{blue}{\log \left(e^{1 - {\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)}\right)} \]

      4. exp-diff 51.2%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \log \color{blue}{\left(\frac{e^{1}}{e^{{\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)}}\right)} \]

      5. log-div 51.2%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \color{blue}{\left(\log \left(e^{1}\right) - \log \left(e^{{\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)}\right)\right)} \]

      6. rem-log-exp 51.2%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(\color{blue}{1} - \log \left(e^{{\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)}\right)\right) \]

      7. rem-log-exp 54.9%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \color{blue}{{\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)}\right) \]

      8. *-commutative 54.9%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\color{blue}{\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

      9. associate-*l* 54.9%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\color{blue}{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

      10. *-commutative 54.9%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \color{blue}{\left(\frac{D}{d} \cdot 0.5\right)}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

      11. associate-*l/ 54.9%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \color{blue}{\frac{D \cdot 0.5}{d}}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

      12. associate-/l* 54.9%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \color{blue}{\frac{D}{\frac{d}{0.5}}}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

   7. Simplified 54.9%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - {\left(M \cdot \frac{D}{\frac{d}{0.5}}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]

   8. Taylor expanded in M around 0 40.3%

      \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \color{blue}{0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}}\right) \]
   9. Step-by-step derivation

      1. associate-*r/ 40.3%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \color{blue}{\frac{0.125 \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\ell \cdot {d}^{2}}}\right) \]

      2. *-commutative 40.3%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \frac{0.125 \cdot \left({D}^{2} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)}{\ell \cdot {d}^{2}}\right) \]

      3. associate-*r/ 40.3%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \color{blue}{0.125 \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{\ell \cdot {d}^{2}}}\right) \]

      4. unpow2 40.3%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - 0.125 \cdot \frac{\color{blue}{\left(D \cdot D\right)} \cdot \left(h \cdot {M}^{2}\right)}{\ell \cdot {d}^{2}}\right) \]

      5. *-commutative 40.3%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - 0.125 \cdot \frac{\left(D \cdot D\right) \cdot \color{blue}{\left({M}^{2} \cdot h\right)}}{\ell \cdot {d}^{2}}\right) \]

      6. associate-*l* 43.1%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - 0.125 \cdot \frac{\color{blue}{D \cdot \left(D \cdot \left({M}^{2} \cdot h\right)\right)}}{\ell \cdot {d}^{2}}\right) \]

      7. *-commutative 43.1%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - 0.125 \cdot \frac{D \cdot \left(D \cdot \left({M}^{2} \cdot h\right)\right)}{\color{blue}{{d}^{2} \cdot \ell}}\right) \]

      8. unpow2 43.1%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - 0.125 \cdot \frac{D \cdot \left(D \cdot \left({M}^{2} \cdot h\right)\right)}{\color{blue}{\left(d \cdot d\right)} \cdot \ell}\right) \]

      9. associate-*l* 45.0%

         \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - 0.125 \cdot \frac{D \cdot \left(D \cdot \left({M}^{2} \cdot h\right)\right)}{\color{blue}{d \cdot \left(d \cdot \ell\right)}}\right) \]

      10. times-frac 50.0%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - 0.125 \cdot \color{blue}{\left(\frac{D}{d} \cdot \frac{D \cdot \left({M}^{2} \cdot h\right)}{d \cdot \ell}\right)}\right) \]

      11. *-commutative 50.0%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - 0.125 \cdot \left(\frac{D}{d} \cdot \frac{D \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}}{d \cdot \ell}\right)\right) \]

      12. unpow2 50.0%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - 0.125 \cdot \left(\frac{D}{d} \cdot \frac{D \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)}{d \cdot \ell}\right)\right) \]

      13. associate-*r* 52.1%

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - 0.125 \cdot \left(\frac{D}{d} \cdot \frac{D \cdot \color{blue}{\left(\left(h \cdot M\right) \cdot M\right)}}{d \cdot \ell}\right)\right) \]

   10. Simplified 52.1%

       \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - \color{blue}{0.125 \cdot \left(\frac{D}{d} \cdot \frac{D \cdot \left(\left(h \cdot M\right) \cdot M\right)}{d \cdot \ell}\right)}\right) \]

   if 26000 < l

   1. Initial program 63.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 63.9%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. unpow1/2 63.9%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval 63.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 63.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative 63.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* 63.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

      7. times-frac 63.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

      8. metadata-eval 63.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Simplified 63.8%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]

   4. Taylor expanded in d around inf 65.0%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   5. Step-by-step derivation

      1. *-un-lft-identity 65.0%

         \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot d \]

      2. *-commutative 65.0%

         \[\leadsto \left(1 \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}}\right) \cdot d \]

   6. Applied egg-rr 65.0%

      \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot d \]
   7. Step-by-step derivation

      1. *-lft-identity 65.0%

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      2. unpow-1 65.0%

         \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \cdot d \]

      3. sqr-pow 65.0%

         \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}}} \cdot d \]

      4. rem-sqrt-square 65.0%

         \[\leadsto \color{blue}{\left|{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}\right|} \cdot d \]

      5. sqr-pow 64.7%

         \[\leadsto \left|\color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)}}\right| \cdot d \]

      6. fabs-sqr 64.7%

         \[\leadsto \color{blue}{\left({\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)}\right)} \cdot d \]

      7. sqr-pow 65.0%

         \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}} \cdot d \]

      8. metadata-eval 65.0%

         \[\leadsto {\left(h \cdot \ell\right)}^{\color{blue}{-0.5}} \cdot d \]

   8. Simplified 65.0%

      \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \cdot d \]
3. Recombined 2 regimes into one program.

4. Final simplification 55.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 26000:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 - 0.125 \cdot \left(\frac{D}{d} \cdot \frac{D \cdot \left(M \cdot \left(h \cdot M\right)\right)}{\ell \cdot d}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \end{array} \]

Alternative 13: 26.5% 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 63.4%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
2. Step-by-step derivation

   1. metadata-eval 63.4%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. unpow1/2 63.4%

      \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. metadata-eval 63.4%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. unpow1/2 63.4%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   5. *-commutative 63.4%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]

   6. associate-*l* 63.4%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]

   7. times-frac 63.4%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]

   8. metadata-eval 63.4%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

3. Simplified 63.4%

   \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)} \]

4. Taylor expanded in d around inf 29.4%

   \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
5. Step-by-step derivation

   1. *-un-lft-identity 29.4%

      \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{1}{\ell \cdot h}}\right)} \cdot d \]

   2. *-commutative 29.4%

      \[\leadsto \left(1 \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}}\right) \cdot d \]

6. Applied egg-rr 29.4%

   \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot d \]
7. Step-by-step derivation

   1. *-lft-identity 29.4%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

   2. unpow-1 29.4%

      \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \cdot d \]

   3. sqr-pow 29.4%

      \[\leadsto \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}}} \cdot d \]

   4. rem-sqrt-square 29.5%

      \[\leadsto \color{blue}{\left|{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}\right|} \cdot d \]

   5. sqr-pow 29.3%

      \[\leadsto \left|\color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)}}\right| \cdot d \]

   6. fabs-sqr 29.3%

      \[\leadsto \color{blue}{\left({\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)} \cdot {\left(h \cdot \ell\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)}\right)} \cdot d \]

   7. sqr-pow 29.5%

      \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{\left(\frac{-1}{2}\right)}} \cdot d \]

   8. metadata-eval 29.5%

      \[\leadsto {\left(h \cdot \ell\right)}^{\color{blue}{-0.5}} \cdot d \]

8. Simplified 29.5%

   \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \cdot d \]

9. Final simplification 29.5%

   \[\leadsto d \cdot {\left(\ell \cdot h\right)}^{-0.5} \]

Reproduce

herbie shell --seed 2023215 
(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)))))